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(3.1)
where e is an infinitesimal parameter and fi(
—/i 2 with /i2 > 0), we enter a different phase of the system. Show that the ground state is now (p = const* = v, A^ = 0. What is the photon mass in this phase? Calculate the potential between two static point charges each of value Q. What sets the scale of the screening length? d) Let us now add an external field to the system, 1 M) ~~* M) T ^rfii/rext
5Criticai whereas for B < JBCriticai it is the phase with
(p'(x') =
(3.2)
13 Symmetries and near symmetries
9
such that in the restriction back to constant e, C becomes invariant and
Use of the equation of motion together with the invariance of the lagrangian under the transformation in Eq. (3.1) yields d^J^ = dC/de(x) = 0 as desired. The Noether charge Q = f dsx JQ is timeindependent if the current vanishes sufficiently rapidly at spatial infinity, i.e.
^ = fd3xdoJo =  f dsxV3 = 0 .
(3.4)
We refer the reader tofieldtheory textbooks for further discussion, including the analogous procedure for constructing Noether currents of spacetime symmetries. Identifying the current does not exhaust all the consequences of a symmetry but is merely the first step towards the implementation of symmetry relations. Notice that we have been careful to use the word 'classical' several times. This is because the invariance of the action is not generally sufficient to identify symmetries of a quantum theory. We shall return to this point. Examples of Noether currents Let us now consider some explicit field theory models in order to get practice in constructing Noether currents. (i) Isospin symmetry: SU(2) isospin invariance of the nucleonpion system provides a standard and uncomplicated means for studying symmetry currents. Consider a doublet of nucleon fields
and a triplet of pion fields TT = {TT*} (i = 1,2,3) with lagrangian C = ip (ip  m) ip +  [d [d^ •d^d^n  ml* • it]+ +igtpT igtpT ir^ip • ir^ip  (TT  (TT TT) • TT)2 l* it] (TT (36)
where m is the nucleon mass matrix (m
0 \
\0
m)
and {r*} are the three Pauli matrices. This lagrangian is invariant under the global SU(2) rotation of the fields ^  > ^ ' = [/^, U = exp (ir  a/2) (3.7)
10
/ Inputs to the Standard Model
for any a1, (i — 1,2,3) provided the pion fields are transformed as
.
(3.8)
In proving this, it is useful to employ the identity 7T • 7T =  Tr ( r • 7TT • 7T) ,
(3.9)
from which we easily see that TTV* is invariant under the transformation of Eq. (3.8). The response of the individual pion components to an isospin transformation can be found from multiplying Eq. (3.8) by r% and taking the trace, ) = i Tr ( r W t / t )
.
(3.10)
To determine the isospin current, one considers the spacetimedependent transformation with a now infinitesimal, j> = (liT>a(x)/2)i/>
,
7ti = 7rieijknjak(x)
.
(3.11)
Performing this transformation on the lagrangian gives
;, jt) = C(i/>, TT) + \ ^ r . a^a^ and applying our expression Eq. (3.3) for the current yields the triplet of currents (one for each oci) ^ = ^
+^ 8 /
.
(3.13)
By obtaining the equations of motion for ip and TT, it is straightforward to verify that this current is conserved. (ii) The linear sigma model: With a few modifications the above example becomes one of the most instructive of all field theory models, the sigma model [GeL 60]. One adds to the lagrangian of Eq. (3.6) a scalar field a with judiciously chosen couplings, and removes the bare nucleon mass,
C = $ifal) + \d^iz  d»n + ^ad^a gip(aiT
2 A 2 7r75) if) + y (a2 + TT2)   (a2 + TT2)2 .
(3
*14)
2
For /i > 0, the model exhibits the phenomenon of spontaneous symmetry breaking (cf. Sect. 15). In describing the symmetries of this lagrangian, it is useful to rewrite the mesons in terms of a matrix field E = a + irn
,
(3.15)
13 Symmetries and near symmetries
11
such that a2 + 7T2 = i T r ( E f E ) .
(3.16)
Then we obtain
C = 4>LipipL + iMftfa + \ Tr (a^Sa^E*) + i / i 2 Tr (E+E)  — Tr
2
(s*s)  g ($LXtl>R + V ^ E ^ L ) ,
where I/JL,R are chiral fields (cf. Eq. (2.3)). The lefthanded and righthanded fermion fields are coupled together only in the interaction with the E field. The meson portion of the lagrangian is obviously invariant under rotations among the
, with Ui, and UR being arbitrary SU(2) matrices,
,
(3.18)
UL,R = exp (ictL,R • r/2) . (3.19) The fermion portions of the transformation clearly involve just the SU(2) isospin rotations on the lefthanded and righthanded fermions. However, the mesons involve a combination of a pure isospin rotation among the Tr fields together with a transformation between the a and n fields
'•5
,fe
=
(uLrk
Tr [UL1 TR)G +
+ _
l
U
Rjir
{OLL OLR) •
•(r*
^ ^ )
j(«i £4)
l Tr (T kUlTel " 2 +
]
\ ag) , (3.20)
case is for infinitesimal a i, a p . For each invariance there is a separate conserved current ad form in each
^1>L ~ \ Tr (rk (3.21)
12
/ Inputs to the Standard Model
These can be formed into a conserved vector current Vk r
—
T^
L.
T
—
I/J/V
ih J e:
^'^rr
r)
(*\
TT^
99^
which is just the isospin current derived previously, and a conserved axialvector current i^ + ^d^aad^
.
(3.23)
(iii) Scale invariance: Our third example illustrates the case of a spacetime transformation in which the lagrangian changes by a total derivative. Consider classical electrodynamics (c/. Sect. II—1) but with a massless electron, £ = ~Fp,F'»' + $ilptl> ,
(3.24)
where if) and A^ are the electron and photon fields, D^ is the covariant derivative of ^ , and F^v is the electromagnetic field strength. We shall describe the construction of both D^ip and F^v in the next section. Since there are no dimensional parameters in this lagrangian, we are motivated to consider the effect of a change in coordinate scale x —• x' = Xx together with the field transformations
,
A^x) > A'^x) = XA^Xx)
. (3.25)
Although the lagrangian itself is not invariant, C(x) > £{x) = X4C(Xx) ,
(3.26)
with a change of variable the action is easily seen to be unchanged,
S = fd4x£(x)
+ fd4xX4C(Xx) = fd4x'£(x')
=S .
(3.27)
There is nothing in this classical theory which depends on how length is scaled. The Noether current associated with the change of scale is %* = *•>*"' > where 9^ is the energymomentum tensor of the theory, v
(328)
F (3.29)
Since the energymomentum tensor is itself conserved, d^v — 0, the conservation of scale current is equivalent to the vanishing of the trace of the energymomentum tensor, V^ie = ^" = 0 • (330) This trace property may be easily verified using the equations of motion.
13 Symmetries and near symmetries
13
Approximate symmetry Thus far, we have been describing exact symmetries. Symmetry considerations are equally useful in situations where there is 'almost' a symmetry. The very phrase 'approximate symmetry' seems selfcontradictory and needs explanation. Quite often a lagrangian would have an invariance if certain of the parameters in it were set equal to zero. In that limit the invariance would have a set of physical consequences which, with the said parameters being nonzero, would no longer obtain. Yet, if the parameters are in some sense 'small', the predicted consequences are still approximately valid. In fact, when the interaction which breaks the symmetry has a well defined behavior under the symmetry transformation, its effect can generally be analyzed in terms of the basis of unperturbed particle states by using the WignerEckart theorem. The precise sense in which the symmetry breaking terms can be deemed small depends on the problem under consideration. In practice, the utility of an approximate symmetry is rarely known a priori, but is only evident after its predictions have been checked experimentally. If a symmetry is not exact, the associated currents and charges will no longer be conserved. For example in the linear sigma model, the symmetry is partially broken if we add to the lagrangian a term of the form £ = aa= ^ T r S ,
(3.31)
where S is the matrix defined in Eq. (3.15). With this addition, the vector isospin SU(2) symmetry remains exact but the axial SU(2) transformation is no longer an invariance. The axialcurrent divergence becomes 0MJ, = an* ,
(3.32)
and the charge is timedependent, dt
= a I dzx 7T* .
(3.33)
In the linear sigma model, if the parameters g, A are of order unity it is clear that the perturbation is small provided 1 » a/fi 3, as /i is the only other mass scale in the theory. However if either g or A happens to be anomalously large or small, the condition appropriate for a 'small' perturbation is not a priori evident. In our example (iii) of scale invariance in massless fermion electrodynamics, the addition of an electron mass (3.34)
14
/ Inputs to the Standard Model
would explicitly break the symmetry and the trace would no longer vanish, ^ O . (3.35) This is in fact what occurs in practice. Fermion mass is typically not a small parameter in QED and cannot be treated as a perturbation in most applications.
14 Gauge symmetry In our discussion of chiral symmetry, we considered the effect of global phase transformations, IJJL,R{%) —* exp (—ictL,R)ipL,R(z)> Global phase transformations are those which are constant throughout all spacetime. Let us reconsider the system of chiral fermions, but now insist that the phase transformations be local. Each transformation is then labeled by a spacetimedependent phase OLL,R{X)<> Such local mappings are referred to as gauge transformations. The free massless lagrangian of Eq. (2.1) is not invariant under the gauge transformation
because of the spacetime dependence of OLL,R (in the final term, the quantity d^aLfi does not generally vanish). In order for such a local transformation to be an invariance of the lagrangian, we need an extended kind of derivative D^, such that L^R{x)
.
(4.3)
under the local transformation of Eq. (4.1). The quantity D^ is a covariant derivative, so called because it responds covariantly, as in Eq. (4.3), to a gauge transformation.
Abelian case Before proceeding with the construction of a covariant derivative, we broaden the context of our discussion. Let Q(x) now represent a boson or fermion field of any spin and arbitrary mass. We consider transformations 9 > U(a)G DfAQ+U(a)DpG
(4.4) ,
(4.5)
with a spacetimedependent parameter, a = a(x). Suppose these gauge transformations form an abelian group, e.#., as do the set of phase trans
I~4 Gauge symmetry
15
formations of Eq. (4.1).* It is sufficient to consider transformations with just one parameter as in Eqs. (4.4)(4.5) since we can use direct products of these to construct arbitrary abelian groups. One can obtain a covariant derivative by introducing a vector field A^x), called a gaugefield,by means of the relation
DIAe = (dfA + ifAIA)e
,
(4.6)
where / is a realvalued coupling constant whose numerical magnitude depends in part on the field 0 . For example, in electrodynamics / becomes the electric charge of ©. The problem is then to determine how A^ must transform under a gauge transformation in order to give Eq. (4.5). This can be done by inspection, and we find A^^A^
+ d^U{a)'U\a) .
(4.7)
The gauge field A^ must itself have a kinetic contribution to the lagrangian. This is written in terms of a field strength, i 7 ^, which is antisymmetric in its indices. A general method for constructing such an antisymmetric second rank tensor is to use the commutator of covariant derivatives, [£>M, A,] 6 = 1/1^,0 .
(4.8)
By direct substitution we find Fp, = dpA,,  duAp .
(4.9)
It follows from Eq. (4.7) and Eq. (4.9) that the field strength F^v is invariant under gauge transformations. A gaugeinvariant lagrangian containing a complex scalar field (p and a spin 1/2 field ip, chiral or otherwise, has the form F^F
+ ( D ^ ) D ^ + i ^ ^ + ... ,
(4.10)
where the ellipses stand for possible mass terms and nongauge field interactions. There is no contribution corresponding to a gauge boson mass. Such a term would be proportional to A^A^, which is not invariant under the gauge transformation, Eq. (4.7).
Nonabelian case The above reasoning can be generalized to nonabelian groups [YaM 54]. First we need a nonabelian group of gauge transformations and a set of fields which forms a representation of the gauge group. Then we must An abelian group is one whose elements commute. A nonabelian group is one which is not abelian.
16
/ Inputs to the Standard Model
construct an appropriate covariant derivative to act on the fields. This step involves introducing a set of gauge bosons and specifying their behavior under the gauge transformations. Finally the gauge field strength is obtained from the commutator of covariant derivatives, at which point we can write down a gaugeinvariant lagrangian. Consider fields 9 = {6*} (i — 1,... , r) which form an rdimensional representation of a nonabelian gauge group Q. The O^ can be boson or fermion fields of any spin. In the following it will be helpful to think of G as an rcomponent column vector, and operations acting on © as r x r matrices. We take group Q to have a Lie algebra of dimension n, so that the numbers of group generators, group parameters, gauge fields, and components of the gauge field strength are each n. We write the spacetimedependent group parameters as the ndimensional vector a = {aa(x)} (a = 1,..., n). A gauge transformation on O is G; = U(a)e ,
(4.11)
where the r x r matrix U is an element of group Q. For those elements of Q which are connected continuously to the identity operator, we can write U(a) = exp(ia a G a ) ,
(4.12)
where G = {G a } (a = 1,..., n) are the generators of the group Q expressed as hermitian r x r matrices. The set of generators obeys the Lie algebra [G<\ G6] = zca6cGc
(a, b, c = 1,..., n) ,
(4.13)
where {ca6c} are the structure constants of the algebra. We construct the covariant derivative D^O in terms of gauge fields B^ — {#"} (a = 1,... ,n) as + igB^)@ .
(4.14)
where g is a coupling constant analogous to / in Eq. (4.6). In Eq. (4.14), I is the r x r unit matrix, and B /1 = G a SJ .
(4.15)
Realizing that the covariant derivative must transform as we infer from Eqs. (4.12)(4.14) the response, in matrix form, of the gauge fields, .
(4.17)
I~4 Gauge symmetry
17
The field strength matrix F ^ is found, as before, from the commutator of covariant derivatives, implying F ^ = dpBv  dJBp + ig[B^ Bu] .
(4.19)
Eqs. (4.17) and (4.19) provide the field strength transformation property, F%, = U(3)F U/ U 1 (5) •
(4.20)
Unlike its abelian counterpart, the nonabelian field strength is not gauge invariant. Finally, we write down the gaugeinvariant lagrangian C =  ^ T r ( F T ^ ) + (D /i $)*D^$ + z * # * + ... ,
(4.21)
where $ and * are distinct multiplets of scalar and spin 1/2 fields and the ellipses represent possible mass terms and nongauge interactions. Analogously to the abelian case, there is no gauge boson mass term. The most convenient approach for demonstrating the theory's formal gauge structure is the matrix notation. However, in specific calculations it is sometimes more convenient to work with individual fields. To cast the matrix equations into component form, we employ a normalization of group generators consistent with Eq. (4.21), TV(GaG6) = ]8ah
(a, b = 1,..., n) .
(4.22)
To obtain the ath component of the field strength F^v = { i ^ } (a = 1,..., n), we matrix multiply Eq. (4.19) from the left by G a and take the trace to find
F;u = d^Bl  dvBl  gc^B^Bl
(a, b, c = 1,..., n) .
(4.23)
The lagrangian Eq. (4.21) can likewise be rewritten in component form, C = \P^F% + (D^iprn^D^knipn + i^Mi^j
+ ''• >
(424)
where a = 1,... ,n and the remaining indices cover the dimensionalities of their respective multiplets. Mixed case In the Standard Model, it is a combination of abelian and nonabelian gauge groups which actually occurs. To deal with this circumstance, let us consider one abelian gauge group Q and one nonabelian gauge group Q' having gauge fields A^ and B*1 = {B%} (a = 1,... ,n) respectively. Further assume that Q and Q' commute and that components of the
18
/ Inputs to the Standard Model
generic matter field 9 transform as an rdimensional multiplet under Q'. The key construction involves the generalized covariant derivative, written as an r x r matrix, )p + ifAIJ)l + igBpG}Q
,
(4.25)
where I is the unit matrix and / , g are distinct realvalued constants. Given this, much of the rest of the previous analysis goes through unchanged. The field strengths associated with the abelian and nonabelian gauge fields have the forms given earlier. So does the gaugeinvariant lagrangian, except now the extended covariant derivative of Eq. (4.25) appears, and both the abelian and nonabelian field strengths must be included. For the theory with distinct multiplets of complex scalar fields $ and spin onehalf fields * , the general form of the gaugeinvariant lagrangian is 1
— — —Tr ( F ^ F ) ~\~ i^lTJbty \tm\IJ 2 4 _l_ (D^$) D <& V($ ) + C,{^ *&) (4.26) where m is the fermion mass matrix, y($ 2 ) contains the $ mass matrix and any polynomial selfinteraction terms, and £(\£, 3>) describes the coupling between the spin onehalf and spin zero fields. £ —
1 piw p
1—5 On the fate of symmetries Depending on the dynamics of the theory, a given symmetry of the lagrangian can be manifested physically in a variety of ways. Apparently all such realizations are utilized by Nature. Here we list the various possibilities. 1) The symmetry may remain exact. The electromagnetic gauge U(l) symmetry, the 577(3) color symmetry of QCD, and the global 'baryonnumber minus leptonnumber' (B — L) symmetry are examples in this class. 2) The apparent symmetry may have an anomaly. In this case it is not really a true symmetry. Within the Standard Model the global axial [/(I) symmetry is thus affected. Our discussion of anomalies begins in Sect. III3. 3) The symmetry may be explicitly broken by terms (perhaps small) in the lagrangian which are not invariant under the symmetry. Isospin symmetry, broken by electromagnetism and lightquark mass difference, is an example. 4) The symmetry may be 'hidden' in the sense that it is an invariance of the lagrangian but not of the ground state, and thus one does not
15 On the fate of symmetries
19
'see' the symmetry in the spectrum of physical states. This can be produced by different physical mechanisms. a) The acquiring of vacuum expectation values by one or more scalar fields in the theory gives rise to a spontaneously broken symmetry, as in the breaking of SU(2)L invariance by Higgs fields in the electroweak interactions. b) Even in the absence of scalar fields, quantum effects can lead to the dynamical breaking of a symmetry. Such is the fate of chiral SU(2)L x SU(2)R symmetry in the strong interactions. The various forms of symmetry breaking in the above are quite different. In particular, the reader should be warned that the word 'broken' is used with very different meanings in case (3) and the cases in (4). The meaning in (3) is literal  what would have been a symmetry in the absence of the offending terms in the lagrangian is not a symmetry of the lagrangian (nor of the physical world). Although the usage in (4) is quite common, it is really a malapropism because the symmetry is not actually broken. Rather, it is realized in a special way, one which turns out to have important consequences for a number of physical processes. The situation is somewhat subtle and requires more explanation, so we shall describe its presence in a magnetic system and in the sigma model. Hidden symmetry The phenomenon of hidden symmetry occurs when the ground state of the theory does not have the full symmetry of the lagrangian. Let Q be a symmetry charge as inferred from Noether's theorem, and consider a global symmetry transformation of the vacuum state ,
(5.1)
where a is a continuous parameter. Invariance of t h e vacuum,
eiaQ\0) = \0)
(alia) ,
(5.2a)
implies that Q0) = 0 .
(5.2b)
In this circumstance, the vacuum is unique and the symmetry manifests itself in the 'normal' fashion of mass degeneracies and coupling constant identities. Such is the case for the isospin symmetric model of nucleons and pions discussed in Sect. 13, where the lagrangian of Eq. (3.6) implies the relations g(ppn0)
= g(nmr°)
=
(
+
) / / 2
( ~ ) / / 2
20
/ Inputs to the Standard Model
with TT* = (TTI =F ^2) /y/%Alternatively, if new states a) 7^ 0) are reached via the transformations of Eq. (5.1), we must have 0 .
(5.4)
Since, by Noether's theorem, the symmetry charge is timeindependent, Q = i[H,Q] = 0 ,
(5.5)
all of the new states a) must have the same energy as 0). That is, if EQ is the energy of the vacuum state, H0) = i?oO)> then we have H\a) = HeiaQ\0) = eiaQH\0) = Eo\a) .
(5.6)
Because the symmetry transformation is continuous, there must occur a continuous family of degenerate states. Can one visualize these new states in a physical setting? It is helpful to refer to a ferromagnet, which consists of separate domains of aligned spins. Let us focus on one such domain in its ground state. It is invariant only under rotations about the direction of spin alignment, and hence does not share the full rotational invariance of the hamiltonian. In this context, the degenerate states mentioned above are just the different possible orientations available to the lattice spins in a domain. Since space is rotationally invariant, there is no preferred direction along which a domain must be oriented. By performing rotations, one transfers from one orientation to another, each having the same energy. Let us try to interpret, from the point of view of quantum field theory, the states which are obtained from the vacuum by a continuous symmetry transformation and which share the energy of the vacuum state. In a quantum field theory any excitation about the ground state becomes quantized and is interpreted as a particle. The minimum excitation energy is the particle's mass. Thus the zero energy excitations generated from symmetry transformations must be described by massless particles whose quantum numbers can be taken as those of the symmetry charge(s). Thus we are led to Goldstone's theorem [Go 61, GoSW 62]  if a theory has a continuous symmetry of the lagrangian which is not a symmetry of the vacuum, there must exist one or more massless bosons (Goldstone bosons). That is, spontaneous or dynamical breaking of a continuous symmetry will entail massless particles in the spectrum. This phenomenon can be seen in the magnet analogy, where the excitation is a spinwave quantum. When the wavelength becomes very large, the spin configuration begins to resemble a uniform rotation of all the spins. This is one of the other possible domain alignments discussed above, and to reach it does not cost any energy. Thus in the limit of
15 On the fate of symmetries
21
infinite wavelength (A —• oo), the excitation energy vanishes (E yielding a Goldstone boson.*
—>
0),
Spontaneous symmetry breaking in the sigma model We proceed to a more quantitative analysis of hidden symmetry by returning to the sigma model of Sect. 13. Let us begin by inferring from the sigma model lagrangian of Eq. (3.14) the potential energy V{*,
ir) =  ^
(<x2 + TT 2 ) + \
(<x2 + TT 2 ) 2
.
(5.7)
With n2 negative, minimization of V(a, TT) occurs for the unique configuration a = 7T = 0. Hidden symmetry occurs for // 2 positive, where minimization of V(a, n) reveals the set of degenerate ground states to be those with /^
(5g)
A
Let us study the particular ground state,
Other choices yield the same physics, but require a relabeling of the fields. For this case, field fluctuations in the pionic direction do not require any energy, so that the pions are the Goldstone bosons. Defining a = av
,
(5.10)
we then have for the full sigma model lagrangian
ir
7T75) ip  Xva (a2 + TT2)   [(a 2 + TT2)2  v4
(5.11) Observe that the pion is massless, while the a and nucleon fields are massive. Thus at least part of the original symmetry in the sigma model lagrangian of Eq. (3.14) appears to have been lost. Certainly the mass degeneracy ma = m^ is no longer present, although the normal pattern of isospin invariance survives. However, the full set of original symmetry currents remain conserved. In particular the axial current of Eq. (3.23), In the ferromagnet case, the spin waves actually have E oc p 2 ~ A 2 for low momentum. In Lorentzinvariant theories, the form E oc p is the only possible behavior for massless single particle states. For a more complete discussion, see [An 84].
22
/ Inputs to the Standard Model
which now appears as ^ ^ ^  v d ^
+
rfd^ad^
,
(5.12)
still has a vanishing divergence, d^A1^ = 0. We warn the reader that to demonstrate this involves a complicated set of cancellations. For a normal symmetry, particles fall into massdegenerate multiplets and have couplings which are related by the symmetry. The isospin relations in Eq. (5.3) are an example of this. In a certain sense, a hidden symmetry likewise gives rise to degenerate states whose couplings are related by the symmetry. The degeneracy consists of a state taken alone or accompanied by an arbitrary number of Goldstone bosons. For example, in the sigma model it can be a nucleon and the same nucleon accompanied by a zeroenergy massless pion which are degenerate. Moreover, the couplings of such configurations are restricted by the symmetry. Later, we shall derive identities (c/. Sect. IV5) like ^
,
(5.13)
where O is some local operator and iV, Nf are nucleons or other states. Such softpion theorems relate the couplings of the N states to those of the degenerate TTN states. This is the analog for hidden symmetry of coupling constant relations for a normal symmetry. To summarize, if a symmetry of the theory exists but is not apparent in the singleparticle spectrum, it still can have a great deal of importance in restricting particle behavior. What happens is actually quite remarkable  in essence symmetry becomes dynamics. One obtains information about the excitation or annihilation of particles from symmetry considerations. In this regard, hidden symmetries are neither less 'real' nor less useful than normal symmetries  they simply yield a different pattern of predictions. Problems 1) The Poincare Algebra a) Consider the spacetime (Poincare) transformations, x** —• h^x"+ a^, where A ^ A ^ = g^v. Associated with each coordinate transformation (a, A) is the unitary operator £/(a, A) = exp(ia /i P /i — ^e^M^). For two consecutive Poincare transformations there is a closure property, J7(a7, A')£/(a, A) = U(...). Fill in the dots. b) Prove that (7(a\0)tf(a',0)t/(a,0) = *7(a',0), and by taking a'^a^ infinitesimal, determine [ ]
Problems
23
c) Demonstrate that (A" 1 )^ = A^A, and then show that [7(0, hl)U{af, A')t/(O, A) = U{K~la', A^A'A). d) For infinitesimal transformations we write AMA ~ <7^A + eMA. Prove that ea\ = —e\(j and hence Ma\ = — M\ a. Upon taking primed quantities in (c) to be infinitesimal, prove £7(0, A~1)P/iC/(0, A) = M*VPV and U{Q,hrl)M^U{Q,K) = A ^ A ^ M ^ . Finally, letting unprimed quantities be infinitesimal as well, determine [M a/3, and [M a ^,M^]. 2) The Meissner Effect in gauge theory [Sh 81] The lagrangian for the electrodynamics of a charged scalar field is \
Tip)  V(
with covariant derivative D^ = 6^ + ieA^ and potential energy, (A>0) . a) Identify the electromagnetic current of the tp field. b) For m2 > 0, show that the ground state is
'
To see that F£x\ indeed acts like an applied field, show that if one disregards the field
•{ °
Again, these correspond to unscreened and screened phases of the electromagnetic field. f) Calculate the energy of the two phases if F^t describes a constant magnetic field. Show that the phase in part (e) with
II Interactions of the Standard Model
A gauge theory involves two kinds of particles, those which carry 'charge' and those which 'mediate' interactions between currents by coupling directly to charge. In the former class are the fundamental fermions and nonabelian gauge bosons, whereas the latter consists solely of gauge bosons, both abelian and nonabelian. The physical nature of charge depends on the specific theory. Three such kinds of charge, called color, weak isospin, and weak hypercharge, appear in the Standard Model. The values of these charges are not predicted from the gauge symmetry, but must rather be determined experimentally for each particle. The strength of coupling between a gauge boson and a particle is determined by the particle's charge, e.g. the electronphoton coupling constant is —e, whereas the Mquark and photon couple with strength 2e/3. Because nonabelian gauge bosons are both charge carriers and mediators, they undergo selfinteractions. These produce substantial nonlinearities and make the solution of nonabelian gauge theories a formidable mathematical problem. Gauge symmetry does not generally determine particle masses. A fermion mass term, although violating chiral symmetry, is consistent with gauge invariance. Although gaugeboson mass would seem to be at odds with the principle of gauge symmetry, the WeinbergSalam model contains a dynamical procedure, the Higgs mechanism, for generating mass for both gauge bosons and fermions alike. I I  l Quantum Electrodynamics Historically, the first of the gauge field theories was electrodynamics. Its modern version, Quantum Electrodynamics (QED), is the most thoroughly verified physical theory yet constructed. QED represents the best introduction to the Standard Model, which both incorporates and extends it. 24
IIl
Quantum Electrodynamics
25
U(l) gauge symmetry Consider a spin 1/2, positively charged fermion represented by field ip. The classical lagrangian which describes its electromagnetic properties is Cem = ^F2
+ ^(Hpm)i/j
.
(1.1)
Here, the covariant derivative is D^ = (d^+ieA^ip^ m and e are respectively the mass and electric charge for ^ , A^ is the gauge field for electromagnetism, F^ is the gaugeinvariant field strength, and F2 = F^VF^V. This lagrangian is invariant under the local t/(l) transformations
tl){x) > e fo (*ty(x) , Ap{x)+All{x)
l
+ e dl]La{x) .
(1.2) (1.3)
The associated equations of motion are the Dirac equation
(ipme4)^ = 0,
(1.4)
and the Maxwell equation e$Yip .
(1.5)
It is worthwhile to consider in more detail the important subject of C/(l) gauge invariance, addressing both its extent and its limitations. (i) Universality of electric charge: The deflection of atomic and molecular beams by electric fields establishes that the fractional difference in the magnitude of electron and proton charge is no larger than 0(1O~ 20). Likewise, there is no evidence of any difference between the electric charges of the leptons e, /x, r. Whatever the source of this charge universality may be, it is not the U(l) invariance of electrodynamics. For example assume that in addition to ^ , there exists a second charged fermion field ip' with charge parameter /3e. It is easy to see that gauge invariance alone does not imply j3 = 1. The electromagnetic lagrangian for the extended system is Cem  ~
F2 + ^ {ilf)  m)
(16)
where DJI/J f = (d^ + i/3eA^(x))^ . The above lagrangian is invariant under the extended set of gauge transformations ,
i//(x) > e  ^ < * y (x) ,
(17)
This demonstration of gauge invariance is valid for arbitrary /?, and thus says nothing about its value. The f/(l) symmetry is compatible with, but does not explain, the observed equality between the magnitudes of the electron and proton charges. We shall return to the issue of charge
26
/ / Interactions of the Standard Model
quantization in Sect. II—3 when we consider how weak hypercharge is assigned in the WeinbergSalam model. (ii) A candidate quantum lagrangian: The quantum version of Cem is in fact the most general Lorentzinvariant, hermitian, and renormalizable lagrangian which is C/(l) invariant. Consider the seemingly more general structure
ZF2 + iZ^\Tp^
+ iZ^Ip^
M$l
M*tfl (1.8)
where Z, ZR^ are constants,^ is the covariant derivative of Eq. (1.1), and M can be complexvalued. This lagrangian not only apparently differs from £em, but seemingly is CPviolating due to the complex mass term. However, under the rescalings
4
e' = Z^e ,
<
L
= 4'SW ,
(1.9)
we obtain
C> = \F* + 09 V  M^l where Mf = {ZRZL)~1^2M.
 M'> L VH ,
(1.10)
A subsequent global chiral change of variable e
il>L,R = ~*a75V>L,i2
( " = constant)
(1.11)
does not affect the covariant derivative term but modifies the mass terms,
^Yi>l^
. (1.12)
Choosing the parameter a so that Im (M'e2ta) = 0 and defining m = Re(M'e2ta), we see that £gen reduces to Cem which appears in Eq. (1.1). (iii) Renormalizability and [7(1)  Renormalizability plays a role in the preceding discussion because [7(1) symmetry by itself would admit a larger set of interaction terms. In principle, J7(l) invariant terms like ^a^F^, WF^F^, ^YYl^^FnvFap, etc. could appear in the QED lagrangian. However, they do not because the condition of renormalizability admits only those contributions which have dimension d < 4. As discussed in App. C3, the canonical dimension of boson and fermion fields is d = 1, 3/2 respectively, and each derivative adds a unit of dimension. Accordingly, the above candidate operators have d = 5, 7, 7 and ^a^F^Fa^, thus are ruled out. There remains an operator, F^F**" = which is gaugeinvariant and has dimension 4. A noteworthy aspect of this quantity is that, unlike the other operators encountered thus far, it is odd under CP. This follows from writing it as —8E • B and realizing that under CP, E —• E and B —• —B. However, a simple exercise shows that we can identify this operator as a fourdivergence F^VF^V = where K» = £e» UOL(3AvdaAp. Thus a contribution proportional to F
IIl Quantum Electrodynamics
27
can be of no physical consequence. Upon integration over spacetime, it becomes a surface term evaluated at infinity. There is nothing in the structure of QED which would cause such a surface term to be anything but zero. QED to one loop The perturbative expansion of QED is carried out about the free field limit, and is interpreted in terms of Feynman diagrams. Two distinct phenomena are involved, scattering and renormalization. The latter encompasses both an additive mass shift for the fermion (but not for the photon) and rescalings of the charge parameter and of the quantum fields. To carry out the calculational program requires a quantum lagrangian £QED t° establish the Feynman rules, a regularization procedure to interpret divergent loop integrals, and a renormalization scheme. One can develop QED using either canonical or pathintegral methods. In either case a proper treatment necessitates modification of the classical lagrangian. As we have seen, the [7(1) gauge symmetry implies a certain freedom in defining the A/i(x) field. Regardless of the quantization procedure adopted, this freedom can cause problems. For canonical quantization, the procedure of selecting a complete set of coordinates and their conjugate momenta is upset by the freedom to gauge transform away a coordinate at any given time. For path integration, the integration over gauge copies of specific field configurations gives rise to specious divergences (cf. App. A6). In either case, superfluous gauge degrees of freedom can be eliminated by introducing an auxiliary condition which constrains the gauge freedom. There are a variety of ways to accomplish this. The one adopted here is to employ the following gaugefixed lagrangian, £QED
=
]F2
  L ( 0 • A)2 + ^ {ip eofi
mo)ip ,
(1.13)
where eo and mo are respectively the fermion charge and mass parameters. The quantity £o is a realvalued, arbitrary constant appearing in the gaugefixing term. This term is Lorentzinvariant but not U(l) invariant. One of its effects is to make the photon propagator explicitly dependent on £o The value £o — 1 corresponds to Feynman gauge, whereas the limit £o —> 0 defines the Landau gauge. The zero subscripts on the mass, charge, and gaugefixing parameters denote that these bare quantities will be subject to infinite renormalizations, as will the quantum fields. This process is characterized in terms
28
/ / Interactions of the Standard Model
of quantities Zi and <Sra,
1> = Zxfyr ,
A^ = Z\I2AI ,
e 0 = ZiZ 2  1 Z 3 " 1/2 e ,
m0 = m  6m ,
(1.14)
where the superscript V labels renormalized fields. The renormalization constants Zi, Z2, and Z3 (associated respectively with the fermionphoton vertex, the fermion wavefunction, and the photon wavefunction) and the fermion mass shift 6m are chosen order by order to cancel the divergences occurring in loop integrals. For vanishing bare charge eo = 0, they reduce to Zi?2,3 = 1, 6m = 0. The Feynman rules for QED are: fermionphoton
vertex.
p
a

(1.15)
fermion propagator iSap(p): P
i (p + rao)Qjg
p
p2 — raj + ie
photon propagator

»»
a
(1.16) iD
fJ>u
(q):
(1.17) In the above e is an infinitesimal positive number. The remainder of this section is devoted to a discussion of the oneloop radiative correction experienced by the photon propagator.* Throughout, we shall work in Feynman gauge.
Fig. II—1 The full photon propagator as an iteration. * We shall leave calculation of the fermion selfenergy to Prob. II—3 and analysis of the photonfermion vertex to Sect. Vl.
IIl Quantum Electrodynamics
29
Let us define a proper or oneparticle irreducible {IPI) Feynman graph such that there is no point at which only a single internal line separates one part of the diagram from another part. The proper contributions to photon and to fermion propagators are called selfenergies. The point of finding the photon selfenergy is that the full propagator iD'^v can be constructed via iteration as in Fig. IIl. Performing a summation over selfenergies, we obtain iD' = —i
where the proper contribution
V
2 0
(1.19)
is called the vacuum polarization tensor. It is depicted in Fig. II2(a) (along with corrections to the photonfermion vertex and fermion propagator in Figs. II2(b)(c)), and is given to lowest order by
=He)2 / A T TV y ^
10!^1
1 •
(1.20) This integral is divergent due to singular high momentum behavior. To interpret it and other divergent integrals, we shall employ the method of dimensional regularization [BoG 72, 'tHV 72, Le 75]. Accordingly, we consider UaP(q) as the fourdimensional limit of a function defined in d spacetime dimensions. Various mathematical operations, such as summing over Lorentz indices or evaluating loop integrals, are carried out in d dimensions and the results are continued back to d = 4, generally expressed as an expansion in the variable* e = 4 — d. Formulae relevant to this procedure are collected in App. C4. For all theories
pq (a)
(b)
(c)
Fig. II—2. Oneloop corrections to (a) photon propagator, (b) fermionphoton vertex, and (c) fermion propagator. We shall follow standard convention is using the symbol e for both the infinitesimal employed in Feynman integrals and the variable for continuation away from the dimension of physical spacetime.
30
// Interactions of the Standard Model
described in this book, we shall define the process of dimensional regularization such that all parameters of the theory (such as e2) retain the dimensionality they have for d = 4. In order to maintain correct units while dimensionally regularizing Feynman integrals, we modify the integration measure over momentum to (1.21) The parameter ji is an arbitrary auxiliary quantity having the dimension of a mass. It appears in the intermediate parts of a calculation, but cannot ultimately influence relations between physical observables. Indeed, there exist in the literature a number of variations of the extension to d ^ 4 dimensions. These are able to yield consistent results because one is ultimately interested in only the physical limit of d = 4. Let us now return to the photon selfenergy calculation to see how the dimensional regularization is implemented. The selfenergy of Eq. (1.20), now expressed as an integral in d dimensions, is
2
e
J (2?r)d
]
)
(1.22) where we retain the same notation 11°^(g) as for d — 4 and we have already computed the trace. Upon introducing the Feynman parameterization, Dirac relations, and integral identities of App. C4, we can perform the integration over momentum to obtain
f I
x(l  x) /
9
9/1
\\*/9
'
[ml  qlx(\  x)flA (1.23) We next expand Tla^(q) in powers of e and then pass to the limit e —• 0 of physical spacetime. In doing so, we use the familiar ae = l + eln a + ... , (1.24) /o Jo
and take note of the combination r(e/2)
2
^
where 7 = 0.57221... is the Euler constant. The presence of e~l makes it necessary to expand all the other edependent factors in Eq. (1.23) and to take care in collecting quantities to a given order of e. To order e2, the
II1 Quantum Electrodynamics
31
vacuum polarization in Feynman gauge is then found to be
6TT 2
(1.26) The above expression is an example of the general property in dimensional regularization that divergences from loop integrals take the form of poles in e. These poles are absorbed by judiciously choosing the renormalization constants. Renormalization constants can also have finite parts whose specification depends on the particular renormalization scheme employed. One generally adopts a scheme which is tailored to facilitate comparison of theory with some set of physical amplitudes. In the minimal subtraction (MS) renormalization, the Z{ subtract off only the epoles, and thus have the very simple form, ^T
(* = 1.2,3).
(127)
n=l
Because the < Z\ H have no finite parts, they are sensitive only to the ultraviolet behavior of the loop integrals, and the c^n are independent of mass. The simple appearance of the MS scheme is somewhat deceptive since further (finite) renormalizations are required if the mass and coupling parameters of the theory are to be asociated with physical masses and couplings. A related renormalization scheme is the modified minimal subtraction (MS) in which renormalization constants are chosen to subtract off not only the epoles but also the omnipresent term ln(47r) — 7 of Eq. (1.25). Minimal subtraction schemes are typically used in QCD where, due to the confinement phenomenon (cf. Sect. II—2), there is no natural renormalization scale that could naturally be associated with the mass of a freely propagating quark. Yet another approach is the onshell (os) renormalization, where the renormalized mass and coupling parameters of the theory are arranged to coincide with their physical counterparts.
32
/ / Interactions of the Standard Model Onshell renormalization of the electric charge
The renormalization scale for electric charge is set by experimental determinations typically involving solid state devices like Josephson junctions. These refer to probes of the electromagnetic vertex — eT J/(p2,pi) of Fig. II2(b) with onshell electrons (p2 = Pi = m e 2 ) an<^ with q2 = (jp\ — P2)2 — 0. The value of the electromagnetic fine structure constant a = e2/4?r obtained under such conditions is given in rationalized units by a" 1 = 137.0359895(61) .
(1.28)
To interpret this in the context of the theoretical analysis performed thus far, recall from Eq. (1.18) how the photon propagator is modified by radiative corrections, 2 ~>
ie2D
'
=
 ^
We display only the g^v piece since, in view of current conservation, only it can contribute to the full amplitude upon coupling the propagator to electromagnetic vertices. The above suggests that we associate the physical, renormalized charge e with the bare charge parameter eo by
4

(1JW
>
In this onshell renormalization prescription, the g^v part of the photon propagator ID' (q) is seen to assume its unrenormalized form in the physical limit q2 —• 0. The appellation 'onshell' means that the physical kinematic point q2 = 0 is used to implement the renormalization condition and by absorbing the singular vacuum polarization in the electric charge, one ensures that the photon has zero mass. Likewise, in the onshell renormalization approach fermion propagators have poles at their physical masses. Next, we show how to infer the form of the renormalization constant Z% i n the onshell scheme. There is a relation, called the Ward identity, that implies Z\ = Z2 as a consequence of the gauge symmetry of the theory. Prom Eq. (1.14), this gives (1.31) Use of the relation e2 = Z3 tion constant to be
e§ then specifies the onshell renormaliza
IIl
Quantum Electrodynamics
33
One can similarly absorb the epole in either the MS or MS schemes by adopting
(1 33)
Eqs. (1.32), (1.33) display how the various renormalization constants differ by finite amounts. The epoles in the fermion selfenergy and the fermionphoton vertex can be dealt with in the same manner and we find, e.g. in MS renormalization (c/. Prob. II—3 and Sect. Vl),
^ m 
+ O(e4) .
(1.35)
Electric charge as a running coupling constant The concept of electric charge as a 'running' coupling constant is motivated by the following consideration. In the perturbative Feynman expansion for a given theory, the hope is that corrections to the lowest order amplitudes will be small. However, potentially large corrections of the form In Q2/QQ can arise if the theory is renormalized at scale QQ but then applied at a very different scale Q2. It is convenient to deal with this problem by absorbing such logarithms into scaledependent or 'running' renormalized coupling constants and masses. To see why scaledependent charge is not an unreasonable concept, consider the vacuum polarization process of Fig. II—3, which depicts virtual production of a fermion of charge Qie together with its antiparticle near a charge source. Due to the source, each such vacuum fluctuation is polarized, and thus the source becomes screened. All charged fermion species contribute to the screening, and the larger the mass of the virtual pair, the closer they lie to the source. The effect is somewhat akin to concentric onion skins, with each virtual pair forming a layer, resulting in an effectively scaledependent source charge.
Fig. II—3
Virtual pair production in the vicinity of a charge.
34
// Interactions of the Standard Model
Let us seek a method for specifying a running fine structure constant a(q) for nonzero momentum transfers, with a(0) to be identified with the a of Eq. (1.28). The interpretation of eg/(l + Re H(q)) as a running charge is appealing since it would maintain the simple —i/q 2 structure of the lowest order photon exchange amplitude. The fact that H(q) is divergent (see Eq. (1.26)) can be circumvented by subtracting off its value at q2 = 0 to define a finite quantity tl(q) = H(q) — n(0) and defining
so that a(q) = e2(g)/47r. It is not difficult to deduce the behavior of from the integral representation of Eq. (1.26), and we find
£{*,
""
(137)
Observe that the arbitrary energy scale // is absent from ft(#), as would be expected since fi(q) is a physically measureable quantity. The above formulae correspond to the loop correction of one fermion of mass m. Generally, loops from all available fermions must be included, although contributions of heavy (m2 » q 2) fermions are seen to be suppressed. Important modern applications of the Standard Model engender phenomena at scales provided by the gauge boson masses M\y,Mz. To obtain an estimate for a(Myy), we can apply Eq. (1.36) to find
 ! ) +  ] • MS) If a sum over quarkloops (each being accompanied by the color factor Nc = 3) and leptonloops is performed, then the mass values in Table 11 yield the approximate determination a~l(M^) ~ 128. The main uncertainty in this approach arises from quarks. It is possible to perform a more accurate evaluation of a(M^r) (cf Sect. XVI5) which avoids this difficulty. Let us return to the question of how to define a momentumdependent coupling. To emphasize the fact that a 'running fine structure constant' is after all a matter of definition, let us consider a somewhat different derivation (and definition) of a(q2). One is able to renormalize the electric charge in a massindependent scheme [We 73] by calculating renormalization constants with m = 0. If we return to the vacuum polarization
IIl
Quantum Electrodynamics
35
diagram, but with m = 0, we find e/2
(1.39)
In order to apply the renormalization program, we must specify the value of the coupling at some renormalization point*, which we choose to be q2 =  / x  , identifying
(1.40) However, if we had chosen a different renormalization point fi2R , we would have obtained a different value,
The functional dependence of the charge on the renormalization scale is embodied in the socalled beta function of electrodynamics [GeL 54],
It can be shown [Po 74] that the leading and nexttoleading terms in a perturbative expansion of /3QED are independent of both renormalization and gauge choices. The quantity e2(/i^) defined by integrating the beta function,
^TT "— • #QED(e)
(143>
fiR
is not exactly the same quantity as the running coupling constant defined in Eq. (1.36), differing by a (small) finite renormalization. For example, the electron contribution to the running coupling in the range ml < ji 2R < M$y is
°\A)\^mi
 «" V S )l i = M s, = ^ In ^ f ,
(1.44)
which contains the dominant logarithmic dependence, but differs from Eq. (1.38) by a small additive term. However, complete calculations of * Note that the renormalization point fiR and the scale factor /x in dimensional regularization need not be identical. They are sometimes confused in the literature, and hence we use a different notation for the two quantities.
36
/ / Interactions of the Standard Model
all corrections to physical observables using the two schemes will yield the same answer. Since the running coupling constant is but a bookkeeping device, one's choice is a matter of taste or of convenience. Regardless of the specific definition employed for a(# 2 ), we see that as the energy scale is increased (or as distance is decreased), the running electric charge grows, as is expected on physical grounds from vacuum polarization. The use of a massindependent scheme is convenient for identifying the high energy scaling behavior of gauge theories. One useful feature is in the calculation of the oneloop beta function. Dimensional analysis requires that the oneloop charge renormalization be of the form, 9 = 90
 9ob ( Z~2 )
( +
finite t e r m s
(1.45)
)
where g is the 'charge' associated with the gauge theory being considered. Choosing the renormalization point as q2 = —^ 2R and forming the beta function as in Eq. (1.42), we see that (3 = bg3. This allows the beta function to be simply identified with the coefficient of e" 1 to this order. II—2 Quantum Chromodynamics Chromodynamics, the nonabelian gauge description of the strong interactions, contains quarks and gluons instead of electrons and photons as its basic degrees of freedom [PrG 72, MaP 78]. A hallmark of quantum chromodynamics is asymptotic freedom [GrW 73a,b, Po 73], which reveals that only in the shortdistance limit can perturbative methods be legitimately employed. The necessity to employ approaches alternative to perturbation theory for longdistance processes motivates much of the analysis in this book. SU(3) gauge symmetry Chromodynamics is the SU(3) nonabelian gauge theory of color charge. The fermions which carry color charge are the quarks, each with field t 3L m , where a = u, d, s,... is the flavor label and j = 1,2,3 is the color index. The gauge bosons, which also carry color, are the gluons, each with field Aa, a = 1,..., 8.* Classical chromodynamics is defined by the lagrangian
£color = ~\ F?F%, + J2 ^VttVjk ~ ™
(Q)
Wia) ,
(21)
* In this section, it will be particularly important to explicitly display color indices. We shall reserve indices which begin the alphabet for gluon color indices (e.g., a,b,c — 1,...,8 ), use midalphabetic letters for quark color indices (e.g., j,k,£ = 1,2,3 ), and employ greek symbols for flavor indices.
II2 Quantum Chromodynamics
37
where the repeated color indices are summed over. strength tensor is F% = d^Al  dvAl  9ifahcA\Al
The gauge field
,
(2.2a)
gs is the SU(3) gauge coupling parameter, and the quark covariant derivative is T>^ = (dfl + igsAa^W
.
(2.2b)
The lagrangian of Eq. (2.1) is invariant under local SU(3) transformations of the color degree of freedom, under which the quark and gluon fields transform as given earlier in Eqs. (14.11), (14.17). Equations of motion for the quark and gluon fields are
(2 3)
*
In its quantum version, the gs —• 0 limit of £coior describes an exceedingly simple world. There exist only free massless spin one gluons and massive spin onehalf quarks. However, the full theory is quite formidable. In particular, accelerator experiments reveal a particle spectrum which bears no resemblance to that of the noninteracting theory. The group SU{3) has an infinite number of irreducible representations R. The first several are R = 1, 3, 3*, 6, 6* 8, 10, 10*, . . . , where we label an irreducible representation in terms of its dimensionality. Quarks, antiquarks, and gluons are assigned to the representations 3, 3*, 8 respectively. We denote the group generators for representation R by {Fa(R)} (a = 1 , . . . , 8). The quantities A/2 are group generators for the d = 3 fundamental representation, i.e., F(3) = A/2. They have the matrix representation 0 0 A4 = . 0 0 1 0 A2 = \ i
0
0
A5 =
0 0 0 0 i 0 0
A3 = I 0
1
0 I
A6 =  0 u 0
0 0u
i> 0 .
^
=
/ —i 0 0 1I j
73 0 0
.
1 0 /
0
73 0
0 0 2
73(2 4^ y
'
J
As generators, they obey the commutation relations [Aa, A6] = 2i/ a6c A c
(a, b, c = 1 , . . . , 8)
(2.5a)
38
/ / Interactions of the Standard Model
where the /coefficients are totally antisymmetric structure constants of SU(3). There exist corresponding anticommutation relations {Aa, Xb} = pab I + 2dabc\c
(a, 6, c = 1,..., 8)
(2.5b)
with dcoefficients which are totally symmetric. Values for fabc and dabc are given in Table II—1. Useful trace relations obeyed by the {Aa} are Tr Aa = 0
(a = l , . . . , 8)
(2.6)
from Eq. (2.4) and TV \aXb = 26ab (a, b = 1,..., 8) from Eq. (2.5). The statement of completeness takes the form, KjKi = —SijSid + 26aSjk
(t, j ,fc,J = 1,2,3) ,
(2.7) (2.8)
where a = 1,..., 8 is summed over. Useful labels for the irreducible representations of SU(3) are provided by the Casimir invariants. For any representation R, the quadratic Casimir invariant C2(R) is defined by squaring and summing the group generators {Fa(i?)}5 8
o=l
There is also a thirdorder Casimir invariant, 8
CS(R)I=
] T dabcFa(R)Fb(R)Fc(R) a,6,c=l
Table II—1. Nonvanishing /, d coefficients abc
123 1 147 1/2 156 1/2 246 1/2 257 1/2 345 1/2 367 1/2 458 V3/2 678 V3/2
dabc
abc
dabc
118 l/>/3 355 1/2 146 1/2 366 1/2 157 1/2 377 1/2 228 1/VS 448 l/2>/3 247 1/2 558 l/2>/3 256 1/2 668 l/2\/3 338 1/V3 778 l/2\/3 344 1/2 888 1/V5
.
(2.10)
II2 Quantum Chromodynamics
39
The quark and antiquark states form the bases for the smallest nontrivial irreducible representations of SU(3). It is possible to use products of them, say p factors of quarks and q factors of antiquarks, to construct all other irreducible tensors in SU(S). Each irreducible representation R is then characterized by the pair (p, q). For example, we have the correspondences 1 ~ (0,0), 3 ~ (1,0), 3* ~ (0,1), 8 ~ (1,1), 10 ~ (3,0), etc. The (p, q) labeling scheme provides useful expressions for the dimension of a representation, dip, q) = (p + l)(q + l)(p + q + 2)/2 ,
(2.11)
and of the two Casimir invariants, C2(p,
3)/l8.
From Eq. (2.12) we find C 2 (3) = C2(3*) = 4/3 for the quark and antiquark representations. Equivalently, upon setting j — k and summing in Eq. (2.8) we obtain
Kj^i = fh
= ^C2(3)6il.
(2.13)
Generators for the d = 8 regular (or adjoint) representation are determined from the structure constants themselves, (Fa(8))bc = ifabc
(a, 6, c = 1 , . . . , 8) .
(2.14)
It follows directly from Eq. (2.14) and from using Eq. (2.12) to compute C 2 (8) = 3 that (215)
facdfbcd = ^2(8) 8ah = 3 6ab • This result, in turn, enables us to determine = ifabcfbcd^d = ^ 2 ( 8 ) Aa .
^
(2.16)
As a final example involving 5C/(3), we evaluate the quantity a b 6b A6]  [A0a, A"]A \b]Xb& + + A Xb6XAb6XAa a + + AXaA"A XX A6AaA" =  ( A6[Aa , A"] 2 V
a
6
c
y
= 4C 2 (3)A + i/ a6c [A , A ] = 4 (C 2 (3)  C 2 (8)) Aa .
(2.17)
Shortly, we shall see how such combinations of color factors arise in various radiative corrections.
40
II Interactions of the Standard Model
Including only gaugeinvariant and renormalizable terms, we can write the most general form for a chromodynamic lagrangian as
(2.18) where the flavor matrices ZL,R are hermitian, color and flavor indices are as before, except that for simplicity we suppress quark color notation. The final contribution to Eq. (2.18) is called the 0term. We can reduce £gen to the form of £Color by first rescaling, A>a = z A
>
£
gz ?
(
)
and then diagonalizing and rescaling with respect to quark flavors, IPL,R = UL,RTPL,R,
UL>RZL,RUIR = AL,R,
<
f l
= A ^ ' ,
(2.20)
where AL,R are diagonal. Finally we diagonalize the mass terms
Anass = J'LaM'a^
 SSWfy? ,
(221)
where M' = A^ ' ULMU^A^ ' , by means of yet another set of unitary transformations on the quark fields. Aside from the 0term, this results in the canonical expression for £coior °f Eq. (2.1). We shall demonstrate later in Sect. IX5 that the above quark mass diagonalization procedure induces a modification in the ^parameter, M' .
(2.22)
This does not imply 0 = 0 because both 0 and the original quark mass matrices are arbitrary from the viewpoint of renormalizability and SU(3) gauge invariance. In fact, the 0term cannot be ruled out by any of the tenets which underlie the Standard Model. Moreover, although the 0term can be expressed as a fourdivergence (2 23)
^ ,
' (2.24)
analysis demonstrates that K^1 is a singular operator and that its divergence cannot be summarily discarded as was done in electrodynamics. This is a curious situation because the 0term is CPviolating. Thus, one is faced with the spectre of large CPviolating signals in the strong interactions. Yet such effects are not observed. Indeed, it has been estimated that the 0term generates a nonzero value for the neutron electric dipole moment de(n) ~ 5 x 10~16 0 ecm, but to date no signal has been
II2 Quantum Chromodynamics
41
observed experimentally, de(n) < 10~25 ecm. This provides the upper bound 9 < 2 x 10"10. Perhaps Nature has dictated 0 = 0, albeit for reasons not yet understood. QCD to one loop To develop Feynman rules for QCD, we must first obtain an effective lagrangian which properly addresses the issue of SU(3) gauge freedom. For the £7(1) gauge invariance of QED, this was accomplished by adding a gaugefixing term to the classical lagrangian. The situation for SU(3) is analogous, but somewhat more complicated due to its nonabelian structure. If we continue to use a Lorentzinvariant gaugefixing procedure, the effective QCD lagrangian (for simplicity, consider just one quark flavor) can be expressed as
Ca + gsfi
Bare quantities carry the subscript '0' and the field strengths and covariant derivative are defined as in Eqs. (2.2a), (2.2b). The quantities {ca(x)} (a — 1,..., 8) are called ghost fields. As explained in App. A6, they are anticommuting cnumber quantities (i.e., Grassmann variables) which couple only to gluons. Ghosts occur only within loops, and never appear as asymptotic states. Each ghost field loop contribution must be accompanied by an extra minus sign, analogous to that of a fermionantifermion loop. Their presence is a consequence of the Lorentzinvariant gaugefixing procedure. In alternative schemes such as axial or temporal gauge, ghost fields do not appear, but compensating unphysical singularities occur in Feynman integrals instead. The Feynman rules for QCD are threegluon vertex: v,b
P^n^r
f
 p)v]
q
S \,c
(2.26)
quarkgluon vertex. n,a
y)jfc
S P,k —»
2
•_
a ,j
(2.27)
42
/ / Interactions of the Standard Model
fourgluon vertex:  Wlfl[(fabefcde(9fi\9v
(2.28) ghostgluon vertex.
•
I
*. _ c
(2.29)
J
quark propagator iS ap(p): itijk U> + mo)ap
P
p2 ml + ie
(2.30) b
gluon propagator iD" u(q):
q2
(2.31)
ghost propagator.
b
m
a
(2.32)
The above rules involve a total of four distinct interaction vertices. Of these, the threegluon and fourgluon selfvert ices, and the ghostgluon coupling have no counterpart in QED. That all four vertices are scaled by a single coupling strength g$ is a consequence of gauge invariance. Also, chromodynamics exhibits a certain coupling constant universality, called flavor independence, in the quarkgluon sector. All fields which transform according to a given representation of the SU(3) of color have the same interaction structure, e.g., all triplets couple alike, all octets couple alike but differently from triplets, etc. Quarks are assigned solely to the color triplet representation. Thus, the quarkgluon interaction is independent of flavor.
//# Quantum Chromodynamics
43
The renormalization constants of QCD are defined by 53,0 = ^ 1 ^ 3 ^—
Zf
4A
53 , o CI'X ^° *
JLJ O
2
€o = ^ 3 € 5
1
Z 3  1/2 < 73 ,
= ^ 1 ^ 3 ^3
(2.33)
53 j
mo = m — 6m . In the following, working in £o — 1 gauge we shall compute the oneloop contributions to the gluon selfenergy and to the quarkgluon vertex, and by absorbing the epoles, thereby obtain expressions for Z3 and Z\F to leading order. Determination of the remaining renormalization constants, which can be computed from loop corrections to the quark and ghost propagators and the threegluon, fourgluon and ghostgluon vertices will be left as exercises. However, it is clear from the definition of #3,0 in Eq. (2.33) that the relations Z\ Z\
Z\ Z3
Z\ Z3
must hold in any consistent renormalization scheme. These are the analogs of the Ward identities in QED. Physically, they ensure that the coupling constant relations which appear in the QCD lagrangian (as a consequence of gauge invariance) are maintained in the full theory. The QCD oneloop contribution to the quarkantiquark vacuum polarization amplitude of Fig. II4(a)*, quark
2
J (2?T) (Z.60)
*
differs from the QED selfenergy only by the group factor (Aa)jfc(A6)jtj = Tr (AaA6) = 26ab (cf. Eq. (2.7)). Comparing with Eq. (1.42), we obtain
+ . . . ] • (2.36) This must be multiplied by the number of quark flavors rif which contribute in the vacuum polarization loops. To avoid notational clutter, we shall not put subscripts on the bare coupling for the remainder of this subsection.
44
77 Interactions of the Standard Model
(a)
(b)
(c)
Fig. II—4. Oneloop corrections to the gluon propagator: (a) quarkantiquark pair, (b) gluon pair, and (c) ghosts.
The contribution from the gluongluon intermediate state of Fig. II4(b) can be written y74 7
A/*^
with N$ = gsf^ig^q + *), + ^(2fc  q){, + 9v0{2q  fc)M] acd t x g3f W a(q + k)v +
]
(2.39)
with Nap = ("5
+ (2d  3) (qakf3 + 9/jfca) + (6  4d)fck Integration of Eq. (2.39) yields
The final contribution to the gluon propagator is the ghost loop amplitude of Fig. II4(c),
=
~ J (2 — [93f
acd
ka]
.
II2 Quantum Chromodynamics
45
The bracketed quantities arise from the gluonghost vertices, and the minus prefactor must accompany any ghost loop. Following the standard steps to a ddimensional form, we arrive at
J (2^
ghost
which becomes to leading order in e,
The sum of gluon and ghost contributions takes the gaugeinvariant form
(2.45) Finally, adding the quark contribution for n/ flavors gives the total result
Renormalizing at q2 = — ji
(2.46)
2
R,
we find
Proceeding next to the quarkgluon vertex, written through first order as
= if
(a)
•••
(b)
Fig. II—5 Oneloop corrections to the quarkgluon vertex.
,
(2.48)
46
// Interactions of the Standard Model
we see from Fig. II—5 that there are radiative corrections from both quark and gluon intermediate states. The quark contribution is 3
/»
J4
7
*
OL3
(2 49)
Aside from the replacement e —• g% and a color factor AbAaAb/8, which is evaluated in Eq. (2.17), the remaining expression is the QED vertex which will be analyzed in detail in Sect. Vl. Thus we anticipate from Eq. (Vl.24) that at p\ = p2 = p and \p\2 » ra 2,
= (C2(S)  \C2(8)) & 1
(^j
(2.50) The twogluon intermediate state, which has no counterpart in QED, has the form
fJ  k  pi)Q + gpa(2k  pi  pi)v + g w (2pi  k [k?  m2 + te][(pi  A;)2 + ze]2[(p2  A;)2 + ie]
(2.51) By a nowstandard set of steps, it is not difficult to extract the epole from the extension of the above to d dimensions, and we find
J^
\
... , (2.52)
implying a total vertex correction of the form,
(K(P,P)hLt = (A7%7, [C2(3) + C2(8)] ^
(^j
(2.53) We thus determine the renormalization constant for the quarkgluon vertex at pf = —i^\ to be
Z1F = 1 [C2(3) + C 2 (8)]i ( ^ y i + ...) .
(2.54)
There remains the task of determining Z2 We shall leave this for an exercise (c/. Prob. II.3) and simply quote the result
i
+
...).
(2.55,
II2 Quantum Chromodynamics
47
Asymptotic freedom and renormalization group A striking property of QCD is that of asymptotic freedom. This is the statement that, unlike the electric charge, the coupling constant QZ{IJLR) of color decreases as the scale of renormalization fiR is increased. To demonstrate this, we proceed just as we did for QED, using the relation .
(2.56)
Prom the coefficient of e"1, we learn that
 ^ = /3QCD =  [jC 2 (8)  ^C 2 (3)] ^
+ O{gl) ,
(2.57a)
which implies
/?QCD =  [ y C 2 ( 8 )  ^

^ (2.57b)
The sign of the leading term in /?QCD is negative for the sixflavor world rif = 6, becoming positive only if the number of quark flavors exceeds sixteen. As we have already seen, the QED vacuum acts as a dielectric medium with dielectric constant CQED > 1 because spontaneous creation of charged fermionantifermion pairs results in screening (i.e., vacuum polarization) of electric charge. The dielectric property CQED > 1 means that the QED vacuum has magnetic susceptibility HQED < 1? &nd thus is a diamagnetic medium. The QCD vacuum is the recipient of similar effects from virtual quarkantiquark pairs, but these are overwhelmed by contributions from virtual gluons. As a result, the QCD vacuum is a paramagnetic medium (//QCD > 1) and antiscreens (CQCD < 1) color charge [Hu 81]. The effect of asymptotic freedom can be displayed most clearly by performing a renormalization group (RG) analysis on the IP I amplitudes of the theory. A connected* renormalized Green's function is defined in coordinate space as G^n»\{x})
= (0\T (f{xx)...
Ar{xn)) 0)conn
(2.58)
where the numbers of quark and gluon fields are rip, TIB respectively, and for convenience we suppress color and Lorentz indices. We employ the * All the fields participating in a connected Green's function are affected by interactions; in a disconnected Green's function, one or more of the field quanta propagate freely.
48
/ / Interactions of the Standard Model
same symbol G^nF'UB^ for the momentum Green's function (2TT)464(PI
+..
i (2.59) where n = np + % . The \PI amplitudes r(" FiTlB) are obtained by removing the externalleg propagators from G^/jr' , (2.60) where unprimed (primed) momenta represent initial (final) states. The relations of Eq. (2.33) imply for any renormalization scheme, which we need not specify yet, that G
Z ~Z2 * _CT° ' (2.61) D — Z3 Do; S = Z2 So , where the zero subscript denotes unrenormalized quantities. Prom this, we have
and the combination of terms ^
(263)
is therefore independent of the renormalization scale //#. Let us now ascertain the behavior of r( n F ' n B ) in the deep Euclidean kinematic limit where all momenta {p} are both spacelike (in order to avoid singularities) and very large compared to any other mass scale in the theory. To keep the situation as simple as possible, we omit the dependence of T^nFTlB^ on both the quarkmass m(/x^) and gauge £(/J>R) parameters. Then from Eq. (2.59) we find in response to a scale transformation p —> Xp that
This behavior is almost that of a homogeneous function occurring in a scale invariant theory. Canonical dimensions of the fields appear in the exponent of the scaling factor along with an additive factor of four arising from the fourmomentum delta function in Eq. (2.59). However in G^nF'nB\ there is also an implicit dependence on A due to the presence of the renormalization scale /i#. The corresponding scaling property of
II2 Quantum Chromodynamics
49
the IP I amplitude is found from Eqs. (2.63), (2.64) to be (2.65)
or
(2.66) B
( ' This functional relationship can be converted to a differential RG equation by taking the Aderivative of both sides and then setting A = 1,
(2.67)
where ^
In Zl212 ,
1B
= / i * /  In Z\12
(2.68)
are called the anomalous dimensions of the respective fields and /3QCD is as in Eq. (2.57). Let us now see how to obtain leading order estimates for the anomalous dimensions and the beta function. To this order, the result for /?QCD is both gauge and renormalization schemeindependent. To start, we can use the result of Eq. (2.55) to determine 7^, ' < 2  69) and analogously for 75. To solve the RG equation, we introduce the scaling variable t = In A and the running coupling constant g3 (£), ^f
,
93(0)=93
(270)
Then it is straightforward to verify that the solution to Eq. (2.67) is where
V(t) = exp ( jT dtf [nBlB (gs(t')) + nFlF {g^t1))} )
(2.72)
is the anomalous dimension factor. The scaling behavior of the IP I amplitude is seen to have field dimensions with anomalous contributions in addition to the canonical values.
50
77 Interactions of the Standard Model
Despite naive expectations, the interaction strength at the scaled momentum is not #3, but rather the running coupling constant ~g% whose magnitude decreases as the momentum is increased. Employing the lowest order contribution Eq. (2.57) for /3QCD> w e c a n integrate Eq. (2.70) over the interval t\ < t < t^ to obtain  2 (11  2n//3) (t2  *I)/16TT2 ,
(2.73)
where nj is the number of quark flavors having mass less t h a n yft^. It is conventional t o express this relation in a somewhat different form. Defining a scale A at which ~g% diverges and letting as(q2) = g^(2 we have ( 2^
12?r
(33  2nf) 6(153  19n/) In (ln(g2/A2)) 1(33  2n/) 2
where rif is the number of quark flavors with mass less than y q2, and for completeness we have included the nexttoleading term. If as{q2) continues to grow as q2 is lowered, any perturbative representation of /3QCD ultimately becomes a poor approximation, and we can no longer integrate Eq. (2.70) with confidence. Although unproven, a popular working hypothesis is that the QCD coupling indeed continues to grow as the energy is lowered, leading to the phenomenon of quark confinement. In QED, the free parameter a(q2 ~ 0) ~ 1/137 is quite small and expansions in powers of a converge rapidly. However QCD behaves differently. In particular it is clear from Eq. (2.74) that as is not really a free parameter, but is instead inexorably related to some mass scale, e.g., A. This phenomenon, called dimensional transmutation, means that an energy such as A can effectively serve to replace the dimensionless quantity as in the formulae of QCD. Specifying QCD operationally requires not only a lagrangian but also a value for A. For example, QCD perturbation theory is useful only if large mass scales M2 ((A/M) 2 << 1) are probed. Because the complexity of low energy QCD has thus far prevented direct analytic solution of the theory, there have been substantial efforts to develop alternative approaches. These include attempts to solve QCD numerically (latticegauge theory), phenomenological study of various theoretical constructs (potential, bag, Skyrme models), exploitation of the invariances contained in £QCD (notably chiral and flavor symmetries), and consideration of the infinite color limit iVc —• oc as a first approximation to QCD (N~* expansion). The first of these topics is beyond the scope of this book [Cr 83], but the others will form the basis for much of our discussion.
II3 Electroweak interactions
51
Table II—2. Determinations of as Experiment
as (34 GeV)
0.14 ±0.02 0.127 ±0.006 DILS T decay 0.123 ±0.009 e+e~ —> jets 0.135 ±0.015 0.120 ±0.016 77 Re+e~
Ag^(MeV)
A^(MeV)
370 ± 350 220 220 ± 60 180 ± 80 330 ± 200 175 ± 125
240 ± 230 140 140 ± 40 120 ± 50 215 ± 130 115 ± 80
Attempts to infer as(q2) from experimental data are typically carried out under kinematic conditions for which a perturbative analysis of QCD presumably makes sense. Systems commonly used for this purpose include deep inelastic lepton scattering (DILS) structure functions, T decay, e+e~ scattering data (both total hadronic cross section (oc R) and jet structure), and photon (77) structure functions. Suppose, as is often the case, a given process is computed to leading and nexttoleading order in QCD perturbation theory and regularized in the MS scheme. If such a theoretical expression is then used to fit the data with a q2 characteristic of the given process employed, an expression such as Eq. (2.74) can be used to infer a value of A and as(q2) can be evolved to different q2. Since this operation depends on both the regularization procedure and the number of quark flavors rif used in Eq. (2.74), a notation like A—^ would be precise. Unfortunately there is no uniformity in the rate of convergence of QCD perturbation theory from process to process. Thus determinations of as(q2) are affected by both theoretical and experimental uncertainties, and a scatter of quoted values results. We display in Table II—2 a collection of values inferred from data collected at moderate energies [Al 89]. Additional determinations from higher energy LEP data have more recently become available [He 92], [ 0.115 ± 0.008
= { 0.141 ±0.017
(jet structure)
(rrViT)
•
(2J5)
Collectively, the LEP data imply as = 0.120 ± 0.007, which corresponds To summarize, determinations of as(q2) are qualitatively in accord consistent with the q2 dependence of Eq. (2.74). Taken over the full range of available data, values in the range 0.1 < A(GeV) < 0.4 are not uncommon.
52
77 Interactions of the Standard Model II—3 Electroweak interactions
The WeinbergSalamGlashow model [Gl 61, We 67, Sa 69] is a gauge theory of the electroweak interactions whose input fermionic degrees of freedom are massless spin onehalf chiral particles. It has the group structure SU{2)L x C/(l), where the SU(2)ii U(l) represent weak isospin and weak hypercharge respectively. The subscript 'L' on SU(2)L indicates that among fermions, only lefthanded states transform nontrivially under weak isospin.
Weak isospin and weak hypercharge assignments First we shall discuss how the fermionic weak isospin (Tw, Tw%) and weak hypercharge (Yw) quantum numbers are assigned. The fermion generations are taken to obey a 'template' pattern  we assume that each succeeding generation differs from the first only in mass. Thus, it will suffice to consider just the lightest fermions for the remainder of this section. The first generation electroweak assignments are displayed in Table II3. For weak isospin, experience gained from charged weak current interactions such as nuclear beta decay dictates that lefthanded fermions belong to weak isodoublets while righthanded fermions be placed in weak isosinglets, as in e
leptons :
£L = (
quarks:
qL = I : I
j
eR u R dR
.
Observe that by assumption there is no righthanded neutrino. Each of the degrees of freedom displayed above must be assigned a weak hyperTable II3. SU(2)L x assignments Particle Tw ue eL eR UL
dL UR
dR
1/2 1/2 0 1/2 1/2 0 0
Tw3
Yw
1/2 1 1/2 1 2 0 1/2 1/3 1/2 1/3 4/3 0 0 2/3
II3 Electroweak interactions
53
charge. There are a priori five in all,*
Y(qL) = Yq, Y(uR) = Yu, Y{dR) = Yd, Y(eR) = Ye.
(3.2) Shortly, we shall see that the {Yi} are related to the fermion electric charges. One might therefore attempt to proceed by using the latter to determine the former. However, given the weak isospin assignments already made, it turns out to be both possible and worthwhile to use the theoretical apparatus of the WeinbergSalam model to predict the {Yi}. Several constraints can be obtained by employing the anomaly cancelation conditions alluded to previously in Sect. 12 and to be discussed in detail in Sect. Ill—3 (see especially Eq. (III3.60b) and subsequent discussion). In particular, the conditions TrF 3 2 F w = 0 , 2 = 0, = 0 ,
(3.3a) (3.3b) (3.3c)
which arise from the cancelation requirement, imply respectively the relationships 2Yq = Yu + Yd ,
6Yq33Yu3 = 3Yd32Ye3 + Ye3 ,
(3.4a)
(3.4c)
where in Eq. (3.3a), F% is the third generator of the octet of color charges, and "Tr' represents a sum over fermions. To obtain further constraints, we note the linear relation between the electric charge Q and the SU(2)L X U(l)y quantum numbers TW3 and Yw, aQ = TwS + bYw ,
(3.5)
where a, b are constants. We can use the freedom in assigning the scale of the electric charge Q to choose a = 1. Ultimately, of course, the lefthanded and righthanded components of the charged chiral fermions must unite to form the physical states themselves. Consistency demands that the electric charges of the chiral components of each such charged fermion * The reason that weak hypercharge engenders so many free parameters in contrast to weak isospin lies in the difference between an abelian gauge structure (like weak hypercharge) and one which is nonabelian (like weak isospin). Thus all doublets have the same weak isospin properties irrespective of their other properties, analogous to flavor independence in QCD. For the abelian group of weak hypercharge, the group structure by itself provides no guidelines for assigning the weak hypercharge quantum number. Like the electric charge, it is a priori an arbitrary quantity.
54
77 Interactions of the Standard Model
be the same, whatever value that charge might have. From this, we learn Yq Yu
Yq Yd+ Ye Ye + (3 bj  ~2b'  2b' ~ 2b • Then, for example, insertion of the above into Eqs. (3.4b), (3.4c) yields
'+ ^ 1 =0 •
(37)
Thus all products bY{ (c/. Eq. (3.5)) are determined, resulting in the observed electric charge values (c/. Table I.I). The above analysis is instructive in several respects. First, it reveals that once the weak isospin is chosen as in Eqs. (3.1), (3.2) and all possible chiral anomalies are arranged to cancel, one obtains the observed quantization of electric charge. Also, we see that any attempt to determine weak hypercharge values from the known fermion electric charges is affected by an arbitrariness associated with the value of 'b\ This accounts for the variety of conventions seen in the literature. For definiteness, we have taken b = 1/2 in Table II—3, so that .
(3.8)
SU(2)ixU(l)y gaugeinvariant lagrangian Having assigned quantum numbers, we turn next to the electroweak interactions. The WeinbergSalam lagrangian divides naturally into three additive parts, gauge (G), fermion (F), and Higgs (J5T), Avs = CG + CF + CB. (3.9) Throughout this section we shall concentrate on establishing the general form of the electroweak sector, referring at times to only a few treelevel amplitudes. We shall return in Chap. V to the subject of electromagnetic radiative corrections, and present the electroweak Feynman rules along with various radiative corrections in Chap. XVI. The gauge boson fields which couple to the weak isospin and weak hypercharge are respectively W^ = (W^, Wj*, Wj*) and By,. These contribute to the purely gauge part of the lagrange density as
\
J
\
,
(3.10)
where F^v (i = 1,2,3) is the SU(2) field strength,
Flv = dliWlduWJlg2eiikWiW* and B^, is the U(l) field strength, B^ = d^Bv  d^
.
,
(3.11) (3.12)
IIS
Electroweak interactions
55
The fermionic sector of the lagrange density includes both the lefthanded and righthanded chiralities. Summing over lefthanded weak isodoublets ipL and righthanded weak isosinglets ipn, we have
Since righthanded chiral fermions do not couple to weak isospin, their covariant derivative has the simple form D^R
= (d^i^YwB^R
.
(3.14)
This expression serves to define the U(l) coupling g\. Its normalization is dictated by our convention for weak hypercharge Yw. The corresponding covariant derivative for the SU(2)L doublet ipL is )
L,
(3.15)
given in terms of the SU{2) gauge coupling constant gi and the 2 x 2 matrices I, r . We shall not display the quark color degree of freedom in this section for reasons of notational simplicity. However, it is understood that all situations in which quark internal degrees of freedom are summed over, as in Eq. (3.13), must include a color sum. Similarly, relations like Eq. (3.14) or Eq. (3.15) hold for each distinct internal color state when applied to quark fields. The above equations define a mathematically consistent gauge theory of weak isospin and weak hypercharge. However, it is not a physically acceptable electroweak theory of Nature because the fermions and gauge bosons are massless. A Higgs sector must be added to the above lagrangians to arrive at the full WeinbergSalam model. Thus we introduce into the theory a complex doublet
(£)
(316)
of spinzero Higgs fields with electric charge assignments as indicated. The quanta of these fields then each carry one unit of weak hypercharge. The Higgs lagrange density Cn is the sum of two kinds of terms, £HG and £HF> which contain the Higgsgauge and Higgsfermion couplings respectively. The former is written as (3.17) where
56
/ / Interactions of the Standard Model
and V is the Higgs selfinteraction, (3 1 9 ) 2
The parameters fi and A are positive but otherwise arbitrary. For simplicity, we write the Higgsfermion interaction in this section for just the first generation of fermions. Denoting the lefthanded quark and lepton doublets respectively as qi and ^£, we have £HF = ML^R
~ fdqL^dR  feh$eR + h.c. ,
(3.20)
where the coupling constants fui /^, and fe are arbitrary and we employ the charge conjugate to 3>, $ = 2T2$* .
(3.21)
There is no term in Eq. (3.20) containing a righthanded neutrino because we assume such particles do not exist. In a sense the Higgs potential V and Higgsfermion coupling £HF lie outside our guiding principle of gauge invariance because neither contains a gauge field. However, there is no principle which forbids such contributions, and their presence is phenomenologically required. Moreover, note that each is written in SU(2)L x [/(I) invariant form. Spontaneous symmetry breaking Mass generation for fermions and gauge bosons proceeds by means of spontaneous breaking of the SU(2)L X [/(I) symmetry. To begin, we obtain the ground state Higgs configuration by minimizing the potential V to give $ (  / i 2 + 2A$ f $) = 0 .
(3.22)
We interpret this ground state relation in terms of vacuum expectation values, denoted by a zero subscript. Eq. (3.22) has two solutions, the trivial solution ($)o = 0 and the nontrivial solution, (3.23) with
v= ][^ •
(3.24)
Let us consider the latter alternative. A nontrivial vacuum Higgs configuration which obeys the constraint Eq. (3.23), respects conservation of electric charge, and describes the spontaneous symmetry breaking of the original SU(2)L x [7(1) symmetry is
°)
.
(3.25)
II3 Electroweak interactions
57
In one interpretation, it is the order parameter for the WeinbergSalam model, playing a role analogous to the magnetization in a ferromagnet. Group theoretically, it is seen to transform as a component of a weak isodoublet. The energy scale, v, of the effect is not predicted by the model and must be inferred from experiment. The fermion and gauge boson masses are determined by employing Eq. (3.25) for the Higgs field everywhere in the lagrange density CH. We first define chargedfieldsWjjf,
j
\
.
(3.26)
corresponding to the gauge bosons W±. By substitution, we find for the mass contribution to the lagrange density
= j=(fuuu + fddd + feee)+(^f^ 9l92\ ' " ' " N 9\ )
'
(3 27)
'
The fermion masses are given by v ma = —f=fa {OL = u, d, e, ...) . (3.28) V2 Although the theory can accommodate fermions of any mass, it does not predict the mass values. Instead, the measured fermion masses are used to fix the arbitrary Higgsfermion couplings. The charged VFboson masses can be read off directly from Eq. (3.27), Mw = ^g2
,
(3.29)
but the symmetry breaking induces the neutral gauge bosons to undergo mixing. Their mass matrix is not diagonal in the basis of VF3, B states. Diagonalization occurs in the basis Z^ = cos0w Wl  sin0w B^ , A^ = sin 0W W* + cos 9W B^ ,
where the weak mixing angle (or Weinberg angle) 6W is defined by tan0 w = ^
.
(3.31)
92
The neutral gauge boson masses are found to be (3.32)
58
/ / Interactions of the Standard Model
and the fields A^ and Z^ correspond to the massless photon and massive Z°boson respectively. Observe that the W±toZ° mass ratio is fixed by .
(3.33)
Electroweak currents Now that we have determined the mass spectrum of the theory in terms of the input parameters, we must next study the various gaugefermion interactions. The traditional description of electromagnetic and low energy charged weak interactions of spin onehalf particles is expressed as /4 /eh
fr> QA\
where Jem is the electromagnetic current
JL = e^e +  « y «  idVd + • • • ,
(3.35)
J^h is the charged weak current (ignoring quark mixing) J
ch = ^ 7 ^ ( 1 + 75)e + 5 ^ ( 1 + 75)d + • • • ,
(3.36)
and GF ^ 1.166 x 10"5 GeV"2 is the Fermi constant (cf. Sect. V2). Alternatively, we can use Eqs. (3.13)(3.15) to obtain the charged and neutral interactions in the SU(2)L X U(1)Y description,  giD^yJem — ^
w3J
,
[6.6()
where J^ 3 is the third component of the weak isospin current, "T
'
,
(3.38)
summed over all lefthanded fermion weak isodoublets. Substituting for B^ and W^ in Eq. (3.37) in terms of A^ and Z^ yields ^ J ^ + £ntlwk ,
(3.39)
775 Electroweak interactions
59
where J ^ = 2 J^ 1 + i 2 is given in Eq. (3.36) and the neutral weak interaction £nti_wk for fermion / is*
= T(f)
_
— ^w3
2
. 2Q
Z S m
Specifically, we have
.(u,c,t) _ I _ f
gin2
OU)
^w V e l
g
(/) ,
5a
= T (/)
—
(tx,c,*) _
" " 2
w3
I
1 2 v
i
1 3
~~
(3'41)
2 '
9v
Observe the structure of the neutral weak couplings g^l. If 0W were to vanish, neutral weak interactions would be given strictly in terms of TW3, the third component of weak isospin. However in the real world, phenomena like low energy neutrino interactions, Mw/Mz, deep inelastic lepton scattering data, etc. all depend on the value of 0W. In addition, we note that because sin2 #w ~ 0.23, the leptonic vector coupling constants g{e,H,T) a r e substantially suppressed relative to the axialvector couplings. Comparison of Eq. (3.34) with Eq. (3.39) yields e = (7icos#w = <72sin#w .
(3.42)
The Fermi interaction of Eq. (3.34) corresponds in the WeinbergSalam model to a secondorder interaction mediated by VFexchange and evaluated in the limit of small momentum transfer (1 » q 2
Together these relations provide a treelevel expression for the Wboson mass, *,2 1 ML = —o
/ 37.281 G e V \ 2 .
™ •=— ~
One should be careful not to confuse Eq. (3.40) with the alternate form
Vf = J^
W
2 sin ^ w cos ^ w which also appears in the literature.
el
£w3 2 sin ^ w cos 0^
(3.44)
60
// Interactions of the Standard Model
Also, Eqs. (3.29), (3.43) imply v = 2" 1/4G~1/2 ~ 246.221(2) GeV .
(3.45)
It is the quantity v which sets the scale of spontaneous symmetry breaking in the SU(2)L X [ / ( 1 ) theory, and all masses in the Standard Model are proportional to it, although with widely differing coefficients. We shall resume in Chaps. XV, XVI discussion of a number of topics introduced in this section, among them the Higgs scalar, the W± and Z° gauge bosons, and phenomenology of the neutral weak current. Also included will be a description of quantization procedures for the electroweak sector, including the issue of radiative corrections. First however, in the intervening chapters we shall encounter a number of applications involving light fermions undergoing electroweak interactions at very modest energies and momentum transfers. For these it will suffice to work with just treelevel W± and/or Z° exchange, and to consider only photonic or gluonic radiative corrections. We shall also neglect the gaugedependent longitudinal polarization contributions to the gauge boson propagators (analogous to the q^qu term in the photon propagator in Eq. (1.21)), as well as effects of the Higgs degrees of freedom. For photon propagators, the q^q" terms do not contribute to physical amplitudes because of current conservation. Although current conservation is generally not present for the weak interactions, both the q^qv propagator terms and Higgs contributions are suppressed by powers of (rrif/Mw)2 for an external fermion of mass rrif. II4 Fermion mixing In our discussion of the WeinbergSalam model, we limited the number of fermion generations to one. We now lift that restriction and consider the implication of having n generations. Although the existing experimental situation supports the value n = 3, we shall take n arbitrary in our initial analysis. Diagonalization of mass matrices To begin, it is necessary to generalize the Higgsfermion lagrangian of Eq. (3.20) to
£HF = ff qtj*ip + ff qitfidju + ff ll^h
+ h'c '
II~4 Fermion mixing
61
where we employ the summation convention a, (3 = 1,..., n , and adopt the notation d' = ( d ' , s ' , b ' ,
...),
vo
The primes signify that the states which appear in the original gaugeinvariant lagrangian are generally not the mass eigenstates. That is, there is no reason why the n x n generational coupling matrices fw, f^, fe should be diagonal. Following spontaneous symmetry breaking, we obtain the generally nondiagonal n x n mass matrices m j , mj, m / from the analog of Eq. (3.28), mQ' = — f
Q
(a = u, d, e)
(4.3)
Although not diagonal in the gauge basis, these matrices can be brought to diagonal form in the mass basis. The transformation from gauge eigenstates to mass eigenstates is accomplished by means of the steps —A\ mass = u£ mj^i? + d£ nijd£ + e£ mje£
+ h.c. ,
= UL mu UR + di md dji + ex, m e en \ h.c. = u mu u + d m.d d + e m e e . The n x n unitary matrices *S£ R (a = u, d, e) relate the basis states, U
'L
— ^L UL 5
d/ = S^ di ,
e/ = S^ e^ ,
d ^i? = S1RUR i R = s i rfi? » and induce the biunitary diagonalizations
e
i? = s /? eR 5
= SUL mu
, (4.6)
thus yielding the diagonal quark mass matrices / mu 0 0 ... \ / rrid 0 0
\
mc 0
0 mt
... I ... '
0 0
m d =
\
0 ms 0
0 0 mi,
(4.7a)
62
/ / Interactions of the Standard Model
and the diagonal lepton mass matrix
/me me =
0 0
0
0
ra» 0
0 mT
\ : : : Although the WeinbergSalam model is first written down in terms of the gauge basis states, actual calculations which confront theory with experiment are performed using the mass basis states. We must then transform from one to the other. This turns out to have no effect on the structure of the electromagnetic and neutral weak currents. One simply omits writing the primes which would otherwise appear. The reason is that (aside from mass) each generation is a replica of the others, and products of the unitary transformation matrices always give rise to the unit matrix in flavor space. Thus, at the lagrangian level, there are no flavorchanging neutral currents in the theory. As an example of this, consider the leptonic contribution to the electromagnetic current, (4.8) where we sum over family index a = 1 , . . . , n and invoke the unitarity of matrices S ^ . Note that there is no difficulty in passing the SeL R through 7^ because the former matrices act in flavor space whereas the latter matrix acts in spin space. The charged weak current of leptons behaves analogously. Transformation from gauge basis to mass basis leaves the form of the charged current essentially unchanged. However, the reasoning differs from that used above, and involves the neutrino sector in an essential way. Let us denote neutrino states (u^) in the gauge basis and (V®) in the mass basis,
H = ("e', < , "T')L ,
*i° =iyl, i/J, u% ,
(4.9)
with a transformation S£ relating the two, "£, a = (SLUVIP
(a,0 = l,...,n).
(4.10)
Upon writing down the charged leptonic current we encounter J&(lept) = 2%>"e L ' i a = 2(P £ °SL t ) a y(Sie) a ,
(4.11)
which we reexpress as J^(lept)  2vL^ea
.
(4.12)
II~4 Fermion mixing
63
That is, we employ a third neutrino basis (Z?L), vL,a = (S^SDapulp
(a, (3 = 1,..., n) ,
(4.13)
which serves to define the set of physical neutrino states. We are allowed mathematically to perform this transformation on the original choice of mass eigenstates provided the neutrinos are assumed to be mass degenerate. Existing data is consistent with all neutrinos having zero mass. Quark mixing Thus far, the distinction between gauge basis states and mass eigenstates has been seen to have no apparent effect. However, mixing between generations does manifest itself in the system of quark charged weak currents. By convention, the mixing is assigned to the Q = —1/3 quarks by )X
<
,
(4.14)
where < a = Va(,dLt(,
(a, (3 = 1,..., n) ,
(4.15)
and
V = S£fS£ .
(4.16)
Thus the Q — —1/3 quark states participating in transitions of the charged weak current are linear combinations of mass eigenstates. The quarkmixing matrix V, being the product of two unitary matrices, is itself unitary. The Standard Model does not predict the content of V. Rather, its matrix elements must be phenomenologically extracted from data. For the two generation case, V is called the Cabibbo matrix [Ca 63]. For three generations, it is called the KobayashiMaskawa matrix [KoM 73] (or sometimes simply the weak mixing matrix) and is often denoted by the abbreviations KM or CKM. We shall analyze properties of such mixing matrices for the remainder of this section. An nxn unitary matrix is characterized by n2 realvalued parameters. Of these, n(n — l)/2 are angles and n(n + l)/2 are phases. Not all the phases have physical significance, because 2n — 1 of them can be removed by quark rephasing. The effect of quark rephasing uL,a > eiB«u L,* ,
dL,a  • e^d^a
(a = 1 , . . . , n)
(4.17)
on an element of the mixing matrix is VcH^Vape^W
(<*,/?= l , . . . , n ) .
(4.18)
Since an overall common rephasing does not affect V, only the 2n — 1 remaining transformations of the type in Eq. (4.17) are effective in
64
II Interactions of the Standard Model
removing complex phases. This leaves V with (n — l)(n — 2)/2 such phases. One must be careful to transform the leftchirality and rightchirality fields of a given flavor in like manner to keep masses real. If so, all terms in the lagrangian other than V are unaffected by this procedure. For two generations, there are no complex phases. The only parameter is commonly taken to be the Cabibbo angle OQ and we write
cos0cJ * A common notation for the n = 2 mixed states is
Within the two generation approximation, weak interaction decay data imply the numerical value, sin 9c — 0.22. The three generation case involves the 3 x 3 matrix /Kd V = \Vcd
\vtd
Ks Vub\ Vcs Vch ,
(4.21)
vts vthj
written here in a form which emphasizes the physical significance of each matrix element. Current experimental data place the matrix element magnitudes in the 90% confidence intervals [RPP 90] ' 0.9738  0.9750 0.218  0.224 0.001  0.007 \ 0.218  0.224 0.9734  0.9752 0.030  0.058 0.003  0.019 0.029  0.058 0.9983  0.9996 /
.
(4.22)
The n = 3 mixing matrix can be expressed in terms of four parameters, of which one is a complex phase. The presence of a complex phase is highly significant because it signals the existence of CP violation in the theory. We shall return to this point shortly. The KobayashiMaskawa (or KM) representation employs three mixing angles 9a (a — 1, 2,3) and a complex phase 6. It can be viewed as an Eulerian construction of three rotation matrices and a phase matrix,
where sa = s i n ^ , ca = cos9a (a — 1,2,3). In combined form this becomes /
C\
— S1C3
— S1S3
\
(4.23)
II~4 Fermion mixing
65
By means of quark rephasing, it can be arranged that the KM angles #1,2,3 all he in the first quadrant. In the limit 62 = #3 = 0, KM mixing reduces to Cabibbo mixing with the identification 9\ = —6cAn alternative approach for describing quark mixing, the Wolfenstein parameterization [Wo 83], assumes that the mixing matrix is endowed with a hierarchical structure. That is, one defines A = Kis — 0.22, and expands a given matrix element of V in powers of A according to the perceived magnitude of the matrix element. One finds to order A3 (A4) in the real (imaginary) couplings, lA2/2 V= A \X3A(lpir)) /
A 1  A2/2  irjA2X4 X2A
X3A(piri(l X2A(1 + ir)X2) 1
(4.24) where A, p, 77 are realvalued parameters anticipated to be of order unity. B meson data (cf. Sect. XIV1) imply the values A = 0.95 ± 0.14 and v/p2 + T72 = 0.45 ±0.14. CPviolation and rephasinginvariants It is clear from the preceding discussion that there is no unique parameterization for three generation quark mixing. Any scheme which is convenient to the situation at hand may be employed as long as it is used consistently and adheres to the underlying principles. There is, however, a somewhat different logical position to adopt, that of working solely with rephasinginvariants. After all, only those functions of V which are invariant under the rephasing operation in Eq. (4.18) can be observable. An obvious set of quadratic invariants are the squared moduli A^j = l^ a/?2 where a, (3 = 1, 2,3. The unitarity conditions V^V = VV* = I constrain the number of independent squaredmoduli to four. They are of course all realvalued. In addition there are quartic functions Aap = V^V^rVXV^, where we suspend the summation convention for repeated indices and, to avoid redundant factors of squaredmoduli, take a, /?, 7 (p, a, r) cyclic. These nine quantities are generally complexvalued. A permutation on either index cycle, e.g. a, /?, 7 —>• a, 7, /?, maps a quartic invariant into its complex conjugate. The Aap are not all independent. If we multiply the unitarity relation V
*rVs*
(v ± r)
by V*V7T and recall the cyclic nature of the indices, we obtain A(4)  A (4) *  W \2\V I2
(4.25)
66
/ / Interactions of the Standard Model
From this line of reasoning, we can determine that there is just one independent quartic term and that all nine have the same imaginary part. There are yet higher order invariants, but they are all expressible in terms of the quadratic and quartic functions. An important consequence of our discussion thus far is that a unique measure of CP violations for three generations is given by the rephasinginvariant ImA^i . It is independent of its indices (which we therefore suppress) and is given in the KM and Wolfenstein parameterizations by Im A' 4 '  s21c1s2c2sscss6
= X6A2rj + O(XS) .
(4.27)
These combinations of mixing parameters will always appear in calculations of CP violating phenomena. To have nonzero CP violating effects, the KM angles must avoid the values #1,2,3 = 0, TT/2, and 6 = 0, TT. The CP violating invariant ImA^4^ achieves its maximum value for c\ — 1/^/3, c2 = c3 = l/>/2, s6 = 1 at which it equals 1/6A/3. This set of circumstances is very unlike the real world where current estimates place Im A ^  I O " 4 . The consideration of rephasinginvariants need not involve just the mixing matrix V, but can also be applied to the Q = 2/3, —1/3 nondiagonal mass matrices m!u, m^ themselves. In particular, the determinant of their commutator is found to provide an invariant measure of CP violations [Ja 85]. If, for simplicity, we work in a basis where m u ', m^ are hermitian, it can be shown that S^ = Su/ = S M . Thus we have m'd] = Su [m,, VmdVt] Swt  SUV [ V W V , md] V+S^ , (4.28) from which it follows that det [m^rn^] = det [mu, VmdV^
= det [V^niuV.md]
.
(4.29)
The two commutators on the right hand sides of this relation are skewhermitian and each of their matrix elements is multiplied by a Q — 2 / 3 ,  1 / 3 quark mass difference respectively. The determinant is thus proportional to the product of all Q = 2/3, —1/3 quark mass differences, and explicit evaluation reveals
det [m^,my = 2i ImA^ JJ(m M?a  mu^)(md^a  md,p) 
(4.30)
This provides a more extensive list of necessary conditions for CP violations to be present. Not only are the mixing angles constrained as discussed above, but also the quark masses within a given charge sector must not exhibit degeneracies. Our discussion of the n = 2, 3 generation cases suggests how larger systems n — 4, . . . can be addressed, although the number of parameters becomes formidable, e.g. four generations require six mixing angles
Problems
67
and three complex phases. However, existing data indicates the existence of just three fermion generations  measurements from Z°decay fail to see additional neutrinos, and there is no evidence from e + e" or pp collisions for additional quarks or leptons lighter than roughly 50 GeV. Moreover, if there were more than three quark generations the full quarkmixing matrix would be unitary, but one would expect to see violations of unitarity in any submatrix. Yet to the present level of sensitivity, the 3 x 3 KM mixing matrix obeys the unitarity constraint. The most accurate data occurs in the (V^y)n = 1 sector, where using the values (cf. Chaps. VIII, XII) Kd = 0.9737 ± 0.0010, Ks = 0.220 ± 0.003, and Kb < 0.01, we find )n = Kd 2 + Ks 2 + Kb 2  0.9965 ± 0.0032 .
(4.31)
Problems 1) SU(3) a) Starting from the general form A^A^ = A6ij (a = 1 , . . . , 8 is summed), determine A, S, C by using the trace relations of Eqs. (2.5ab), etc. b) Determine TrAaA6Ac. c) Determine e^ewA^Aj^A^A^.. d) Consider the 8 x 8 matrices (Fa)&c = —if abc, where the {fabc} are SU(3) structure constants (a,6,c = 1,...,8). Show that these matrices (the regular or adjoint representation) obey the Lie algebra of 577(3), and determine Tr FaFb. 2) Gauge invariance and the QCD interaction vertices a) Define constants / , g such that the covariant derivative of quark q is D^q = (d^ + ijA^)q and the QCD gauge transformations are A^ —• UA^U~l + ig^UdyJJ'1 and q —> Uq, where A^ are the gauge fields in matrix form (c/. Eq. (14.15)). Show that qJp q is invariant under a gauge transformation only if / = g. b) Define a constant h such that the QCD field strength is F^v — d^Av — dyA^ + ih[A^ Av], Let the gauge transformation for A^ be as in (a). Show that F^v transforms as F^v —» UF^U~l only if h = g. 3) Fermion selfenergy in QED and QCD a) Express the fermion QED selfenergy, —iS(p), of Fig. II2(b) as a Feynman integral and use dimensional regularization to verify the forms of Z{2MS), 6m,(Msr> appearing in Eqs. (1.37), (1.38). b) Proceed analogously to determine Z^ for QCD and thus verify Eq. (2.55).
68
/ / Interactions of the Standard Model
4) Gravity as a gauge theory The only force which remains outside of the present Standard Model is gravity. General relativity is also a gauge theory, being invariant under local coordinate transformations. The full theory is too complex for presentation here, but we can study the weak field limit. In general relativity the metric tensor becomes a function of spacetime, with the weak field limit being an expansion around flat space, g^v{x) — g^v + Wv\x), with 1 ^> Wv. Let us consider weak field gravity coupled to a scalar field, with lagrangian C = £grav + ^matter defined as
£matter =
\
where h^p = h^v — g^vh\j2^ all indices are raised and lowered with the flat space metric g^v and GN is the Cavendish constant. a) Show that the action is invariant under the action of an infinitesimal coordinate translation, x^ —• x/fl — x^ + e^{x) (1 ^> e /i(x)), together with a gauge change on
Ill Symmetries and anomalies
Application of the concept of symmetry leads to some of the most powerful techniques in particle physics. The most familiar example is the use of gauge symmetry to generate the lagrangian of the Standard Model. Symmetry methods are also valuable in extracting and organizing the physical predictions of the Standard Model. Very often when dealing with hadronic physics, perturbation theory is not applicable to the calculation of quantities of physical interest. One turns in these cases to symmetries and approximate symmetries. It is impressive how successful these methods have been. Moreover, even if one could solve the theory exactly, symmetry considerations would still be needed to organize the results and to make them comprehensible. The identification of symmetries and near symmetries has been considered in Chap. I. This chapter is devoted to their further study, both in general and as applied to the Standard Model, with the intent of providing the foundation for later applications. IIIl Symmetries of the Standard Model The treatment of symmetry in Sects. 14, 16 was carried out primarily in a general context. In practice, however, we are most interested in the symmetries relevant to the Standard Model. Let us briefly list these, reserving for some a much more detailed study in later sections. Gauge symmetries: As discussed in Chap. II, these are the SU(3)C x SU(2)L x U(l)y gauge invariances. It is interesting to compare their differing realizations. SU(3)C is unbroken but evidently confined, whereas SU(2)L x U(l)y undergoes spontaneous symmetry breaking, induced by the Higgs fields, leaving an unbroken U(l)em gauge invariance. Fermion number symmetries: There exist global vector symmetries corresponding to both lepton and quark number. These are of the form ^  e^Va 69
(LI)
70
777 Symmetries and anomalies
for fields of each chirality. The index a refers to either the set of all leptons or the set of all quarks, and the conserved charges Qa are just the total number of quarks minus antiquarks and the total number of leptons minus antileptons. In addition, if the neutrinos are all massless, there are separate conserved numbers for each type of lepton (e, i/e), (/i, i/^), (r, ur). Conservation of baryon number B is violated due to an anomaly in the electroweak sector, but B — L remains exact. Global vectorial symmetries of QCD: If the quarks were all massless, there would be a very high degree of symmetry associated with QCD. Even if m ^ 0, symmetries are possible if two or more quark masses are equal. Three of the quarks (c, 6, t) are heavy compared to the confinement scale AQCD and widely spaced in mass, so they cannot be accommodated into a global symmetry scheme.* However the ix, d, and s quarks are light enough that their associated symmetries are useful. The best of these is the isospin invariance, which consists of field transformations
where {T1} (i = 1,2,3) are SU(2) Pauli matrices and {#*} are the components of an arbitrary constant vector. Associated with the SU(2) flavor invariance are the three Noether currents Jf=i>i^
•
(13)
Isospin symmetry is broken by the updown mass difference, £mass =
^
\uu + dd)
(uu  dd) ,
(1.4)
and by electromagnetic and weak interactions. Inclusion of the strange quark extends isospin to SU(3) flavor transformations %j)' = exp(i0 • A)^ ,
(1.5)
where {Aa} (a = 1,2,...,8) are the SU(3) GellMann matrices. The SU(3) flavor symmetry is broken significantly by the strange quark mass, and to a lesser extent by other effects. Predictions of isospin symmetry work at the 1% level, whereas SU(3) predictions hold only to about 30%. It is occasionally convenient to employ a particular SU{2) subgroup of See, however, the discussion of the dynamical heavyquark symmetries in Sects. XIII3, XIV2.
IIIl Symmetries of the Standard Model
71
*SC/(3), called Uspin, which corresponds to the transformations .
(1.6)
[/spin is also a symmetry of the electromagnetic interaction, since its generators commute with the electric charge operator. The [/spin symmetry is broken by the large dquark, squark mass difference. Approximate chiral symmetries of QCD: The vectorial symmetries are valid if quark masses are equal. If the masses vanish, there are additional chiral symmetries, because in this limit the lefthanded and righthanded components of the fields are decoupled (cf. Sect. 13), m=0
A 4
;
,
(1.7)
i.e. the lefthanded and righthanded fields have separate invariances. For massless up and down chiral quarks, the symmetry operations are ipL + exp(WL • r)ipL = L^L ,
^R + exp(iOR • r)ipR = RipR , (1.8)
where ^L,R are chiral projections of the ij) doublet in Eq. (1.2). These can also be expressed as vector and axialvector isospin transformations, , (1.9) with Oy = ( 0 £ + 0 R ) / 2 , and 0A — {0L—0 R)/2. This invariance is variously referred to as chiral#C/(2), SU{2)L x SU{2)R or SU(2)V x SU(2)A. In QCD, it is broken by quark mass terms, J
Anass = muuumddd
= mu(uLuR+uRUL)rnd(dLdR+dRdL).
(1.10)
Thus if mu = rrid ^ 0, separate lefthanded and righthanded invariances no longer exist, but rather only the vector isospin symmetry. The generalization to three massless quarks defines chiral 5/7(3) (or SU(3)L X SU(3)R) and is a straightforward extension of the above ideas. More discussion of these chiral symmetries will be given in Chaps. VI,VII. Discrete symmetries: Since the Standard Model is a hermitian and Lorentzinvariant local quantum field theory, it is invariant under the combined set of transformations CPT. Both QCD (given the absence of the 0term) and QED conserve P, C, and T separately. By contrast, the electroweak interactions have maximal violation of P and C in the chargedcurrent sector. If a nonzero phase resides in the quarkmixing matrix, there will exist a breaking of CP, or equivalently of T, invariance. Otherwise the weak interactions are invariant under the product CP. In addition to the above exact or approximate symmetries of the Standard Model, there are some important 'nonsymmetries' of QCD. By
72
/ / / Symmetries and anomalies
these we mean invariances of the underlying lagrangian which might naively be expected to appear as symmetries of Nature but which, for a variety of reasons, do not. These include the following. Axial U(l): The QCD lagrangian would have an axial U(l) invariance of the form
if the u, d, s quarks were massless. However, this turns out not to be even an approximately valid symmetry, as it has an anomaly. We shall return to this point in Sect. Ill—3. Scale Transformations'. If quarks were massless, the QCD lagrangian would contain no dimensional parameters. The lagrangian would therefore be invariant under the scale transformations
^X) _> A3/fy(As) ,
A%{x)  \Al{\x)
,
(1.12)
where ip and A^ are respectively the quark and gluon fields. This invariance is also destroyed by anomalies (see Sect. Ill—4). 'Flavor Symmetry': Because the gluon couplings are independent of the quark flavor, one often finds reference in the literature to a flavor symmetry of QCD. Unless the specific application is reducible to one of the above true symmetries, one should not be misled into thinking that such a symmetry exists. For example, flavor symmetry is often used in this context to relate properties of the pseudoscalar mesons 7/(549) and 7/(960) (or analogous particles in other nonets). However the result is rarely a symmetry prediction. Rather, this approach typically pertains to specific assumptions about the way quarks behave, and is dressed up by incorrectly being called a symmetry. In group theoretic language, this may arise by assuming that QCD has a U(3) symmetry rather than just that of SU(3).
Ill—2 Path integrals and symmetries The transition from classical physics to quantum physics is in many ways most transparent in the path integral formalism. In this chapter we use these techniques to provide a quantum description of symmetries, complementing the treatment at the classical level of Sects. 14, 16. A brief pedagogical introduction to those path integral techniques which are important for the Standard Model is provided in App. A.
III2 Path integrals and symmetries
73
The generating functional In order to implement a quantum description of currents and current matrix elements, one studies the generating functional, W, of the theory. For a generic field (p, we have W[j] = eiZ^ = J[dtp] expij
d4x (£(, dtp)  j(p) ,
(2.1)
where j(x) is an arbitrary classical source field whose presence allows us to probe the theory by studying its response to the source. The symbol [dip] indicates that at each point of spacetime one integrates over all possible values of the field
6n In
6j(xk)...6j(xp)
(2.2) 3=0
where n is the number of fields in the matrix element. If there is more than one field, i.e. the set {
(2.3)
In this case all matrix elements involving J^ can be obtained by functional derivation with respect to v^, J»{x) = i
6\nW
(2.4)
where the bar in J^ indicates that it is a functional describing matrix elements of the current J^. Specific matrix elements are obtained by further derivatives, as in (2.5) This device allows one to discuss all possible matrix elements of the current J*\ As an example, consider the vector and axialvector currents of QED. We define
= AdVWH^K^^^
. (2.6)
74
/ / / Symmetries and anomalies
A threecurrent (connected) matrix element is obtained then as Tfuxpix, y, z)conn = (0 \T (4>(x)jtl'y5ip(x) $(y)yatp(y) tp(z)^0ip(z))  0)
lnW
[ J
^
)
(2.7) where the axialvector quantity J^ is defined in analogy with Eq. (2.4).
Noether's theorem and path integrals Returning to the general case, let us consider an infinitesimal transformation of a set offields\jp%} (2.8) such that the current under discussion is
If this is a symmetry transformation, one has up to a total derivative, C {tp\ dip') = C (
.
(2.10)
If e(x) is a constant, the lagrangian is invariant under the transformation. This is the statement of the classical symmetry condition. In order to study the consequences of this situation, we rewrite our previous definition of the current matrix elements ]
(2.11)
0Vfi{X)
in integral form by noting
8\nW[v^] = lnW[v^ + 6vfl}lnW[v^\
= i fd4xJ^{x)6v^x)
, (2.12)
which is just the inverse of Eq. (2.11). Now choosing the particular form for Svp, 6v^x) =d^(x)
,
we have 6€ ln W[v»] = ln W[v^  d^e]  ln W[v^}
= i f dAx J»(x)d^(x) = i f d4x e(x)dfiJfi(x) .
(2.13)
7773 The U(l) axial anomaly
75
With this procedure we can isolate a divergence condition for J^. If v^ — d^e] = Wfr;^], then d^J^{x) = 0. To check this, consider
expijd4x
(£fo, d
. (2.15)
If we can change integration variables so that
J[dhpi] =
(2.16)
with $ given by Eq. (2.8), then we obtain W[v^d^e)
= Jid^expiJ'd*x(£{
and therefore dPJ^x) = 0 .
(2.18)
This change of variables seems reasonable and in most cases is perfectly legitimate. After all, the symbol [d(fi(x)] means that we integrate over all values of the field cpi separately at each point in spacetime. Shifting the origin of integration at point x by a constant, (fi(x) = (f[(x) — e(x)/j, and then integrating over all values of (p[ should amount to the original integration. Given this shift, we have obtained in Eq. (2.18) by Noether's theorem a quantum conservation law involving matrix elements. The expression dfJ/JfI(x) = 0 means that all matrix elements of JM, obtained via further functional derivatives (as in Eq. (2.5)), satisfy a divergenceless condition, i.e. the current J^ is conserved in all matrix elements. It was Fujikawa who first pointed out the consequences if the change of variables, Eq. (2.16), is not a valid operation in a path integral [Fu 79]. Certainly, many procedures involving path integrals need to be examined carefully in order to see if they are welldefined. We shall explicitly study some examples in which the change of variable is nontrivial and can be calculated. In such cases one finds dflJfJ'(x) ^ 0, which implies that the classical symmetry is not a quantum symmetry. In these situations it is said that there exists an anomaly. III3 The U(l) axial anomaly For massless quarks mu = rrid = ms = 0, the lagrangian of QCD contains an invariance £QCD —> £QCD under the global U(l) axial transformations (3.1) In this limit, which we shall adopt until near the end of this chapter, Noether's theorem can be applied to identify the classically conserved
76
77/ Symmetries and anomalies
axial current,
4°]=uWu
+ dWd + sw5s,
d*40J=0 ,
(3.2)
where the superscript on J^J denotes an SU(3) singlet current. We shall see that this is not an approximate symmetry of the full quantum theory in that the current divergence has an anomaly. This can be demonstrated in various ways. For a direct 'handson' demonstration, the early discussion [Ad 69, BeJ 67, Ad 70] of Adler and of Bell and Jackiw, which we recount below, has still not been improved upon. However for a deeper understanding, Fujikawa's path integral treatment [Fu 79], also described below, seems to us to be the most illuminating. The effect of an anomaly is simply stated, although one must go through some subtle calculations to be convinced that the effect is inescapable. An anomaly is said to occur when a symmetry of the classical action is not a true symmetry of the full quantum theory. The Noether current is no longer divergenceless, but receives a contribution arising from quantum corrections. It is this contribution which is often loosely referred to as the anomaly. The Ward identities relating matrix elements no longer hold, but rather, are replaced by a set of anomalous Ward identities which take into account the correct current divergence. There are two applications of the axial anomaly which have proved to be of importance to the Standard Model. One is in connection with the SU(3) singlet axial current described above. Here the anomaly will end up telling us that the current is not conserved in the chiral limit, but rather that
d"JjS = ^Fa^F^
{F^ = d^FS0)
.
(3.3)
This will serve to keep the ninth pseudoscalar meson, the 7/, from being a pseudoGoldstone boson. The other application is in the decay TT° —> 77, which is historically the process wherein the anomaly was discovered. The quantity of interest here is an isovector axial current J^ which transforms as the third component of an SU(3) flavor octet,
(3.4)
J
Without the anomaly, one would expect that the current J§ } would be conserved in the chiral 5/7(2) limit even in the presence of electromagnetism. This follows from the apparently correct procedure
= u\(f+ iQ4) 75 + 75 (fi +
} (3.5)
III3 The U(l) axial anomaly
77
However explicit calculation shows that the current has an anomaly, such that
43 J
^
,
(3.6)
where F^v is the electromagnetic field strength. This will be important in predicting the TT° —> 77 rate and serves as a test for the number of quark colors.
Diagrammatic analysis To review the work of Adler and of Bell and Jackiw, we first consider the Ward identities for the coupling of the C/(l) axial current to two gluons. We define
, q)=ij
d4x d4y e*fcV™ (o \T (4°J(z) J°(y) J>(0)) o)
(3.7)
where J^ is a flavorsinglet (coloroctet) vector current coupled to gluons
l = £
«7«7a y<7
•
(38)
q=u,d,s
It is important to understand that the SU(3) matrices pertain here to the color degree of freedom and should not be confused with analogous matrices which operate in flavor space. The amplitude T°^Qis related to the vacuumtodigluon matrix element by a
{\u q) G\X2, k  q)
, q) .
(3.9)
There are two Ward identities, representing the conservation of axial and vector currents. The vector Ward identity, corresponding to color current conservation, is
<7 a O*;,
(3.10)
The axial Ward identity is derived in a similar fashion using the assumed conservation of the 17(1) axial current in the massless limit,
Jx) = 0 ,
(3.11)
to yield b
,q)=0
.
(3.12)
In order to reveal the anomalous behavior of this coupling, we calculate the vertex in lowest order perturbation theory via the triangle diagram of Fig. Ill—1. With the momenta as labeled in the figure, this produces
III Symmetries and anomalies
78 the amplitude
Aa
\ bl
(3.13)
where the prefactor of 3 arises from the three massless quarks, each of which contributes equally. Observe that these integrals are linearly divergent, and so may not be welldefined. In particular, there exists an ambiguity corresponding to the different possible ways to label the loop momentum. An example will prove instructive, so we consider the integral
J
P
[P4
(314)
(p
This is evaluated by transforming to Euclidean space, where po = ip± and p2 — —p\ — p 2 — —p\ In order to perform the integration, one may note that for a general function, F(p), whose fourdimensional integral is linearly divergent (i.e. one with p3F(p) ^ 0, but psF'(p) = p3F"(p) = ... = 0 for p —> oc), one finds by Taylor expanding and using Gauss' theorem that
J dAPE [F(p)  F(p  £)] =
+ ...
dzS»F{p) (3.15)
kq
kq
Fig. Ill—1 Triangle diagram associated with the axial anomaly.
III3 The U(l) axial anomaly
79
where dsS^ indicates integration over a threedimensional surface at p —• oo.* Applying this result to the case at hand, we obtain a surface integral 7
\P
\P ~ £) J
V P (3.16) Note that from euclidean covariance we can replace p^p^ by ^ 7 p 2 / 4 , to yield o
r
J
1
P
J
^2o
(3.17) where the last step uses the surface area of a threedimensional surface in fourdimensional euclidean space, S4 = 2ir2R3. In the case of T ^ , consider the effect of shifting the integration variable of the first term in Eq. (3.13) from p to p + b\q + b2(—k — q). In order to maintain the Bose symmetry of TJ^Q (i.e. symmetry under the interchange a <» (3 at the same time as q <> (—k — q)) we must shift the second integration from p to p+b\(—k — q) + b
(3.18) induced by the shift of the original integration variable jf. This is an indication that there may be trouble in the calculation of this diagram, but it is not yet proof of any violation of the Ward identities. Let us now check the Ward identities. In both cases, use can be made of identities similar to qa = pa — (pa — qa) in order to change the result into a difference of integrals. We find for the vector Ward identity
36ab r d4P
J
* Note that this is just the fourdimensional generalization of the onedimensional formula
f
dx [f(x + y) f(x)] = ^
dx \yf(x) + \y2f"{x)
= y[/(oo)/(oo)
valid for /(±oo) ^ 0 but /'(±oo) = /"(±oo) = ... = 0.
+ .. .1
III Symmetries and anomalies
80
= 6i6a
J (2*)_
k)2(pq)2
Y
(3.19)
{p
while for the axialvector case, d4p
ab
= *S ea0paj
f
757/?"
k)P(pq)c
dd4p
1
^
qfp* {pq)pp°
(3.20)
It is easy to see that if one could freely shift the integration variable, each expression would separately vanish. However, direct calculation using Eq. (3.14)Eq. (3.17) yields ) = Y^ellPpak^
and WT£0(k,q) = g ^  ^ f c V • (3.21) If, on the other hand, the original integration variable were shifted as in Eq. (3.18) one would obtain S6ab 36ab 8^"
(3.22)
Thus either one of the original Ward identities may be regained by a particular choice of 61 — 62? but both expressions cannot vanish simultaneously. The discussion of the manipulations of the Feynman diagrams should not obscure the main physical fact illustrated above, i.e., despite the claim of Noether's theorem that there are two sets of conserved currents (vector 5(7(3) of color and axialvector (7(1)), oneloop calculations indicate that only one can in fact be conserved. On physical grounds, we know that in Nature the vector current is conserved, as its charge corresponds to QCD color charge. Thus it must be the axial current which is not conserved. This phenomenon is at first sight quite surprising and it deserves the name 'anomaly' by which it has come to be called. Noether's theorem has misled us, and it is only by direct calculation of the quantum corrections that the true symmetry structure of the theory has been "ex
1/73 The U(l) axial anomaly
81
posed. Note that the situation is not the same as spontaneous symmetry breaking, where the symmetry is hidden by dynamical effects. There the currents remain conserved, as demonstrated in Sect. 16. Here current conservation has been violated. In particular, the calculation described above (with b\ — b
^4} = ^ ^ ^
( 3  23 )
•
Both sides of this equation have the same twogluon matrix elements. It is clear from this that the apparent Z7(l) symmetry predicted by Noether's theorem is not a symmetry of the quantum theory after all. Path integral analysis In a path integral treatment [Pu 79], the symmetry of the theory can be tested by considering the generating functional, as described in Sect. Ill— 2. In particular, if we consider a functional of the gluon field Ab^ and an axial current source a^,
J
(3.24)
then the steps leading to Eq. (2.14) produce
i f dAxP(x)&*J$(x)
= InW [a^  8^(3, Ab^ \nW [aM, Aj] (3.25)
where JL (x) denotes the matrix elements of the current J§ ,
J }
S
J k l n w K ^ ] L =° •
(3 26)

In particular, the twogluon matrix described above is given by
A< =0 au=0
\
(3.27)
In order to solve for d^J^J, we note that the 8^(3 term can be absorbed into a redefinition of the fermion fields. This can be seen from the identity (for infinitesimal /?), $ifhl> + 8^(3 ^ 7 5 ^ = ^ (1  i/?75) ip (1  Wis) ^ • The following quantities are invariant under this transformation:
(3.28)
82
III Symmetries and anomalies
Mass terms would not be invariant, but we are presently working in the massless limit. Therefore, if we define ^ ' = (1  i / W = e  ^ —.
—
—
+ O(f32) ,
a
l
n
if/ = V(l  i07i>) = V'e^ 7 5 + C(/32) , we see that the lagrangian can be written in terms of ip', £QCD(, & AD + &>0 J<J> = £QCD(V>', ^', A%)
.
(3.31)
1
Furthermore, we would like to change from ip to if) in the path integration. To be general, we allow for the possibility of a jacobian J accompanying this change of variables, viz.,
JW][dfi}J .
(3.32)
If, as will be shown later, the jacobian J is independent of xp and ?/>, it can be taken to the outside of the path integral, resulting in J
Thus the test for the symmetry, Eq. (3.25), depends entirely on J, lnj=i
I d4x f3{x)d^jf^{x) .
(3.34)
The lesson learned is that if the lagrangian and the path integral measure are invariant under the U(l) transformation, then there exists a [/(I) symmetry in the theory, with d^JU = 0. However if the lagrangian is invariant, as it is in this case, but the path integral is not (i.e. J / 1), then the [/(I) transformation is not a symmetry of the theory, i.e., We shall show below that the jacobian, when properly regularized, has the form
J = exp(2itr/3 75) = exp \i Jd4xf3(x)3^F^Fa^]
,
(3.35)
so that the current divergence has the form given in Eq. (3.3),
Functional differentiation using Eq. (3.27) yields the same result for q^Tf^p as obtained in ordinary perturbation theory. The nontrivial transformation of the path integral measure has prevented the axial U(l) transformation from being a symmetry of the theory. We now turn to the calculation of the jacobian.
IIIS The U(l) axial anomaly
83
The jacobian in fact diverges, and a regularization is needed in order to make it finite. In Fujikawa's original calculation the regularizer was introduced early into the procedure, allowing each step to be welldefined. We will be slightly less rigorous by introducing the regularizer somewhat later. In order to calculate the jacobian we need to review the properties of integration over Grassmann numbers (which are described in more detail in App. A6). The anticommuting nature of the variables requires that any function constructed from them terminates after linear order in each variable. Thus a function of two Grassman numbers z\, z2 {z\z2 = —z 2z\, zf = 0) becomes f(zuz2)
= /o + fizi + h*2 + fi2Ziz2 ,
(3.36)
where /o, / i , /2? /12 are real numbers. The primary property of an integral to be transferred to Grassmann numbers is completeness, i.e.
fdzf(z)=
fdz f(z + z') .
(3.37)
where zf is a constant Grassmann number. Expanding both sides we have
J dz (/o + hz) = J dz (/0 + hz + flZ')
.
(3.38)
For this to be true, the condition
fdz = 0
(3.39)
is required. Now consider a change of variables z\ = cnz[ + c12z'2 ,
z2 = C2\z[ + C22Z2 ,
(3.40)
involving a matrix of coefficients C. The jacobian is defined by / dzx dz2 f(z) = jfdz[
dz'2 /(Cz') .
(3.41)
Application of Eq. (3.36) leads to the consideration of only the f\2 term, /12 / dzi dz2 z\z2 = Jf 12 / dz[ dz2 (cnz[ + ci2z2)(c2iz[ + c22z2) = Jfi2{cnc22
 C12C21) / dz[ dz2 z[z2 , (3.42)
and hence the identification of the jacobian, ^^[detC]"1
.
(3.43)
Although derived in the simple 2 x 2 case, Eq. (3.43) generalizes to arbitrary dimension. Note that this is the inverse of the usual jacobian, due to the Grassmann nature of the variables.
84
777 Symmetries and anomalies
Turning now to the path integral, we temporarily consider ip(x) as a finite number of Grassmann variables corresponding to four Dirac indices at each point of spacetime (i.e., imagine that the spacetime label is discrete and finite). At each point, the transformation is from \p —> xj)f rp(x) = el^x^^{x)
,
$(x) = $'(x) e*^ 75 ,
(3.44)
so that the overall jacobian has the form
J = [det (e i/375 )] ~* [det (e^ 7 6 )] ~*
(3.45)
with one factor from each of the ip and jp variables. The determinant runs over the 4 x 4 Dirac indices, the three flavors, colors and also the spacetime indices. This is a rather formal object, but can be made more explicit by using = etrlnC ,
(3.46)
valid for finite matrices, to write The symbol tr denotes a trace acting over spacetime indices plus Dirac indices, flavors and colors, tr/3 7 5  Tr' f d*x (x\/3^\x)
,
(3.48)
with Tr7 indicating the Dirac, color and flavor trace. This will become clearer through direct calculation below. The jacobian still is not regulated. Fujikawa suggested the removal of high energy eigenmodes of the Dirac field in a gaugeinvariant way. Consider, for example, the simple extension
J=
lim exp[2itr f/?75 e~^^2)]
,
(3.49)
where Ip is the QCD covariant derivative. The insertion of a complete set of eigenfunctions of Ip exponentially removes those with large eigenvalues. There has been an extensive literature demonstrating that other regularization methods produce the same results as Fujikawa's, provided that the regulator preserves the vector gauge invariance. In order to complete the calculation we employ the following identity:
(350)
£>„£>"
III3 The U(l) axial anomaly
85
In this case the expression
(x\exp(ff)/M)2\x)
(3.51)
has the same form as given in Eqs. (Bl.l), (B1.9), (Bl.1718) with the identifications <*„ = !>„,
a = ^XaF^
,
r = ^
.
(3.52)
Applying the calculation done there to our present situation yields
J= = lim
lim
M —00
ei^/
M+OO
The notation is defined in App. Bl. The first two traces vanish, leaving only the factor with two a^v matrices in (22. Prom the result Tr ( 7 5 ^ ^ ) =  Tr 757 V T V = ^^uap
,
(3.54)
it is easy to calculate J exp
=
= exp (j±;Jd*x0{x) 3 • 2<5a6 • 4 ie^&F^F^
(3.55)
where the trace Tr' has produced factors for three flavors, color and the Dirac trace. Although the calculation of the jacobian has been somewhat involved, we have succeeded in making sense out of what seemed to be a rather abstract object. The fact that it is not unity is an indication that the U(l) transformation is not a symmetry of the theory. Applying Eq. (3.34) we see that
In J = iJd4xP(x)d»J^(x) = iJd4xf3(x)^F^Fr
, (3.56)
or once again
The choice of a regulator which preserves the vector SU(3) gauge symmetry is important. Whereas in the Feynman diagram approach, we had the apparent freedom to shift the integration variable to preserve either the vector of axialvector symmetries, the corresponding freedom in the path integral case is in the choice of regularization.
86
III Symmetries and anomalies
If quark masses are included, the operator relation becomes
4°J
mdd^d + mss75s) + ^F^F^
.
(3.58)
Masses do not modify the coefficient of the anomaly, basically because it arises from the ultraviolet divergent parts of the theory, which are insensitive to masses. One does not have to go through these lengthy calculations for each new application of the anomaly. The anomalous coupling for currents
4 where Ty\T^
l
(3.59)
are matrices in n the space of quark flavors, is of the th form r\bcd
=j Dbcd
^ eF^FZp
_^
+ mass terms ,
£rW  r W , T f } )
,
(3.60a) (3.60b)
where Nc is the number of colors. In particular, for the electromagnetic coupling to the isovector axial current we have (361) leading to the result already quoted in Eq. (3.6). The full content of the anomaly was given by Bardeen [Ba 69]. Consider a fermion with rj internal degrees of freedom (flavor or color) coupled to vector and axialvector currents v^a^
C = ^{ip1)fa)xP
.
(3.62)
These currents are in an rj x 77 representation Vfi
= vp + v*\k,
aM = a0/ + ajAfc .
(3.63)
Thus the axial current is J5 = ^7^75 A ^ , and the anomaly equation becomes
2i
2i
8
+ i[v^ vv\ + i[a^ av\ ,
+ i[vfJLi av\ + i[vv, aM] .
(3.64)
IIIS The U(l) axial anomaly
87
This may also be expressed in terms of the lefthanded and righthanded field tensors i^ and r^v by using the identities,
\
±a^aaf3 = 1 ( V ^ + 7>rn + 1 ( ^ + *
(3.65) In the language of Feynman diagrams, one encounters the anomaly contributions not only in the triangle diagram, but also in square and pentagon diagrams (e.g. from the a^a^^a^ term). Our previous result, Eq. (3.57), is obtained for aM = 0, v^ = g^A^Xk/2, with three flavors and three colors of quarks. We have seen that symmetries of the classical lagrangian are not always symmetries of the full quantum theory. This is the general situation when there are anomalies. These appear in perturbation theory and are associated with divergent Feynman diagrams. This sometimes gives the mistaken impression that the dynamics has 'broken' the symmetry, and hence one might expect a massless particle through the application of Goldstone's theorem. In the path integral framework the impression is different. There the symmetry never exists in the first place, as the calculation performed above is simply the path integral test for a symmetry, generalizing Noether's theorem. Hence there is in general no expectation for a Goldstone boson. Can anomalies cause problems? When the anomaly occurs in a global symmetry, such as the above U(l) example, the answer is 'no'. They just need to be taken properly into account, e.g. as in Eq. (3.61). Given the specific form of the anomaly operator relation, there exist 'anomalous Ward identities' which contain terms attributable to the anomaly [Cr 78]. These anomalies can even be associated with a variety of specific phenomena. For example, in Sect. VI3 we shall see how the decay TT°—> 77 is attributed to the axial anomaly. Another example, the anomaly in the baryonnumber plus leptonnumber current */B+L, c o u ld in principle lead to an even more dramatic kind of experimental signal. However, the proper physical conditions, if any, for it to be observed have not yet been agreed upon at this time. The presence of anomalies in gauge theories is far more serious because they destroy the gauge invariance of the theory and wreak havoc with renormalizability. Thus, one attempts to employ only those gauge theories which have no anomalies. In some cases this can be arranged by ensuring, through the group or particle content of the theory, that the coefficient Dbcd of Eq. (3.60b) vanish. For example, in the Standard Model it must be checked that this occurs for all combinations of the
88
/ / / Symmetries and anomalies
SU(3)C x SU(2)L x U(l)y generators. These were already compiled in Eqs. (II3.3ac) and were seen to lead to the quantized fermion charge values observed in Nature.
Ill—4 Classical scale invariance and the trace anomaly If the fermion masses were zero in either QED or QCD, these theories would contain no dimensional parameters in the lagrangian, and they would exhibit a classical scale invariance. The associated quark and gluon scale transformations would be ip{x) —» A3/2/0(Ax) and Aa(x) —• \Aa^{\x) for arbitrary A. We saw in Sect. 16 that this leads to a traceless energymomentum tensor, with conserved dilation current ^ ^
=0 ,
(41)
where 6^ is the energymomentum tensor. Such a situation would have drastic consequences on the theory, since all single particle states would be massless. This can be seen as follows. For any hadron H, the matrix element of the energymomentum tensor at zero momentum transfer is (H(k)\e^\H(k))=2k»k 1/
,
(4.2)
where the normalization of states is chosen in accordance with the conventions defined in App. C2. A vanishing trace would imply zero mass, i.e. (H(k)\e^\H(k))
= 0 = 2M2H .
(4.3)
This is most obviously a problem in QCD where the quark masses are small compared to most composite particle masses.* We would not expect the proton mass to vanish if the quark masses were set equal to zero, yet the scale invariance argument implies that it must. A resolution is suggested by the method which is used to renormalize the theory. In practice, renormalization prescriptions introduce dimensional scales into the theory. Most commonly, there is the momentum scale at which one specifies the running coupling constant to have a particular value, e.g. as(9l GeV) = 0.12. This in turn defines a scale A which enters the formula for the running coupling constant, Eq. (II—2.74). Thus, to fully specify QCD one needs to specify not only the lagrangian, but also a scale parameter, and the full quantum theory is not scale invariant. Although this argument does not, at first sight, seem to nullify the reasoning based on Noether's theorem, it turns out that the trace of the energymomentum tensor has an anomaly [Cr 72, ChE 72, CoDJ 77], and the specification of a scale and the coefficient of the anomaly are in fact related. * As can be justified, we neglect here the existence of very heavy quarks, c, b and t.
III4 Classical scale invariance and the trace anomaly
89
In the following, let us start directly with the path integral treatment [Fu 81], again in the framework of QCD, concentrating on the effect of a single quark. We can introduce an external source coupled to 9^ into the generating functional
l
J
J
[
l
)
+ h(x)e^] , (4.4)
where
e^ = yh^rl) .
(4.5)
As in the case of the chiral anomaly, we can use this as a starting point to explore the nature of the trace 9^. The key is that if one makes the change of variables ip(x) = e~ a ( : r ) / y(z)
,
(4.6)
one obtains for infinitesimal a
r
= / dAx [£QCD (', Al) + a()mlj'l'
(4.7)
+
ifi^'d^a]
The last term vanishes after an integration by parts. The focus of our calculation can thus be shifted to a jacobian J ' by a change of variable,
(4 g)
=f Thus we obtain the identity i f d4x 0^a{x) =lnj
+ i f d*x m^
a(x) .
(4.9)
The form of the jacobian which follows from the work done in Sect. Ill—3 is 12
J = [det ( W 2 ) l
= Km e*'!** <*l««PW/JO a*>
(4 . 10 )
where we have adopted the same regulator as used previously. The final result is easily obtained from the general heat kernel calculation of App. Bl, again using the identity of Eqs. (B1.17), (B1.18).
90
/ / / Symmetries and anomalies
After some algebra this becomes TV'(zexp
4TT2
Qp/M)2\x) «A*
48TT 4 8 22
""
a
[D^ ° r *fi + a0 PF F
b
a
(4.11) Here we have found both a term which is a divergent constant, and one which involves twogluon field strengths. The divergent constant corresponds to the infinite zeropoint energy of the vacuum. This can be seen by noting that if the zero point energy is defined by the vacuum matrix element (4.12) then Lorentz covariance requires a nonzero trace 0*01 0) = ^ gT = > (0 \d»(x)\ 0) = 4 ^ .
(4.13)
Thus, a constant in the vacuum matrix element of the trace is just four times the zero point energy density. It is standard practice to subtract off this zero point energy, and we shall do so by dropping the constant term. This is similar to the procedure of normal ordering the energymomentum tensor. If we now combine these results using Eq. (4.9), we obtain
i f d*x 0>(z) = ijd4x ^A^F^Fr + m^] a(x)
(4.14)
which is equivalent to the operator relation ^
.
(4.15)
One may also derive the trace anomaly via the calculation of a Feynman diagram, the triangle diagram of Fig. Ill—1 but with the axial current replaced by the energymomentum tensor. The trace anomaly is different from the chiral anomaly in that it also receives contributions from gluons. In the Feynman diagram approach, this arises from the replacement of quark lines by gluons, while in the path integral context it occurs when one considers scale transformations of the gluon field. A full calculation yields 0" = P9™F"F^ + muuu + mddd + msss + ... ,
(4.16)
7/75 Chiral anomalies and vacuum structure
91
where /?QCD is the beta function of QCD (cf. Eq. (II2.58)). The result of our previous calculation, Eq. (4.15), corresponds to the lowest order contribution of a single quark to the beta function. A feeling of why the beta function enters can be obtained from an extremely simple, but heuristic, derivation of the trace anomaly. Let us rescale the gluon field to Aa^ = g^A^, such that the massless action becomes
£ = ^»Fr
+ iWD^
.
(4.17)
The coupling constant g% now enters only as an overall factor in the first term. However in renormalizing the coupling constant, we need to introduce a renormalization scale. If we interpret this coupling as a running parameter, the action is no longer invariant under scale transformations. Instead, taking A = 1 + £A, we find
ML  [d4x— (—*—]2 F a F^  fdf 4x a SX ~ J dX V 4g 3(X)J ^ " J
2gs where we have changed back to the standard normalization of A^ in the final term. By Noether's theorem, the scale current is no longer conserved, and Eq. (4.16) is reproduced. The need to specify a scale in defining the coupling constant has removed the scale invariance of the theory. The trace anomaly occupies a significant place in the phenomenology of hadrons because it is the signal for the generation of hadronic masses. Returning to the discussion of masses which began this section, we see that the mass of a state is expressible as a matrix element of the energymomentum trace. For example, we find for the nucleon state that mNu(p)u(p) = msss + muuu + mddd\N(p)) The terms containing the light quark masses mu,md are expected to be small, and indeed the 'crterm' determined in TTN scattering (cf. Sect. XII3) implies that they contribute about only 45 MeV. This leaves the bulk of the nucleon's mass to the gluon and squark terms in Eq. (4.19), of which the Ftf1'F^v part is expected to be dominant. Although this presents a conceptual problem for the naive quark model interpretation of the proton as a composite of three light quarks, it is nevertheless a central result of QCD. Ill—5 Chiral anomalies and vacuum structure There is a fascinating connection between the axial anomaly described previously in this chapter and the vacuum of QCD. This has important
92
777 Symmetries and anomalies
phenomenological consequences for both the rj mass and the strong CP problem. Here we present an introductory account of this topic [Pe 89].
The 0vacuum One is used to considering the effect on gluon fields of 'small' gauge transformations, i.e. those which are connected to the identity operator in a continuous manner. There also exist 'large' gauge transformations which change the color gauge fields in a more drastic fashion. For example the gauge transformation [JaR 76] generated by A
,
x
Al(x)
x 2 d2
+
2idr x
/r_N
(o)
=^T^ ^T^ '
where d is an arbitrary parameter and r is an SU(2) Pauli matrix in any SU(2) subgroup of SU(3), transforms the null potential A(x) = 0 into
2X,(T • x)  2d(x x T)i]
.
(5.2) Here, we are using the matrix notation A
^
This potential lies in an SU(2) subgroup of the full color 577(3) group, and is 'large' in the sense that it cannot be brought continuously into the identity. The r • x factor couples the internal color indices to the spatial position such that a path in coordinate space implies a corresponding path in the SU(2) color subspace. All gauge potentials AM carry a conserved topological charge called the winding number,
n=^Jd3xTr
(M*)Aj(*)A*(*))^ •
(54)
As can be demonstrated by direct substitution, the gauge field of Eq. (5.2) corresponds to the value n = 1. Fields with any integer value of the winding number n can be obtained by repeated applications of Ai(x), viz., A (V) = fAi (x)]n
(5 5)
All gauge potentials can be classified into disjoint sectors labeled by their winding number. The existence of these distinct classes has interesting consequences. For example, consider a configuration of the gluon field that starts off at t = — oo as the zero potential A(x) = 0, has some interpolating A{x, t) for
III5 Chiral anomalies and vacuum structure
93
intermediate times, and ends up at t = +oc lying in the gauge equivalent configuration A(x) = A^\x).* Then the following integral can be shown to be nonvanishing: /
.
(5.6)
This is surprising because the integrand is a total divergence. As noted previously in Eq. (II2.23), FF can be written as
and thus the integral can be written as a surface integral at t — ±00. For the field configuration under consideration, this reduces to the winding number integral 2
f
^3
2 A
/ d TF
64TT^ J
a
F
afJll/
— ^
**
f
3
/ d4rd
647T^ J ^—
2
r
On
I I
"^7r
K^ t=o
°
o CL X TYf)
^
<=oo
9l i I dzx eijk Tr (Af}(x)A*1'(a;)A^(a;))
24TT 2
= 1 . (5.8) More generally, the integral of F F gives the change in the winding number
64TT 5
f r14r Fa Fa^ 
9
*
f
t=oo
(5.9)
between asymptotic gauge field configurations. Thus, the vacuum state vector will be characterized by configurations of gluon fields which fall into classes labeled by the winding number. Moreover, there will exist a corespondence between the gauge transformations {An} and unitary operators {Un} which transform the state vectors. For example, a vacuum state dominated by field configurations in the zero winding class ('near' to A^ = 0) would be transformed by U\ into configurations with a dominance of n = 1 configurations, or more generally, Ui\n) = n + l) . * Such configurations are known to exist [Co 85].
(5.10)
94
III Symmetries and anomalies
This implies that a gaugeinvariant vacuum state requires contributions from all classes, such as the coherent superposition \6) = Ye~ine\n)
,
(5.11)
n
where 9 is an arbitrary parameter. It follows from Eq. (5.10) that this 9vacuum is gaugeinvariant up to an overall phase U1\e)=e»\d) .
(5.12)
The QCD vacuum must contain contributions from all topological classes. The 0term
Given this nontrivial vacuum structure, one requires three ingredients to completely specify QCD: (1) the QCD lagrangian, (2) the coupling constant (i.e. AQCD), and (3) the vacuum label 9. How can we account for the diflFerent vacua corresponding to different choices of 91 In a path integral representation, the 9 = 0 vacuum would imply generic transition elements of the form out<0
= 0\X\9 = 0) in  [ ^
n,m
The presence of a nonzero 9 leads to an extra phase, n)
* ont{m\X\n)in .
(5.13) (5.14)
n,m
However, this phase can be accounted for in the path integral by the addition of a new term to SQCD In particular we have, through the use of Eq. (5.9), X out(mXn) out n,m
where X is some operator. We see that the quantity (ra — n) given by the winding number difference of the fields contributing to the path integral is equivalent to a new exponential factor containing F^Fa^v. Thus a correct procedure for doing calculations involving 9vacua is to follow the ordinary path integral methods but with a QCD lagrangian containing the new term {
°$
^
.
(5.16)
7/75 Chiral anomalies and vacuum structure
95
The parameter 6 is to be considered a coupling constant. Since the operator FF is Podd and Todd, a nonzero 0 can induce measurable T violation. In Sect. IX5, we shall show how to connect 6 to physical observables. There is an important distinction between the various 9 vacua of QCD and the many possible vacuum states of a spontaneously broken symmetry such as the Higgs sector of the electroweak theory. In the latter case, the various possible vacuum expectation values of the Higgs field label different states within the same theory. In contrast, each value of 6 corresponds to a different theory, just as each value of AQCD would label a different theory. Specifying 0 and AQCD then specifies the content of the version of QCD used by Nature. Connection with chiral rotations There is a connection between the axial anomaly and the presence of a #vacuum ['tH 76a,b]. It involves the matrix element of FF as follows. Consider the limit of Nf massless quarks. The U(l) axial current Nf
is not conserved due to the anomaly,
PJ$!! = ^FSJ°r
.
(5.18)
However, because of the fact that FF is a total divergence, one can define a new conserved current
While J^^ does form a conserved charge, Q5 = fd3x
J5,o(z) ,
(5.20)
neither Q5 nor J^ is gaugeinvariant. In fact, under the gauge transformation Ai of Eq. (5.1), it follows from Eq. (5.8) that the operator Q5 changes by a cnumber integer UxQhU{x = Qh2Nf
.
(5.21)
This tells us that in the world of massless quarks, the different 0vacua are related by a chiral 17(1) transformation, ^
^
^
)
e
^
5

0
)
,
(5.22)
96
/ / / Symmetries and anomalies
or from Eq. (5.12), eia^\6) = \02Nfa)
,
(5.23)
where a is a constant. Therefore, in the limit of massless quarks, when Q5 is a conserved quantity, all of the 6vacua are equivalent and one can transform away the 6 dependence by a chiral U(l) transformation. The same is not true if quarks have mass, as the mass terms in £QCD are not invariant under a chiral transformation. We shall return to this topic in Sect. IX5. To summarize, one finds that the existence of topologically nontrivial gauge transformations, and of field configurations which make transitions between the different topological sectors of the theory, leads to the existence of nonvanishing effects from a new term in the QCD action. Chiral rotations can change the value of 0, allowing it to be rotated away if any of the quarks are massless. However for massive quarks, the net effect is a measurable CP violating term in the QCD lagrangian. Problems 1) Currents and anomalies a) Show that all currents coupled to gauge bosons in the Standard Model are anomaly free. b) Show that the baryon number current B^ has an anomaly, but that the current B^ — L^ (L^ is the lepton number current) is free from anomalies. 2) Trace anomaly in QED In d dimensions, the trace of the energymomentum tensor does not vanish classically, except at d = 4. For example, in massless QED the energymomentum tensor,
has trace 0^ = ^^FXcTF\a. In the renormalization of the operator FX(J F\a, one encounters a renormalization constant which diverges as d —• 4. Use this feature to calculate the QED trace anomaly using dimensional regularization.
IV Introduction to effective lagrangians
The purpose of the effective lagrangian method is to represent in a simple way the dynamical content of a theory in the low energy limit, where effects of heavy particles can be incorporated into a few constants. The basic plan of attack is to write out the most general set of lagrangians consistent with the symmetries of the theory. At sufficiently low energies only one, or perhaps a few, of the lagrangians are relevant, and it is straightforward to read off the predictions of the theory. Effective lagrangians are used in all aspects of the Standard Model and beyond, from QED to superstrings. Perhaps the best setting for learning about them is that of chiral symmetry. Besides being historically important in the development of effective lagrangian techniques, chiral symmetry is a rather subtle theory which can be used to illustrate all aspects of the method, viz. the low energy expansion, nonleading behavior, symmetry breaking and loops. In addition, the results can be tested directly by experiment since effective lagrangians provide a framework for understanding the very low energy limit of QCD. IV1 Nonlinear lagrangians and the sigma model The linear sigma model, introduced in Sects. 13, 15, provides a 'user friendly' introduction to effective lagrangians because all the relevant manipulations can be explicitly demonstrated. The Goldstone boson fields, the pions, are present at all stages of the calculation. By contrast, low energy QCD is far less transparent, involving a transference from the quark and gluon degrees of freedom of the original lagrangian to the pions of the physical spectrum. Nevertheless, we expect the low energy properties of the two theories to have many similarities. 97
98
IV Introduction to effective lagrangians
First let us illustrate what an effective lagrangian is by simply quoting the result. Recall the sigma model of Eq. (13.14), C = ipipip + \dpir
^ (L1)
1 A 2 a  IT • 7r75) \j) + /x2 {a2 + TT2)   (a 2 + TT2)
.
This is a renormalizable field theory of pions, and from it one can calculate any desired pion amplitude. Alternatively, if one works at low energy, then it turns out that all matrix elements of pions are contained in the rather different looking 'effective lagrangian' £eff = ^  Tr (dpUdvU^ ,
U = expzr • n/F ,
(1.2)
where F = v = \/fi2/\ at tree level (cf. Eq. (15.9)). This effective lagrangian is to be used by expanding in powers of the pion field ...
, (1.3)
and taking treelevel matrix elements. This procedure is a relatively simple way of encoding all the low energy predictions of the theory. Moreover, with a modest increase in effort, corrections to the low energy results can also be treated in the effective lagrangian formulation.
Representations of the sigma model In order to embark on the path to the effective lagrangian approach, let us rewrite the sigma model lagrangian as + ^  Tr (^Y)  — \ C = \ Tr (dj:d^\ 4 V M / 4 V / 16 L + ^hip^L + ^RifiipR  9 {^PL^R + $
2
with E = a + ir • Tr. The model is invariant under the SU(2)L transformations
X
SU(2)R
(1.5) for L,i? in SU{2). This is the linear representation.* After symmetry breaking and the redefinition of the a field,
* A number of distinct 2 x 2 matrix notations, among them E, U, and M, are commonly employed in the literature for either the linear or the nonlinear cases. It is always best to check the definition being employed and to learn to be flexible.
IV1 Nonlinear lagrangians and the sigma model
99
the lagrangian reads^ £ = i (dJfr&'a  2fj2a2) + \d^ A 2   (a2 + 7T2) +i>{ip
• d»n  Xva (a 2 + TT2) gv) ip  gi\)(a  ir • 7775) ^ ,
(L7)
indicating massless pions and a nucleon of mass gv. All the interactions in the model are simple nonderivative polynomial couplings. There are other ways to display the content of the sigma model besides the above linear representation. For example, instead of a and TT one could define fields 5 and <£>, S=\/(a
+ V) +7T2V
— C7 + . . . ,
(fi =
= = 7T + . . . ,
V (a + vf + ^ (1.8) where one expands in inverse powers of v. For lack of a better name, we can call this the square root representation. The lagrangian can be rewritten in terms of the variables S and cp as c =
„ 9^1 . 1 fv + S^2
1 [,n ^2 2
" " • • " • •

g
+
()
(1.9) Although this looks a bit forbidding, no longer having simple polynomial interactions, it is nothing more than a renaming of the fields. This form has several interesting features. The pionlike fields, still massless, no longer occur in the potential part of the lagrangian, but instead appear with derivative interactions. For vanishing 5, this is called the nonlinear sigma model. Another nonlinear form, the exponential parameterization, will prove to be of importance to us. Here the fields are written as E = a + iT'n such that nf — 7T +
c = \ Ud.sf
= (v + S)U ,
U = exp (ir • n'/v)
Using this form, we find
2ns} +
!i
 XvS3   S 4 + ikpt/;  g(v + S) (i>LUi(>R + Here, and in subsequent expressions for C, we drop all additive constant terms.
(1.10)
100
IV Introduction to effective lagrangians
The quantity U transforms under SU(2)L does £, i.e.
X SU(2)R
in the same way as (1.12)
This lagrangian is reasonably compact and also has only derivative couplings.
Representation independence We have introduced three sets of interactions with very different appearances. They are all nonlinearly related. In each of these forms the free particle sector, found by looking at terms bilinear in the field variables, has the same masses and normalizations. To compare their dynamical content, let us calculate the scattering of the Goldstone bosons of the theory, specifically TT+TT0 —> TT+TT0. The diagrams that enter at tree level are displayed in Fig. IV1. The relevant terms in the lagrangians and their treelevel scattering amplitudes are as follows. 1) Linear form: b =~
4
2 2 vI* )7
 \vair2 ,
{2i\vf
2
q
o
= ~2i\
T 1
%
g
(1.13)
m(J 2 2\v 1 iq2 =z 2 1 = \ 2 + ... , q 2\v \ v\
where q = p+ — p + = po — Po and the relation ml — 2Xv 2 = 2/J,2 has been used. The contributions of Figs. IV1 (a), l(b) are seen to cancel at q2 = 0. Thus, to leading order, the amplitude is momentumdependent even though the interaction contains no derivatives. The vanishing of the amplitudes at zero momentum is universal in the limit of exact chiral symmetry. 2) Square root representation:
(a) Fig. IV1 Contributions to TT+TT0 elastic scattering
IV1 Nonlinear lagrangians and the sigma model
101
For this case, the contribution of Fig. IVl(b) involves four factors of momentum, two at each vertex, and so may be dropped at low energy. For Fig. IVl(a) we find
./,
^ .2
a15)
3) Exponential representation: d = (V + ^
Tr (dpUWU^ + ... .
(1.16)
Again Fig. IVl(b) has a higher order (C?(p4)) contribution, leaving only Fig. IVl(a),
r ' ' *
The lesson to be learned is that all three representations give the same answer despite very different forms and even different Feynman diagrams. A similar conclusion would follow for any other observable that one might wish to calculate. The above analysis demonstrates a powerful field theoretic theorem, proved first by R. Haag [Ha 58, CoWZ 69, CaCWZ 69], on representation independence. It states that if two fields are related nonlinearly, e.g. ip = xF(x) with ^(0) = 1, then the same experimental observables result if one calculates with the field
102
IV Introduction to effective lagrangians
one uses the nonlinear representations, whereas for the linear representation more complicated calculations involving assorted cancelations of constant terms are required to produce the correct momentum dependence. In addition, the nonlinear representations allow one to display the low energy results of the theory without explicitly including the massive a (or S) and xj) fields.
IV2 Integrating out heavy fields When one is studying physics at some energy scale i£, one must explicitly take into account all the particles which can be produced at that energy. What is the effect of fields whose quanta are too heavy to be directly produced? They may still be felt through virtual effects. When using an effective low energy theory, one does not include the heavy fields in the lagrangian, but their virtual effects are represented by various couplings between light fields. The process of removing heavy fields from the lagrangian is called integrating out the fields. Here, we shall explore this process.
The decoupling theorem There is a general result in field theory, called the decoupling theorem, which describes how the heavy particles must enter into the low energy theory [ApC 75, OvS 80]. The theorem states that if the remaining low energy theory is renormalizable, then all effects of the heavy particle appear either as a renormalization of the coupling constants in the theory or else are suppressed by powers of the heavy particle mass. We shall not display the formal proof. However, the result is in accord with physical expectations. If the heavy particle's mass becomes infinite, one would indeed expect the influence of the particle to disappear. Any shift in the coupling constants is not directly observable because the values of these couplings are always determined from experiment. Inverse powers of heavy particle mass arise from propagators involving virtual exchange of the heavy particle. In the Standard Model, the most obvious example of this is the role played in low energy physics by the W± and Z gauge bosons. For example, while VF±loops can contribute to the renormalization of the electric charge, the effect cannot be isolated at low energies. Also, the residual form of W^exchange amplitudes is that of a local product of two weak currents (Fermi interaction) with coupling strength GF Its effect is suppressed because GF OC M^ 2 . However, in the Standard Model there is an example where the heavy particle effects do not decouple. For a heavy top quark, there are many loop diagrams which do not vanish as mt —* oc, but instead behave as
IV2 Integrating out heavy
fields
103
m2 or \n(m2). This can occur because the electroweak theory with the tquark removed violates the SU(2)i symmetry, as the full (£) doublet is no longer present. Without the constraint of weakisospin symmetry, the theory is not renormalizable and new divergences can occur in flavor changing processes. These wouldbe divergences are cut off in the real theory by the mass m*. Note that at the same time as mt —• oc, the top quark Yukawa coupling also goes to infinity, and hence induces strong coupling which can also lead to a violation of decoupling. In the sigma model, all the low energy couplings of the pions are proportional to powers of 1/v2 oc 1/raj*, the simplest example being Eq. (1.9). Hence the effective renormalizable theory is in fact a free field theory, without interactions. The interactions have been suppressed by powers of heavy particle masses. We shall use the energy expansion of the next section to organize the expansion in powers of the inverse heavy mass. Integrating out heavy fields at tree level The name of this procedure comes from the path integral formalism, where the process of integrating out a field H and leaving behind light fields £i is defined in terms of an effective action Ze^[£i], eiZev[ti]
= I
[dH] eif#xC(H(x)A(*))/
I
[dH]
However, the procedure is equally familiar from perturbation theory, in which the effect of the path integral is represented by a sum of Feynman diagrams. Let us proceed with a path integral example. Consider a linear coupling of H to some combination of fields J, with the lagrangian C = \ (d^Hd^H  m2HH2) + JH .
(2.2)
One way to integrate out H is to 'complete the square', i.e. we write
fd4xC(H,J)=
f d4x \~HVH
+ JH\
=  i fd4x [(H  V1 J) V(H
V~1J)  JV~1J]
=  i I d4x [H'VH'  JV~XJ] , (2.3)
104
IV Introduction to effective lagrangians
where we have used the shorthand notations, V = H + m2H ,
V~lJ =  J dAyAF{xy)J{y) , (Dx + m2H) AF{x y) = 8*(x  y) ,
H\x) = H(x) + Jd4y AF(x  y)J(y) ,
f d*x JV~lJ
=  j dAxdAy J(x)AF(x  y)J(y) ,
and have integrated by parts repeatedly. Since we integrate in the path integral over all values of the field at each point of spacetime, we may change variables [dH] = [dHf] so
= f[dH'}ei _Afd4xJV1J from which we obtain
ZeS[J] = ^Jd4xd4yJ(x)AF(xy)J(y)
.
(2.6)
Finally one may obtain a local lagrangian by noting that the heavy particle propagator is peaked at small distances. Over these scales, we may Taylor expand J(y) as „=*+ •••
•
(27)
Keeping the leading term and using
we obtain r
i
+ ... ,
(2.9)
where the ellipses denote terms suppressed by additional powers of Outside the path integral context, this result is familiar from VFexchange in the weak interactions. We can apply this procedure to the lagrangian for the sigma model where the scalar field 5 is heavy with respect to the Goldstone bosons. Thus, considering the theory in the low energy limit, we may integrate out
IV3 The low energy expansion
105
the field S. Referring to Eq. (1.11) and neglecting the 5 2 interactions, it is clear that we should make the identifications H —• S and J —• vTv{dyJJd^U^)/2. The effective lagrangian then takes the form
£eff = j IV (dpUVrf) + ^
[Tr (dlxUd^)]\
...
,
(2.10)
where the second term in Eq. (2.10) is the result of integrating out the 5field and gives rise to the diagram of Fig. IVl(b). Additional treelevel diagrams are implied by the sigma model when one includes the S3 and 5 4 interactions. Since these carry more derivatives, the above result is the correct treelevel answer with up to four derivatives. Upon including the S2 Tr (d^Ud^U^) terms, loop diagrams become possible. Our purpose is not to calculate these explicitly, but merely to indicate the nature of the effective lagrangian. The net result will be a low energy effective lagrangian which is written as an expansion in powers of Tr (d^Ud^W). In calculating transitions of pions, this is used by expanding the U matrix in terms of the pion fields and taking matrix elements. Interested readers may verify that this procedure applied to Eq. (2.10) reproduces the first two terms in the TT+TT0 scattering amplitude previously obtained in Eq. (1.13).
IV3 The low energy expansion The expansion in energy is the essential feature which allows the effective lagrangian framework to be useful. We have already seen some aspects of it in the sigma model example. Let us now explore this topic in more generality.
Expansion in energy What would happen if, instead of having a straightforward theory like the linear sigma model, we were dealing with an unknown or unsolvable theory with the same SU{2)L X SU(2)R chiral symmetry? In this case there would exist some set of pion interactions which, although not explicitly known, would be greatly restricted by the SU(2) chiral symmetry. Once again we could choose to describe the pion fields in terms of the exponential parameterization U, with a symmetry transformation (3.1) for L, R in SU(2). Not having an explicit prescription, we would proceed to write out the most general effective lagrangian consistent with the chiral symmetry. In view of the infinite number of possible terms contained in such a description, this would appear to be a daunting process. However, the energy expansion allows it to be manageable.
106
IV Introduction to effective lagrangians
It is not difficult to generate candidate interactions which are invariant under chiral SU{2) transformations. For the purpose of illustration, we list the following twoderivative, fourderivative and sixderivative terms in the exponential parameterization,
There can be no derivativefree terms in a list such as this because Tr (C/C/t) = 2 is a constant. It is clear that one can generate innumerable similar terms with arbitrary numbers of derivatives. The general lagrangian can be organized by the dimensionality of the operators, C = C2 + £4 + A, + Cs + . . . +ai[Tr (d^Ud^U^]2
(3.3)
a2Tr (dpUdvU^ • Tr The important point is that, at sufficiently low energies, the matrix elements of most of these terms are very small since each derivative becomes a factor of the momentum q when matrix elements are taken. It follows from dimensional analysis that the coefficient of an operator with n derivatives behaves as 1/Mn~4, where M is a mass scale which depends on the specific theory. Therefore the effect of an nderivative vertex is of order qn/Mn~4, and at an energy small compared to M, largen terms have a very small effect. At the lowest energy, only a single lagrangian, the one in Eq. (3.3) with two derivatives, is required. We shall call this an 'O(E2y contribution in subsequent discussions. The most important corrections to this involve four derivatives, and are therefore 'C?(i?4)\ In practice then, the infinity of possible contributions is reduced to only a small number. The coefficients of these terms are not generally known, and must thus be determined phenomenologically. However, once fixed they can be used to allow predictions to be made for a variety of reactions.
Loops It would appear that loop diagrams could upset the dimensional counting described above. This might happen in the calculation of a given loop diagram if, for example, two of the momentum factors from an O(E4) lagrangian are involved in the loop and are thus proportional to the loop momentum. Integrating over the loop momentum apparently leaves only two factors of the 'low' energy variable. It would therefore seem that for certain loop diagrams, an O(E4) lagrangian could behave as if it were
IV3 The low energy expansion
107
O(E2). If this happened, it would be a disaster because arbitrarily high order lagrangians would contribute at O(E2) when loops were calculated. As we shall show, this does not occur. In fact, the reverse happens. When O{E2) lagrangians are used in loops, they contribute to O(E4) or higher. Before we give the formal proof of this result, let us demonstrate how it works by example. Consider a loop diagram for TT+TT0 —• TT+TT0, as in Fig. IV2. Using the lowest order vertex for this process, we find
de
 /
^) 2 i
a2
i
(eP+po)2
A*i/)2
{3A)
where / is the loop integral with the factor v 4 extracted. Counting powers of energy factors is most easily done in dimensional regularization. The loop integral contains no dimensional factors other than p+, po and j / + . Since, in four dimensions it has the overall energy unit E4, it must therefore be expressible as fourth order in momentum. Despite the loop integration, the end result is expressed only in terms of the external momenta. These momenta are small, and hence all the energy factors involved in power counting are taken at low energy. In dimensional regularization, there can also be a dependence on the arbitrary scale //,
f d4£^ fi4~d I dd£ ,
(3.5)
but in the limit d —• 4 this occurs only in dimensionless logarithms such as ln(E2/fji2). Thus, the order of momentum can be found by counting the factors of 1/v2 which occur for every vertex from the lowest order lagrangians. We shall soon see (cf. Eq. (5.3)) that the parameter u, defined in the sigma model, is equal to the pion decay constant Fn. Each factor of 1/v2 must be accompanied by momenta in the numerator in order to produce a dimensionless amplitude. Each vertex in a diagram contributes powers of 1/v2, and higher order loop diagrams require more vertices. Thus every time a loop is formed, the overall momentum power of the amplitude must increase rather than decrease.
Fig. IV2 Loop contribution to TT+TT0 elastic scattering
108
IV Introduction to effective lagrangians
Weinberg's power counting theorem To prove this result [We 79], consider some diagram with a total of Ny vertices. Then letting Nn be the number of vertices arising from the subset of effective lagrangians which contain n derivatives (e.g. N4 is the number of vertices coming from fourderivative lagrangians), we have Ny = T,nNn. The overall energy dimensionality of the coupling constants is thus MNc with
where M is a mass scale entering into the coefficients of the effective lagrangian (e.g. the quantity v in the sigma model). Each pion field comes with a factor of 1/v, so that associated with NE external pions and Ni internal pion lines is an energy factor (1/M)2NI+NE . (Recall that two pions must be contracted to form an internal line.) However, the number of internal lines can be eliminated in terms of the number of vertices and loops (NL), n
l
.
(3.7)
Any remaining dimensional factors must be made up of powers of the energy E times a dimensionless factor of E/fi where // is the scale employed for renormalizing the coupling constants. (When using dimensional regularization, these factors of E/fi enter only in logarithms.) Thus the overall matrix element is composed of energy factors M ~ (M)£n^4) K
]
L—
EDF(E/fi)
MNE+2NL+2^N"2
K
'
}
(3.8)
4
~ (Mass or E n e r g y ) " ^ , where the second line is the overall dimension of an amplitude with external bosons. The renormalization scale \i can be chosen of the order of E so no large factors are present in F(E/[/,). Overall the energy dimension is then
^2
.
(3.9)
A diagram containing NL loops contributes at a power E2NL higher than the tree diagrams. This theorem is of great practical consequence. At low energy, it allows one to work with only small numbers of loops. In particular, at O(E4) only oneloop diagrams generated from £2 need to be considered. The astute reader will have noticed that the loop example used above raises another difficulty, viz., the loop integral is badly divergent. Of
IV~4 Symmetry breaking
109
course divergences are generally present in a quantum field theory, and here they can be handled in the usual way, by renormalizations of the parameters in the theory. If the original effective lagrangian which we have written down is indeed the most general one consistent with the given symmetry, then it must have enough parameters of the right form to encompass any divergences which occur. In particular, our power counting argument tells us that when £2 is used in oneloop diagrams, the divergences are of order E4 and should be capable of being absorbed into the parameters of that order. Since the parameters are generally unknown and are to be determined phenomenologically, the only difference this makes is to cast physical results in terms of the renormalized parameters instead of the bare ones. The end result is a very simple rule for counting the order of the energy expansion. The lowest order (E2) behavior is given by the twoderivative lagrangians treated at tree level. There are two sources at the next order (E4): (i) the O(E2) oneloop amplitudes, and (ii) the treelevel O(E4) amplitudes. When the coefficients of the E4 lagrangians are renormalized, finite predictions result. IV—4 Symmetry breaking Effective lagrangians can be used not only in the limit of exact symmetry but also to analyze the effect of small symmetry breaking. Let us first return to the sigma model for an illustration of the method, and then consider the general technique. The SU(2)L X SU(2)R symmetry of the sigma model is explicitly broken if the potential V(a, TT) is made slightly asymmetric, e.g. by the addition of the term breaking = CL<J =  TV (E + E + )
(4.1)
to the basic lagrangian of Eq. (1.4). To first order in the quantity a, this shifts the minimum of the potential to
and produces a pion mass
"4 = 2 .
(43)
Although the latter result can be found by using the linear representation and expanding the fields about their vacuum expectation values, it is
110
IV Introduction to effective lagrangians
easier to use the exponential representation, taking = \{v + S)Tr(U + &) = \{v + S)TV (2  ( ^ )
2
+ ...
2
= a(v + S) ^n  7T + ... = a(y + S)  ^ n •
+ ... . (4.4) The chiral SU(2) symmetry is seen to be slightly broken, but the vectorial SU(2) isospin symmetry remains exact. As we have seen, the O(E2) Lagrangian is obtained by setting 5 = 0, 2
TT
2
C2 = V Tr (dJJ&lfi) + ^v2 Tr(U + rf) .
(4.5)
Higher order terms will contain products like l Tr(U + [/+)]\ ml Tr (U + /7f) • Tr (d,AUdtAtf), ... ,
(4.6)
and can be obtained by integrating out the field 5 as was done in Sect. IV2. It is important to realize that the symmetry breaking sector also has a low energy expansion, with each factor of m^ being equivalent to two derivatives. If ra^ is small, the expansion is a dual expansion in both the energy and the mass. If we encounter a theory more general than the sigma model, the effect of a small pion mass can be similarly expressed in low orders by, ^breaking = axm\ TV (U + tf) + a2 [ml Tr(U l Tr(U + U]) TV (d^Ud^)
+ a4ml TV \(U
(4.7) with coefficients that are generally not known. An important consideration is the symmetry transformation property of the perturbation. The symmetry breaking term of Eq. (4.1) is not invariant under separate lefthanded and righthanded transformations but only under those with L — R. All the terms in Eq. (4.7) have this property. Other symmetry breakings can be analyzed in a manner analogous to the treatment just given of the mass term. One identifies the symmetry transformation property of the perturbing effect and writes the most general effective lagrangian with that property. Most often the perturbation is treated to only first order, but higher order behavior can also be studied. We shall encounter another example of this in our study of weak decays.
IV5 PCAC
111
IV5 PCAC In addition to effective lagrangian techniques for dealing with chiral symmetry, there also exist current algebra methods which go by the name of Partial Conservation of the Axial Current or PCAC [AdD 68]. These yield the same results but are often more cumbersome. Let us again use the sigma model to illustrate this subject, now with a pion mass included and the 5field integrated out. The lagrangian of Eq. (4.5) gives rise to the vector and axialvector currents V
2
Vj = i  TV \rk (u%U + Ud^uA] , 4
L
4 = i""TV [[r [ 2
= i"2 TV
V
n
(51)
)]
(
with k = 1, 2, 3. The equation of motion is found to be
d^ (ifidJJ) H \
/
^ (C/ C/+) — 0 . 2 V
(5.2)
/
and two important matrix elements are
(p)) = ivp^
(o\ dMj  it* (p)) = vml6ki .
,
(5.3)
0 The former allows the identification v = Fn, where Fn is the pion decay constant Fn ~ 92 MeV, while the latter follows either from Eq. (5.1) directly, or by use of the equation of motion for A*, v2m2
0M£ = i
r
k
/
f
M
 Tr \ r \ U  C/ J = F ^ m 2 ^ + ...
.
(5.4)
This last equation forms the heart of the PCAC method. It describes a situation covered by Haag's theorem (recall Sect. IV1), and says that we may use either ?rfc or d^A^ (properly normalized) as the pion field. It is more general than the sigma model which we used to motivate it. This, plus certain smoothness assumptions, gives rise to a softpion theorem for the following matrix element of a local operator O, lim (7rk(c[)(3\O\a) = ~4r (p\ \Q\IO\ qf+0 \
I
Fn \
L
\<X) J /
,
(55)
k
where /?, a are arbitrary states and Q = f d?x AQ(X) is an axial charge. The softpion theorem The proof of this starts with the LSZ reduction formula. We consider the matrix element for the process a —> (3+7r k(q) as the pion fourmomentum
112
IV Introduction to effective lagrangians
q is taken off the massshell,
Uk{q)j3\O{Q)\a\ = i Id4xeiqx
( • + m\) ((3\T Uk{
= i fd4xeiqx(q2
+ ml) ((3\T (7r*(x)0(O)) \a) ,
(5.6) The pion field can be replaced by using the PC AC relation (valid in the sense of the Haag theorem),
"* leading to
(7rk(q)(3\O(0)\a) = i{ml~J*] f dAx e^x (f3\T (&>A*(x)O(0)) a) . (5.8) The derivative can be extracted from the timeordered product by using (ilJ(x)O(O))  a )
a) + 6(x0) (/3\ [4(x), O(0)] \a) ,
(5 9)
where the last term arises from differentiating the functions 6(±xo) which occur in the timeordering prescription. Upon integrating by parts, we find
x   ( 0  An(x),O(0)] \a} 6(x0)  itf (/3\T (A*( (5.10) Up to this stage all the formulae are exact for physical processes, even if appearing rather senseless, since d^A^ has the same singularity for q2 —• ml as does the field 7rfc. However, to obtain the softpion theorem one assumes that the matrix element does not vary much between its onshell value and the point where the pion's fourmomentum vanishes. In that circumstance, we have [NaL 62, AdD 68] lim {7rk(q)/3\O\a) = —— (/3\ \QLO(0)\ q»+o \ v J ' ' / Fn \ L °
la) + lim iq^R* M, J / q^^o
(5.11) v y
where i*
r
.
.
i
/
\
\
a>
.
(5.12)
The remainder term of Eq. (5.11) vanishes unless Rk has a singularity as q11 —> 0. Such a singularity can occur if there are intermediate states in
IV6 Matrix elements of currents
113
R^ which are degenerate in mass with either a or f). This last statement can be proven by inserting a complete set of intermediate states in the timeordered product in i2*, and taking the q** —• 0 limit. This caveat should be kept in mind as it is sometimes relevant. The softpion theorem relates to the intuitive picture for dynamically broken symmetries mentioned in Sect. 16. Since a chiral transformation corresponds in the symmetry limit to the addition of a zero energy Goldstone boson, we expect the states (f3\ and (TT£ = O /? to be related by the symmetry, and indeed, the softpion theorem expresses this. Although the softpion theorem is exact in the symmetry limit, a smoothness assumption is needed in the real world to pass from q^ — 0 to q 2 = ra^, implying that corrections of order q^ or of order m%. can be expected. In the Standard Model, the charge commutation rules are commonly abstracted from those of the quark model. Upon expressing charge operators in terms of quark fields,
Qk = J tfx jno^
, Q\ = jdzxi>lQl^,
(5.13)
one obtains the algebra
[Q\ V*] = if**Vt , [Ql V£\ = if*A* , i k [Q ,AJl]=if* A$, [Ql^]=ifjkVk . These commutation rules can be extended to equaltime commutators which contain a charge density, e.g.
[V$(x), 4(y)] lO=y o = ifikAk6®(x
 y) .
(5.15)
However, commutators which involve two spatial components can be more problematic [AdD 68]. Sometimes, in the PC AC approach, if the matrix element is assumed to be strictly constant, the various softpion limits turn out to be contradictory. If so, the amplitude must be extended to include momentum dependence, as happens in nonleptonic kaon decay. By contrast, the effective lagrangian approach automatically gives the appropriate momentum dependence, and its predictions follow in a straightforward manner. Moreover, effective lagrangians are especially useful in identifying and parameterizing corrections to the lowest order results. They allow a systematic expansion in terms of energy and mass. IV—6 Matrix elements of currents There is an elegant technique which allows one, at a minimal increase in complexity, to calculate matrix elements of currents from a chiral effective lagrangian [GaL 84,85a]. The idea is to add to the lagrangian
114
IV Introduction to effective lagrangians
terms containing external sources coupled to the currents in question. Construction of the effective lagrangian, including source terms, then allows the current matrix elements to be easily identified. We shall explain this technique here, and use it extensively in our discussion of QCD in subsequent chapters. First consider how current matrix elements are identified in a path integral framework. We have seen in Chap. Ill (see also App. A) that by adding a source coupled to the desired current, matrix elements can be obtained from differentiation of the path integral, e.g. Eqs. (Ill—2.1), (III— 2.4). For example, we can modify threeflavor QCD by adding sources to obtain
C = \F%F\? + $Hp1> ^ p ^
^lj^r^
where £^ rM, s, p are 3 x 3 matrix source functions expressible as
^ = ^+^A a ,
r^rl+r^W
s = s°+sa\\
p = p°+pa\a , (6.2)
with a = 1,..., 8. The lagrangian in Eq. (6.1) reduces to the usual QCD lagrangian in the limit l^ — r^ = p = 0, s = m, where m is the 3 x 3 quark mass matrix. The electromagnetic coupling can be obtained with the choice l^ = r^ = eQA^ where A^ is the photon field and Q is the electric charge operator defined in units of e. Various currents can be read off from the lagrangian, such as the lefthanded current J {X) =
^
"a)= ^)^^P^V(z)
(63)
or the scalar density ^
•
(
6

4
)
Moreover, matrix elements of these currents can be formed from the path integral by taking functional derivatives. The simplest example is i=r=p=O
while other examples appear in Sect. Ill—2. Matrix elements and the effective action A low energy effective action for the Goldstone bosons of this theory will be a functional of the external sources. One way to define the connection of the effective action with QCD is to consider the effect of the sources
IV6 Matrix elements of currents
115
in QCD,
At low energy, all heavy degrees of freedom can be integrated out and absorbed into coefficients in the effective action Z. However, the Goldstone bosons propagate at low energy, and they must be explicitly taken into account. One then writes a representation of the form
_ f
(6.7)
where as usual U contains the Goldstone fields. This form then allows inclusion of all low energy effects while maintaining the symmetries of QCD. The lagrangian of Eq. (6.1) has an exact local chiral SU(3) invariance if we have the external fields transform in the same way as gauge fields. In particular, the transformations ipL  •
if>R • R{x)^R , + id,L(x)L\x) , + idfiR{x)R\x) , 0 + ip) ^ L(x)(s + ip)R\x) provide an invariance for any L{x), R(x) in SU(3). In constructing the effective action, these invariances must be included. This is easy to do if l^ and r^ enter in the same way as gauge fields. In particular, upon defining a covariant derivative DJJ = O^U + i^U  iUr^ , (6.9) L(X)I/JL
,
and field strength tensors V = d^  dj^ + %[ip, 4 ] , c a a , r l
^6 °^
we obtain the following covariant responses to local transformations: U > L(x)UR\x) ,
D,U v L{x)D,UR\x) ,
L^ ^ L(x)L^(x) , R^ f JR(x)i?^i?t(a;) . The effective action is then expressed in terms of these quantities. At order E2, there are only two terms in the effective lagrangian, £ 2 = ^L TV (DpUiyrf}
+ ^ Tr (xU] + UX])
(6.12)
+ ip),
(6.13)
where
116
IV Introduction to effective lagrangians
and Bo is a constant with the dimension of mass. In the limit l^ = r^ — p = 0, s = ra, this is the same effective lagrangian with which we have been dealing in the SU(2) examples, with the identification m^ — (mu + rrid)Bo. Note that this usage requires Bo to be positive. Having constructed the effective action, we can obtain a number of interesting matrix elements. For example, use of Eq. (6.5) provides the identification of the vacuum scalardensity matrix element as (6.14) to this order in the effective lagrangian. Similarly, use of Eq. (6.3) reveals the lefthanded current to be LJ =
_i^L ^ ^\kUd^
.
(6.15)
One other advantage of the source method is to allow the use of the equations of motion. The standard Noether procedure for identifying currents does not work if the equations of motion are employed in the lagrangian. To become convinced of this, one can consider the following exercise. We examine the response of the two trial lagrangians, A =
IV1 Heavy particles in effective lagrangians
117
We shall proceed by first working out an example, the fermionic sector of the linear sigma model, £f = $[ift
g(a ir
 TT75)] I/;
We shall drop reference to the scalar field S in the following. The above lagrangian is invariant under the chiral transformations ^L^L^L,
II>R+RII>R,
U^LUrf
,
(7.2)
with L in SU(2)L and R in SU(2)R. AS always, we are free to change variables via contact transformations. In this instance, a useful choice of field redefinitions turns out to be
£/ = ££ ,
(7.3)
where £ = exp(zr • ir/2Fw). This is seen, after some algebra, to convert the fermion lagrangian to
which is a theory of fermions of mass M = gv having pseudovector coupling. The new fields transform as
M
 idpV* • V) V* ,
Ap 
^
,
M
^
(7.5) For purely vector transformations we have L = R = V. For L ^ i?, the property of V is more complicated, and Eq. (7.5) implies that it cannot be a simple global transformation, but must be a function of TT(X) and hence a function of x. At first sight, the need to express an 5(7(2) transformation matrix like V as a function of TT(X) appears unnatural. However, it is in fact consistent with physical expectations. Recall from the general discussion of dynamical symmetry breaking in Sect. 16 that in the symmetry limit, axial transformations mix the proton not with the neutron (as in isospin transformations), but rather with states consisting of nucleons plus zero momentum pions. Mathematically, the important point is that iV^ and NR transform in an identical fashion. This corresponds to the fact that heavy fields do not transform chirally, but have a common vectorial 5/7(2) transformation. It can be directly verified that Eq. (7.5) is a symmetry of the lagrangian. Thus we have obtained the expected result that the baryons can have a vectorial SU(2) invariance, while maintaining a chiral invariance for pion couplings.
118
IV Introduction to effective lagrangians
We see in the above example the ingredients of a general procedure for adding heavy fields to effective chiral lagrangians. The heavy fields are assumed to have an SU(2) (or SU(n), if desired) transformation described by the matrix V. A derivative d^ acting on a heavy field must be incorporated as part of a covariant derivative V^ in order to maintain this invariance. Couplings to pions are described by the matrices £ and [/, with £ having the same transformation as in Eq. (7.5). It is usually straightforward to combine factors of £ and U in such a way that the overall lagrangian is invariant. In the general case, each invariant term will have an unknown coefficient which must be determined phenomenologically. For example, the ip fi ip term in Eq. (7.4) would be expected to have a coefficient different from unity; the unit coefficient is a prediction specific to the linear sigma model. Effects which break the symmetry in an explicit fashion, like mass terms or electroweak interactions, can be added by using appropriate external sources. To date, heavy field lagrangians have been used in applications primarily at tree level. The feature which is essential for their application is that the pion momenta are small, and hence the heavy fields are essentially static. We conclude this discussion with an application to B meson decay. Consider the quark semileptonic decay, b —> u + e + ve, induced by the weak current J» = h»(l + 7b)u .
(7.6)
This quark process appears at the hadronic level in decays like i?(+'°) —>• (TT'S) + ez/e, etc. For most of the kinematically allowed region, the pions have too much energy to contemplate using chiral symmetry methods. However, in that corner of phase space where the lepton pair eve carries off most of the available energy, chiral symmetry can be used to restrict the allowed couplings, and an effective lagrangian analysis is justified. Let us obtain an expression for the hadronic weak current in B decay by constructing a chirallyinvariant lagrangian which contains a weak bquark current as an external source. We first define an isospindoublet B of heavymeson fields and an isospindoublet source current J^,
Then noting that the source currents are purely lefthanded, we have the transformation laws ^
,
,
U > LUB) ,
B^VB, <£  L£V]
V£R)
IV8 Effective lagrangians in QED
119
A chirally invariant effective lagrangian, containing up to one derivative acting on the pion fields, is then
£ = faJl&rB +foJl {dPU) &B +foJl (dyU) £V»V»B .
(7.9)
Finally, the hadronic weak current can be found by differentiating with respect to the source current J^ as described in Sect. IV6. Observe that a derivative acting on a heavy field is not a small quantity in the energy expansion, but in this corner of phase space a derivative acting on the pion fields is small. An additional example of this approach, but applied instead to baryons, appears in Sect. XII3. IV8 Effective lagrangians in QED We have explored in some detail the structure of effective lagrangians by using chiral symmetry as an example. However, this is not meant to imply that effective lagrangians are useful only in that one context. In fact, they can be applied to a wide variety of situations. Here, we apply the technique to QED. Consider situations in which the photon's fourmomentum is small compared to the electron mass. In such cases, the electron and other fermions cannot be produced directly, but instead influence the physics of photons only through virtual processes. The lowest order diagrams, i.e. those which contain a single electron loop, with increasing numbers of external photon legs are shown in Fig. IV3. Note that the oneloop diagram containing three photons, or indeed any odd number of photons, vanishes by virtue of charge conjugation invariance. This is true to all orders in the coupling e, and is refered to as Furry's theorem. Diagrams like those in Fig. IV3 have effects at low energy which are typically calculated in perturbation theory. The associated amplitudes have coefficients which scale as some power of the inverse electron mass. They can be generated by means of an effective lagrangian, as we shall now discuss. Let us seek a description which eliminates the electron degrees of freedom. That is, we wish to write a lagrangian which involves only photons, but nevertheless includes effects like the ones in Fig. IV3. The result
(a)
< (b)
Fig. IV3 Photon amplitudes containing a single fermion loop.
120
IV Introduction to effective lagrangians
must of course be gauge invariant. The procedure may be defined by the path integral relation / [dAp] exp \i / dAx £eff(Au) p][di/;][dx/j] exp [i JdAx £ Q E D ( A ^ VO] exp [i f d±x
(8 1)
where CQED is the full QED lagrangian, and Co is the free fermion lagrangian. Thus £eff has precisely the same matrix elements for photons as does the full QED theory. Specifically, it includes the virtual effects of electrons. The techniques described in App. A5 enable us to formally express the content of Eq. (8.1) as [Sc 51],
J
In [ ^ ^ 1 •
(8.2)
This form, although formally correct, does not readily lend itself to physical interpretation. However, we can determine various interesting effects directly from perturbation theory. For example, the vacuum polarization of Fig. IV3(a) modifies the photon propagator, i.e. the twopoint function. From Eqs. (II—1.26), (II—1.29), we determine the result for a photon of momentum q to be 2
^
^
+ ...
.
(8.3)
The essence of the effective lagrangian approach is to represent such information as the matrix element of a local lagrangian. In the present example, we find that the term in Eq. (8.3) corresponds to the interaction
where • = d^d^. The calculation of Fig. IV3(b) is a somewhat more difficult, but still straightforward, exercise in perturbation theory. We shall lead the reader through a calculation using path integrals in a problem at the end of this chapter. It too can be represented as a local lagrangian, and is usually named after Euler and Heisenberg [ItZ 80]. One finds the full result to oneloop order to be 4
607rm 2 J
^ a2
r
. 7
_
.1
(85)
IV9 Effective lagrangians as probes of new physics
121
where F^v = t^apF  Corrections to this effective lagrangian can be of two forms, (i) even at one loop there are additional terms of higher dimension
involving either more fields or more derivatives, or (ii) the coefficients of these operators can receive corrections of higher order in a through multiloop diagrams. We see here an example of the energy expansion which we have discussed at length earlier in this chapter. In this case it is an expansion in powers of q2/m2. The effective lagrangian of Eq. (8.5) can be used to compute aspects of low energy photon physics such as the low energy contribution of the vacuum polarization process or the matrix element for photonphoton scattering.
IV9 Effective lagrangians as probes of new physics One of the most common and important uses of effective lagrangians is to parameterize how new physics at high energy may influence low energy observables. The general procedure can be abstracted from our earlier discussion. Remember that one is trying to represent the low energy effects from a 'heavy' sector of the theory. This is accomplished by employing an effective lagrangian (9.1) where the {Oi} are local operators having the symmetries of the theory and are constructed from fields that describe physics at low energy. There need be no restriction to renormalizable combinations of fields. Most often the operators can be organized by dimension. The lagrangian itself has mass dimension 4, so that if an operator has dimension d{ the coefficient must have mass dimension a~MAdi
.
(9.2)
The mass scale M is associated with the heavy sector of the theory. It is clear that operators of high dimension will be suppressed by powers of the heavy mass. To leading order, this allows one to keep a small set of operators. Some applications will involve phenomena for which the dynamics is well understood. If so, the coefficients of the effective lagrangian can be determined through direct calculation as in the preceding sections. Other examples occur in the theory of weak nonleptonic interactions (c/. Sect. VIII3) and in the interactions of Wbosons (c/. Sect. XV4). Even more generally, effective lagrangians can also be used to describe the
122
IV Introduction to effective lagrangians
effects of new types of interactions. In these cases, dimensional analysis supplies an estimate for the magnitude of the energy scales of possible new physics. We shall conclude this section by using effective lagrangians to characterize the size of possible violations of some of the symmetries of the Standard Model. Given certain input parameters, the Standard Model is a closed, selfconsistent description of physics up to at least the mass of the Z°, and is described by the most general renormalizable lagrangian consistent with the underlying gauge symmetries. What would happen if there were new interactions having an intrinsic energy scale of several TeV or beyond? In general, such new theories would be expected to modify predictions of the Standard Model. The modifications would be described by nonrenormalizable interactions, organized by dimension in an effective lagrangian description as £eff = £SM + jCb + ^ £
6
+ ...
(9.3)
where Cn has mass dimension n and A is the energy scale of the new interaction. Consider the contribution £5 of dimension 5. There is, in fact, only a single operator of this dimension which contains the fields of the Standard Model and is consistent with its symmetries, i.e., C5 = c s e ^ f g ) ^
(i,j,k,e= 1,2) ,
(9.4)
where C5 is a constant, (^L)& and $& are respectively members of the lepton and Higgs doublets, and (4C')fe = ^ ) * 7 2 7 o
(9.5)
is the conjugate lepton representation. Replacing the Higgs fields by the vacuum expectation value v/y/2 produces neutrino Majorana masses of order
^fx •
(9 6)

Taking C5 = O(l) and A = (9(lTeV) would yield a neutrino mass of about 60 GeV. Phrased differently, existing upper bounds on Majorana masses of order 10 eV require that A > 1010 TeV. Turning next to possible contributions of dimension 6, there are eighty distinct operators consistent with the gauge symmetries of the Standard Model [BuW 86]. These can generate a variety of effects which deviate from the Standard Model. For example, the operator (*t*) W W^ ,
(9.7)
Problems
123
containing the Higgs field
a2
The current level of precision, p = 1.003 ± 0.004 (for mt = 100 GeV), requires A > 4.0 TeV for d = 1. Another possibility concerns the violation of flavor symmetries in the Standard Model. The operator c" C = ^ ^ ( 1 + 75 )/i 57^(1 + 7s)d + h.c. conserves generational or family number but violates the separate lepton number symmetries. It leads to the transition KL —> e~/x+ such that (99)
The present bound, B r ^ o ^ g < 2.2 x 10~10 at 90% confidence level, requires A > 250 TeV for c" ~ 1. In a similar manner, constraints on other physical processes imply bounds on their corresponding energy scales A, generally in the range 5 —> 3000 TeV. Of course, if there is new physics in the TeV energy range, it need not generate all eighty possible effective interactions. The ones actually appearing would depend on the couplings and symmetries of the new theory. In addition, the coefficients of contributing operators could contain small coupling constants or mixing angles, diminishing their effects at low energy. However, the effective lagrangian analysis indicates that the continued success of the Standard Model is quite nontrivial and places meaningful bounds on possible new dynamical structures occurring at TeV, and even higher, energy scales.
Problems 1) Effective lagrangian for fi —» e + 7 In describing the decay /i —* e + 7, one may try to use an effective lagrangian £3,4 which contains terms of dimensions 3 and 4, £3,4 = as(efi + fie) + ia± where D^ = d^ + ieQ^A^ and as, a^ are constants. a) Show by direct calculation that £3,4 does not lead to /x —• e + 7. b) If £3,4 is added to the QED lagrangian for muons and electrons, show that one can define new fields / / and e1 to yield a lagrangian which is diagonal in flavor. Thus, even in the presence of £3,4, there are two conserved fermion numbers.
124
IV Introduction to effective lagrangians c) At dimension 5, // —> e + 7 can be described by a gaugeinvariant effective lagrangian containing constants c, d, £ 5 = e
2) The EulerHeisenberg lagrangian: Constant magnetic field Consider a charged scalar field
= det(C 2 + m2)/det(D2 + m2) ,
The operation 'Tr In' applied to a differential operator is not a trivial one and the purpose of this problem is to evaluate this quantity for the case at hand. a) Demonstrate that SeS(B) = i b) In order to evaluate the trace we require a complete set of solutions to the equations D2fin(x, y, z, t) = \n
SeS(B) = t £ f ° e' m n
Jo
2s
(e—
s
S
c) With the gauge choice A^ = (0, Bxj ) show that the eigenstates are (f(x, y, z, t) = exp i(kxx + kyy + kzz  ktt) ,
,
where ^n(^) is an eigenstate of the harmonic oscillator hamiltonian, and the eigenvalues are Kn = —k 2 + k2 + k2 + k2, Xn = k2 + k2z + eB(2n + l)
Problems
125
d) Rotate to euclidean space and evaluate the trace using box quantization. Taking a box with sides Li, 1/2,1/3 and a time interval T, we have , r°° d*k (2TT)4 '
J—c
L2L3T /
A:
dky I
JO
Joo
n=Q
where the integration on ky is over all values with x' = x — ky/eB positive, e) Evaluate the effective action
2n+l)s _ f°° dk xdky ,(*S+*j))« 2TT
loo W2
n=0
and show that Seff(B) = 1
/o
eBs 1 \sinh eBs
s°
Expand this in powers of 5 , finding the (divergent) wavefunction renormalization and the B4 piece of the EulerHeisenberg lagrangian. f) Show that the corresponding result of a constant electricfieldcan be found by the substitution B —> iE so that roo
ds
 l
C3
g) Demonstrate that, although Im Setf(B) = 0, one nonetheless obtains Im n=\
and discuss the meaning of this result [Sc 51].
V Leptons
From the viewpoint of probing the basic structure of the Standard Model, leptons constitute an attractive starting point. Since effects of the strong interaction are generally either absent or else play a secondary role, the theoretical analysis is relatively clean. Moreover, a great deal of high quality data has been amassed involving these particles. Thus leptons serve as an ideal system for defining our renormalization prescription, and for investigating the effects of various radiative corrections. V  l The electron Some of the most precise tests of the Standard Model (or more exactly of QED) occur within the elementary electronproton system. The renormalization program for the theory has been introduced in Sect. II—1, where it was shown how ultraviolet divergent contributions to such calculations can be removed by means of subtraction from a finite number of suitably constructed counterterms. Here we examine the finite pieces which remain after such subtractions and compare theory with experiment. BreitFermi interaction The electromagnetic properties of the electron are studied by use of a photon probe. To lowest order, the eej vertex has the structure (e(p'e,X'e)\J^m\e(pe,Xe)) = eu(p'e,X'e)fu(pe,\e)
,
(1.1)
and the interaction between two charged particles is governed by the exchange of a single virtual photon. An important example is the electron126
Vl The electron
127
proton interaction, which has the invariant amplitude* MeP
= e2u{p'e1 A'e
(p e , Xe)^u(Pp, \phpu(
p,
Xp) ,
(1.2)
where p e , pfe and p p , p^, are respectively electron and proton momenta and q = pe—p' e is the fourmomentum transfer. In the following, we shall demonstrate how the above single photon exchange amplitude is associated with wellknown contributions in atomic physics. Denoting proton twospinors with tildes, we begin by reducing the amplitude of Eq. (1.2) in the small momentum limit to
p^+p?
1  Pp+Pp
X
Pe
XX
'
2mp
x
2me
(1.3) where me,mp are respectively the electron and proton masses. The various terms in the above expression can be interpreted physically by recalling that in Born approximation the transition amplitude and interaction potential are Fourier transforms of each other,
VeP(v) = J  ^
(1.4)
where r = r e — ry From the relation 5)
we recognize the leading (velocityindependent) term, (16)
X']X
as the Coulomb interaction between electron and proton. The identity f ddzzqq ee22 iia • p'e x pe 3 2
J (27r) q
4 m
2
e
_iq,r
e2 a • r x p e 4m2
4?rr3
^ 
We work temporarily with spinors normalized as u^ (p, s) w(p, s)=l and consider only ideal 'Dirac' protons.
128
V Leptons
allows us to recognize an additional piece of Eq. (1.3) as the spinorbit potential, which is often expressed as
but evaluated in this instance with Vo — — e2/47rr. Combining the remaining 0(p 2 /ra 2 ) terms in Eq. (1.3), we can cancel the q2 term in the denominator to obtain the socalled Darwin potential, ^ X ^ X X
4
X
•
(19)
This term has its origin in the electric interaction between the particles, and by employing the Gauss' law relation, ,
(1.10)
it can be reexpressed in the equivalent form VD = ^
V • E C o u l x'jX X']X •
(l.H)
The spinorbit and Darwin potentials, together with the 0(p 2 /ra 2 ) relativistic corrections to the electron kinetic energy, give rise to atomic fine structure energy effects. The remaining terms in the photon exchange interaction of Eq. (1.3) are effects produced by electron and proton current densities, the terms (Pe + Pe)/2me a n d ~i°"x (Pe — Pe)/<^rne representing convection and magnetization contributions respectively. In particular, the interaction between magnetization densities is equivalent to the dipoledipole potential
£

( ^ f l ) .
(1.12)
Recognizing that the magnetic field produced by the magnetic dipole moment of a (point) proton is Bproton = V X (— * ' t ^ X
XVrM ,
(1.13)
we can interpret the hyperfine energy as the interaction between the electron magnetic moment and the spininduced proton magnetic field. Upon dropping the proton and electron spinors and using the identity ,
,U4)
the dipoledipole interaction may be written as a sum of hyperfine and tensor terms, Vdpledple — ^hyp + Vtensor j
Vl The electron
Censor =
129
^ ^ [3(se  f)( Sp • r)  Se  Sp]
Denoting the total electronproton spin as Stot^Se + Sp, it follows that the hyperfine interaction splits the hydrogen atom ground state into components with Stot — 1 and stot = 0 The frequency associated with this splitting is one of the most precisely measured constants in physics and is the source of the famous 21 cm radiation of radioastronomy. As seen in Table Vl, the experimental determination is about six orders of magnitude more precise than the theoretical value. Precision in the latter is limited by the nuclear force contribution (about 3 parts in 105). Let us gather all the terms discussed thus far. In addition, we treat the proton and electron on an equal footing, since it will prove instructive when we discuss models of quark interactions in Chaps. XIXIII. We then obtain the full onephoton exchange potential (BreitFermi interaction) for the electronproton system, a 
a [serxpe r 3 [ 2m
sp r x pp 2m^
(
s p • r x p e — s e • r x pp m e mp
pe • p p + r(r • p e ) • pp] ,
(1.16)
where we recall r = r e — rp and note that a spinindependent orbitorbit interaction has been included as the final term. The singlephoton exchange interaction is seen to include a remarkable range of effects, all of which are necessary to understand details of atomic spectra. QED corrections Also important in precision tests of atomic systems are the higher order QED corrections. We have just demonstrated how the simple q2 piece of the photon propagator leads to the BreitFermi interaction between electron and proton. The vacuum polarization correction discussed in Sect. II—1 produces an additional component of the eP interaction called the Uehling potential. From Eq. (II—1.37), we recall that in the onshell renormalization scheme the subtracted vacuum polarization ft behaves in the small momentum limit m^ ^> q2 as
130
V Leptons
By the process of Fig. II—3, this yields the contribution
)  f fS_e^r  x —  J ^ ^ e
£  —6^(r)
^ Xg 0 7 r 2 ^ 
15ma
d
(1 17)
W •
(117J
The presence of the delta function implies that Swave states of the hydrogen atom are shifted by this potential while other partial waves are not. Contributions from the Uehling potential have been observed in recent scattering experiments despite its O(a) suppression relative to the dominant Coulomb scattering [Ve et al. 89]. The photonelectron vertex is also affected by radiative corrections. Let us write the proper (IPI) electronphoton vertex through first order in oc as ieTv{p'e,pe) = iejv + ieK(p'e,Pe) + . . . ,
(1.18)
where, referring to Fig. II2(b), we have in Feynman gauge
Note that a small photon mass A has been inserted to act as a cutoff in the small momentum domain, and we take both incoming and outgoing electrons to obey p2e — p^ = m2e. With a modest effort, the integral in Eq. (1.19) can be continued to d spacetime dimensions,
dx J ——
(2TT)«
(£
O l jf»/^/ lr
I A.lr(T)
I nrJ \
Am
d
i»
4lT>
I /ri

] (1.20) where px = xpe + (1 — x)p'e, and the result of performing the fcintegration can be expressed as p'e,Pe) = (h)u + (h)u ,
(121)
where (ii)t, is singular in the e —> 0 limit, .
e3
r(e/2) (e2) 2 /x e Z"1
Z"1
j/ (1.22)
Vl The electron
131
and (h)v is not, ""
(4TT)/2
\^y 3
y0
2
y0 f
V J « + A 2 (I
^ — 2/ (e — 2)ftxjl/'fix + 4y [(pe + p e)v$x — Tne(px)u — (pe
(1.23) The singular term (h)i,, which arises from the # 7 ^ term in Eq. (1.20), is infrared finite, and thus the photon mass A can be dropped from it. Upon expanding (Ii)v in powers of e and performing the yintegral, we obtain
{h)v = ieiu^
[ \ + ln(47r) y1Jdxhi
{^j
+ O{e)\ . ((1.24)
Because (I2)v is not multiplied by any quantity which is singular in e, we can immediately take the e —• 0 limit to cast it in the form dx
JO
JO
dy
y2p 2 + \2{ly) V
' *"
(1.25)
 rne(px)v  (pe + Pg) • p ^ ] • The photon mass A can be dropped from the terms in Nv proportional to y2 and ys since they are nonsingular even if A = 0. Performing the yintegration then yields the result, O3
ri rx
V * % ]v"x
i~ ^h^c
re, J
+4[(p e + pfe)vj)x — rne(px)v — (pe + Pg) • p^7i/]) • The identities VxivVx — 2m 2
Je{px)u
2
— Vr ^iv i
2
p j = m e
q x(l
\Ve + Pf>) ' Px — 2vn
p
—a
2 ,
 x)
allow (/2)i/ to be expressed in terms of #2 = (pe — Pg)2, and the dependence of p2 on the symmetric combination x{l — x) implies / dx (Px)J(p2) = \(pe +p'e), / dx f(p2) Jo * Jo
.
(1.28)
These steps, plus use of the Gordon decomposition of Eq. (C2.8) finally lead to the expression
r PP
5 Pe)
~
^OV
e2 1
' ~A o
4TT 2
r i i 7T~
2e
2+7
132
V Leptons
'ml  q2x(l  x) \
2
f
dx
2
1 f1
\
/i
~9
3m 2  q2
T^,
d •
(129)
Jo
In the onshell renormalization program, the electronphoton vertex
is constrained to obey 11 TY"1 1 & I
11111
ICl
^
' [ 1T\
j.
1 7^^ 1 &^S1
ITk
\irP^ir€,)
*"^ /l^
I
5
I
•{ I
I
V *••*'•*• 1
Q—*0
so that
(o_s) 1
e 2 [" 1
7 — ln(4?r)
2
4TT [26
y(MS)
e
2
f
1
{™>l\ 1 2
4
4
\ /i /
2
7 — ln(4?r)
1
fml\
1,
]
[ml
\ A2
fmV '(1.32)
The onshell renormalized vertex is thus given by (1.33) where F\{q2) is given by a complicated expression which we do not reproduce here, and
8TT2 y 0
dx
2— 
In addition to its original spin structure 7^, the electromagnetic vertex is seen in Eq. (1.33) to have picked up a contribution proportional to (JvpqP. The 7^ and GV^ contributions are called the Dirac and Pauli terms respectively, and F\(q2) and F2(q2) are the Dirac and Pauli form factors of the electron. The vertex correction turns out to have several important experimental consequences.
Vl The electron
133
Consider the interaction of an electron with a classical electromagnetic field for very small q2. Using Eq. (1.33) and the Gordon identity, we have nmt = eAv{x){e(p'e)\ J» m(x)\e(pe))
= eAu{x)u(p'e) ]^f  %^^\
u(pe)e^ + O(q2)
(1.35) The first term describes the coupling of the photon to the convective current of electron, but it is the second term which interests us here. Ignoring the convective term and integrating by parts, we obtain to lowest order in q, P2
T)
i
^
. (1.36) Noting that in the nonrelativistic limit a^vFpv/2 —> — a • B, we see that this is the coupling of a magnetic field to the electron magnetic moment. The result is usually expressed in terms of the gyromagnetic ratio ge\, where /xe = —eg e\se/2me, and to order e2 we have (1 +
) M ( } ) e ) e
Clearly, the radiative corrections have modified the Dirac equation value, g^ irac) = 2. The factor a/27r, which arises from the Pauli term, is but the first of the anomalous contributions. Terms of higher order in a are sufficiently large to be measurable [KiY 90], 2(^1 y  2 ) = d = ,
C2 = 0.328 478 965 ... ,
C4 = 1.472(152) ,
6 ~ 4.46 x 10~12 ,
C3 = 1.175 62(56) ,
(1.38) where uncertainties in those coefficients which were determined numerically are placed in parentheses and 6 represents contributions from sources like muon loops, tau loops, and hadrons. The theoretical values of the gfactors for both electron and muon are displayed in Table Vl, and are seen to be in accord with the experimental determinations. These represent an even more stringent test of QED than the hyperfine frequency in
134
V Leptons
hydrogen. In this case, theory is far less influenced by hadronic effects, and is thus about a factor of 104 more accurate. Radiative corrections also modify the form of the Dirac coupling. One effect of this vertex correction is to contribute to the Lamb shift which lifts the degeneracy between the 2£ 1 / 2 and 2P\j2 states of the hydrogen atom. Recall that the fine structure corrections, computed as perturbations of the atomic hamiltonian, give a total energy contribution (AE)fine str — (AE')Darwin + (A£% p i n _ O rbit + (AE)Te\
_
kin en
4
7.245 x !Q eV / 1 _ _3_\ rfi \j + 1 / 2 4nJ
(1.39)
which depends only upon the quantum numbers n and j . Thus the 2Si/2 and 2Pi/2 atomic levels are degenerate to this order, and in fact to all orders. However, the vertex radiative correction breaks the degeneracy, lowering the 2Px/2 level with respect to the 2Si/2 level by 1010 MHz. When the anomalous magnetic moment coupling (+68 MHz), the Uehling vacuum polarization potential (—27 MHz), and effects of higher order in a/ir are added to this, the result agrees with the experimental value (c/. Table Vl). Since the entire Lamb shift arises from fieldtheoretic radiative corrections, one must regard the agreement with experiment as strong confirmation for the validity of QED and of the renormahzation prescription.
The infrared problem Viewed collectively, the results of this section point to a remarkable success for QED. Yet there remains an apparent blemish  the theory still contains an infinity. When the photon 'mass' A is set equal to zero, the vertex modification of Eq. (1.29) diverges logarithmically due to the presence of terms logarithmic in A2. The resolution of this difficulty lies in realizing that any electromagnetic scattering process is unavoidably accompanied by a background of events containing one or more soft photons Table Vl. Precision tests of QED Experiment i/£yp [geY  2}a [0M  2]a AS (Lamb)6 a
1420.405 751 767(1) 1 159 652 193.(10) 11 659 230.(84) 1057.851(2)
Actually O.b(g  2) x 1012. In units of MHz.
6
Theory 1420.403(1) 1 159 652 133.(29) 11 659 194.7(143) 1057.867(11)
Vl The electron
135
whose energy is too small to be detected. For example, consider Coulomb scattering of electrons from a heavy point source of charge Ze. The spinaveraged cross section for the scattering of unpolarized electrons in the absence of electromagnetic corrections is dtt
"  P e 2/32 s i n 4 '
4
where (3 — \pe\/E is the electron speed. Radiative corrections modify this result. Using the onshell subtraction prescription and neglecting the anomalous magnetic moment contribution, one has in the limit m 2e^> q2, LIU
,
~~^
_^~
^
.
# • • "C
1
v
'
from the QED vertex correction. This diverges if we attempt to take A>0. However, we must also consider the bremsstrahlung process, in which the scattering amplitude is accompanied by emission of a soft photon of infinitesimal mass A and fourmomentum fc^. Forfcosufficiently small, the inelastic bremsstrahlung process cannot be experimentally distinguished from the radiatively corrected elastic scattering of Eq. (1.41). To lowest order in the photon momentum k the invariant amplitude MB for bremsstrahlung is* KA MB
Ze
~l
'\\l
i
•
ie
— ~T (PP) \\~ i)~r, U
q2
•
Ti
& + 1kme
L
\}
(~2e7oJ
J
(
u
\
\Pe)
_ r i ] 1 (ief)\ u(Pe) + u(Pe) (~ie70)i j L p e — p — me »* K
pe 
and has the corresponding cross section da1 = da(0> e2 / —  ^ — > 7 J (2TT)3 2fc0 ^
^—  ^—. \p'e • k pekj
(1.43)
v
J
The prime on the integral sign denotes limiting the range of photon energy, A < fco < A£*, where AE is the detector energy resolution. The polarization sum in Eq. (1.43) is performed with the aid of the complete* For simplicity, we shall take the photon as massless in this amplitude, and at the end indicate the effect of this omission.
136
V Leptons
ness relation for massive spinone photons 1
+
^ ,
(1.44)
pol
t o yield f ' dzk dH ~
cM
e
1 /
m2e
2pep'e
m2e
(2TT) 33 2k 2k~QQ \j/ee • k pee • k ~ (p e • k)2
JJ
{p'e • k)2 )
(1.45) Performing the angular integration in Eq. (1.45) with the aid of
"*°
a
4
"i
<146)
rhf!* I we find
Adding this to the nonradiative cross section of Eq. (1.41), we obtain the finite result, da
+
da^
da® I",
1+
dn £ = ~dn i
2a q2 / ,
ln
/
me \
+
5\1
uk ( ( 2 ( A E ) j 8)\ •
,
JoN
(L48)
Thus, the net effect of soft photon emission is to replace the photon mass A by the detector resolution 2A2?, leaving a finite result.* V2 The muon The analysis just presented for the electron can just as well be repeated for the muon. However, the muon has the additional property of being an unstable particle, and in the following we shall focus entirely on this aspect. The subject of muon decay is important because it provides a direct test of the spin structure of the charged weak current. It is also important to be familiar with the calculation of photonic corrections to muon decay, as they are part of the process whereby the Fermi constant Gn is determined from experiment. * As anticipated, the result quoted in Eq. (1.48) is not quite correct, since although we have given the photon an effective mass A we have not consistently included it, as in Eq. (1.42). In a more careful evaluation the constant  is replaced by the value ^ .
V2 The muon
137
Muon decay at treelevel The transition /i(pi,s) —• ^(P2) + e(p3) +Ve(p4) is the dominant decay mode of the muon. In the Standard Model, this process occurs through W^boson exchange between the leptons. However, since the momentum transfer is small compared to the Wboson mass, it is possible to express muon decay in terms of the local Fermi interaction, ^Fermi = ~^ ^
^ ^ ( 1 + T s ) ^ ^7a(l ^
a
( l +
) ^ ^ ( l
+T*)^
(2.1)
+
(2.2)
)/) ,
where the coupling constant G^ is to be considered a phenomenological quantity determined from the muon lifetime. At treelevel, G^ is related to basic Standard Model parameters as in Eq. (II3.43). The orderings in Eqs. (2.1)(2.2) are called respectively the charge exchange and charge retention forms of the interaction, and are related by the Fierz transformation of Eq. (C2.11). Let us consider the decay of a polarized muon, with restframe spin vector s, into final states in which spin is not detected. For simplicity, we set the electron mass to zero. The muon decay width is given in terms of a threebody phase space integral by
5
2^3,S4
(2.3)
where in chargeexchange form,
The muon polarization is described by a fourvector s^ which equals (0, s) in the muon rest frame. In computing the squared matrix element, we employ ^(pi,5i)U a (pi,5i) =  [(ra^+^iXl 75^)]0 a
(2.5)
to obtain \M\2 = 64GJ (pi P4P2P3
m^pt • s p2 • p 3 ) •
(2.6)
The neutrino phase space integral is easily found to be n Pi)P

^
=
l l ^
2
+ 2QV), (27)
138
V Leptons
where Q = p\ — p%. For the electron phase space, it is convenient to define a reduced electron energy x = Ee/W, where W = ra^/2 is the maximum electron energy in the limit of zero electron mass. The standard notation for the electron spectum involves the socalled Michel parameters p, <5, £ whose values depend on the tensorial nature of the beta decay interaction,
(—
 1J J j j lx2dxsin0d9
. (2.8)
For the VA chiral structure of the Fermi model, we find p = <5 = 0.75 ,
f = 1.0 ,
(2.9)
in good agreement with the current experimental values, p = 0.7518 ± 0.0023 ,
6 = 0.755 ± 0.009 ,
£P^/p
> 0.99677 , (2.10)
where P^ is the longitudinal muon polarization from pion decay (P^ — 1 in VA theory). In making comparisons between Eq. (2.8) and Eq. (2.10), one shoud first subtract from the data corrections due to radiative effects. Additional possible terms in Eq. (2.8), which would arise from taking electron mass into account or allowing for electron polarization, are also found to support the Fermi model. Upon integration over the electron phase space, Eq. (2.10) gives rise to the wellknown formula,
This relation is often used to provide an estimate for decay rates of heavy leptons and quarks.
Photon radiative corrections Thus far, we have worked to lowest order in the local Fermi interaction and have assumed massless final state particles. A more precise determination which includes lowest order corrections for the small electron mass, the large VFboson mass, and O{a) QED radiative effects yields 2+
5
G 2 ra 5 ^  [1 + (4203.85  187.12 + 1.05) x 10~6 + . . . ] , (2.12) where the magnitude of each correction is exhibited in the second line. The QED radiative correction is by far the largest of these, whereas the
V2 The muon
139
Table V2. Determinations of Fermimodel couplings Factor
Determination
GJJ, G(3
Muon decay GM plus QED theory f Nuclear beta decay \ f Hyperon beta decay Pion beta decay (TT^) Kaon beta decay (K12)
Wboson mass correction is near the present limit of sensitivity. Taking these corrections into account one obtains from the muon lifetime the value GM = 1.16637(2) x 10"5 GeV"2. One can improve the O(a) calculation by incorporating a summation of leadinglog photonic effects. This again yields Eq. (2.12), but with a replaced by the 'running' fine structure constant a{mfy. In particular, the electronpositron loop contribution to Eq. (II1.36) leads to oTl[ml)  136. The above analysis serves to define G^ as the Fermi model coupling constant extracted from muon decay, including photon radiative corrections but not other possible electroweak radiative contributions. The latter are certainly present, and will be discussed in Sect. XVI5. The coupling G^ is also commonly used to describe weak leptonic transitions of the r lepton, provided lepton universality is assumed. However, for weak semileptonic transitions of hadrons {e.g. nuclear beta decay) the photonic corrections are not identical to those in muon decay because quark charges differ from lepton charges. Such processes define instead a quantity called G/?, and we shall present in Sect. VI1 a calculation of Gp for the case of pion decay. As seen in Table V2, determinations involving G/3 generally contain quark mixing factors and also meson decay constants.
r\ (a)
(b)
(c)
(d)
(e)
Fig. Vl. Contributions to muon decay from (a) vertex, (bc) wavefunction renormalization, and (de) bremsstrahlung amplitudes.
140
V Leptons
Computation of the lowest order electron and Wboson mass corrections appearing in Eq. (2.12) is not difficult, and is left to Prob. Vl. However, the QED radiative correction is rather more formidable, and it is to that which we now turn our attention. Rather than attempt a detailed presentation, we summarize the analysis of [GuPR 80]. We shall work in Feynman gauge, and employ the charge retention ordering for the Fermi interaction. There is an advantage to performing the calculation as if the muon existed in a spacetime of arbitrary dimension d. Working in d = 4 dimensions entails factors which are logarithmic in the electron mass and which would forbid the simplifying assumption me = 0. Dimensional regularization frees one from this restriction, and such potential singularities become displayed as poles in the variable e = 4 — d. Although there would appear to be difficulty in extending the Dirac matrix 75 to arbitrary spacetime dimensions, this turns out not to be a problem here. The set of radiative corrections consists of three parts which are displayed in Fig. Vl, (i) vertex (Fig. Vl (a)), (ii) selfenergy (Fig. Vl(bc)), and (iii) bremsstrahlung (Fig. Vl(de)). We shall begin with the bremsstrahlung part of the calculation and then proceed to the vertex and selfenergy contributions. The amplitude for the bremsstrahlung (B) process /i(pi) —• e(P3) + ^e(p4) + 7(P5) is given by
x«(P3) [hd + 7,)^ _ I _ mJ
+ ih +1
_ ^7.(1 + 75)] u(Pl) ,
(2.13) where e is the photon polarization vector. The spinaveraged bremstrahlung transition rate for d spacetime dimensions in the muon rest frame is then given by* 1
f
5
2«V J A j=2l
x
J\
'
'
spins
(2.14)
* Since the result that we seek is finite and scale independent, in this section we shall suppress the scale parameter /i introduced in Eq. (II—1.21).
V2 The muon
141
where Q' = Q — p§ = p\ — ps — p$. A lengthy analysis yields a result which can be expanded in powers of e = 4 — d to read
_
r(2f)
I92TT3 4
2> 2
(2.15)
w
6
Observe that singularities are encountered as e —> 0. The radiative correction (R) contribution to the muon transition rate is given by
r
«=AI n ( j=2
x
J^
'
'
spins
(2.16) where A^?nt is the interference term between the Fermi model amplitudes which are respectively zeroth order (M^) and first order (M^) in e 2, \M&t = M<®*MW + MW*M<® .
(2.17)
The firstorder amplitude can be written as a product of neutrino factors and a term (MR) containing radiative corrections of the charged leptons,
i^
(2.18)
The quantity M^ is itself expressible as the sum of vertex (V) and selfenergy (SE) contributions, MaR = u(p3)(M$
+ MaSE)u(Pl)
.
(2.19)
The vertex modification of Fig. Vl(c) (27T)4
A;2 (p,  fc)2 [ ( p i  k)2 _
TO2]
'
^ ^
has the same form as the electromagnetic vertex correction for the electron discussed previously (c/. Eq. (1.19)) except that it contains the weak vertex 7 a ( l + 75). Upon employing the Feynman parameterization in Eq. (2.20) and using the muon equation of motion, the extension of the vertex amplitude to d dimensions can ultimately be expressed in terms of hypergeometric functions,
(2.21)
142
V Leptons
where
_F(3f,l;f;Q 1
(d3)(d2)
(d 3 ) F ( 2  j , 1; j ; Q (d4)(d2)
f,l;gl,fl '(d3)
=
2F(3 1,2; 1 + 1;fl
'
( 2 2 2 )
2F(3  f, 1; f;Q
d(d2) (d2)(d3) " For the muon selfenergy (SE) amplitude of Fig. (Vl(d)), we write
remembering that a factor of e2 has already been extracted in Eq. (2.18). Implementing the Feynman parameterization and integrating over the virtual momentum yields an expression,
;
2 ;
f / dx [md+x(2d)^1
^ ( 2 . 2 4 )
which with the aid of App. C4 can be written in terms of hypergeometric functions,
f) i — d
d+ 2 p
When the selfenergy is expanded in powers of j£ — m^, the leading term is just the mass shift, which is removed by mass renormalization. We require the /^derivative of T,(p) evaluated at jj = mM. Being careful while carrying out the differentiation to interpret p2 factors as /£/£, we find 1
r(2)
id
—IZA,
I2"26)
n •
It is this quantity multiplied by the vertex 7 a (l + 75) which ultimately gives rise to M^. However, in addition to mass renormalization there is also wavefunction renormalization, whose effect is to reduce the above quantity by a factor of 2, yielding 1
^ » )
7
°(
l + i > )

<2 27>

V3 The r lepton
143
In principle, there also exists the electron selfenergy contribution. As can be verified by direct calculation, this vanishes because the electron is taken as massless. Thus, we conclude that
V
1 d1 2 "~4(4d)(d3) *
'
The net effect of the selfenergy contribution is to replace A\ in the vertex amplitude of Eq. (2.21) by A = Ax + A2. Insertion of the radiatively corrected amplitudes into Eq. (2.16) leads to a transition rate TR which expanded to lowest order in e = 4 — d, has the form
_Gjm» 3a (( ml \ " € r(2  f) * I92TT3 4 4^ ^3 322^^ rr()r(f)r(5) ()r(f)r(5) 24
10127
,.
571 2
o2
"
(229)
5
Like the bremsstrahlung contribution, the radiatively corrected decay rate is found to be singular in the e —> 0 limit. However, the final result which is gotten by adding the radiative correction of Eq. (2.29) to that of the bremsstrahlung expression of Eq. (2.19) is found to be free of divergences,
^s 
1
^
  j
,
(2.30)
and we obtain the finite QED correction term appearing in Eq. (2.12). V3 The r lepton The heaviest known lepton is r(1784), having been discovered in e + e~ collisions in 1975. There exists also an associated neutrino vr with current mass limit mVr < 35 MeV. Like the muon, the r can decay via purely leptonic modes, M" + Pfj, + vT _> e~ + ue + uT T
(31)
However a new element exists in r decay, for numerous semileptonic modes are also present [Ts 71],
144
V Leptons Inclusive decays
It is possible to obtain a simple estimate of the relative amounts of the leptonic and semileptonic decays as follows. Because the r is lighter than any charmed state, semileptonic decay amplitudes must involve the weak quark current J& = Kd dfil + %)u + Ks sY(l + 75)M ,
(33)
where VU(\ and VQS are KM mixing elements. Neglect of both all final state masses and also effects associated with quark hadronization (an assumption only approximately valid at this relatively low energy) implies the simple estimates
r emileptonic s
_
^
1 leptonic
^
B
B
2
~ j—j^
NC=S
^
,~ , x
0.2 ,
where Nc is the number of quark color degrees of freedom. The predicted ratio of semileptonictoleptonic rates is somewhat smaller than the existing experimental value, J semileptonic 1 leptonic
= 1.85 ±0.02.
(3.4b)
expt
In order to improve upon the above naive analysis, one must include additional effects. We shall limit our discussion to a consideration of electroweak and QCD perturbative corrections on the predicted semileptonictoleptonic ratio in Eq. (3.4a). Although there exists an argument that QCD perturbation theory is justified at the relatively low scale of mT [Br 88], this claim is controversial. In the following, we shall simply quote the result of working respectively to second and third order in the electroweak and strong perturbations,
r,semileptonic * leptonic
theo
l
L
7T
\mT 2
X
(3.5)
V3 The r lepton
145
where the term containing the Z° mass accounts for the different electroweak radiative corrections of leptonic and semileptonic processes* and the QCD correction is computed with the aid of Eq. (II2.74). The qualitative effect of such corrections is seen to improve agreement between theory and experiment. Exclusive leptonic decays The r affords an opportunity to test the principle of lepton universality, i.e. the premise that the only physical difference among the charged leptons is that of mass. In particular, all the charged leptons are expected to have identical weak couplings. Thus, the muon and electron branching ratios should be equal, aside from a relative phase space suppression for the former,
^r _ei> T
eJ / T
 18 f^V12 feV In 4m\+  •• 0.973 . (3.6) \m ) \m ) T
T
From the measured leptonic branching ratios [Dan 92], 0.1789 ±0.0014
(r> ePci/T) ,
we obtain i?expt = 0.969 ± 0.019, which is consistent with the prediction of Eq. (3.6). Another noteworthy consequence of lepton universality is the observation that the rates for r^e+ue+uT and /i—>e+P e+z^ must have the same form (cf. Eq. (2.12)). Thus, their lifetimes should obey the scaling relation
Using the experimental value for BrT^e^eZ,r given in Eq. (3.7) as input, this can be used to predict the tau lifetime, which we display together with the measured value, r r  thy = (2.942 ± 0.024 ± 0.038) x 10"13 s , i9
(3.8b)
r r  expt = (2.95 ± 0.03) x 10"13 s , where the two uncertainties which we associate with r T  thy arise respectively from BrT+ejyel/T and mT. Agreement between theory and experiment * This point is considered in detail in Sect. VI1.
146
V Leptons
in Eq. (3.8b) is somewhat unsatisfactory. Since the data base continues to expand, a sharper test of universality should be forthcoming. The momentum spectra of the electron and muon modes also probe the nature of r decay. The Michel parameter p of Eq. (2.8) should equal 0.75 for the usual VA currents, zero for the combination V+A and 0.375 for V or A separately. The observed value p — 0.70 ± 0.05 is in accord with the VA structure. Exclusive semileptonic decays Matters are somewhat more complex for the hadronic final states, due in part to the large number of modes. However, for many of these we can make detailed confrontation of theoretical predictions with experimental results. We begin by noting that the semileptonic decay amplitude factorizes into purely leptonic and hadronic matrix elements of the weak current J^k = J£ pt + j£ adr ,
a Msemilept = "7= ^
Hfr
,
p = (hadronsj£ adr 0)  (hadronsCl where the flavor SU(3) components l+i2 (4M5) induce AS = 0 (AS = 1) transitions respectively. It is easiest to analyze the modes containing a single meson, T~+Meson + */r
(Meson = TT, K~, p"(770), if*(892)) . (3.10)
Weakcurrent matrix elements which connect the vacuum with Jp = 0~,l + hadrons are sensitive to only the axialvector current, whereas Jp = 0 + ,l~ states arise from the vector current. In each case, the vacuumtomeson matrix element has a form dictated up to a constant by Lorentz invariance, (7r(q) Ajr i2 (0) 0) = iyftF^
,
(3.11a)
(if"(q) 4~ i 5 (0) 0) = iV2FKq,
,
(3.11b)
2
(p"(q,A)  ^ ( 0 )  0 ) = V 2 ^ ( q , A) ,
(3.11c)
iT(q, A) ^fwhere the quantities gp and QK* are the vector meson decay constants. These quantities contribute to the transition rates for pseudoscalar (P)
V3 The r lepton
147
and vector (V) emission, and we find from straightforward calculations
(312) 2
mT
where rap, ray are the meson masses, 77KM — c? for AS = 0 decay and rfcM = c^s\ for AS = 1 decay. It is possible to imagine using the above formulae to extract the quantities F ^ , . . . , gx* from tau decay data. However, they are obtained more precisely from other processes and in practice, one employs them in tau decay to make branching ratio predictions. In Chaps. VI, VII, we shall show how the values Fn = 92 MeV and FK/F^ = 1.22 are found from a careful analysis of pion and kaon leptonic weak decay. Interestingly, the hadronic matrix elements which contribute there are just the conjugates of those appearing in Eqs. (3.11a), (3.11b). By contrast, the quantity gp is obtained not from weak decay data, but rather from an electromagnetic decay such as p°—• ee. That the same quantity gp should occur in both weak and electromagnetic transitions is a consequence of the isospin structure of quark currents. That is, the electromagnetic current operator is expressed in terms of octet vector current operators by J^m — Vjf + 7?^?Since the latter component is an isotopic scalar whereas the p meson carries isospin one, it follows from the WignerEckart theorem that
<0J£V(P)> = (0^>0(P)> = ^
• (3.13)
The transition rate for the electromagnetic decay p°—> ee is given, with final state masses neglected, by (3.14) from which we find gp/mp = 0.198 ± 0.009. The K* vT mode can be estimated by using the flavor SU(S) relation gK*—9p The predictions for single hadron branching ratios are collected in Table V3, and are seen to be in satisfactory agreement with the observed values. A somewhat different approach can be used to obtain predictions for strangeness conserving modes with Jp = 1~. Since matrix elements of the vector charged current can be obtained through an isospin rotation from the isovector part of the ee annihilation cross section into hadrons,
148
V Leptons Table V3. Some hadronic modes in tau decay. Mode T~ —• 7T~
Hadronic input + VT
r
T" > K~ + I/T r " > p " 4 vT
A:
T~ —•
K*~ \ Vr T~ —• 7T~7r~7r 7r°iyr T~
a
—> 7T~7T 7T 7r°i/r
C%Sl9l
a(e+e~ —^ • h a d r ) cr(e + e~ —J • h a d r )
Br[thy]°
Br [expt]a
11.4 0.8 23.4 ± 0.8 1.1 ±0.1 4.9 0.98
10.8 ± 0.6 0.78 ± 0.08 22.2 ±1.0 1.33 ±0.13 4.8 ± 0.4 0.98 ±0.14
Branching ratios are given in percent,
we can write for a given neutral 7 = 1 hadronic final state / ° ,
where we have defined / The r
(3.16) transition into the isotopically related charged state / ,
in
(h
is governed by the same function Uf(q2) of hadronic final states as occurs in Eq. (3.15). Including the lepton current and relevant constants, and performing the integration over vT phase space yields a decay rate
tf {ml  q2)Hml + 2
f*
(m 2  S ) 2 (m 2 + 2 s ) s ^^ / O ( S ) . (3.19)
Thus, we find, e.g. for 4?r final states, the results listed in Table V3. Yet another hadronic channel of interest is that with Jp = 1 + , which is expected to be dominated by the axialvector meson ai(1260). In view of the decay chain a{
 > p°n~  • 7T+7r~7r" ,
(3.20)
this resonance should appear in the 7r+7r~7r~ spectrum in r decay. In this instance, r decay data is being put to the somewhat novel use of helping
V~4 The neutrinos
149
to pin down the a\ mass and width values. The determination of these quantities has been a longstanding problem in hadron spectroscopy due in part to the presence of substantial background in the data, whereas r decay data is expected to relatively free of such problems. The current world averages are [RPP 90] mai = 1260 ± 30 MeV ,
r fll = 330 > 500 MeV ,
(3.21a)
to be compared with the values inferred from r decay [Dan 92], mai = 1211 ± 70 MeV ,
r a i = 446±? MeV ,
(3.21b)
There exist numerous additional hadronic decay modes of the r lepton. Examples include final hadronic states containing KK,KKTT, etc., and it is possible to analyze each of these with various degrees of theoretical confidence. Another interesting use of the r semileptonic decay has been to confirm by inference the fundamental structure of the weak quark current from the absence of the mode r~ —> 7r~r]uT. This mode, proceeding through the vector current, would violate Gparity invariance. Here Gparity refers to the product of charge conjugation and a rotation by TT radians about the 2axis in isospin space, G = Ce~i7rl2 .
(3.22)
A weak current which could induce a AG ^ 0 transition is referred to as a second class current. Such currents do not occur naturally within the quark model. The 7r~r) VT mode has not been detected, with an existing sensitivity of Br r _>,,.^ < 0.9%. This result, consistent with the absence of second class currents, fits securely within the framework of the Standard Model. V—4 The neutrinos Throughout this book, we assume that neutrino mass is zero. However, if neutrinos do have mass [KaGP 90, BoV 87, MoP 91], the phenomenon of neutrino mixing will arise, analogous to the quark mixing of Sect. II—4. For simplicity, let us consider the case of just two neutrino generations, v e and i/r, which are defined by the weak interaction. The weak interaction eigenstates are related to mass eigenstates v\ and v
\vr)\
Vsin0 v
where 6y is the vacuum mixing angle. How is it possible to determine the mass of neutral and weakly interacting particles like neutrinos? One probe involves measurement of the electron energy spectrum in nuclear beta decay, (A, Z) —» (A, Z + l) + e~ + ve.
150
V Leptons
If, for the sake of argument, we assume rai < 7712, then the maximum electron energy (endpoint of the electron energy spectrum) in nuclear beta decay occurs at ml
.
(4.2)
Studies of tritium beta decay yield the best bound, mVf, < 9 eV. With the birth of neutrino astronomy via detection of a neutrino burst from supernova 1987A, an independent check of the ve mass limit has yielded a consistent, although slightly weaker limit, mVe < 18 eV. Useful information from nuclear beta decay data is also contained in the electron energy spectrum N(Ee) away from the endpoint. It follows from Eq. (4.1) that N(Ee) should be the sum of two components, corresponding respectively to detection of v\ and v
Neutrino oscillations Despite the lack of direct evidence for neutrino mass, there is a vigorous ongoing experimental effort to search indirect signals like neutrino oscillations. In the twogeneration description, an electrontype neutrino created at time t = 0 and propagating with momentum p will have at subsequent time t the state vector \ue(t)) = i/i) cos0 v e~iElt + \u2) sin0 v e~iE2t = i/ e )(cos 2 0 V e~iElt + sin 2 6 V e~iE^) + i/r) sin(9v cos(9v (e~
iE2t
iElt
 e~ )
(4.3) ,
where E\ = >/p 2 + m 2 and E
PUe(t) = 1  PVT(t) . (4.4)
Since neutrino mass is small compared to neutrino momentum in typical situations, we approximate
^i
... ,
(4.5)
so that PVT (t) ~ sin2 20V sin 2 [Am 2 L/4^] ,
P^ (t) = 1  PVT (t) .
(4.6)
where Am 2 = ml — mf. This is the expression that is usually employed in twoflavor analyses of neutrino oscillation experiments. It contains the
V~4 The neutrinos
151
two theoretical parameters 6y and Am 2. Observe that we have written the expression in terms of the path length L ~ t/c instead of the time t. Sometimes, L is expressed in units of a 'vacuum length' Lv, PUr(t) ~ sin2 2<9V sin2[7rL/Lv]
(Ly = AnE^/Am2)
.
(4.7)
For Am2 = 2.48 eV2 and Ev = l MeV, one has L v = l m . Terrestial searches for neutrino mixing There are two categories of terrestial searches for neutrino oscillations. The first consists of 'disappearance' experiments in which detectors are placed outside a reactor that emits an intense ve flux, and reaction rates for processes like uep —> e+n are measured as a function of the reactordetector separation L. If no oscillation is present, the measured rates should scale like L~2 as L is varied. However, neutrino mixing superimposes the additional factor P^e(L), and it is this modulation which signals the existence of the mixing phenomenon. The second category involves 'appearance' experiments. Suppose an intense 7r± flux is generated at a proton accelerator by directing the proton beam onto a massive target (such an operation is called a 'beamdump'). Escaping from the target will be a high flux of v^, v^ and ve particles resulting from the decays TT+ —> /i + v^ /i + —• e + z/ e ^. Where are the ue? They are in fact present, but since most of the \i~ flux from TT~ decay will be absorbed by the target before decay, there will be a suppression of ve particles leaving the target. Thus, one probes the generation of ve from v^ve mixing by measuring the rate for ve interactions at the detector. Results of disappearance and appearance experiments are compiled in [RPP 90]. No positive signals from either type of experiment have yet been observed, and the region of parameter space with sin2 26y &nd Am2 both 'large' has been ruled out, e.g. vef+ve disappearance experiments imply the constraints sin2 20V^O.14 (for large Am2) and Am2^0.014 eV2 (for sin2 2#v — !)• Of course, this need not indicate that neutrino mixing does not occur, since regions of very small Am 2 and #v can never be ruled out. In particular, taking (Eu) ~ 10 MeV and L ~ 100 m, we estimate a sensitivity for terrestial experiments at the level of Am 2 > 10"1 eV2 for reasonable values of the mixing angle. This can be improved with the study of solar neutrinos, where L ~ 1.5 x 1011 m. In addition, one encounters the interesting phenomenon of neutrino propagation in matter. Solar neutrinos As a result of various nuclear processes in the Sun's core, solar neutrinos are created with a spectrum of energies. For the sake of simplicity, we
152
V Leptons
can classify energies as high (H), intermediate (/), and low (L), EH>6
MeV :
Ei ~ 0.8 + 2 MeV :
Bs > (5e 8 )* + e+ + ue J5e7 + e" > Li7 + i/e
0 15
EL ~ 0.2 + 0.5 MeV :
_^ ^15
+
e+ +
^
+
p + p  * d + e + z/e ,
and solar nuclear reactions which contribute to each class are also shown. There are a number of existing and proposed solar neutrino detectors which probe different parts of the neutrino spectrum. Those which have published solar neutrino data are the Homestake mine chlorine detector [Da 89], the Kamiokande water Cerenkov detector [Hi et.al. 90], and the SAGE gallium detector [Ga 90]. It is common to summarize the data in terms of R = /£ b s //^ S M , the ratio of observed solar neutrino flux to that predicted on the basis of the Standard Solar Model [Bah 90], and to express R in terms of the survival probability PUe for solar neutrinos as measured on Earth. Letting P^e (k = H, / , L) represents the survival probability of z/e's in the high, intermediate, and low parts of the energy spectrum, we can approximate the energy dependence of each type of detector as ifamst ^ 0.78 P ^ + 0.22 Ple
[ye + Cl37 > Ar37 + e~)
ifaam ^ 0.14 + 0.86 P%
(i/c + e~ > ve + e~)
{ye + Ga71 > Ge71 + e~) , (4.9) where reactions used in each detector are also shown. To date, the detected flux of high energy neutrinos is found to be less than that predicted, i?Gai ^ 0.1 P? + 0.36 Pi + 0.54 P£
#Hmst = 0.27 ± 0.05 ,
#Kam = 0.46 ± 0.08 .
(4.10)
Taken at face value, Eqs. (4.94.10) indicate the suppression of high energy neutrinos is accompanied by an even stronger suppression of intermediate energy neutrinos. However, because of the strong sensitivity to experimental uncertainties and details of solar structure, the interpretation of these results is not yet decisive. Fortunately, gallium detectors have begun to observe events, and if one assumes values P^ ~ 0.38, P j ~ 0, sensitivity to the less model dependent flux of lower energy neutrinos is predicted, i?Gai ^ 0.04 + 0.54 P£ .
(4.11)
In addition, heavy water [Ea et. al. 86] and borex [RaP 88] detectors will provide sensitivity to charged and neutral current neutrino scattering
V~4 The neutrinos
153
events, „
i
x
Heavy water:

^
• •
(charged)
(neutral) V ; Borex:
.
<
(charged)
(4.12)
(neutral) , where i/a is a neutrino of arbitrary flavor. Comparison of neutral and charged current events within a given detection is insensitive to calculations of absolute solar fluxes, thus permitting a direct test of the oscillation scenario. Several possibilities for attributing the solar neutrino deficit to neutrino properties are: 1) 2) 3) 4)
neutrino neutrino neutrino resonant
decay into noninteracting ('sterile') particles, oscillation in the vacuum, spinflip induced by magnetic effects within the Sun, neutrino oscillation within the Sun.
Of these, let us consider the last, the MSW effect [Wo 78, MiS 86], since it is perhaps the most natural explanation and has attracted a good deal of attention. Considering the twoflavor model and absorbing any common phase factors into the state vectors, we have for neutrino propagation within the Sun l
,d_ I" i/c) 1 _ Am2 /  cos 20V + 4 ^ p / A m 2
dt [i/ T )J ~~ AEV V
sin20 V
sin 20V \ I We)'
cos20 v y [\vT)
 (4.13) where p = \/2G^Ne and Ne = Ne(r) is the solar electron number density. The factor p arises from the vee charged current interaction,* and modifies the ve propagation relative to any other neutrino flavor, thus giving rise to an additional phase in \ve(t)). The vacuum mixing of Eq. (4.1) is thus altered to
K)l
\vT)
=
/ cos0M
\ — sin0M
where 0M, the mixing angle in matter, obeys tan 20M =
cos 20 V 
* Neutral weak current effects are common to all neutrino flavors.
154
V Leptons
with the critical electron number density N£ given by ^rr
N s
Ara 2 cos20 v
_ Am2(eV2)
3
 JTiG^r   ' " s s W 
94
*
on
10 cm
o
•
Since Ne will decrease from the solar center to the surface, provided Ne ~ 1026 cm" 3 ^> N£ at or near the solar center, interesting effects are expected within the Sun at Ne ~ iVec [RoG 86], [Hax 87], [BaB 90]. Two solutions having the required suppression of high energy solar neutrinos are found [GeR 86], an adiabatic solution* corresponding to large mixing (sin2 20 ~ 1) with little suppression at intermediate and low energy, and a nonadiabatic solution which can be approximated as log10 Am2(eV2) + log10 sin2 20V ^ 7.5 ,
(4.17)
with stronger suppression at lower energies. Although existing results appear to support the latter solution, data from low energy neutrinos will be needed to provide a stringent test of the MSW mechanism. Dirac mass and Majorana mass In a theory containing both lefthanded and righthanded massless neutrinos of a given flavor, the neutrino fields are expressible as tp^/ = F L V ^ \ R
R
where the projectors FL = ^(1 ± 75) ensure that the masslessfieldsij)\/ R R . have just two nonzero components. It is possible to construct a Hermitian, Lorentzinvariant, leptonconserving interaction which couples the two chiralities £Dira, =  m i ) ( # # + # # ) =  m D ^ ^ , (4.18) where the field ip^ is now a fourcomponent entity.^ Such an interaction is said to give rise to a Dirac contribution to fermion mass, and in the Standard Model it is this mechanism (via the Higgs effect) which gives rise to fermion mass. Of course, if the field ip^ does not exist, then no such construction is possible. This is why neutrinos are massless in the Standard Model. A logically independent source of neutrino mass, called a Majorana contribution, is available even in the absence of righthanded neutrinos. In this instance, a chiral field such as ^L *S c o u Pl e d to its conjugate ^ * Neutrino propagation within the Sun is 'adiabatic' if the fractional change in Ne is small per neutrino oscillation cycle. t Note that lepton conservation is ensured by the invariance of Z^Dirac under the field transformations ?/>L —• e~iotipL and ^R, ~> ^ R e i a .
Problems
155
(cf. Eq. (IV9.5)), ^Majorana = ™>M{$^
C
^Y, + h x  ) *
( 4 19)
Although ^Majorana is Lorentz invariant and Hermitian, it does not conserve lepton number. One can use this feature to seek experimental evidence for Majorana mass in nuclear double beta decay, A,Z + 2) + 2e+2Pe A,Z + 2) + 2e"
(m,.=0) (Majorana) .
The first case in Eq. (4.20), which occurs in the usual description with massless neutrinos, corresponds to a secondorder weak interaction transition, and has been observed in Se82 [E1HM 87] with halflife T 1 / 2 = 1.1^0*3 x 1020 yr This constitutes the slowest transition which has been detected by a laboratory measurement. The second case (neutrinoless double beta decay) can occur for Majorana neutrinos, which upon being emitted from a beta decay vertex can propagate to initiate a second beta transition. Neutrinoless double beta decay has not been observed in Ge76 at the level of T 1 / 2 > 5 x 1023 yr [Ca etal. 87]. This places the limit ^M < 0.7 —• 1.8 eV on the effective Majorana mass. The range of values indicates the uncertainty in evaluation of nuclear matrix elements and 'effective' mass is a linear combination of Majorana mass eigenstates, each weighted by the absolute square of the corresponding mixingmatrix element. Problems
1) Muon Decay a) Obtain the leading (9(ra 2 /ra 2 ) correction to the Fermi model expression Eq. (2.17) for the muon decay width. b) Do the same for the leading O{m2tl/M^) correction. 2) Vacuum polarization and dispersion relations The vacuum polarization II(g2) associated with a loop containing a spin onehalf fermionantifermion pair, each of mass m, can be written as the sum of a term containing an ultraviolet cutoff A and a finite contribution ft2
U(q2) =  \l In ^  2 f dx x(l  x) In (1  ^x(l  x) a , A2 i
156
V Leptons a) Show that flf(q2) is an analytic function of q2 with branch point at q2 = 4m2 and with Im tlf(q2) = aRf(q2)/3, where
is related to the rate for radiative pair creation via
p/)(f\JZaWWL\o)
= (flV + 9 ^
b) Use Cauchy's theorem and the result of (a) to express 3TT J4m2
s(s  q
2
 le)
2
c) The form of tif{q ) given in part (a) can be reexpressed in a dispersion representation. First change variables in (a) to y = 1 — 2x and integrate by parts to obtain
(V " I") Then, change variables again to s = 4m2/(1—y 2) and demonstrate that the dispersion result of (b) obtains.
VI Very low energy QCD — pions and photons
At the lowest possible energies, the Standard Model involves only photons, electrons, muons and pions, as these are the lightest particles in the spectrum. We give a separate discussion of this portion of the theory because it can be done with a higher level of rigor than most other topics. The main tool is chiral symmetry, in particular the effective chiral lagrangians. Pion physics gives us a chance to see how these techniques work and to test them in a relatively clean setting. VI1 QCD at low energies The SU(2) chiral transformations AA i>L,R =
,
> exp (iOL,R' T)I\)L,R
(1.1)
WL,R
almost give rise to an invariance of the QCD lagrangian for small rau, m^, but do not appear to induce a symmetry of the particle spectrum. This is because the axial symmetry is dynamically broken (i.e. hidden) with the pion being the (approximate) Goldstone boson. Vectorial isospin symmetry, i.e., simultaneous SU(2) transformations of ipL a n d ipR, remains as an approximate symmetry of the spectrum. Isospin symmetry is seen from the near equality of masses in the multiplets (T^TT 0 ), (K+,K0), (p, n), etc. In the language of group theory, we say that SU(2)L X SU(2)R has been dynamically broken to SU(2)y> What is the evidence that such a scenario is correct? Ultimately it comes from the predictions which result, such as those which we detail in the remainder of this chapter. There is as yet no rigorous proof that QCD does indeed undergo dynamical symmetry breaking with the pattern observed in Nature. Lattice Monte Carlo techniques [Cr 83] are coming close to such a proof, and perhaps need only the inclusion of dynamical fermions to become convincing. However, our confidence in this symme157
158
VI Very low energy QCD  pions and photons
try realization has historically been developed mainly through theoretical selfconsistency and phenomenological success. The effective lagrangian for pions at very low energy has already been developed in Chap. IV. In particular we recall the formalism of Sect. IV6 which includes couplings to lefthanded (righthanded) currents ^ ( x ) (r^x)), and scalar and pseudoscalar densities s(x) and p(x), with the resulting O(E2) lagrangian,
C2 = *f TV (D,UD^) U = exp(ir • TT/F^) ,
+ *f TV \ = 2B0(s + ip) ,
where D^U = d^U + ilJU — iUr^ and BQ is a constant. QCD in the absence of sources is recovered with l^ = r^ — p = 0 and s = m, where m is the quark mass matrix.
Vacuum expectation values and masses With a dynamically broken symmetry, the lagrangian is invariant but the vacuum state does not share this symmetry. A useful measure of this noninvariance in QCD is the vacuum expectation value of a scalar bilinear, ( 0  ^  0 > = <0ViLVi?0> + <0^VL0>
.
(1.3)
Up to small corrections, isospin symmetry implies (0im0) = (0
.
(1.4)
Such matrix elements, if nonzero, cannot be invariant under separate lefthanded or righthanded SU(2) transformations. Indeed it is evident from Eq. (1.3) that the vacuum expectation value couples together the lefthanded and righthanded sectors. One way that the vacuum expectation values of Eq. (1.4) affect phenomenology is through the pion mass. If u and d quarks were massless, the pion would be a true Goldstone boson with m^ = 0. The part of the QCD lagrangian which explicitly violates chiral symmetry is the collection of quark mass terms Hm = —C
m
= muuu + mddd .
(1.5)
To first order in the symmetry breaking, the pion mass is generated by the expectation value of this hamiltonian, ml = (n \muuu + rriddd\ TT) .
(1.6)
This quantity can be related to a vacuum expectation value by using softpion techniques (see Sect. IV4) to remove both pions, resulting in
VI1 QCD at low energies
159
where qq represents either uu or dd. Equivalently, the effective chiral lagrangian can be used to find the vacuum expectation value ( 0  ^  0 ) . Using the notation of Sect. IV6, we have ml = (mu + md)B0,
(0 \qq\ 0) = —^
= F^B
0
,
(1.8)
which has the same content, to this order in the energy expansion, as Eq. (1.7). Thus since both BQ and the quark masses are required to be positive, consistency requires that (0gd independently. As an aside, we note that for Goldstone bosons there is a clear answer to the perennial question of whether one should treat symmetry breaking in terms of a linear or quadratic formula in the meson mass. For states of appreciable mass, the two procedures are equivalent to first order in the symmetry breaking since 6(m2) = (mo + 6m) — TTIQ = 2rriQ 6m + . . . .
(19)
However when the symmetry expansion is about a massless limit, the m vs m2 distinction becomes important. Because pions are bosonic fields we require their effective lagrangian to have the properly normalized form, (0^7r^7rra£7r7r)+... .
(1.10)
The prediction for the pion mass must then have the form m\ = (mu + md)B0 + (mu + md)2C0 + {mu  md)2D0 + . . . .
(1.11)
In principle, Nature could decide in favor of either m\ ex mq orra^ex m2 depending on whether the renormalized parameter BQ vanishes or not. However, the choice Bo = 0 is not 'natural' in that there is no symmetry constraint to force this value. Since one generally expects a nonzero value for Bo, the squared pion mass is linear in the symmetry breaking parameter mq. There is every indication that BQ ^ 0 in QCD. When we add the kaons and the eta to the analysis, this will imply that SU(3) symmetry breaking and the GellMannOkubo relation must also be treated with squared masses.
Pion leptonic decay and Fn Throughout our previous discussion of chiral lagrangians, the pion decay constant Fn has played an important role. It is defined by the relation »Fi r p^* ,
(1.12)
160
VI Very low energy QCD  pions and photons
where the axial vector current A^ is expressible in terms of the quark
fields V = 0 as
4 = V^75y •
(113)
This operator is probed experimentally in the decays n —> eve and TT —> • ^ which are induced by the weak hamiltonian
(1.14)
The decay TT+ —• /i"^^ has invariant amplitude
where the Dirac equation has been used to obtain the second line. An analogous expression holds for TT4" —> e+ve. We see here the wellknown helicity suppression phenomenon. That is, the weak interaction current contains the lefthanded chiral projection operator (1 + 75) which in the massless limit produces only lefthanded particles and righthanded antiparticles. However, such a configuration is forbidden in the decay of a spinzero particle to massless fiP^ or eve because the leptons would be required to have combined angular momentum Jz = 1 along the decay axis. Thus the amplitudes for n —> /iP^, eve must vanish in the limit m^ = me = 0. Since the neutrino is always lefthanded, the /x+, e + in pion decay must have righthanded helicity to conserve angular momentum. It is helicity flip which introduces the factors of m^, me. The decay rate is found to be
^
[j
.
(1.16)
However, before using this expression to extract the pion decay constant, one must include radiative corrections. We shall do this in some detail as similar considerations will be used to obtain the kaon decay constant in Sect. VII2 and in the analysis of nuclear beta decay in Sect. XII4. Since a complete analysis would be overly lengthy, we present a simplified argument which stresses the underlying physics. A more formal presentation, with identical results, appears in [Si 72]. In Chap. V we found that the radiative correction to the muon lifetime is ultraviolet finite even in the approximation of a strictly local weak interaction. However, this is not the case for semileptonic transitions, as
VI1 QCD at low energies
161
Fig. VI1 Photonic radiative corrections to the weak quarkquark interaction can be easily demonstrated. Consider the photon loop diagrams shown in Fig. VI1. We divide the photon integration into hard and soft components. The former, which determine the ultraviolet properties of the diagrams, have short wavelengths A < < i?, where R is a typical hadronic size, and are sensitive to the weak interaction at the quark level. In Landau gauge (i.e. £=0), the ultraviolet divergences arising from the wavefunction renormalization and vertex renormalization diagrams depicted in Fig. VI1 vanish. For example, the vertex term is r(u.v.)
iGF
f d4k 1
2
k ^7A(1 + 7
where Qie is the electric charge of the ith particle. Using J
(1.17)
k* ~ 4 J (27r)4fc4
(2TT)4
we find that /Vertex = 0 as claimed. It is clear, employing a Fierz transformation, that photon exchange between particles 4,1 and 2,3 is also ultraviolet finite. This result simply represents the nonrenormalization of the vertex of a conserved current found in Chap. V. The only ultraviolet divergences then arise from final state and initial state interactions, i.e. photon exchange between particles 4,2 and 3,1 , for which d k1 r(u.V.) —^G JP 2/I
f *
(
j (2^^ ( x «4y^ ^F
C
7
A
( l + 7 5 ) ^ 2 7 ^ 7 A ( 1 + 75)«3 + (2,4 > 1,3)
/^ yr^ ! _
A r_
_
/.,
, _ x
_ _.a
a X
V2 327r 2 ^ 4 ^ ' l + 7 5 ) u i ] + (2,4> 1,3) .
(1.19)
162
VI Very low energy QCD  pions and photons
Using the identity in Eq. (C1.5) for reducing the product of three gamma matrices, Eq. 1.19 becomes 1 ^
= M{0) x —(Q4Q2 + QsQi) ln(A/// L) ,
(1.20)
where M.^ is the lowestorder vertex. However, the full calculation of the radiative corrections must include the propagator for the H^boson as well. When the contact weak interaction is replaced by the ^exchange diagram and is added to that with the photonexchange replaced by Zexchange, one obtains a finite result at the ultraviolet end with A = mzThe integral is cut off at the lower end at some point /J,L ~ rri£ below which the full hadronic structure must be considered. In the case of muon decay we have
Thus as found in Chap. V, there is no divergence. On the other hand, for beta decay we obtain QeQu + Qv Qds =
(1.22)
We observe that there exists an important difference between the betadecay effective weak coupling (Gp) and the muondecay coupling (G^) (1.23) This hard photon correction must be added to the softphoton component which can be found be evaluating the radiative corrections to a structureless ('point') pion with a high energy cutoff //#. These were calculated long ago with the result [Be 58, Ki 59],
') .
r (0)
a24)
where
^z l n x l l x — 1
\\n(x2l)2\nxA
J[ , ) \nx
4 3 lOz ?, 15z2  2 1 + 4 ^ L ( l x + z\wx + ^ , 4 x2 — 1 4 (x2 — I)2 4(x2 — 1) (1.25) with L(z) = JQ J ln(l — t) being the Spence function and x = m^/m^. Adding the hard and softphoton contributions with //# = JIL — mp, we 2
VI2 Chiral perturbation theory to one loop
163
find the full radiative correction,
l + f (B{x) + 3In ™* + In ^ m
2?r V
Taking Vud
7r
 6In ^
rnp
m^
(1.26) from Sect. XII4, and T ^ + i / = 3.841 x 10 s" , we find Fv = 92.4 ± 0.2 MeV , (1.27) {
7
1
where we have appended an uncertainty associated with possible radiative effects O(a/27r) which are not included in Eq. (1.26). For chiral symmetry applications in this book we shall generally employ the value Fv~92MeV
.
(1.28)
A clear indication of the importance of radiative corrections can be seen in the ratio R = I 7+
which is strongly suppressed by the helicity mechanism discussed earlier. Application of the lowestorder formula given in Eq. (1.16) leads to a prediction
in disagreement with the measured value Rexpt = (1.218 ± 0.014) x 10"4 .
(1.31)
However, when the full radiative correction given in Eq. (1.26) is employed, the theoretical prediction is modified to become Rth
= R
+ .. ) = 1.235 x 10"4 ,
(1.32)
which is consistent with the experimental value.
VI2 Chiral perturbation theory to one loop Let us summarize the development thus far. Interactions of the Goldstone bosons can be expressed in terms of an effective lagrangian having the correct symmetry properties. To lowest order in the energy expansion, i.e. to order E2, it suffices to use the minimal lagrangian of Eq. (1.2) at treelevel. In the SU(2) theory, this involves just the known constants Fn and m^. At the next order, one encounters both the general O(E4) lagrangian, given below, and also oneloop diagrams [ApB 81, GaL 84,85a]. The O(E4) lagrangian introduces new parameters, which must be determined
164
VI Very low energy QCD  pions and photons
from experiment. It is also necessary to give a prescription which allows one to handle the loop calculations. The general method is described in this section. The program is called chiral perturbation theory. If one works to order E4 in the energy expansion (this is the present state of the art), there are typically three ingredients: 1) the general lagrangian £2 (of order E2) which is to be used both in loop diagrams and at treelevel, 2) the general lagrangian £4 (of order E4) which is to be used only at treelevel, 3) the renormalization program which describes how to make physical predictions at oneloop level. The general O(E2) lagrangian has already been given in Eq. (1.2). Now we shall turn to the construction of the chiral SU(n) lagrangian to order E4. The order E4 lagrangian The O(E4) lagrangian can involve either fourderivative operators or twoderivative operators together with one factor of the quark mass term, X ~ 2mqBo (which itself is of order ra^ or m2K) or products of two quark mass factors. There are four possible chiral invariant terms with four separate derivatives, TV (
V
Tr
D
^
U
( D ^ U D V U * )
D
^
U
]
J
D
' T r( D ^ U D ^ U ^
V
U
V
,
[Tr
D
V
U (
^
^
^
J
]
) , T r ^ (2.1) 2
Other structures, such as
[Tr (x'tfDpU) Tr (Vt/tZ^f/)]
,
(2.2)
can be expressed in terms of these by using SU(n) matrix identities. For the case of SU(3), the operators in Eq. (2.1) are not linearly independent. The identities quoted in Eq. (II—2.17) can be used to show that Tr + Tr IDUUDVIP) • Tr iLPUirW)
 2 TV
(2.3) leaving only three independent operators in this class. In 5C/(2), a further identity, 2Tr [D^UD^U]DVUDVU^
= [Tr
(D^UD^U^)2
,
(2.4)
V
)
VI2 Chiral perturbation theory to one loop
165
leaves us with only two independent O(E^) terms. Another conceivable class of operators could have at least two derivatives acting on a single chiral matrix, such as • Tr (u^DuDvU^
Tr
.
(2.5)
However, since the E 4 lagrangian is to be used only at treelevel, all states to which it is applied obey the equation of motion,
D" (U^D^U) + i (xfC/  tfX) = 0 •
(2.6)
This can be used to eliminate all the double derivative operators in favor of those involving four single derivatives or with factors of \. The remaining operators are reasonably straightforward to determine, and the most general O(E4) 5/7(3) chiral lagrangian is,* 10
= c*i [Tr (D^UD^U^ + a3 Tr
+ a2Tv (D^UD^
• Tr
(D^UD^D (
^
Tr
UX]))
a7[Tr +ai0Tr
^
(2.7) where L ^ , R^v are the fieldstrength tensors of external sources given in Eq. (IV6.10). This is a central result of the effective lagrangian approach to the study of low energy strong interactions. Much of the discussion in the chapters to follow will concern the above operators and involve a phenomenological determination of the {a^}. In chiral 5C/(2), three operators become redundant. * We are using the operator basis first set down by Gasser and Leutwyler [GaL 85a]. However, in order to avoid confusion with the symbol for lagrangian, we shall call the coefficients ai instead of ^ .
166
VI Very low energy QCD  pions and photons Table VI1. Renormalized coefficients in the chiral lagrangian £ 4 given in units of 10~3 and evaluated at renormalization point ji = nirj.
Coefficient a\
«S
Origin
0.65 ± 0.28 1.89 ±0.26 3.06 ± 0.92
7T7T scattering
2.3 ±0.2
«5 r
Value
r
2a 7 + a 8 "10
K)° K)6 («S) 6
and Ki4 decay FK/FK
0.4 ±0.1
meson masses
7.1 ±0.3 5.6 ±0.3
rare pion decays
0.4±? 0
~0
a
Determined only with the additional assumption of 777/ mixing. b Vanishes in the 7VC —> 00 limit.
For completeness, we note that there may also exist two combinations of the external fields, £ext = Pi TV (L
+ R^BT) + 02 TV (x f x) ,
which are chirally invariant without involving the matrix U. These do not generate any couplings to the Goldstone bosons and hence are not of great phenomenological interest. However, if one were to use the effective lagrangian to describe correlation functions of the external sources, these two operators can generate contact terms. Finally, we summarize in Table VI1 the values of the low energy constants {cti} as obtained phenomenologically. The determination and application of these coefficients will be discussed in subsequent sections of the book. They provide a characterization of the low energy dynamics of QCD. The renormalization program The renormalization procedure is as follows. The lagrangian, £2, when expanded in terms of the meson fields, specifies a set of interaction vertices. These can be used to calculate treelevel and oneloop diagrams for
VI2 Chiral perturbation theory to one loop
167
any transition of interest. This result is added to the contribution which comes from the vertices contained in the O(E4:) lagrangian £4, treated at treelevel only. At this stage, the result contains both bare parameters and divergent loop integrals. One needs to determine the parameters from experiment. The first step involves mass and wavefunction renormalization, as well as renormalization of Fn. In addition, the parameters entering from £4 need to be determined from data. If the lagrangian is indeed the most general one possible, relations between observables will be finite when expressed in terms of physical quantities. All the divergences will be absorbed into defining a set of renormalized parameters. This fundamental result is demonstrated explicitly in App. B2. There exists always an ambiguity of what finite constants should be absorbed into the renormalized parameters a\. This ambiguity does not affect the relationship between observables, but only influences the numerical values quoted for the low energy constants. Similarly, the regularization procedure for handling divergent integrals is arbitrary.* We use dimensional regularization and the renormalization prescription, •
(2 8)

where the constants 7$ are numbers given in Table Bl. When working to O(E4:) the following procedure is applied. One first computes the relevant vertices from £2 and £4. There are too many possible vertices to make a table of Feynman rules practical. In practice, the needed amplitudes are calculated for each application. The vertices from £2 are then used in loop diagrams, including mass and wavefunction renormalizations. The results may be expressed in terms of the renormalized parameters of Eq. (2.8). If these low energy constants can be determined from other processes, one has obtained a welldefined result. Including loops does add important physics to the result. The low energy portion of the loop integrals describes the propagation and rescattering of low energy Goldstone bosons, as required by the unitarity of the 5matrix. Oneloop diagrams add the unitarity corrections to the lowest order amplitudes and in addition contain mass contributions and other effects from low energy. The effective lagrangian may be used in the context either of chiral 5*7(2) or of chiral SU(3). Because SU(2) is a subgroup of 517(3), the general SU(3) lagrangian of Eq. (2.7) is also valid for chiral SU(2). However, the SU(2) version has fewer low energy constants, so that only * Care must be taken that the regularization procedure does not destroy the chiral symmetry. Dimensional regularization does not cause any problems. When using other regularization schemes, one sometimes needs to append an extra contact interaction to maintain chiral invariance [GeJLW 71]. The problem arises due to the presence of derivative couplings which imply that the interaction Hamiltonian is not simply the negative of the interaction lagrangian. The contact interaction vanishes in dimensional regularization.
168
VI Very low energy QCD  pions and photons
certain combinations of the a\ will appear in pionic processes. If one is dealing with reactions involving only pions at low energy, the kaons and the eta are heavy particles and may be integrated out, such that only pionic effects need to be explicitly considered. This procedure produces a shift in the values of the low energy renormalized constants OL\ such that the a\ of a purely 5/7(2) chiral lagrangian and an 5/7(3) one will differ by a finite calculable amount. In this book, we shall use the SU(3) values as our basic parameter set. The SU{2) coefficients can be found by first performing calculations in the SU(3) limit and then treating ra^ra^ as large. Equivalently, all may be calculated at the same time using the background field method [GaL 85 a]. The results are
3 = 2«\ + a\  \t lK 2
,
«)r
 \lK  ±fa ,
=
 _
"if = cTw + tK
^ ^ l n 4 >
(29)
where we use the superscript (2) to indicate constants in the SU(2) theory and define £, = [ln(mf//x2) + l] /384TT2. In practice these shifts are much smaller than the magnitude of the low energy constants, so that v/e always simply quote the SU(3) value. Let us now calculate the mass and wavefunction renormalization constants to O(E4) in chiral 5(7(2). Setting % = (mu + VTI^BQ = m2,, we may expand the basic lagrangian as
(2.10)
where Fo denotes the value of Fn prior to loop corrections. When this lagrangian is used in the calculation of the propagator, the terms of O((p4) in £2 will contribute to the selfenergy via oneloop diagrams which in
VI2 Chiral perturbation theory to one loop
169
volve the following ddimensional integrals, 6ikI(m2) =iA F j k (0) = T,
2N
4
4
f ddk
m
y (a^M
2
(2.11) These contributions can be read off from £2 by considering all possible contractions among the O((p4) terms, and result in the quadratic effective lagrangian,
£ eff = I
8a»]]  ^
• *»f [32a? .
(2.12)
To oneloop order, there are no other contributions to the self energy. Observe that we have changed mo, FQ into ra^, Fn in all of the O(E4) corrections, as the difference between the two is of yet higher order in the energy expansion. If we expand in powers of d — 4 and define the —1/2
renormalized pion field as (pr = Zn '
C/t) which forbids the transition of an even number of mesons to an odd number. However this is not a symmetry of QCD. More importantly, there are a set of lowenergy relations, the WessZumino consistency conditions [WeZ 71], which must be satisfied in the presence of the anomaly and which involve hadronic reactions. The effect of the anomaly was first analyzed by Wess and Zumino who noted that the result could not be expressed as a single local effective lagrangian, and gave a Taylor expansion representation for it.* Witten [Wi 83a] subsequently gave an elegant representation of the WessZumino contribution as an integral over a fivedimensional space whose boundary is physical fourdimensional spacetime. Since the considerations leading to the WessZuminoWitten action can be rather formal, it is best to adopt a direct calculational approach. Fortunately, we are able to employ the familiar sigma model (with fermions) because it contains the same anomaly structure as QCD. That is, it is the presence of fermions having the same quantum numbers as quarks which ensures that the anomaly will occur. The absence of gluons in * Note that if Eq. (2.13) is used to express Eq. (2.14) solely in terms of «7, the result is explicitly scaleindependent because otj does not depend on /z. * For a textbook treatment, see [Ge 84].
]
(2.13) then the lagrangian assumes the canonical form £eff = \dtf>T • &><pr ~ \m\ipr • ifr
.
(2.14)
Note that, using the definitions of the renormalized parameters, the physical pion mass is identified as (2)r
A
(2)r
o
(2)rl .
^4
m (2.15)
170
VI Very low energy QCD  pions and photons
The quantity £eff i n Eq. (2.14) is the quadratic portion of the oneloop effective lagrangian. Since loop effects have already been accounted for, it is to be used at tree level. This is a simple application of the background field renormalization discussed in App. B2. An alternative approach to the renormalization is the subject of Prob. VI5.
VI3 Interactions of pions and photons In order to understand the nature of chiral predictions and the range of validity of the energy expansion, let us work out several examples. At first, these will seem to be rather obscure processes, but they are the simplest hadronic reactions of QCD. As the bosonic interactions of the Goldstone bosons of the theory, they are the cleanest processes for demonstrating the dynamical content of the symmetries and anomalies of QCD.
The pion form factor The electromagnetic form factor of charged pions is required by Lorentz invariance and gauge invariance to have the form* where q** = (pi — pif and (7^(0) = 1. The electromagnetic current may be identified from the effective lagrangian of Eq. (1.2) by setting fji _ r/i _ eQAy^ x — 2i?om, where Q is the quark charge matrix and m is the quark mass matrix. To (9(i?4), we then find
8ai2>l^ (3.2)
0 71
>Y > (a)
(b)
Fig. VI2 Radiative corrections to the pion form factor The neutral pion form factor is required to vanish by charge conjugation invariance.
VIS
Interactions of pions and photons
171
The renormalization of this current involves the Feynman diagrams in Fig. VI2. That of Fig. VI2(a) is simply found using the integral previously defined in Eq. (2.11), (3.3) (2a)
Evaluation of Fig. VI2(b) is somewhat more complicated. Using the elastic TT+TT" scattering amplitude given by £2? (7r + (k 1 ) 7 r(k 2 )7r + (p 1 ) 7 r(p 2 )) (3.4) we compute the vertex amplitude to be
*_ f
A k ddk
 ^ J {2ix)
1 (k
1 it (k  \q)  ml 2
(3.5) Upon integration, most terms drop out because of antisymmetry under k^ —> — fc^,and we find
+ hf  ml) ((&  h)2 ~ ™l)
(3.6)
We can evaluate this integral using dimensional regularization, 2 fj,A~d f , \ 1 (Pi + P 2 ) T (l — 2) Wem>(26) = ~J^ ^ d / 2 J dx\~2 ( m 2 _ 2X n _ x\) ld/2
"(3.7) where as usual fi is an arbitrary scale introduced in order to maintain the proper dimensions. Onshell, we can disregard the term in q^ since q • (pi +P2) = ml — ml = 0. For the remaining piece, we expand about d = 4 to obtain (Pi
(3.8) x
172
VI Very low energy QCD  pions and photons
and the xintegration then yields (Jem) (2b) —
d4
+ 7 — 1 — In 4TT
(39) where
H(a) =  /
Jo
dx\n(lax(lx))
22\l V a
(0 < a < 4)
= < In
m^(a  4)
(otherwise) .
(3.10) Now we add everything together. The treelevel amplitude is modified by wavefunction renormalization, Sml ,„ (2) C2)^
ml
f2
4TT2F2 \ d  .
7 — 1 — In 4?r + In —^ (2)
(3.11) while Figs. VI2(a),2(b) contribute as
f 2
^ < —r + 7  1  In47r + In 2
(2a)
(26)
16^2^2
{d4
^~lq
) lrf4
71 
m2
(3.12)
VI3
Interactions
of pions and
photons
173
respectively. Summing Eqs. 3.11,3.12 we see t h a t all terms independent of q2 cancel, a^ ' becomes a$ and the final result is
(313) T h e divergences have been absorbed in a^ , while the imaginary part required by unitarity is contained in H(q2/m^). Note t h a t the loops also induce a nonpowerlaw behavior in Gn(q2). However, numerically this t u r n s out to be small and is unobservable in practice. A simple linear approximation,
2a {2)r
TTI"
+ ...
(3.14)
is obtained by Taylor expanding about q2 = 0. The corresponding result for chiral SU(3) is
The pion form factor is generally parameterized in terms of a charge radius, G,{q2) = 1 + \{rl)q2 + • • • •
(3.16)
Thus, for any given value of the energy scale //, the parameter a^ can be determined from the experimental charge radius. Prom the present experimental value (r£) = (0.44 ± 0.02) fm2, we obtain ag(fj, — m v) = (7.1 ±0.3) x 10 3 . The scale fj, enters the calculation in such a way that, had we used a different scale // but kept the physical result invariant, we would have had
V' 2
(317)
In fact, if we look back to the origin of In fi in the transition from Eq. (3.7) to Eq. (3.8) using 2
2
'
°
~"
4) ,
(3.18)
174
VI Very low energy QCD  pions and photons
we see that the scale dependence is always tied to the coefficient of the divergence.* The general result is then (3.19) where {7^} are the constants of Table Bl of App. B, used in the renormalization condition of Eq. (2.8). This calculation also nicely illustrates the range of validity of the energy expansion. The pion form factor is welldescribed by a monopole form, G
*(q2) " 1 
2/m2
= 1 +
^
+
'"
^ 3 ' 20 )
'
with m ~ ™> p(77o) The energy expansion is then in powers of q2/m2.
Rare pion processes The calculation above is clearly nonpredictive as it contains a free parameter, ag, which must be determined phenomenologically. However, predictions do arise when more reactions are considered because relations exist between amplitudes as a consequence of the underlying chiral symmetry. In particular there exists a set of reactions which are described in terms of two low energy constants. These pionic reactions are shown in Table VI2. With the additional input of FK/FK, the kaonic reactions shown there are also predicted. Each case contains hadronic form factors which need to be calculated. This section briefly describes the procedure for relating such reactions in chiral perturbation theory. All calculations follow the pattern described above, so that we shall only quote the results [GaL 85 a, DonH 89]. In the processes involving photons (?r+ —» e+^ e 7, TT+ —• e+i/ee+e~ and 7?r+ —> 77r+), there are always Born diagrams where the photon couples
« \ (a)
(b)
(c)
Fig. VI3. Born diagrams for (a) TT+ —> e + i/ e 7, 7r+ —> e^~uee^e~, and (b)(c) +
+
We have chosen to keep the low energy constants {a^} dimensionless in the extension to d dimensions. In [GaL 85a], the constants have dimension / i d ~ 4 . However, the resulting physics is identical in the limit d —>• 4.
VI3 Interactions of pions and photons
175
Table VI2. The radiative complex of pion and kaon transitions. Pions 7 —•
Kaons
7 > K  ^ +
7T + 7T~
^ + _7e+^7
TT"*" —* e~^~i/e'y +
n+ »7r°e i/ e e~
K > ite+ve K+ + e+vee+e~
to hadrons through the known 7T7T7 coupling. These are shown in Fig. VI3. In addition, there can be direct contact interactions associated with the structure of the pions. These introduce new form factors. For the decays TT+ —• e+z/e7, e+vee+e~, the matrix elements are
^
j (
P
, q )  ^
(3.21)
X u{p2)l^v{pi)u{pv)Y{^]rlb)v{Pe)
,
where the hadronic part of the quantity M^v has the general structure
,q) = J ((p  ?)M9i/  g^q  (p  q)) a p^ .
(3.22)
The first line represents the Born diagram and in subsequent lines the subscripts V and A indicate whether the vector or axialvector portions of the weak currents are involved. The form factor rA in Eq. (3.22) can only contribute with virtual photons, i.e., as in TT+ —» e + i/ e e + e~. The 7?r+ —» 7TT+ reaction is analyzed in terms of the pion's electric and magnetic polarizabilities, C*E and (3M, which describe the response of the pion to electric and magnetic fields. In the static limit, electromagnetic fields induce the electric and magnetic dipole moments, PE = 4naEE
,
» M = 4TT/3MH ,
(3.23)
which correspond to an interaction energy V =2n
(aEE2 + (3MH2)
.
(3.24)
176
VI Very low energy QCD  pions and photons
These forms emerge in the nonrelativistic limit of the general Compton amplitude
,p',qi) =  i J d*X € (2pf + g l ) ^(2p  q2) ~mi
(PQ2)
mi
 g^qi ' 92) + • • • , (3.25) where <J is a coefficient proportional to the polarizability and q^q2 are the photon momenta, taken as outgoing, with p = pr + q\ + q2. The first three pieces are the Born and seagull diagrams. The last contains the extra term which emerges from higher order chiral lagrangians, and the ellipses indicate the presence of other possible gauge invariant structures which we shall not need. The chiral predictions are obtained in the same manner as used for the pion form factor. The results at q2 ~ 0 are hy =
N, 12y/2 ir2 F* Nc=3
= 0.027 m;; '
(3.26) where t = (q\ + qz)2 and •^*****
X
i V "^ / "" /
7
i r» ~T
l»a;
* * *
*
lO.^I
I
12
The prediction for hy is especially interesting since hy is related by an isospin rotation to the amplitude for TT0 —• 77 (c/. Prob. VI2). As shown in Sect. VI5, this is absolutely predicted from the axial anomaly. The presence of a[0 implies that one of the above measurements must be used to determine it. We use the precisely known value for hAJhy to yield a[0(fjL = rrtri) = (5.6 ± 0.3) x 1(T3 .
(3.28)
The results are compared with experiment in Table VI3. We see that with one exception, the chiral predictions are in agreement with experiment. That exception, the electric polarizability in 7TT+ — > • 7TT+, comes from a single difficult experiment using a pion beam on a heavy Z atom [An et al. 85]. The coulomb exchange in n+A —> ^n+A is used to provide the extra photon (this is called the Primakoff effect),
VI~4 Pionpion
scattering
177
Table VI3. Chiral predictions and data in the radiative complex of transitions. Reaction 7 —> 7r
+
7r~
7 » * : + # K+ > e + ^ e 7 • e
Quantity
Theory
Experiment
0.44a
0.44 ± 0.02
<^> (fm2) ^(m1) hA/hv (hy + hA)(rn~1)
0.44 0.027 0.46a 0.038
0.34 ± 0.05 0.029 ± 0.017 0.46 ± 0.08 0.043 ± 0.003 2.3 ±0.6 1.4 ±3.1
rA/hv
(a£+/?M)(10 4 fm) a£;(104fm)
X _>
7re+Zye
f = /_(0)// + (0) A+ (fm2) Ao (fm2)
a
2.6 0 2.8
0.13 0.067 0.040
6.8 ±1.4 0.20 ± 0.086 0.065 ± 0.005fe 0.050 ± 0.0126
Used as input. There axe experimental inconsistencies in these measurements. We use average values.
b
and the Born diagram must be carefully subtracted off. Before being concerned with this discrepancy, it would be preferable to have the experiment carefully redone. We have also listed the known results on kaonic processes predicted by the same constants. The analyses for 7 —» K+K~ and K —> e+uej are identical to the above results. For K —* n°e+uei we defer a presentation until the next chapter. VI4 Pionpion scattering The process na + TT^ —> TT7 + TT6 is, of course, not directly measurable in the laboratory. However, information about it can be inferred indirectly [MaMS 76]. For example, in the decay K —• 7T7reue the rescattering of the pions in the final state leads to an observable interference between the 5wave and Pwave TTTT amplitudes. A somewhat more complex procedure can be used in TTN —> TTTTTV. Here the longest range scattering will be due to pion exchange, as in Fig. VI4. If the amplitude, defined as a function of q2, were to be measured at the pion pole (q2 = ra2), the pion scattering amplitude could be unambiguously identified. Unfortunately, the pion pole cannot be reached in this reaction, because q2 is negative. In principle, however, the data can be extrapolated outside the physical region to determine the value at the pion pole. This is the procedure used in most studies of TTTT scattering. In practice, some caution should be exercised in that additional theoretical assumptions are often introduced
178
VI Very low energy QCD  pions and photons
in the extrapolation procedure. These make the experimental error bars smaller than is justified by experiment alone, and this has led to the situation where experiments are often in conflict outside of their quoted errors. However, the general trend of these experiments is reasonably agreed upon. In terms of the Mandelstam variables S= (Pa+ Ppf ,
P7)2 ,
t=(pa
U= (pa Psf
,
(4.1)
the 7T7T scattering amplitudes are determined in terms of a single function A(s,t,u) as = A ( s , t, u ) 6 a p 6 7 s + A ( t , s , u ) 6
a i
6p
6
+ A ( u , t,
s
)
^
(4.2) Moreover, they can be decomposed into amplitudes of definite isospin, T°(s, t, u) = 3A(s, t, u) + A(t, s, u) + A{u, t, s) , Tl(s, t, u) = A{t, s, u)  A(u, t, s) ,
(4.3)
2
T (s,t,u) =A(t,s,u) + A{u,t,s) , and the partial wave amplitudes can be projected out, 7
«,t,u)
.
(4.4)
Below the threshold for inelastic reactions such as TTTT —• 4TT, KK, etc., the 5matrix is given in terms of the phase shift 6j by Sj = e2tSt for each isospin and partial wave. In the following, we work with Tmatrix elements, defined by
In practice this form is useful up to about 5 = 1 GeV2, by which point inelastic effects have become sizeable J.
_
_
_
_
_
_
_
A
K
N
Fig.
n N
VI4 Pionpion scattering from irN —> TTTTN.
VI~4 Pionpion scattering
179
Table VI4. The pion scattering lengths and slopes. Lowest Ordera First Two Ordersa
Experimental
a°0
0.26 ± 0.05
0.16
0.20
b°o
0.25 ± 0.03
0.18
0.26
al
0.028 ± 0.012
0.045
0.041
bl
0.082 ± 0.008
0.089
0.070
a\
0.038 ± 0.002
0.030
0.036
b\
—
0
0.043
0
20 x 10"4
0
3.5 x 10"4
al al a
(17 ±3) x 10"
4
(1.3 ±3) x 10"4
Predictions of chiral symmetry.
At low energies, the partial wave amplitude can be expanded in terms of a scattering length a\ and slope b, defined by
where q2 = (s — 4ra^) /4. Since the chiral expansion is similarly a power series in the energy, a\ and b\ provide a useful set of quantities to study. In practice they are extracted from data by using dispersion relations and crossing symmetry to extrapolate some of the higher energy data down to threshold. The only accurate very low energy data is that from K —> 7T7reue. The experimental values are given in Table IV4. At lowest order in the energy expansion, the amplitude for TTTT scattering [We 66b] can be obtained from £2 with the result A(8,t,«) = — ^
•
(4.7)
This produces the scattering lengths and slopes 0
"
2
m
l
u2 _
m
l
*
(4.8)
with the numerical values shown in the table. It is remarkable that the lowest energy form of a scattering process may be determined entirely from symmetry considerations.
VI Very low energy QCD  pious and photons
180
The nature of the chiral expansion also becomes evident from these formulae. The lowest order predictions are real and grow monotonically. As such, they must eventually violate the unitarity constraint at some point. The worst case is the / = £ = 0 amplitude Isml)
,
(4.9)
which violates the simplest consequence of unitarity, a4ro* ,
Z 2
(4.10)
<1 ,
below y/s = 700 MeV. In addition, there are no imaginary terms, which must be present due to the unitarity constraint, (4.11)
Im T =
These drawbacks are remedied order by order in the energy expansion. Note that since T/ starts at order £"2, Im T/ starts at order E4. When one works to order J54, loop diagrams generate an imaginary piece given by Eq. (4.11) with the lowest order predictions for T/ inserted on the righthand side. This process proceeds order by order in the energy expansion.* The O(E4) predictions modify the scattering lengths slightly [GaL 84]. Aside from the renormalization of mn and Fn, the corrections depend only on the low energy constants (2a\ + ajj) and ar2. We shall determine the low energy constants in Chaps. VII, VIII, with the most important
1.00.8T1 1
i
0.6
0.40.20.0
200
400
600
800
1000
Fig. VI5 Pionpion scattering data. One can invent a variety of schemes for imposing unitarity to all orders, but these all involve additional assumptions and depart from the controlledframeworkof chiral perturbation theory.
VI5 The axial anomaly and TT°—> 77
181
influence being the K —> 7T7reue form factors. The formulae are not illuminating, and we quote the results in Table VI4. The overall agreement is good. More instructive is a pictorial representation of the result. One may see the order EA improvement and the nature of the chiral expansion by considering the I = 1, £ — 1 channel, where some of the higher energy data are shown in Fig. VI5. The resonance structure visible is the p(770). The chiral prediction is
with loops having a negligible effect. The lowest order result is given by the dashed line. It clearly does not reflect the presence of the p(770) resonance. The solid line represents the result at order E4 and starts to reproduce the low energy tail of the p(770). It is, of course, impossible to represent a full BreitWigner shape by two terms in an energy expansion  all orders are required. The chiral predictions at O(E4) may reproduce the first two terms, with the resulting expansion being in powers of q2/m^ in agreement with Eq. (3.20).
VI5 The axial anomaly and TT0
77 The description of pions and photons presented thus far does not include the decay TT° —> 77. This process is important in QCD, because to understand it one must include the anomaly in the axial current. The TT°—> 77 amplitude has the general structure fkv^y0
,
(5.1)
as required by Lorentz invariance, parity conservation and gauge invariance, and leads to the decay rate
g
\2
(5.2)
.
Prom the experimental value, T = 7.7 ± 0.6 eV, we find A 77  0.025 ± 0.001 GeV"1 . The effective lagrangians that we have been using thus far do not contain the totally antisymmetric tensor e^VOL^', so it is clear that they cannot generate TT°—> 77. However, there exists another class of contributions. These are associated with the anomaly content of the theory. Let us defer to Chap. VII the derivation of such contributions because there are important features which first appear only for chiral SU(3). Here we simply quote the restriction to Srf/(2), CA = 4 ^ 2
[ ^
p
p
^
p
(5.3)
182
VI Very low energy QCD  pions and photons
with
Tp=Tr (Q2LP + Q2RP + ^ where Aa is the photon field, F^v is the photon field strength, and Nc = 3 is the number of colors. A crucial aspect of this expression is that it has a known coefficient. In this respect, it is unlike other terms in the effective lagrangian, which have free parameters that need to be determined phenomenologically. This is because it is a prediction of the anomaly structure of QCD. A corollary of this is that CA must not be renormalized by radiative corrections. This was proven at the quarkgluon level by Adler and Bardeen [AdB 69]. The 7T° —• 77 amplitude is found by expanding CA to first order in the pion field, yielding
(5.5) where we have integrated by parts in the second line. This produces a Tj0 _> ^j matrix element of the form rvN
t
°^GV> ,
(5.6)
in excellent agreement with the experimental value. This is widely recognized as an important test of QCD, both as a measurement of the number of colors and also as a reflection of the symmetries and anomalies of the theory. It is a remarkable result. What would have happened if the axial anomaly were not present? The decay TT0 —> 77 could still occur, but it would be suppressed. The 7T°—> 77 transition must be at least of order i? 4 , as it must involve the dimension 4 operator FF. The anomaly occurs at this order. However, nonanomalous lagrangians leading to this transition can be constructed at order E6. This result was first derived by Sutherland and Veltman using a softpion technique [Su 67, Ve 67]. Following the steps employed in Sect. IV4, one uses the PC AC identification dxJ§x = rrv^F^ip^o to calculate the matrix element
Id'x 2
_
m
2\
J
(5.7)
VI6 The physics behind the QCD chiral lagrangian
183
In going from the first to the second line, we have integrated by parts and have used the fact that the commutator terms vanish. In the third line, the form which appears for the matrix element is required by the gauge invariance of the electromagnetic currents together with parity. Because of the overall factor of q2, the twophoton amplitude A 77 would seem to vanish in the softpion limit. It could exist at order ra2, but this would lead to a strong suppression of the decay, roughly of order (ml/1 GeV2) ~ 3 x 10~4 in rate. However, the flaw in this argument is the use of the PC AC relation, as will be demonstrated in Chap. VIII. At what level would we expect corrections to the anomaly prediction for 7T°—> 77? It has been checked that ml In ml corrections, which in principle can occur when meson loops are present, do not in fact modify the lowest order result when it is expressed in terms of the physical decay constant F^. However, there still could be corrections of order m 2 /A 2 , where A is the scale in the energy expansion, i.e. A ~ mp —> 1 GeV. These could amount to modifications of order a few percent or perhaps ±0.0008 in A 77. VI6 The physics behind the QCD chiral lagrangian For the most part, we have been using chiral lagrangians as our primary tool for making predictions based on the symmetry structure of QCD. In this section, we pause to examine which features of QCD are important in determining the structure of chiral lagrangians. The general strategy can perhaps be appreciated by a comparison of low energy and high energy QCD methodology. At high energies, due to the asymptotic freedom of QCD, hard scattering processes can be calculated in a power series expansion in the strong coupling constant. However, some dependence on 'soft' physics remains in the form of structure functions, fragmentation functions, etc. These are not calculable perturbatively and must be determined phenomenologically from the data. At high energy then, the predictions of QCD are relations among amplitudes parameterized in terms of various phenomenological structure functions and the strong coupling constant. At very low energies, because of the symmetries of QCD, low energy scatterings and decays can be calculated in a power series expansion in the energy. However, some dependence on 'harder' physics remains in the form of the constants {a£}. These are not calculable from the symmetry structure and must be determined phenomenologically from the data. At low energy then, the predictions of QCD take the form of relations among amplitudes whose structure is based on symmetry constraints but which are parameterized in terms of empirical constants. Nevertheless, QCD should in principle also predict the very structure functions and low energy constants which are employed by these techniques. The trouble at present is that we do not have techniques of
184
VI Very low energy QCD  pions and photons
comparable rigor with which to calculate these quantities. Nevertheless, by using models plus phenomenological insight we can learn a bit about the physics which leads to the chiral lagrangian. The low energy constants Fn and m^ which occur at order E2 do not reveal much about the structure of the theory. All theories with a slightly broken chiral 5/7(2) symmetry will have an identical structure at order E2. The pion decay constant Fn will be sensitive to the mass scale of the underlying theory, while the pion mass m^ will be determined by the amount of symmetry breaking. However, approached phenomenologically, these are basically free parameters and do not differentiate between competing theories. The situation is different at order E 4 . Here the chiral lagrangian contains many terms, and the pattern of coefficients is a signature of the underlying theory. The linear sigma model without fermions provides us with an example of how one can compare a theory with the real world. In Sects. IV2,4 we calculated the treelevel terms in £4 which would be present in the linear sigma model, and obtained a result expressible as F2 1 2ai + as = 2a 4 + a 5 = 8a 6 + 4a 8 = — ^ = rr , ^2,7,9,10 = 0 . (6.1) This pattern is quite different from the structure obtained phenomenologically. It appears that the linear sigma model is not a good representation of the real world. Unfortunately, it is harder to theoretically infer the {c^} directly from QCD. However, a look at phenomenology indicates that we should consider the effects of vector mesons, in particular the p(770). This is the most clear in the pion form factor which shows a dramatic p resonance in the timelike region. Indeed, the whole form factor can be well understood in a simple model as being a BreitWigner shape due to the p resonance
G G
tf)
*tf)
=
m2 P 2  4m2) ' q2 _ m 2 + impTp(q)0(q
(62) (6 2) *
where the normalization is chosen to enforce the condition Gn(0) = 1. This works even in the timelike region. Comparison with the chiral lagrangian approach implies that this model would predict 17.2
a9 = £=•  7.2 x 10" 3 , (6.3) 2 2ra in good agreement with the value obtained earlier, a$ = (7.1 ±0.3) x 10~3. This analysis can be extended to aio This enters into the W+TT+J vertex which occurs in TT+ —• e+ue^f. Here both vector and axialvector mesons can generate corrections to the basic couplings. Explicit calcula
VI6 The physics behind the QCD chiral lagrangian
185
tion yields [EcGPR 89] = 5.8 x 1(T3 .
(6.4)
Here a\ refers to the lightest axialvector meson ai(1260) (c/Sect. V3), and Fai and Fp are the couplings of a\ and p to the W+ and the photons respectively. Again the result is close to the empirical value aio = (5.6±O.3) x 1(T3. The phenomenological low energy constants are scale dependent, and their analysis includes loop effects, while those in Eqs. (6.3), (6.4) are constants, to be used at tree level. Nevertheless, there is some sense in comparing them. The effect of loops in processes involving #9,0:10 is small, and the scale dependence only makes a minor change, ag(fi = 300 MeV) = 7.7 x 10~3 vs ar9(n = 1 GeV) = 6.5 x 10"3. Presumably the appropriate scale is near /i = 771^770). The p(770) provides a much more important effect here than any other input. Finally it also turns out that the use of vector meson exchange leads to a good description of TTTT scattering [DoRV 89, EcGPR 89]. This is not too surprising in light of the need for the chiral lagrangians to reproduce the tail of the p(770), as described in Sect. VI4. As a consequence of crossing symmetry, the p(770) must also influence the other scattering channels. To a large extent, the chiral coefficients 0^,0:2,03 are dominated by the effect of p(770) exchange. We see from these examples that phenomenology indicates that the exchange of light vector particles is the most important physics effect behind the chiral coefficients which we have been discussing. The idea that vector mesons play an important dynamical role is not new. It predates the Standard Model, originating with Sakurai [Sa 69], in a form called vector dominance. The vector dominance idea has never been derived from the Standard Model, but nevertheless enjoys considerable phenomenological support. Put most broadly, vector dominance states that the main dynamical effect at energies less than about 1 GeV is associated with the exchange of vector mesons. The use of a chiral lagrangian with parameters described by pexchange, is compatible with this idea and puts it on a firmer footing. These considerations suggest that for chiral lagrangians the prime ingredient of QCD is the spectrum of the theory. The linear sigma model has a quite different spectrum, with a light scalar and no p, and hence does not agree with the data. QCD, however, seems to predict that deviations from the lowest order chiral relation must be in such a form as to reproduce the low energy tails of the light resonances, in particular the p. At present, we cannot rigorously prove this connection. However, it remains a useful picture in estimating various effects of chiral lagrangians.
186
VI Very low energy QCD  pions and photons
Problems
1) Radiative corrections and TT^ decay To bring Tir^eVe /FV^i^ into agreement with experiment requires a radiative correction whose dominant contribution is the socalled seagull component (y/^F^g^) of M^ (cf. Eq. (3.22)). a) Verify that gauge invariance requires
and show that the seagull term is required in this regard to cancel the pion pole contribution. b) Use the seagull term in Feynman gauge to calculate the radiative correction to TT^ decay. Introduce a photon cutoff via 1 k2
1
A 2
k2 k2  A 2
so that
X w(Pi/)7A(l + 75)
—fr e + fa — rr
and show that \
2?r rri£
where
\
L
is the lowest order amplitude for the 71^2 process. This then is the origin of the lepton mass dependent radiative correction. 2) Radiative pion decay and the anomaly Writing the TT°decay amplitude as
= ie^*e? J
Problems
187
and the vector current amplitude in radiative TT+ decay as
^Lt^a
= *< J d*xe^
demonstrate that isotopic spin invariance requires y/2hy = A77. 3) Pion polarizability and pion decay The relationship given in Eq. (3.26) between the charged pion static electric polarizability and the axial structure function for radiative pion decay HA was first obtained in [Te 73], not via chiral lagrangian methods, but rather using softpion PC AC techniques. a) Take the limit p1 —• 0 in the Compton amplitude of Eq. (3.25) and show using currentalgebra/PC AC methods that lim
T^(p,p',qi)
where M^i/(p, g) is the axialvector component of the radiative pion decay amplitude given in Eq. (3.22) and p — pf + q\ + ^2b) Using the explicit kinematic forms of these amplitudes demonstrate that the relation a(t = 0) = y/2hA/Fn obtains, and that this is equivalent to the relationship between pion polarizability and radiative pion decay given in Eq. (3.26). 4) Unitarity and the pion form factor a) Verify that the pion form factor given in Eq. (3.13) obeys the strictures of unitarity, i.e.
 «&) P«7rT7r+TT
)
where T is the twoderivative (treelevel) pionpion scattering amplitude and is given in Eq. (3.4). b) How does this result change if the K+K~ intermediate state is added to TT+TT"? 5) Background field renormalization Verify that the effective lagrangian developed in Eqs. (2.10)(2.15) yields the same renormalization constant and renormalized mass for the propagator as a standard onshell renormalization prescription carried out in momentum space.
VII Introducing kaons and etas
As the energy is increased, the next degree of freedom which becomes excited is the strange quark, s. The new mesons encountered are kaons, which carry the extra quantum number strangeness of the s quark, and even heavier mesons 7/(549) and 7/(960) with no net strangeness. The kaons are stable against strong and electromagnetic decays, and must decay weakly. In this chapter we describe the basic structure of the TT, K, 77, 7/ system, deferring a discussion of weak decays to Chaps. VIII, IX. Our treatment will continue to be based on effective lagrangians, although quark model ideas enter into the discussion of the rf'. Important results include information on quark mass ratios and the derivation of the WessZuminoWitten anomaly action. VII1 Quark masses The addition of an extra quark adds to the number of possible hadrons. If the strange quark mass is not too large, there are additional low mass particles associated with the breaking of chiral symmetry. Including the quark mass terms, 'mu 0 0 Anass = fami/jR + $RPMI>L ,
m=[
0
md
0 I ,
(1.1)
the QCD lagrangian has an approximate SU(3)L X SU(3)R global symmetry. If the u, d, s quarks were massless, the dynamical breaking of x SU(3)L SU(3)R to vector £77(3) would produce eight Goldstone bosons, one for each generator of £1/(3). These would be the three pions TT*1, TT0, four kaons IT*, K°, K°, and one neutral particlerjs with the quantum numbers of the eighth component of the octet. Due to nonzero quark masses, these mesons are not actually massless, but should be light if the quark masses are not 'too large'. 188
VII1 Quark masses
189
What should the K, rjs masses be? Unfortunately, QCD is unable to answer this question, even if we were able to solve the theory precisely. This is because the quark masses are free parameters in QCD, and thus must be determined from experimental input. This means that the TT, K and 778 masses can be used to determine the quark masses rather than vice versa. The discussion is somewhat more subtle than this simple statement would indicate. Quark masses need to be renormalized, and hence to specify their values one has to specify the renormalization prescription and the scale at which they are renormalized. Under changes of scale, the mass values change, i.e. they 'run.' However, quark mass ratios are rather simpler. The QCD renormalization is flavor independent, at least to lowest order in the masses. In this situation, mass ratios are independent of the renormalization. There can be some residual scheme dependence through higher order dependence of the renormalization constants on the quark masses. However to first order, we can be confident that the mass ratio determined by the TT, K, % masses is the same ratio as found from the mass parameters of the QCD lagrangian. The content of chiral 577(3) is contained in an effective lagrangian expressed in terms of U = exp[i(A •
0
i
1 ~>
4
TS+
K
\
°
(1.3)
as expressed in terms of the pseudoscalar meson fields. If we choose the parameters in Eq. (1.3) to correspond to QCD without external sources, viz. s=m ,
p=0,
D^U = d^U ,
(1.4)
the meson masses obtained by expanding to order cp2 are ml = Bo(mu + md) , m2K0 = B0(ms + md) ,
m2K± = Bo(ms + mu) , m28 = B0(4ms + mu + md) . o
Upon temporarily neglecting isospin breaking, we obtain from Eq. (1.5) the mass relations,
m_ =
ms
ml 2m\m\
=
J_ 26 '
y
' '
190
VII Introducing kaons and etas
where rh = (mu + rrid)/2. Eq. (1.6a) demonstrates the extreme lightness of the u, d quark masses. Despite the difficulties of defining an absolute magnitude for quark masses, most estimates put that of the strange quark in the range ms ~ 150300 MeV. This in turn suggests rh ~ 612 MeV, a range of values significantly smaller than the scale AQCD Of course, the existence of very light quarks in the Standard Model is no more (or less) a mystery than is the existence of very heavy quarks. Both are determined by the Yukawa couplings of fermions to the Higgs boson, which are unconstrained (and not understood) input parameters of the theory. In any case, the small values of the u, d masses are responsible in QCD for the light pion, and for the usefulness of chiral symmetry techniques. The mass relation of Eq. (1.6b) is the GellMannOkubo formula as applied to the octet of Goldstone bosons [GeOR 68]. As mentioned in Sect. VI1, this relation is quadratic in the meson masses. It predicts mVs — 566 MeV, not far from the mass of the 77(549). The small difference between these mass values can be accounted for by second order effects in the mass expansion. In particular, mixing of the r)% with an SU(3) singlet pseudoscalar produces a mass shift of order (ms — rh)2. The difference between the predicted and physical masses is then an estimate of accuracy of the lowest order predictions. The use of the full pseudoscalar octet allows us to be sensitive to isospin breaking due to quark mass differences in a way not possible using only pions. This is because, to first order, the A / = 2 mass difference mn± — ra^o is independent of the A / = 1 mass difference m^ — mu. In contrast, the kaons experience a mass splitting of first order in ra^ — ra u. In particular, the quark mass contribution to the kaon mass difference is
(1.7) In addition there are electromagnetic contributions of the form (m2Kom2K+)em = mloml+
.
(1.8)
This result, called Dashen's theorem [Da 69], follows in an effective lagrangian framework from (i) the vanishing of the electromagnetic selfenergies of neutral mesons at lowest order in the energy expansion, (ii) the fact that K+ and ?r+ fall in the same [/spin multiplet and hence are treated identically by the electromagnetic interaction, itself a [/spin
VII2 Higher order analysis of decay constants and masses
191
singlet.* By isolating the quark mass and electromagnetic contributions to the kaon mass difference, we can write a sum rule ™>d ~ mu 1 2 n md + mu\ / =
2
V^K0
2 ~
m
K+)
\
(1.9) /
~
2
V^TT0 ~~ mK+
2\ )
'
which yields md  mu = Q Q 2 3 m d  mu = Q ^ ra m
VII—2 Higher order analysis of decay constants and masses The purpose of this section is to describe what is presently known about the analysis of masses and decay parameters at the next order in the expansion in energy and/or 5/7(3) breaking. Our emphasis will be on the * Recall that {/spin is the SU(2) subgroup of 5(7(3) under which the d and s quarks are transformed.
192
VII Introducing kaons and etas
effects of quark mass, as it is important to document the level of understanding of these basic parameters. The pion mass (or equivalently the mass of u, d quarks) is not sufficiently large to modify most observables in a major way. However, the kaon mass (influenced by the strange quark mass) has a more significant impact. It is important to examine the effect of this mass on low energy observables. Working to O(E^), the effective lagrangian consists of the O(E2) terms already given in Eq. (VI1.2), the WessZuminoWitten anomaly lagrangians to be derived in Sect. VII3 plus a set of fourth order terms given in Eq. (VI2.7). In Chap. VI on pionic physics, we saw that there were several reactions sensitive to the fourderivative terms and to the photon couplings. This allowed us not only to determine the parameters but also to make predictions. However, there was no phenomenological information on the mass terms in the effective lagrangian. With the addition of the K and 77 particles, we are now sensitive to such terms. Unfortunately, the data set is not sufficiently rich to allow use of these new parameters in a predictive way. Ambiguities in mass parameters Before proceeding, we wish to point out an important feature of the chiral SU(3) effective lagrangian, and indeed of all effective lagrangians. We have constructed the form of the mass terms from symmetry considerations alone, using the fact that \ (= 2i?o(s + w)) and m transform as & ( 3 L 5 3 # ) representation under chiral transformations. However, there exists a continuous family of forms having the same transformation property [KaM86, DoW91], X * XA = X + A det[xf]x ( x ^ ) " 1 ,
m > mA = m + A det[m] m 1 , (2.1) where A is an arbitrary parameter and A = 2i?oA. This means that in any effective lagrangian based on symmetry considerations, one can equally well use xx o r mA instead of \ or m. This algebraic reparameterization invariance can be explicitly seen in the chiral lagrangian with the aid of an identity involving any 3 x 3 matrix M, detM = M 3  M2TTM
 M [Tr (M2)  (TrM) 2 ]
,
(2.2)
which can be derived [Go 87] through use of the CayleyHamilton theorem. In our case this yields
Tr (XAf/f + UX\) = Tr (xtf* + UX])  \ Tr ( V X W + tf W x )
c/xtC/tx)]2} • (2.3)
VII2 Higher order analysis of decay constants and masses
193
The additional term of Eq. (2.1) is equivalent to a combination of O(E4) operators. Any piece of phenomenology obtained with \ o r m a n ( i set {#6, a*?, a$} can be obtained identically with x\ o r mA a n d the set 0 6 0 6 + ^ , ID
«7
a
7 +
T7T ,
O$
ID
<*8
.
(2.4)
O
Thus, the family m^ of quark mass matrices leads to the same phenomenology, and one cannot distinguish between any individual members by symmetry techniques. The same ambiguity can occur whenever quark masses occur in other effective lagrangians. The reparameterization invariance corresponds in terms of the quark masses to mu = mu + Xrudrris , m^ — rrid + Xmums , ms = ms + Xmurrid . (2.5)
Note that this change is of second order in the quark mass and hence is important only if one analyzes quark masses beyond leading order. The above ambiguity is an algebraic feature of the effective lagrangian. However, it also represents a possible form of mass renormalization in QCD. To see this, consider a set of 'bare' quark mass values having mu = 0, but morris ^ 0. In such a world, the chiral 5/7(2) and SU(3) symmetries are explicitly broken (by rad,ms), and axial U(l) is not a global symmetry (because of the axial anomaly and of the vacuum structure of QCD). For an upquark moving in the vacuum fields of QCD, the value mu — 0 is not 'protected', and thus it can be renormalized to a nonzero value. Presumably mu will then be given by some function of rrid and ms. Since the original vanishing ixquark mass would be protected by a chiral SU(2) symmetry in the limit rrid —> 0 or ms —> 0, the dependence must be proportional to the product rridms to accommodate such limits properly. An implication of this discussion is that masses need not be multiplicatively renormalized beyond leading order, but may mix in a way consistent with the underlying symmetries. This is what the transformations of Eq. (2.5) are expressing. In principle, different renormalization schemes could involve differing mass matrices m\. As regards phenomenology, the ambiguity has a simple resolution  one can equally well work with any of the family of HIA since all yield the same phenomenology. Most applications of quark masses occur at first order in the mass, and these must use the lowest order mass matrix of Eq. (1.1). In order to define the mass matrix to next order, one must propose an extra constraint which fixes the definition of the mass. We shall encounter an example of this shortly. Decay constants One effect of higher order lagrangians is to distinguish between the various pseudoscalar decay constants. Thus the decay constants are no longer
194
VII Introducing kaons and etas
equal for the full octet of pseudoscalar mesons. Of the various terms in £4, the one proportional to 0:4 leads only to an overall shift common to Fin FK, FV8. The only tree level contribution to the axial current which differentiates between Fn, FK and Fm comes from a§. In addition, the loop diagram of Fig. Vlll(a) can introduce some mass dependence, as well as renormalizing the tree level parameters. The result is [GaL 85a] FK
,
I
3
5
F
S o re N
16
So
where 9
mf
9
, mf
or numerically, ^
= 1  0.01
^ = f
+ 0.02 .
The experimental value of FK can be determined from the measured K+ —> / i + ^ decay rate, analogous to the extraction of Fn in Sect. VT1. The experimental ratio Y~ = 1.23 ±0.02
(2.9)
implies the value ag (mv) = (2.3 ± 0.2) x 10"3. Masses The GellMannOkubo mass relation is also modified at O(E4) due to higher order quark mass effects. A calculation of the masses of the pseudoscalars including the loop diagrams of Fig. Vlll(b) results in
n (a)
(b)
Fig. VII1 Loop diagram contributions to (a) decay constants, (b) masses.
VII2 Higher order analysis of decay constants and masses mz,
ml K
195
(2.10)
3 ms + m
with
8 (m\  mj)
128
T
(2.11)
{mlml?
After some rearrangement, the violation of the Gell MannOkubo relation can be cast in the form K
,
* ,
2 = 0.21 ,
(2.12a)
compared to
(
2
2\ r r
r
(2.12b)
+2 At this stage, one cannot make a prediction for m, because of the inclusion of new low energy constants. Instead, we can reverse the process to find 2ar7 + ar8 = 0.4 x 10  3
(2.13)
evaluated at the scale /i = m^. Since this combination of coefficients is expressible in terms of physical data, it is naturally invariant under the reparameterization discussed at the start of this section. Finally, we return to the quark mass ratio, m 77i, + m
10.49 + 0.17 0.4 x 10 3 26 1 1  0.32  0.17 x 26
(2.14)
196
VII Introducing kaons and etas
The result depends on the relative contributions of 07, a% to Eq. (2.13),* and illustrates the ambiguity of the quark mass beyond leading order, as given in Eqs. (2.1)(2.5). If we insist on obtaining a mass ratio, we need extra input. One possibility is to assume that a? is given by the 77 — rf mixing model as described in Prob. VII3 which results in
ar7 =  0 . 4 x 10" 3 ,
_A^ = I • ms + m 26
(215)
This choice, which goes beyond the limits of pure chiral symmetry, serves to define a mass matrix at second order in the masses. However, other definitions of mass are possible [DoW 92]. We are not presently able to calculate the u — d mass difference beyond leading order due to a lack of knowledge of the electromagnetic contributions to kaon and pion masses at next order in the quark mass. VII3 The WessZuminoWitten anomaly action At this stage one must also include the effect of the axial anomaly. The anomaly influences not only processes involving photons, such as ir® —> 77, but also purely hadronic processes. For example the reaction KK —> 7r+7r~7r°,allowed by QCD, is not present in any of the chiral lagrangians appearing in previous sections. Its absence is easy to understand because the hadronic part of the lagrangian, with external fields set equal to zero, has the discrete symmetry
VII3 The WessZuminoWitten anomaly action
197
the sigma model is not a problem since, according to the AdlerBardeen theorem[AdB 69], the inclusion of gluons would not modify the result. Since the sigma model involves coupling between mesons and fermions, we can also observe directly the influence of the anomaly on the Goldstone bosons. Although somewhat technically difficult, our approach will clearly illustrate the connection with treatments of the anomaly based on perturbative calculations. Consider as a starting point the lagrangian, Eq. (IV1.11), of the linear sigma model
C = &pip  gv (i>LUipR + ^ C / V L ) + ... .
(3.1)
We have displayed neither the term containing Tr (d^Ud^U^) and nor any term containing the scalar field S. Such contributions are not essential to our study of the anomaly and will be dropped hereafter. In order to simulate the light quarks of QCD, we shall endow each fermion with a color quantum number (letting the number Nc of colors be arbitrary) and assume there are 3 fermion flavors, each of constituent mass M — gv. Although the original linear sigma model has a flavor SU(2) chiral symmetry, Eq. (3.1) is equally well defined for flavor SU(3). Our analysis begins by imposing on Eq. (3.1) the change of variable
$'L = £tyL , ^ = WR, tf = U ,
(3.2)
like that appearing already in Eq. (IV7.3). This yields C = $'(i lp 
(?d»z  td^) .
(3 3)
'
For this change of variable the jacobian is not unity, and thus we must write the effective action as eiT(U)
=
f[
= f[d^][df]J
elfd4x P'WMW
(3.4)
= exp(lnj r ) exp(tr \n(iI/>  M)) . For large M, it can be shown that the tr ln(z Ip — M) factor does not produce any terms at order EA that contain the e^aP dependence char
198
VII Introducing kaons and etas
acteristic of the anomaly.* Hence the effect of the anomaly must lie in the jacobian J, and it is this we must calculate. It is possible to determine the jacobian by integrating a sequence of infinitesimal transformations. Thus we introduce the extension £ —• £ r , gr = expi T
^ = expiry
,
(3.5)
where r is a continuous parameter and £ = £r=i Transformations induced by the infinitesimal parameter 6r will give rise to the infinitesimal quantities £$r and 6J, (36)
Prom Eqs. (III3.44), (III3.47), we find 6J to be 8J = exp(—2i<5rtr ^75) ,
(37)
or dlnj dT
(3.8)
r=0
This result should be familiar from our discussion of the axial anomaly in Sect. Ill—3. There remain two steps, first to calculate the regularized representation of tr (^75), and then to integrate with respect to r. To regularize the trace, we employ the limiting procedure tr (^75) = lim tr (^75 exp [e fT fT\)
{V» = d"+iV^T+iA^)
, (3.9)
with AT and V1^ as in Eq. (3.3), except now constructed from £T and £}. For arbitrary r, we make use of the identities
vT =
{
„i m j
to express Tf)r Tf>T in the form
Tprfr = d^
+a ,
dli = dlt + iVTll + a^Xis = d^ + FTfi , a = 2ATIX + i[(dM + iVTtl) ,X] 75 •
(3.11)
This can be verified by expanding as tr ln(i#  M) = tr ln(M(l  Up/M)) = tr ln(M)  tr (z#) 2 /2M 2 + ... . The first term can be regularized as in the text and directly calculated using the techniques described in App. B. The remaining terms vanish for large M.
VII3 The WessZuminoWitten
anomaly action
199
Prom the heat kernel expansion of App. B, we have*
Carrying out the 'TV' operation, which involves some application of Dirac algebra, yields
= 2iNcTr (^e^lpXXXX)
+•• • ,
(3.13)
where the ellipses denote contributions not involving e^va^ and the factor Nc comes from the sum over each fermion color. Combining the above ingredients, we have for the regulated action
where we recall that Tp = A • (p/(2Fn). This result expresses the effect of the anomaly on the Goldstone bosons. Unfortunately there is no simple way to integrate the entire expression of Eq. (3.14) in closed form. In principle, we could represent each of the axialvector currents therein (e.g. A^) as a Taylor series expanded about r = 0 and perform the integrations to obtain a series of local lagrangians. Alternatively, however, one can simply express Eq. (3.14) as an integral over a fivedimensional space provided we identify r with a fifth coordinate x§ (defined to be timelike). In this case, we use i *
•
(315)
plus the cyclic property of the trace to write )
,
(3.16)
where i , . . . , m = 5,0,1,2,3 with e50123 = +1. This is Witten's form for the WessZumino anomaly function. The r = 1 boundary is our physical spacetime, and the fifth coordinate is just an integration variable. Since each term in the Taylor expansion can be integrated, the result depends * Note the distinction between 'tr' and 'Tr', as in Eq. (Ill—3.48).
200
VII Introducing kaons and etas
only on the remaining four spacetime variables. Observe that vanishes for U in SU(2) due to the properties of Pauli matrices. For chiral 5/7(3), the process K+K~ —• 7r+7r~~7r° is the simplest one described by this action and after expanding Fwzw? it is described by the lagrangian, ^
,
(3.17)
with if = A • cp given in Eq. (1.3). The above discussion has concerned the impact of the anomaly on the Goldstone modes. We must also determine its proper form in the presence of photons or W± fields. For this purpose, we can obtain the maximal information by generalizing the fermion couplings to include arbitrary lefthanded or righthanded currents t^r^ 1 ^
(3.18)
r The calculation of the jacobian then involves the operator V
d
+U
1 + 75
I ir
1  7 5
which generalizes Eq. (3.3). It is somewhat painful to work out the full answer directly, but fortunately we may invoke Bardeen's result of Eq. (Ill— 3.64) for the general anomaly. Using the identities
where ^v,r^v
are given in Eq. (Ill—3.65), we obtain
Fwzw = ^J
drjd4x
+ Va V) 0) ++ (l +^M M^ + 2i,
— —{a^ajyVa^
f^lap)
(3.21)
+ a^Vvadp + v
Note that the first term corresponds to our previous calculation of Eq. (3.14). The WZW anomaly action contains the full influence of the anomalous low energy couplings of mesons to themselves and to gauge fields. By construction, it is gauge invariant. The r integration can be explicitly performed for all terms but the first in Eq. (3.21). However, in the general nonabelian case the result is extremely lengthy [PaR 85]. For
VII4 The rf (960)
201
the simpler but still interesting example of coupling to a photon field A^, the result is Twzw (U, A^) = Twzw(f^)
+
a/3 d4a; iv(Q(
^^ / K
^
— ie FuVAa Tr I Q (LR + R3) + — \ 2
' (3.22) where R^ = (d^U^)U, L^ = Ud^U^.* We have already used the twophoton portion of this result in Sect. VI5 for the decay TT° —• 77. We have seen here that whereas the anomalous divergence of the axial current represents the response to an infinitesimal anomaly transformation, the WZW lagrangian represents the integration of a series of infinitesimal transformations. In our analysis of the sigma model, the anomaly has forced the occurrence of certain couplings, among them Tr0 —> 77, 7 —> 3TT and KK —• 3TT. AS noted earlier, although these results are based on an instructional model, the result has the same anomaly structure as QCD because the answer must depend on symmetry properties alone. Indeed, such conclusions were originally deduced from anomalous Ward identities [WeZ 71] without any reference to an underlying model. We regard such predictions as among the most profound consequences of the Standard Model. VII4 The
T/(960)
Thus far, we have been dealing with hadrons whose presence could be inferred from the symmetry properties of QCD. We have not needed to invoke the quark model to describe the meson spectrum. This procedure is less clear for the rf (960), which is not a Goldstone boson for any of the symmetries of QCD. The least complicated description of the rf is with the quark model. Here one notes that the eight particles (TT"1", TT~, TT°, K+, K~, K°, K° and rjs) havethe same quantum numbers as the respective quarkantiquark pairs (ud, du, (uu—dd), us, su, ds, sJ and (uu+dd—2ss)). However, such QQ pairing suggests a ninth state, the SU(3) singlet 770 ~ (uu+dd+ss). A look through the Particle Data Tables reveals that the lightest candidate for a ninth pseudoscalar is 7/(960). In fact, the 7/(549) and 7/(960) turn out to involve mixtures of 7/8 and 770 (and in principle of other states as well), with the 7/(960) being treated as predominantly the SU(3) singlet. * Witten's original result did not conserve parity, and this was subsequently corrected [PaR85, KaRS84].
202
VII Introducing kaons and etas
One immediate puzzle involving the rf would appear to be its mass, which is almost twice that of the kaon. Of course, the TT, K, r)$ masses must vanish in the limit of zero quark mass, while because of the axial U(l) anomaly there is no such requirement for the TJQ. Moreover, there is the presence of the quark annihilation diagram, Fig. VII2. This can keep the 770 mass finite in the chiral limit. The exact value of this mass contribution is difficult to calculate within the quark model. At any rate, the proper expectation for the lightest pseudoscalar masses is ml = 2mBo , 2 m 5 m 3 ((22m* + + ™) ) 5 o ,
m2K = (ms + m)Bo , 2 m200 + << = m + (ms + 2m)B' ,
(4 1)
where rao is the 770 mass in the chiral limit and B1 is in general a constant different from Bo. Here we have temporarily ignored 770778 mixing. The discussion becomes more interesting when the connection to the axial anomaly is considered. The SU(3) singlet axial current
= ^F^F^
+ 2imuybu + 2imdd7bd + 2ims^s
where Fa" = e ^ ^ i 7 ^ , has the 770tovacuum matrix element * ,
(4.3)
in which F^ is the 770 decay constant. By taking the divergence of this equation we obtain
 ^
J, .
(4.4)
i=u,d,s
If the anomaly were not present, there would exist an almost conserved current for the £^4(1) symmetry. If this were dynamically broken along with the SU(3) chiral symmetry, the (pseudo) Goldstone boson would be the 7/o. The above relation, without the FF term, would then express the result that miQ = O(mq). However, the anomaly contribution (O\FF\r)o) does not in general vanish, so that m^Q remains nonzero in the chiral limit of zero quark mass.
u,d,s
u,d,s u,d,s •
u,d,s G
Fig. VII2 Annihilation diagram
VII4 The rjf (960)
203
In fact, it can be argued that the largest contribution to the 770 mass is due to the axial anomaly. The argument is rather subtle, and the 770 mass was long considered to be a problem despite the anomaly. The difficulty arises because there does exist a conserved J7(l) current in the chiral limit. FF is expressible as a total divergence, FF = d^K^1 with K*1 given in Eq. (Ill—5.7). The conserved current is formed by subtracting K^ from the usual current T(0) _
T(0)
K
B
T(0)/Z _
n
(
,
x
where quark masses are being neglected here. The associated charge will generate UA(X) transformations, and, since UA(1) is not an observable symmetry of the spectrum, it must be dynamically broken. Such reasoning leads one to expect a Goldstone boson. In the matrix element analysis of Eqs. (4.3), (4.4), the same difficulty arises because the identification of FF as a total divergence would seem to imply, by integrating by parts and discarding the surface term at infinity, that fd4xFF
= 0 or
f d4x (0\FF r?o(p)) = 0 .
(4.6)
The latter matrix element would imply the possibility of a zero momentum state (if the quark masses vanished), apparently producing a Goldstone boson despite the anomaly. The resolution of this apparent paradox is twofold. In the first place, the conserved current JL is gauge dependent due to K^. While this does not invalidate either its use as a U(l) generator or the need for a Goldstone boson, it does allow the possibility that the Goldstone boson need not appear in the physical gaugeinvariant spectrum. It can appear rather as a gaugedependent effect [KoS 75] and need not couple to gaugeinvariant operators. (However, neither is it known to be forbidden to do so.) In the matrix element argument, the resolution occurs because there exist topological effects in QCD such that the surface term at the spacetime boundary can prevent Eq. (4.6) from vanishing ['tH 76b] (see the discussion in Sect. Ill—5). This allows the simple analysis of the 7/ mass, which was given in the preceding paragraphs, to be valid. While the proof that precisely these phenomena do occur in QCD is not without controversy, it would appear that this is the route by which Nature has chosen to avoid a ninth Goldstone boson. As a result, the couplings and decays of the rjf are not related by symmetry conditions to those of the octet of pseudoscalars. That is, in QCD there is no 'nonet symmetry' in the standard context of what is meant by symmetry.
204
VII Introducing kaons and etas 770  rjs mixing
If the u, d, s quarks were massless, the picture developed thus far would imply an octet of massless pseudoscalars plus the massive SU(3) singlet 7/0 When quark mass is introduced in perturbation theory, each squaredmass of the pseudoscalar octet becomes linear in quark masses, and the mass of the T/O is shifted slightly. In addition, SU(S) breaking in the quark mass matrix induces a mixing between the singlet and octet, i.e. between rjo and rjg. This mixing yields the physical eigenstates rj and rf \rj) = cos0\r]g)  sin0 r/o) ,
T/) = sin0 T/8) + cos(9 r/o) .
(4.7)
The goal is then to calculate 6 theoretically and measure it experimentally. There are two crucial assumptions which go into the mixing scheme described here. One is that it is possible to neglect the other pseudoscalar 7 = 0 states which could mix with both r/g and 770. A check with [RPP 90] shows that the next possible candidate is 77(1279). This is distant from the 77s mass, and hence might not mix significantly with it, but is not extremely far from the 770 mass. The second, somewhat related assumption is that the mixing parameters do not depend significantly on the energy of the state. If they did, a more complicated scheme for finding the mass eigenstates would be required. These assumptions have not yet been justified in QCD. The utility of this scheme seems to be founded on its success in phenomenological applications [GiK 87]. The 777/ mixing angle may be determined from experiment. In our opinion, the cleanest phenomenological analysis occurs in the two photon decays of TT0, 77, 7/. All of these decay amplitudes have the same Lorentz structure,
7(k2)
= §
(4.8)
where the convenient normalization CM — (l> l/>/3, 2>/2/3 ) has been chosen for the unmixed states M = (TT°, 7/8, 7/0). The parameters FM are not a priori known. However, a prediction of the axial anomaly analysis is that Fn = F^, where Fn is the usual pion decay constant. Similarly, the WessZuminoWitten lagrangian of Sect. VII3 predicts Fm = Fn. We shall explore the sensitivity of this parameter to SU(3) breaking. Finally Fm is not predicted by any symmetry. Quark model prejudice, described below, also suggests Fm ~ Fn. Allowing for 77 — 7/ mixing as in Eq. (4.7),
205
VII4 The 77'(960)
the decay rates become
1 m* fi^costf 3^4 Fm
vo
a svn.9
(4.9)
Fvcos6
+
3 ml
The data is very sensitive to 7777' mixing, largely because of the factor y/8 in the 77 decay amplitude, and in fact requires its presence. In particular, the 'reduced' rates are 3.0 ±0.3 ,
3 ml I V /V = 8
i^sinfl n0

F^cosO
(4.10) = 0.61 ±0.05 .
Using the WessZuminoWitten value for FVs, Eq. (4.10) implies 6 =  1 7 ° ±3° ,
F o = (1.08 ±0.04)2^ .
The dependence on SU(3) breaking in Fm is not large. If we use instead the value JF^ = Fm, where Fm is the usual axial current decay constant calculated in Eq. (2.8), one obtains the compatible values 6 = 22° ±3° ,
~F0 = (1.05 ± 0 . 0 4 ) ^ .
(4.11)
Twophoton decays then require a mixing angle which is around —20°. What is the effect of this mixing? In the quark model, the 778 and 770 states are octet and singlet qq states respectively. The physical mixtures would then be 77
/ _

_x
sin^/ _

_x
—(uu + dd + ss) v 6 (uu + da — 2ss) ~ 0.58(uu + dd)  0.5755 , 3 j^[uu + da — 2ss) 2ss) HH ^ v6 v3 ~ 0.40(im + dd) + 0.8255 .
77 =
(4.12)
+ dd + ss)
The mixing is seen to decrease the 55 content of the 77 and increase it for the 77'. It is the added uu content of the 77 which enhances the transition V ~^ 77 due to the larger u quark charge. It is interesting that the parameter FVo is found to be so nearly equal to Fn. In the quark model, if one assumes that the spatial wavefunctions of the quarks in the TT, 778 and 770 are identical, the twophoton amplitudes would be proportional
206
VII Introducing kaons and etas
to the squared quark charge, i.e., + Q2d + Q2s\M°) .
(4.13)
This would produce relative amplitudes TT°: 778 : 770 = 1 : l / \ / 3 : 2^/2/3, which corresponds in the normalization of Eq. (4.8) to Fn = Fm — Fm. This need not work as well as it seems to. The theoretical aspect of mixing is addressed by considering the mass matrix, with offdiagonal element mls = (r/o \Hmass\ Vs) •
(4.14)
This can be estimated by considering the model where we use the quark model description of 770 and 773, with identical wavefunctions. In that case we would calculate 5
Bo =
—rm\
~ 0.9m2K
.
(4.15)
Although this is not a definitive analysis, it does serve to illustrate how mass mixing can be generated. In an obvious notation, the overall 2 x 2 mass matrix is then m 08 m 0 The mass rn^ is completely unknown, but m\ should be close to the GellMannOkubo value (4mKml)(l
+ 6) ,
(4.17)
where 6 parameterizes those deviations from the GellMannOkubo relations which can occur even in the absence of 777/ mixing. The dependence on 6 is rather strong. If 6 = 0, diagonalization of the matrix gives the values 0 = 10°, mg8 = 0.44ra^, ra0 = 0.95 GeV, while for 6 = 0.25, one finds 6 = 24°, mg8 = 0.93 m\, m0 = 0.90 GeV. These results lie in a reasonable range, consistent with theoretical expectations. Alternatively, one can work backwards from the observed 7]r]f mixing to determine the parameters in the mass matrix. The choice 9 = —20° ± 2° produces 6 = 0.16 ± 0.04, and mg8 = (0.81 ± 0.05)ra^. Again, the quark model prediction Eq. (4.15) is found to work surprisingly well.
Problems 1) Other worlds Describe changes in the macroscopic world if the quark masses were slightly different in the following ways:
Problems
207
a) mu = rrtd = 0, b) mu > ra^, c) mu = 0, rrid — m8. 2) r] decay a) Prove that the amplitude for rj —> 3TT vanishes in the limit of isospin symmetry. b) Using the lowest order chiral lagrangian, calculate the rj —• 3TT decay amplitude and slope, A A
K A
1
i
I
u
yv477_).7r+7r7ro = Alo 1 + ^ I in terms of the quark mass difference rrid ~ mu, where To is the TT°kinetic energy and Q = m^ — Zm^. Compare your results with experiment. This decay remains in disagreement even when a full O(E4) calculation is performed [GaL 85c].
3) T/?/ mixing and chiral lagrangians
Although the 5/7(3) singlet 770 does not occur directly in the SU(3) chiral lagrangian, its effect can appear indirectly through virtual processes. a) Show that the only lowest order chiral coupling of an 770 to the chiral octet is iV6Fnm20S
_
° X)
with a normalization constant chosen to reproduce the 77 — 7/ mixing of Eq. (5.10). b) Integrate out the 7/0 field and show that one thereby obtains a term in the chiral lagrangian with 7
64m20(r4_m2)2 •
Evaluate this numerically. Thus the effect of rj — 7/ mixing is represented in the chiral lagrangian by the value of a.7.
VIII Kaons and the AS = 1 interaction
The kaon is the lightest hadron having a nonzero strangeness quantum number. Since the weak interactions do not conserve strangeness, the kaon is unstable and decays weakly into states with zero strangeness, containing pions, photons and/or leptons. There are in fact dozens of allowed modes, and many of these have been carefully studied. In lieu of detailing all of these possibilities, we shall instead concentrate on some of the primary decay channels. VIII1 Leptonic and semileptonic processes Leptonic decay The simplest weak decay of the charged kaon, denoted by the symbol K^, is into purely leptonic channels K+ —» /x + ^, K+ —• e+z/e. Such decays are characterized by the constant FK, As discussed previously, because of SU(3) breaking FK is about twenty percent larger than the corresponding pion decay constant Fn. As with the pion, but even more so because of the larger kaon mass, helicity arguments require strong suppression of the electron mode relative to that of the muon. The ratio of e+z/e to / i + ^ decay rates, as in pion decay, provides a test of lepton universality [RPP 90],
r*K . e ^
v
*
= (2.44 ±0.11) x 10"5 ,
(1.2)
K++H+V14 expt
in good agreement with the suppression predicted theoretically, m
ml
1  ml/ml •til •"•K 208
(1 + 8) = 2.41 x 1(T5 ,
(1.3)
VIII1 Leptonic and semileptonic processes
209
where 6 = —0.04 is the electromagnetic radiative correction including the bremsstrahlung component.* The notation F' indicates that the experimenters have subtracted off the large structure dependent components of > £+i/£^y but have included the small bremsstrahlung component.
Kaon beta decay and V^ The kaon beta decay reactions K+ —> n o£+iy£ and K° —• TT~^ + I^, called g} and K^ respectively, also are important in Standard Model physics. They are each parameterized by two form factors,
[ (14) Isospin invariance implies f± = f± * = f±. SU(3) symmetry can be invoked to relate these matrix elements to the strangeness conserving transition TT+ > iPl+vi, resulting in /+(0) =  1 and /_(0) = 0. The deviation of /+(0) from unity is predicted to be second order in SU(3) symmetry breaking, i.e. of order (ms — rh)2. This result, the AdemolloGatto [AdG 64] theorem, is proved by considering the commutation of quark vector charges, n
[Q^iQ™]
=QUu~~ss
,
(1.5)
where
j
(
1
.
6
)
Taking matrix elements and inserting a complete set of intermediate states gives
(
2
r
)
•
d7)
Finally, we isolate the single n~ state from the sum and note that in the SU(3) limit the charge operator can only connect the kaon to another state within the same SU(3) multiplet. This implies^ (1.8) * The dominant term here is the simple contact contribution —3(a/7r)ln(m Sect. VIL t This is easiest to obtain in the limit p ^ —• oo
At/me)
discussed in
210
VIII Kaons and the AS — 1 interaction
where e is a measure of SU(3) breaking, and we thus conclude that ,
(1.9)
which is the result we were seeking. It is interesting that the SU(2) mass difference mu ^ rrid can modify f+ n (0) in first order despite the AdemolloGatto theorem. This can be seen by considering a K+ in the formulae of Eq. (VII1.1). Now there exist two intermediate states in the same octet as the kaon, i.e. TT°and 7?°,and it is their sum which obeys the AdemolloGatto theorem, 1 (0) =1 + O(e2) . (1.10) 4 In the isospin limit, each term must separately obey the theorem because n ~. However, when mu ^ rrid each of the isospin relation f+*n° = f+° form factor in Eq. (VII1.10) can separately deviate from unity to first order in mu — rrid as long as the first order effect cancels in the sum. Indeed this is what happens, yielding (cf. Prob. VIII. 1)
/f(0) where IKK — 0.004 arises from chiral corrections at O(E 4) [GaL 856]. This number can also be easily extracted from experiment by using the ratio of K+ and K° beta decay rates, with the result ^ = 1.029 ± 0.010 , J+
"
(1.12)
\y)
in agreement with the prediction. One of the first applications of current algebra/PC AC techniques was made to K& systems, yielding [CaT 66] <7r°(p) 57Mii tf+(k)) _ ,   L (o I [Ql
(1.13)
The corresponding condition on the form factors, [f+(q2) + f(q%2=m2K
= ^
,
(1.14)
K
is called the CallanTreiman relation. Interestingly, this does not imply that /(0) = 1 — FK/FK, as was originally assumed. We can demonstrate this by using the chiral lagrangian of Eq. (VI2.7) at tree level (loop effects
VIII1 Leptonic and semileptonic processes
211
are small and will be added later). Expansion of the vector current matrix element yields
(1.15)
This form explicitly satisfies the CallanTreiman relation offshell, yet at q2 = 0 yields a positive value for /(0),
= f+(m2K,ml,0)=l
,
mj)  0.13 ,
^ K
§F
(L16)
where we have used the previously determined value Og = (7.7±0.2) x 10~3 (which includes the effects of loops). This prediction is in agreement with the results of K^ experiments
f (0)
f
a35 ± 15 = uw 1 "0.20 °' ± 0.08 =
{K ]
^' (combined) ,
(L17)
which clearly require the presence of ag contributions as contained in the chiral lagrangian formalism. The q2 variation of the form factors ,2\ _
/o( 9 2 ) =
£ /J2\
,
9 171
K
/
/.2\ ..
.
A
2
(1*18)
~
follows from the same input parameter as was used to predict rare pionic processes, with the addition of FR The agreement with experiment as displayed in Table VI2 is good. The prime importance of the K& process is that it provides the best determination of the weak mixing element V^. Because of the AdemolloGatto theorem, the reaction is protected from large symmetry breaking corrections. In addition, the use of chiral perturbation theory allows a reliable treatment of the reaction. The above study of the form factors indicates that the theory is under control within the limits of experimental precision. The value [LeR 84] Vus = 0.2196 ± 0.0023 +
follows from an analysis of the K° and K decay rates.
(1.19)
212
VIII Kaons and the AS = 1 interaction The decay K —>
7nreue
The final semileptonic process which we consider is K —> TTTT^, labeled X^4. This reaction has the particular interest in providing the only known constraint on the large Nc assumptions sometimes made concerning chiral lagrangians, and is important in determining the low energy constants. As will be explained more fully in Chap. X, the large Nc limit imposes certain relationships between some of the terms in the chiral lagrangian. The particular relationship that is able to be tested in the K& system is the prediction OL
x [(p+ + p_)M h + (P+  PV h + (k  P+  P)M /a] , <7r + (p + )7r(p_)  l 7 ^ l ^ + ( k ) > = g ^ W ^ ^ + Z
(1.20)
9 •
The form factor f% is essentially unobservable because its effect is proportional to me in ife4, and we shall not consider it further. We have chosen the normalization such that the lowest order chiral predictions are f1 = f2 = g = l [We 66a]. The vector current form factor g = 1 is a prediction of the axial anomaly, as it is related by an SU(3) rotation to 7 —> 3?r. Experimentally, the magnitudes and slopes of the form factors at threshold
+ Ar^]
with fc2 = i ( ( p + + p _ ) 2  4 n 4 )
Table vrai. Experiment and theory for Ku decays. Data /i(0)
/ 2 (0) Ai
A2 9(0) a
1.47 ± 0.04 1.25 ± 0.07 0.08 ± 0.02 0.08 ± 0.02 0.96 ± 0.24
Lowest order0 Order (E4)a 1.00 1.00 0.00 0.00 0.00
1.45 1.24 0.08 0.06 1.00
The lowest order and second order predictions of chiral symmetry.
(1.21)
VIII2 The nonleptonic weak interaction
213
have been measured with the results [Ro et al 77] given in Table VIII1. No dependence on the variable q2 = (k — p+ — p)2 was observed. The enhancement in the threshold form factors and slopes is determined at O(E4:) primarily by 0:1,2,3, which also enters into TTTT scattering. A very consistent pattern emerges. The chiral KM predictions become [Bi 90, RiGDH 91] /i(0) = 1 + Xi + ^
+ {m2K
[32mjal + 4(m2K + 4ml
l
l
\
/2(0) = 1 + X2  A [(m2. *IT
Ai = n + 8 ^  [16aJ + 4a£ + 5ar3] ,
X2 = Y2 8^ar3
,
where the quantities X\ = 0.127, X2 = 0.00, Fi = 0.076, Y2 = 0.045 describe the net effects of loops when evaluated at \x = m^. Since the dependence on a^ is small because of the factor of m^, we shall proceed with c*4 = 0. Phenomenologically, one may either fit /i(0),/2(0), Ai to determine ai?2,3 and then predict A2 and the seven TTTT scattering lengths and slopes, or provide the best determination of ai?2,3 by simultaneously considering all eleven of the TTTT and K^ observables. The results are very similar either way, and therefore we present only the second case. The parameters are those shown in Table VI1, and the predictions for the observables are contained in Table VI4 for TTTT physics and in Table VIII1 for Ku decay. The overall description of the two systems is excellent. We note also that the analysis results in the constraint
which verifies the suppression predicted by the large Nc limit. VIII2 The nonleptonic weak interaction Thus far in this chapter we have discussed leptonic and semileptonic processes. For these, at most one hadronic current is involved. There exist also nonleptonic interactions, in which two hadronic charged weak currents are coupled by the exchange of W± gauge bosons,
= f f
214
VIII Kaons and the AS = 1 interaction
with V being the KM matrix, given in Eq. (II4.26). Such interactions are difficult to analyze theoretically because the product of two hadronic currents is a complicated operator. If one imagines inserting a complete set of intermediate states between the currents, all states from zero energy to Mw are important, and the product is singular at short distances. Thus one needs to have theoretical control over the physics of low, intermediate, and high energy scales in order to make reliable predictions. Because this is not the case at present, our predictive power is substantially limited. Let us first consider the particular case of AS = 1 nonleptonic decays. These are governed by the products of currents dT^u uT^s ,
dT^c cT^s ,
dTH iT^s ,
(2.2)
where F^ = 7^(1 + 75) and color labels are suppressed. The first of these would naively be expected to be the most important, because kaons and pions predominantly contain u, d, s quarks. However, the others also contribute through virtual effects. Some properties of the AS = 1 nonleptonic interactions can be read off from these currents. The first product contains two flavorST/(3) octet currents, one carrying I = 1/2 and one carrying / = 1, 27 , ( 8 0 8)symm 1 1 ^ 3 1(5 = *2" 2 "2 '
SU(3) : Isospin :
(2.3a) (2.3b)
where the symmetric product is taken because the two currents are members of the same octet. The singlet SU(3) representation is excluded from Eq. (2.3a) because a AS = 1 interaction changes the SU(3) quantum numbers and hence cannot be an 5(7(3) singlet. The other two products in Eq. (2.2) are purely SU(3) octet and isospin 1/2 operators. The currents are also purely lefthanded. Thus the nonleptonic hamiltonian transforms under separate lefthanded and righthanded chiral rotations as (8L,1R) and (27 L,1R) These symmetry properties, valid regardless of the dynamical difficulties occurring in nonleptonic decay, allow one to write down effective chiral lagrangians for the nonleptonic kaon decays. The hamiltonian is a Lorentz scalar, charge neutral, AS = 1 operator, and has the above specified chiral properties. At order E2, there exist two possible effective lagrangians for the octet part, viz. £Octet = C$ + C%, where in the notation of Sect. IV6, £ 8 = g8 Tr (^DJJD»U^
,
£ 8 = 98 Tr (A 6X t/t) + h.c. .
(2.4)
It can easily be checked that both £% and C% are singlets under righthanded transformations, but transform as members of an octet for the lefthanded transformations. The barred lagrangian in Eq. (2.4) can in
VIII2 The nonleptonic weak interaction
215
fact be removed, so that it does not contribute to physical processes. This is seen in two ways. At the simplest level, direct calculation of K —• 2TT and K —> 3?r amplitudes using Cs, including all diagrams, yields a vanishing contribution. Alternatively, this can be understood by noting that in QCD the quantity x appearing in Eq. (2.4) is proportional to the quark mass matrix, x — 2B$mq. Thus the effect of £g is equivalent to a modification of the mass matrix, mq —• mq = mq + g%\§mq
.
(25)
This new mass matrix can be diagonalized by a chiral rotation Tr (m'qU) + Tr (Rm'qLU) = Tr (mDU)
,
(2.6)
with mo diagonal. The transformed theory clearly has conserved quantum numbers, as it is flavor diagonal. This means that the original theory also has conserved quantum numbers, one of which can be called strangeness. When particles are mass eigenstates, even in the presence of £8? the kaon state does not decay. Hence this Cs can be discarded from considerations, leaving only Cs as responsible for octet K decays [Cr 67]. This octet operator is necessarily A / = 1/2 in character. Another allowed operator, transforming as (27^,1^), contains both A / = 1/2 and A / = 3/2 portions,
where /»(l/2)
jL/97
27
: =
(1/2) x^1/2 rri
5^97
= ^27
^
/> ^
 \ Ot O r r TT\ \bcs TT TT^ I
i "U
\ ^ OurJ U *• Ou*J C/ I "+" n.C. ,
^ a 6 ^^ l ^ ^ ^ C/' A a^C/ (7 ' ) + h.C. \ /
/O O \
iZ.odi)
(z.ODJ
The coefficients are given by C ly/2
=1
^3/2
_
°fii«7/9 3 —
x
J
C1/2
= — \/2
^3/2
_ J_
°4+t5/2,lt2/2 ~~
/o
^1//2 *
The complete classification at order E4 is difficult, but has been obtained [KaMW 90]. We shall apply these lagrangians to the data in Sects. VIII4, XII6. There we shall see that gs » g^, whereas naive expectations would have octet and 27plet amplitudes being of comparable strength. This is part of the puzzle of the A / = 1/2 rule. The reliable theoretical calculation of the nonleptonic decay amplitudes, which is tantamount to (1/2)
(3/2)
predicting the quantities gs, #27 and g^7 , is one of the difficult problems mentioned earlier. It has not yet been convincingly accomplished.
216
VIII Kaons and the AS = 1 interaction
The best we can do is to describe the theoretical framework of the short distance expansion, to which we now turn. VIII3 Short distance behavior At short distances, the asymptotic freedom property of QCD allows a perturbative treatment of the product of currents. The philosophy is to use perturbative QCD to treat the strong interactions for energies Mw > E > /i. The result is an effective lagrangian which depends on the scale fi. Ultimately matrix elements must be taken which include the strong interaction below energy scale /i and the final result should be independent of //. Short distance operator basis The outcome of the short distance calculation can be expressed as an effective nonleptonic hamiltonian expanded in a set of local operators with scale dependent coefficients [Wi 69], .
(3.1)
As in any effective lagrangian, those operators of lowest dimension should be dominant. If the operator O{ has dimension d, its coefficient obeys the scaling property Q ~ M^d. Let us first see how this hamiltonian is generated in perturbation theory. We can later use the renormalization group to sum the leading logarithmic contributions. The lowest order diagrams renormalizing the current product are given in Fig. VIII1. The process in Fig. VIII1 (a) corresponds to a lefthanded, gaugeinvariant operator of dimension 4, O (d=4) = dlp{\ + 75)5 .
(3.2)
This operator can be removed from consideration by a redefinition of the quark fields (cf. Prob. IV1). The remaining operators are of dimension 6. Simple ^exchange with no gluonic corrections gives rise in the short distance expansion to the local operator OA = d7fi (1 + 7 5 ) ^ 7 ^ (1 + 75) s ,
(3.3)
u,c,t
W W (a)
s
u (b)
u
Q
Q (O
Fig. VIII1 QCD Radiative corrections to the AS = 1 nonleptonic hamiltonian
VIII3 Short distance behavior
217
with a coefficient CA = 2 in the normalization of Eq. (3.1). The gluonic correction of Fig. VHIl(b) generates an operator of the form J7 M (l+75)A a u«7 / '(l+75)A o s ,
(3.4)
a
where the {A } are color SU(3) matrices. However, use of the Fierz rearrangement property (see App. C) and the completeness property Eq. (II2.8) of SU(3) matrices allow this to be rewritten in colorsinglet form h» (1 + 75) Aa« «
7
M
(1 + 75) Xas = \oA
+ 20B ,
+ jb)ud^(l
.
where OB = u^(l
+ j5)s
(3.5)
The strong radiative correction is seen to generate a new operator OBPerturbative analysis Consider now the oneloop renormalizations of the fourfermion interaction Fig. VIIIl(b). In calculating Feynman diagrams we typically encounter integrals such as (neglecting quark masses) 
J
16TT2M^
K2
r iv±w
(3.6)
where we evaluate the integral at the lower end using a scale /x. Clearly presents a natural cutoff in the sense that
The modification of the matrix element to first order in QCD is then ,
(3.8)
where gs is the quarkgluon coupling strength. The gluonic correction to OB must also be examined, and a similar analysis yields OB  OB  JL
In (^)(3OA
O B) .
(3.9)
We observe that the operators, O± = ^(OA±OB)
,
are forminvariant, O± —• c±O±, with coefficients c±,
(3.10)
218
VIII Kaons and the AS = 1 interaction
where e/+ = —2 and GL = +4. The isospin content of the various operators can be determined in various ways. Perhaps the easiest method involves the use of raising and lowering operators [Ca 66, Li 78], /_i_d — u ,
I+u = —d ,
IU = d ,
Id = —u ,
(3.12)
to show that I+O = 0, implying that O_ is the Iz = 1/2 member of an isospin doublet. With repeated use of raising and lowering operators, one can demonstrate that O_ is purely A / = 1/2 whereas O+ is a combination of A / = 1/2 and A / = 3/2 operators. Prom Eq. (3.11), we see that under oneloop corrections the operator O is enhanced by the factor
fL^!2.1
(3.13)
where we use a8(fi) ~ 0.4 (AQCD — 0.2 GeV) at // ~ 1 GeV. Similarly O+ is accompanied by the suppression factor
Renormalization group analysis Choosing an even smaller value of // would lead to an even larger correction. However, maintaining just the lowest order perturbation in the QCD interaction would then be unjustified. It is possible to do better than the lowest order perturbative estimate by using the renormalization group to sum the logarithmic factors [GaL 74, AIM 74]. In a renormalizable theory physically measurable quantities can be written as functions of couplings which are renormalized at a renormalization scale /iR. Physical quantities calculated in the theory must be independent of /iR. Denoting ail arbitrary physical quantity by Q, this may be written Q = /(03(MR)>MR)
>
(3.15)
where / is some function of //R and 53 is the strong coupling constant of QCD. Differentiating with respect to /XR, we have O ,
(3.16)
which is the renormalization group equation. It represents the feature that a change in the renormalization scale must be compensated by a modification of the coupling constants, leaving physical quantities invariant. In order to see how this program can be carried out for the effective weak hamiltonian, consider the following irreducible vertex function
VIII3 Short distance behavior
219
which represents a typical weak nonleptonic matrix element, (0 \
( ^
(y^f (0 where the {<&} are quark fields carrying momenta {pi}. Z2 is the quark wavefunction renormalization constant for the fermion field, and subscripts 'ren', 'unren' denote renormalized and unrenormalized quantities. Choosing the subtraction point yy\ = —/J*^ we require that unrenormalized quantities be independent of //R, r
unren
= 0.
(3.18)
This implies ) \ o)™ = 0,
0
where g$r is the renormalized strong coupling constant, /?QCD is the QCD beta function of Eq. (II2.57) and JF is the quark field anomalous dimension of Eq. (II2.69). As we have seen, QCD radiative corrections generally mix the local operators appearing in the short distance expansion, (0 \T (Onqiq2q3q4)\ 0)£n  £
Xnn, (0 \T (On
and the mixing matrix can be diagonalized to obtain a set of multiplicatively renormalized operators (0 \T (OkQlq2q3q4)\ 0>™ = Zk (0 \T {Om
•
(320)
If Zk has anomalous dimension 7^, Z f c ~ l + 7fclnMR + .. ,
(3.21)
then the coefficient functions Ck(^Rx) must satisfy — + /SQCD^— + Ik ~ 47F I ck(fjLRx) = 0 . /iR dgsr ) Prom the above, we have for the operators O± &±
•
(3.22)
(3 23)

Having specified the anomalous dimensions of the operators O±, we can solve Eq. (3.22) with methods analogous to those employed in Sect. II—2. That is, because QCD is asymptotically free and we are working at large
220
VIII Kaons and the AS = 1 interaction
momentum scales, we can use the perturbative result (cf. Eq. (II2.57)),
where 6 = 1 1 —  n / , rif being the number of quark flavors. Upon inserting the leading term in the perturbative expression for as (cf. Eq. (II2.74)), 12TT M
(325)
one can verify that the solution to Eq. (3.22) is given by / C±(»R)/C ±(MW)
n2
M2
\d±/b
= ^1 + ^ f t l n  f J
.
(3.26)
Note that in the perturbative regime where 1 ^> as, we have C±(/*R)
.
93
M ]n W
c±(Mw) which agrees with our previous result, Eq. (3.11). It is the renormalization group which has allowed us to sum all the 'leading logs'. Of course, at scale Mw one must be able to reproduce the original weak hamiltonian, implying c+(Mw) = c(Mw) = 1. Taking //R ~ 1 GeV and as = 0.4 as before, we find with c_(/iR) ~ 1.5 ,
c+(/iR) ^ 0.8 .
(3.29)
We observe then a A / = 1/2 enhancement of a factor of 2 or so, which is encouraging but still considerably smaller than the experimental value of A0/A2 ~ 22 discussed in the next section. Two additions to the above analysis must now be addressed. One is the proper treatment of heavy quark thresholds. In reducing the energy scale from Mw down to /iR, one passes through regions where there are
Fig. VIII2
Penguin diagram.
VIII3 Short distance behavior
221
successively six, five, four or three light quarks, the word 'light' meaning relative to the energy scale //R. The beta function changes slightly from region to region. A proper treatment must apply the renormalization group scheme in each sector separately. This is a straightforward generalization of the procedures described above. The other addition is the inclusion of penguin diagrams of Fig. VIIIl(c) [ShVZ 77, ShVZ 79b, BiW 84], whimsically named because of a rough resemblance to this antarctic creature. The gluonic penguin is noteworthy because it is purely A/ = ^, thus helping to build a larger A/ =  amplitude, and because it is the main source of CP violation in the AS — 1 hamiltonian. The electroweak penguin, wherein the gluon is replaced by a photon or a Z° boson, also enters the theory of CP violation. The CP conserving portion of the penguin diagrams involves a GIM cancelation between the c, u quarks and hence enters significantly at scales below the charmed quark mass. On the other hand, in the CP violating component, the GIM cancelation is between the t, c quarks and thus this piece is short distance dominated. At lowest order, before renormalization group enhancement, one obtains the following effective interactions for the diagram of Fig. VIII2, ^Vus In ^
L ^
ln
+ V;dVts In ^ 
m? + ^
^ r^
^R
m
^
^
)
^
c\
(3.30) We have used a scale /iR instead of the up quark mass and have quoted only the logarithmic mt dependence. The quarks q = u,d,s are summed over and Qq is the charge of quark q. Note that since the vector current can be written as a sum of lefthanded and righthanded currents, this is the only place where righthanded currents enter Hw. The gluonic penguin contains the righthanded current in an SU(3) singlet, hence retaining the (8L, lfl) property ofHw. However, the electroweak penguin introduces a small (8^,8^) component. The full result can be described with the fourquark AS = 1 operators,
2HC + 2HD , O 5 = d^(l
+
7b)X
2HC  3HD , O 6 = J 7 / i ( l + 75)5 O7 = g>y^(l + 75)d
(3.31)
222
VIII Kaons and the AS = 1 interaction
3 O8 =  2 ^ 7 / i ( l + 75)4/ qj'fil
 ^fb)QqQi i
where q = u,d,s are summed over in 05,6,7,85 * a n d J is the charge of quark q and HA = <*7/i(l + 75)^ ^ ( 1 + 75)5 ,
are
color labels, Qq
# c = d7/x(l + 75)5 ^ ( 1 + 75 K
rf7/i(l + 75)5 vr/Hl + 75)ix , JETD = ^7/i(! + 75)5 57^(1 + 75)5. (3.32) The operators are arranged such that Oi52,5,6 have octet and A / = \ quantum numbers, 03(04) are in the 27plet with A7 = ^(A7 =  ) , while OY$ arise only from the electroweak penguin diagram. The full hamiltonian is (3 33)
•

A renormalization group analysis of the coefficients [BuBH 90] yields = 1.90  0.62r ,
c5 = 0.011  0.079r ,
c2 = 0.14  0.020r ,
ce = 0.001  0.029r ,
c3 = c 4 /5 ,
c7 = 0.009  (0.010  0.004r)a ,
c4 = 0.49  0.005r ,
c8 = (0.002 + 0.160r)a ,
Cl
(3.34)
with A  0.2 GeV, /i R ~ 1 GeV, mt = 150GeV and r = V^Vts/V*^. The number multiplying r has a dependence on mt whereas (within the leading logarithm approximation) the remainder does not if mt > MwThe r dependence in C4 arises only because of the electroweak penguin diagram. This hamiltonian summarizes the QCD short distance analysis and is the basis for estimates of weak amplitudes.
VIII4 The A/ = 1/2 rule
Phenomenology In the decays K —• TTTT, the 5wave twopion final state has a total isospin of either 0 or 2 as a consequence of Bose symmetry. Thus, such decays can be parameterized as
^ o
= Ao ei6°  V2A2ei6*
,
(4.1)
VIII4 The AI = 1/2 rule
223
where the subscripts 0,2 denote the total TTTT isospin and the strong interaction 5wave TTTT phase shifts <5j enter as prescribed by Watson's theorem (c/. Eq. (C2.15). There are, in principle, two distinct 1 — 2 amplitudes A2 and Af2. These are equal if there are no AI = 5/2 components in the weak transition, as is the case in the Standard Model if electromagnetic corrections are neglected. The experimental decay rates themselves imply r+Tr1 = 5.56 x W7mK , ^O«O\ = 5.28 x W7mK , \AK++n+*o\ = 3.72 x 10smK .
(4.2)
The test to see whether Af2 = A2 depends on the imprecisely known TTTT phase difference 80 — 82 For the value 8$ — 82 — 45°favored by fits to 7T7T scattering there appears to exist some AI = 5/2 effect, which would presumably arise from electromagnetic corrections, while for 80 — 82 — 57° there is none. We shall neglect this possibility from now on, and employ just the isospin amplitudes \A0\ = 5.46 x 107mK ,
\A2\ = 0.25 x 10" 7 m^ .
(4.3)
The ratio of magnitudes, \A2/Ao\~l/22 ,
(4.4)
indicates a striking dominance of the AI = 1/2 amplitude (which contributes to Ao) over the AI = 3/2 amplitude (which contributes only to A2). This enhancement of Ao over A2, together with related manifestations to be discussed later, is called the AI — 1/2 rule. As we have seen in previous sections, a naive estimate (and even determinations which are less naive!) do not suggest this much of an enhancement. However, the factor 20 dominance of AI = 1/2 effects over those with AI — 3/2 is common to both kaon and hyperon decay. A similar enhancement of AI — 1/2 is found in the K —• TTTTTT channel. In this case, it is customary to expand the transition amplitude about the center of the Dalitz plot. For the decay amplitude K(k) —• 7r(pi) n(p2) ft(ps), the relevant variables are Si
= (k
(4.5) x
y so
, s
o
where S3 labels the 'odd' pion, i.e. the third pion in each of the final states n+n~n°,nonO7T+,ir+Tr+n~. The large A / = 1/2 amplitudes are
224
VIII Kaons and the AS = 1 interaction
considered up to quadratic order in these variables while the A / = 3/2 amplitudes contain only constant plus linear terms, O^+VKO
=
OI 
2a3
+
(6i

( W +
^ 
= 2(ai + a 3 ) 
^j iS
(4.6)
, i6
+ 2c (V + ^
where ai,6i,c,d are A/ = 1/2 amplitudes, 03,63,623 are A7 = 3/2 amplitudes, and the phases {61} in <5MI (= &M — S\) and 621 (= 62 — f)\) refer to finalstate phase shifts in the 7 = 1 , 2 and mixed symmetry 7 = 1 states respectively. Because of the relatively small Qvalue for such decays (QnTTTT = ™
a3 = 0.37 ± 0.018 , 63 = 0.646 ± 0.125 , d = 3.39 ± 0.33 .
623 = 2.3 ± 0.31 ,
(4.7) Dominance of the A7 = 1/2 signal is again clear in magnitude and in slope terms, e.g. we find at the center of the Dalitz plot, las/ail1/26 .
(4.8)
In SU{3) language, the dominance of A7 = 1/2 effects over A7 = 3/2 implies the dominance of octet transitions over those involving the 27plet. This is a consequence, within the Standard Model, of the fact that the A7 = 1/2 27plet operator contributes relative to the A7 = 3/2 27plet operator with a fixed strength given by the scaleindependent ratio of coefficients C3/C4 ~ 1/5 (viz. Eq. 3.34). The 27plet operator then gives only a small contribution to the A7 = 1/2 amplitudes, with the major portion coming from the octet operators. We shall therefore ignore the A7 = 1/2 27plet contribution henceforth.
VIII4 The A/ = 1/2 rule
225
Chiral lagrangian analysis The lefthanded chiral property of the Standard Model may be directly tested by the use of chiral symmetry to relate the amplitudes in K —> TTTTTT to those in K —> TTTT. We have already constructed the effective lagrangians for (8/,, 1#) and (27^,1#) transitions. Dropping g^7 , the nonleptonic decays are described by the two parameters gs and g^ at O(E2). Let us see how well this parameterization works, and afterwards add O(E4) corrections. The two free parameters may be determined from AQ and A^ in K —> TTTT decays. Prom the chiral lagrangians of Eqs. (2.4), (2.8b), we find (3/2) (
m2}
<
(4
g)
which yields upon comparison with Eq. (4.3), gs ~ 7.8 x l O " 8 ^ ,
5g
/2)
 0.25 x l ( T 8 i ^ .
(4.10)
The K —• TTTTTT amplitude may be predicted from these. Because there are only two factors of the energy, no quadratic terms in Eq. (4.6) are present in the predictions, 1 (1/2)
v
UA
,(3/2)
_
^ 4
(3/2)
_
A 2 m^
'"A
i^
fi
'
7T
5
[
which correspond to the numerical values (again in units of 10~ 7),
+7r+jr _
= 0.47 + 2 . 3 y .
These are to be compared to the experimental results, T+n_n0
= 9.15 + 14.1 y  4.85 f + 0.88 x2 ,
^ ^  ^ = 0.71 + 1 . 3 y , +7r+7r _
(4.12)
= 0.71 + 2.9 y ,
with error bars given previously in Eq. (4.7). This comparison can be seen in Fig. VIII3, where a slice across the Dalitz plot is given. Also
226
VIII Kaons and the AS = 1 interaction
shown are the extrapolations outside the physical region to the 'softpion point' where either p+ —> 0 or j£ —> 0. Predictions at these locations are obtained by using the softpion theorem. The chiral relations clearly capture the main features of the amplitude and demonstrate that the K —• 3?r A/ = 1/2 enhancement is not independent of that observed in K —• 2?r decay. However, for the A/ = 1/2 transitions, knowledge of the parameters c and d which accompany the quadratic kinematic terms (cf. Eqs. (4.6),(4,7)) allows us to do somewhat better. The kinematic dependence of x2 or y2 can come only from a chiral lagrangian with four factors of the momentum, and only two combinations are possible: Aquad = 7i& • PoP+ • P + 72 (k • k+Po ' P + k • pp0  p+)
•
(4.13)
Such behavior can be generated from a variety of chiral lagrangians, (4.14)
g'lTr
+
However the predictions in terms of 7$ are unique. Fitting the quadratic terms to determine 71,72 yields the full amplitude, *O = (9.5 ± 0.7) + (16.0 ± 0.5) y4.85 y +0.88 x2 , (4.15) which provides an excellent representation of the data. Final state interaction effects also provide an important contribution and must be included in a complete analysis [KaMW 90]. Note that in the process of determining the quadratic coefficients, the constant and linear terms have also become improved. This process cannot be repeated for A/ = 3/2 amplitudes due to a lack of data on quadratic terms.
54321
0
1 2
3 4
5 6
Fig. VIII3 Dalitz plot
VIII5 Rare kaon decays
227
Vacuum saturation The discussion of direct calculations of the nonleptonic amplitudes is beyond the scope of this book. Suffice it to say that no treatment is presently adequate. Let us give the simplest estimate, called vacuum saturation, as a convenient benchmark with which to compare the theory. For simplicity we consider only O\ (the largest A/ = 1/2 operator) and O4 (the A/ = 3/2 operator), ,
(4.16)
with c\ ~ 1.9 and C4 ~ 0.5. The vacuum saturation approximation consists of inserting the vacuum intermediate state between the two currents in any way possible, e.g. <7r+(p+)7r(p_) \drf (1 + 75) uuf (1 + 75) s = <7T(P_) \d>fw\ 0) <7T+(P+) \u*f8 + <7r+(p+)7r(p_) \upfual 0> (0 14 n(p_  P + In obtaining this result the Fierz rearrangement property den* (1 + 75) uaup^ (1 + 75) sp = daY (1 + 75) spup^ (1 + 75) ua has been used, where a, j3 are color indices which are summed over. In addition, the color singlet property of currents is employed, (0 \dal^sp\ K°(k))= iy/2FKk^
.
(4.18)
Within the vacuum saturation approximation, we see that the amplitudes are given completely by known semileptonic decay matrix elements. Putting in all of the constants, we find that AQ = —V* dVwFn ( m i  ml) cx = 0.84 x 10~7ra# , (4.19) ^2 =
5—KdV^F,. (m^  m£) c
4
7
= 0.42 x 10" r
We see that the above estimate of A2 works reasonably well, but that Ao falls considerably short of the observed A/ = 1/2 amplitude. This demonstrates that vacuum saturation is not a realistic approximation. However, it does serve to indicate how much additional A/ = 1/2 enhancement is required to explain the data.
228
VIII Kaons and the AS = 1 interaction VIII5 Rare kaon decays
Thus far, we have discussed the dominant decay modes of the kaon. There are, however, many additional modes which, despite tiny branching ratios, are the subject of intense experimental activity. We can divide these into three main categories. 1) Forbidden decays  These include tests of the flavor conservation laws of the Standard Model. Positive signals would represent signals of physics beyond our present theory. 2) Rare decays within the Standard Model  These include decays which occur only at oneloop order. Such processes can be viewed as tests of chiral dynamics as developed in this and preceding chapters (e.g., radiative kaon decays) or as particularly sensitive to the particle content of the theory (e.g., K+ —• TT + Z/^ (£ = e^fi^r) probes the top quark mass and the number of light neutrinos). 3) CP violation studies  As will be discussed in the next chapter, the kaon system has thus far provided the only positive information on CP violation.* Any of these have the potential to yield exciting physics. We shall content ourselves with discussing only a small sample of the many possibilities. Consider first the rare decay K+ —• n+U£U£. This mode is often called 'K+ to TT+ plus nothing', in reference to its unique experimental signature. This process can take place only through loop diagrams, such as those in Fig. VIII4. Calculation of these Feynman diagrams leads to an effective lagrangian of the form [ImL 81, HaL 89]
< • ( ) y^
()••
Fig. VIII4 The decay The observed baryonantibaryon asymmetry of the universe also requires the existence of CP violation within the standard cosmological model.
VIII5 Rare kaon decays
229
and
where Xj =
(4z) 3 x x D(x) = (5.2) 2 4 4 1— x (1x) {lxf 8 In this formula we have neglected mj/rriyy (£ = e,/x, r). The matrix element in this case is simple, being related by isospin to the known charged current amplitude
(?r+(p) [sj^d] K+(k)) = V2
(k)) = f+{q2) {k +p)^ (5.3) with /+(0) = —1, and yields a straightforward prediction for the branching ratio in terms of the number Nu of light neutrinos, Brf
+
(TT°(P) \S^^U\ K
7
(5.4)
W'
= 0.7 x 10T 6 D{xc) + s2 The precise branching ratio depends on the KM angle and top quark mass. Although present estimates suggest only a range of branching ratios, 0.58.0 x 10~10, future determinations of the KM angles and mt can provide a very nontrivial test of the Standard Model at oneloop order. This rate seems to be reasonably insensitive to long distance physics. A different class of rare decays consists of the radiative processes K$ —> 77 and KL —» TT°77. These transitions provide interesting tests of chiral perturbation theory at oneloop order. In this case, the long distance process, Fig. VIII5, is dominant. An important feature is that there is no tree level contribution at order E2 or E4 from any of the strong or weak chiral lagrangians because all of the hadrons involved are neutral. Thus the decays can only come from loop diagrams, or from lagrangians at O(EG). There is also an interesting corollary of this result concerning the renormalization behavior of the loops. Since there are no tree level counterterms at O(E4) with which to absorb divergences from the loop diagrams, and recalling that we have proven all divergences can be han
a*
(a)
(b)
Fig. VIII5 Long distance contributions to radiative kaon decays.
VIII Kaons and the AS — 1 interaction
230
died in this fashion, it follows that the sum of the loop diagrams must be finite. This is in fact born out by direct calculation. For Ks —> 77, the prediction of chiral [D'AE 86, Go 86] loops is given in terms of known quantities such as a2m\ i j S f _> 7 7
1
167T 3 2
Tnz,
mK
mK
In2 Q{z)  2m In Q(z)]
F(z) = lz[ir 1 Q(z) =
(5.5)
where g$ is the nonleptonic coupling defined previously in Eq. (2.4b). The theoretical oneloop branching ratio,
g S n = 20 x 1(T6 ,
(5.6a)
compares favorably with the recent measurement +77
=(2.4±1.2)xlO6
(5.6b)
without the need to consider possible contributions at O(Ee). The case of KL —> n0/yj is also instructive. Again, oneloop contributions are finite and [EcPR 88] unambiguous. Indeed, we know that KL —• TT°77 and KL —* 77 are related by the softpion theorem in the limit"pfc—»• 0. Explicit evaluation yields ,1/2
dT dz / X
where z =
z—
ml
mK
V
m2
\
rn?KJ
(5.7)
F(z)
and X(a, b) = 1 + a2 + b2  2(o + b + ab)
(5.8)
Integration over 2 yields the branching ratio r
(loop)
 6.8 x 10
(5.9)
For this reaction, there is also an O(E6) correction which could be important [Se 90], viz. the diagram of Fig. VIII5(c) which shows the effect of pexchange. Such a contribution would have a very different spectrum from that of Eq. (5.7). However, experiment [Ba et al. 90] reveals a photon energy distribution dT/dz which matches that of Eq. (5.7), but with the rate B r ^ o ^ o ^ ^ (2.1 ± 0.6) x 10~6. No evidence exists for a pexchange contribution.
Problems
231
Problems 1) Ke3 decay The ratio f+^n° (0)/f+O/ir (0) of semileptonic form factors is a measure of isospin violation. Part of this quantity arises from 7r°r/g mixing. a) By diagonalizing the pseudoscalar meson mass matrix, show that rrid ^ mu induces the mixing TT°) = cose\(fs) + sine\(fs) where e ~ V3(md — mu)/[4t(m8 — TO)]and TO = (mu + TO^)/2. b) Demonstrate that this leads to the result (c/. Eq. (1.11))
2) Current algebra/PCAC and K > 3?r decay The results derived in Sect. VIII4 with effective lagrangians can also be obtained by means of current algebra/PCAC methods. a) Using PC AC, show that the softpion limit of the K —> 3?r transition amplitude is given by qa*0
a
c
b {
Fn
where Q^ = f d3x Afifa t) is the axial charge, b) Demonstrate that this may be also written as
)h^(rfyqbK°c\Hw(0)\K2)
= y(7riKcqc\[Qa,Hw(0)]\K%) ,
where Qa is an isotopic spin operator, and hence that
1 where / = 5,  signifies question. c) Use a linear expansion Eq. (4.6) with c=d=0) to corrections of order
+ 0)
the isospin component of the quantities in of the K + 3n transition amplitude, (i.e. to reproduce the results of Eq. (4.11), up ml.
IX Kaon mixing and CP violation
In our discussion of the electroweak interactions in Chap. II, we saw that the KM matrix contains imaginary couplings which have the potential to violate CP invariance. These arise originally in the Yukawa couplings of quarks to the Higgs bosons, but after the definitions of mass eigenstates, they appear only as a single phase in the W± gauge couplings. In this chapter, we focus on the system of neutral kaons to describe how this phase gives rise to CP violating observables. IX1 K°K° mixing It is clear that K° and K°should mix with each other. In addition to less obvious mechanisms discussed later, the most easily seen sourceof mixing occurs through their common TTTT decays, i.e. K°<> TTTT <> K°. We can use secondorder perturbation theory to study the phenomenon of mixing. Mass matrix phenomenology Writing the wavefunctions in twocomponent form \
,
)
we have the time development
232
(1.1)
IX1 K°K° mixing
233
where, to second order in perturbation theory, the quantity in parentheses is called the mass matrix and is given by*
2 Uj
2mK
(K?\Hw\n (1.3) Here the absorptive piece F arises from use of the identity ^—^
= p(^=r)in6(Enu;)
u — t,n + xe
\u) —
,
(1.4)
bjn)
and hence involves only physical intermediate states 1
n> (n\Hw\K°) 2n8{En  mK)
.
(1.5)
Because M and T are hermitian, we have M21 = M*2 and F2i = T\2> The diagonal elements of the mass matrix are required to be equal by CPT invariance, leading to a general form A p2 where A, p2 and g2 can be complex. The states K° and K° are related by the unitary CP operation,
CP\K°)=tK\K°)
(1.7)
with \£K\2 = 1. Our convention will be to choose £# = — 1. The assumption of C P invariance would relate the offdiagonal elements in the mass matrix so as to imply p = q,
{K°\neS\K°) = (K°KCPJ^CP HeS (CP)1CP ^°> = (K°Weff K°), (1.8) where (K° \Hes\ K°) is defined in Eq. (1.3). Combined with the hermiticity of M and F, this would imply that Myi and Fi2 are real. In the actual CPnoninvariant world, this is not the case and we have for the eigenstates of the mass matrix, \KL ) = —=j s
V\P\
2
+ kl
2
\p\K°)± q\K0)} ,
(1.9)
* The factors l/2ra/< are required by the normalization convention of Eq. (C2.7) for state vectors.
234
IX Kaon mixing and CP violation
where from the above discussion, we have p_
Mn  Fi2
The difference in eigenvalues is given by 2qp = (mL  ms)  ~{TL  Ts) ( i \1/2/ i \1/2 = 2 ( M12  ^ r i 2 Mx*2   H 2 ~ 2ReM12  i
(1.11) where the final relation is an approximation valid if CP violation is small (1 » ImMi2/ReMi2). The subscripts in KL and Ks, standing for 'long' and 'short', refer to their respective lifetimes, which differ by a factor of 580. To understand this large difference, we note that if CP were conserved (p = q), these states would become CP eigenstates K± (not to be confused with the charged kaons K±\), \Ks) ^
\K°+) ,
]KL)
0
\K°±) =  L [\K°)T ^ )] ,
_^
]Ko_} f
P=Q
CP\Kl) = ±\K%) .
n 22)
In this limit, which well approximates reality, Ks would decay only to CPeven final states like TTTT, whereas KL would decay only to CPodd final states. Since the phase space for the former considerably exceeds that of the latter at the rather low energy of the kaon mass, Ks has much the shorter lifetime. The states KS,L, expanded in terms of CP eigenstates, are
 Ts)
(113) K°K° mixing can be observed experimentally from the time development of a state which is produced via a strong interaction process, and therefore starts out at t = 0 as either a pure K° or K°,
(1.14)
9±(t) = I
IX1 K°K° mixing
235
where AF = T$ — TL and Am = m^ — 7715, each defined to be a positive quantity. From such experiments, the very precise value Am = (3.522 ± 0.016) x 10"12 MeV has been obtained. Box diagram contribution The Standard Model predicts K°K° mixing at about the observed level. However, it is difficult to precisely calculate the mass difference Am. There are two main classes of effects, shortdistance box diagrams of Fig. IX1 (a),(b) and longdistance contributions like those in Fig. IX2, which can generate the mixing amplitude. We turn now to a study of the first of these. When calculated at the quark level, one can envision several Feynman diagrams which mix K° and K° (see Fig. IX1). The box diagrams in Fig. IX1 (a),(b) lead to an effective hamiltonian dKs)2 H(xc)m%
+ (V^Vts)2
H(xt)m2m
+ 2 (V&VtaV&Vn) G(x« x t)m2cm] OAS=2 + h.c. , where X{ = mf/M^ and [InL 81] l
9
1
3
1
,
4 + 4T^2(l^J2(T^ l n a ; ' (1.16)
v
* ' 4(ls)(ly) The factors 771, 772 and 773 are QCD shortdistance renormalization factors analogous to the renormalization group coefficients for the AS = 1 hamiltonian discussed in the previous chapter. For rrtt > Mw,
u.c.t
u,c,t
u,c,t
W (a)
<j
s
w s
u,c,t d (b)
Fig. IX1 Box (a),(b) and other contributions to CP violation.
236
IX Kaon mixing and CP violation
0.2 GeV, they have the values [BuBH 90], 7/i ~ 0.85 ,
% ^ 0.62 ,
773 ~ 0.36 .
(1.17)
Note that the GIM mechanism is at work here, in that the u, c, t intermediate states would exactly cancel each other if their masses were equal. The mass of the ?xquark has been neglected with respect to heavy quark masses in writing Eq. (1.15). Given present bounds on the KM elements and the t quark mass, the most important contribution to the real part of H^ox is that of the cquark. In view of this, and noting that H(xc) ~ 1, we then have Re H^T * ^2^Re(F c * d F c s ) 2
Vl
OAS=2 .
(1.18)
At this stage one can provide a rationale for neglecting Fig. IXl(c). In this diagram the GIM cancelation is logarithmic rather than power law, so no factor of m2 appears. Evidently, all matrix elements of other diagrams should be suppressed compared to the box diagram by factors like fJ?/rn% (where /i is a hadronic scale) or m^/m2. This has been verified for Fig. IXl(c), and will be discussed below as a 'longdistance' effect. The double penguin diagram of Fig. IXl(d) has a longdistance part and a shortdistance component, obtained when the loop momenta are high. Separating the two is difficult, but the shortdistance piece has been studied [EeP 87]. To make contact with phenomenology, one must evaluate the matrix element of OAS=2 between K° and K° states. It is conventional to express the results in terms of the socalled Bparameter, ^
,
(1.19)
where B = 1 corresponds to the simple vacuum saturation approximation described in Sect. VIII4. A wide variety of models have been applied to determine this amplitude, with estimates for B which vary considerably [PaT 89]. One approach attempts to extract B directly from experiment by using chiral symmetry and SU(3) [DoGH 82]. The point is that both OAS=2 and the A/ = 3/2, AS = 1 operator O4 (see Eq. (VIII3.31)) are formed from a pair of SU(3) octet lefthanded currents, and because of their quantum numbers, each belong to the same 27dimensional representation of SU(3). This implies the SU(3) relation, = v^2 .
(1.20)
The righthand side of the above can be related to the K —• 2n matrix element with the aid of chiral symmetry. Using either soft pion techniques
IX1 K°K° mixing
237
or the chiral lagrangian given in Eq. (VIII2.7), we find
(n°(k)\O4\K0(k))=2VlF«
/
+ + 2(n n°\OAI=3/2\K )
, (1.21)
or with onshell kaons (k2 = rrv^)^ 8y/2F,
ml
where rji is any one of the QCD factors 771, 772, 773. Expressed this way the result is independent of the scale chosen in QCD shortdistance corrections, as it cancels in the ratio of rji/c^. This occurs because the QCD corrections are SU(3) symmetric and treat all members of the 27plet in an equal fashion. Written in terms of the 5parameter the result is* B = 0.33[as(fi)]2/9
.
(1.23)
We stress that any model which respects both chiral symmetry and 5f/(3), and also obtains the correct K+ —• TT+TT0 amplitude must lead to this result. Often, model dependent estimates get an answer different from Eq. (1.23) because they incorrectly predict K+ —> TT+TT0. The only significant issues which can affect the above determination of B are the possible breaking of SU{3) symmetry and the contribution of higherorder chiral corrections. One attempt [BiSW 84] to calculate the higherorder effects has examined the leading logarithmic corrections obtained when one calculates loops in chiral perturbation theoryt. In the language of Chap. VI, this consists of keeping the quantities m In(ra 2//i2) which occur in loop contributions, but dropping all factors of ra4 and all effects of the low energy constants occurring at O(E4). The results depend strongly on the 'chiraF scale, which we may write as /xchin that occurs in the chiral logarithmic factor  no corrections are obtained if Mchir — 1/2 GeV whereas a 50% enhancement obtains if /xchir — 1 GeV. Unfortunately, this logarithmic approximation has not been accurate in other settings, and a full chiral perturbation theory calculation cannot yet be done because one does not have sufficient information to pin down the needed lowenergy constants at O(E4). A final resolution will require a better understanding of the origins and structure of the weak chiral lagrangian. Lattice QCD computer studies in the quenched approximation (i.e. no fermion loops) presently indicate a value B ~ 0.7, but are also presently obtaining values for the K+ —• TT+TT0 amplitude which are about twice as large as the empirical value. * Here a slight correction has been included for the effects of 77°7r°mixing which can lead to a simulated A/ = 3/2 contribution to the K —> 2ir decay process. This is discussed in Sect. IX3. t The paper [BiSW 84] contains an error in the normalization of the B factor, which we have corrected for in the quoted result.
238
IX Kaon mixing and CP violation
If we piece together all of the ingredients to the shortdistance predictions, we obtain from the charm contribution to the box diagrams the mass difference, (Am)^ o r y ~r 7 l B(Am) e x p t .
(1.24)
This result is clearly of the right order of magnitude, but there also exist potentially sizable but incalculable longdistance corrections, which preclude a clean prediction of the mass difference. The longdistance contributions are generically of the form of Fig. IX2, with the AS — 1 weak hamiltonian acting twice. As discussed above, a naive analysis suggests that such contributions should be suppressed by a factor ~ m2K/m2c ~ 1/10 with respect to the shortdistance effect. However, empirically the AI = 1/2 portion of H^s=1 has a factor of twenty enhancement over the simplest estimates (see Sect. VIII4), such that the naive factor should perhaps be modified to be about (20 7n#/ra c ) 2 ~ 40. Although the effect is not quite that large in reality, estimates of K° <• (TT0, TJ) <> K° and K° <»TTTT <• K° intermediate states yield longdistance effects comparable to the experimental mass difference. It is unfortunately not possible to be precise about these, but they can be large. For example, the K —• rj —> K° intermediate state by itself yields a contribution to Am of about 1.1 times the experimental result! This is estimated by using SU(3) plus chiral symmetry, and if one allows a 30% uncertainty in the prediction, it produces a large uncertainty by itself in the predicted mass difference. The TTTT contribution also has significant uncertainty. We can only conclude that, while a precise prediction of Am is not possible, the observed size is roughly compatible with that expected in the Standard Model. IX2 The phenomenology of kaon CP violation The TTTT final state of kaon decay is even under CP provided the strong interactions are invariant under this symmetry. For the 7r°7r° system, this is clear since TT0 is itself a CP eigenstate, CPTT0) = — TT°), and the two pions must be in an 5wave (£ = 0) state, CPITTV)
= (  l f (  l ) V V ) = +7r°7r°).
K°
(a)
(b)
Fig. IX2 Longdistance contributions.
(2.1)
IX2 The phenomenology of kaon CP violation
239
The corresponding result for charged pions reflects the fact that ?r+ and 7T~ are CPconjugate partners, CPTT ± ) = — TTT). We have seen that if CP were conserved, the two neutral kaons would organize themselves into CP eigenstates, with only Ks decaying into TTTT. Alternatively, KL decays primarily into the TTTTTT final state, which is CPodd if the pions are in relative 5waves. The observation of both neutral kaons decaying into 7T7T is then a signal of CP violation. There can be two sources of CP violation in KL —> TTTT decay. We have already seen that KK mixing can generate a mixture of the CP eigenstates in physical kaons due to CP violation in the mass matrix. There also exists the possibility of direct CP violation in the weak decay amplitude, such that the CPodd kaon eigenstate \K^_) makes a transition to 7T7T. These two mechanisms are pictured in Fig. IX3. The Kirn decay amplitudes have already been written down in Eq. (VIII4.1) in terms of realvalued moduli Ao, A2 This decomposition is a consequence of Watson's theorem, and relies in part upon the assumption of time reversal invariance. However, if direct CP violation occurs, Ao and A
,
A2 = \A2\e^
,
(2.2)
with CP violation in the decay amplitude being characterized by the phases £o and £2 Consequently, the KQ —• TTTT and KQ —> TTTT decay amplitudes assume the modified form
+r = \A0\e*°eiS°+ i ^ e ^ e * ,
Using the definitions of KL and Ks in Eq. (1.13), a straightforward calculation leads to the following measures of CP violation: _
KL
K+
(a)
/
K,
^
(b)
Fig. IX3 Mechanisms for CP violation.
240
IX Kaon mixing and CP violation
where e — e + z£o ? ~~
A/2
iei(62*o)
A2 Ao
v/2
A2
/1I m A2
Im Ao
Ao VRe A2
Re A o
(2.5) The expression for e can be simplified by approximating the numerical value Ara/Ar = 0.478 ± 0.003 by Am/AT ~ 1/2. This yields the approximate relation, e i7r / 4 1 e y/2 Am '
i 1 Am+^AT
which we shall use repeatedly in the analysis to follow. In addition, since the rate for K —• TTTT is much larger than that for K —> TTTTTT, and K° —• TTTT is in turn dominated by the 7 = 0 final state because of the A / = \ rule, we have .
(2.7)
The above relations allow us to write i
Im M12
\m M12 , ^ _ e'f ^ Im M12 , Im Ap\ Am + *>) ~ y/2 V2Re M 12 + Re Ao) '
l
'
where UJ = Re^/ReAo ~ 1/22. We see that e is sensitive to CP violation in the mass matrix, while ef is a unique manifestation of direct A 5 = 1 CP violation. All CPviolating observables must involve an interference of two amplitudes. In Eq. (2.8), the quantity e expresses the interference of K° —> 7T7T with K° —• K° —• 7T7T, while ef involves interference of the 7 = 0 and 7 = 2 final states. The formulae for e and ef exhibit an important theoretical property. Since the choice of phase convention for any meson M is arbitrary, its state vector may be modified by the global strangeness transformation \M)  • e iA5 M). For the K° and K° states, this becomes \K°)  • eiX\K°) ,
iX \K°) > e\K°)
,
(2.9)
which has the effect,
Im Ar _^ Im A/ Re A/ ""* Re A/
'
Im Mu _^ Im M i2 _ Re M i2 ~" Re M i2
'
l
. . ''
We see that the values of e and e' are left unchanged. Various phase conventions appear in the literature. In the WuYang convention, A is
IX2 The phenomenology of kaon CP violation
241
chosen so that the Ao amplitude is realvalued. This is always possible to achieve by properly choosing the phase of the kaon state. However, it is inconvenient for the Standard Model, where the Ao amplitude naturally picks up a CPviolating phase. We shall therefore employ the convention in which no such phases occur in the definitions of the kaon states. It was in the K —» TTTT system that CP violation was first observed. At present, the status of measurements is \c\ = (2.263 ± 0.023) x 10~3 , f 0.0023 ± 0.0007 [Ba 92] ,
(2.H)
={
[ 0.0006 ± 0.0007 [Wi 92] , f (46.9 ± 2.2)° h_ = phase(r7+_) = < [(43.2 ±1.5)°
[Wi92] ,
= phase(r/00) = (47.1 ± 2.8)°
[Ca et al 90] .
[Ca et al 90] ,
As this book goes to print, there remains a discrepancy as to whether a nonzero value of e; has been found. This is a critical issue in the study of CP violation whose resolution is eagerly awaited. A violation of CP symmetry has also been observed in the semileptonic decays of KL and Kg. These are related to matrix elements of the weak hadronic currents. Since K° must always decay into e^ue7r~ while K° goes to e~Pe7r+, we have
16
If the semileptonic decays proceed as in the Standard Model, there is no direct CP violation in the transition amplitude, so that ~ l + 4Ree .
(2.13)
Since Re e = Re e, the above asymmetry is sensitive to the same parameter as appears in the KL —• TTTT studies. Here, measurement gives Re e = (1.635 ± 0.060) x 10"3 = e cos(44 ± 3)° ,
(2.14)
which is consistent with the experimental values from K —> TTTT. Finally, although one ordinarily emphasizes the phenomenon of CPviolation in studies of kaon mixing, precision experiments also probe the CPT transformation. For example, a prediction of CPT invariance, the equality of phases <^+_ = <^oo (up to very small corrections from e;), is
242
IX Kaon mixing and CP violation
seen to be consistent with existing data (0.2 ±2.9)°
[CaetoZ.90],
(2.15)
In addition, kaon data can be used to provide a sensitive test of the wellknown CPT prediction that the mass of a particle must equal that of its antiparticle. Specifically, the impressive bound \(mKo —m^o)/m Ko\ < 4x 10~18 has been obtained [Ca et al. 90]. Further study of CPT invariance is left to Prob. XI3. IX3 Kaon CP violation in the Standard Model After diagonalization, there can remain a single phase in the KM matrix. This phase generates the imaginary parts of amplitudes which are required for CP violation. It is a physical requirement that results be invariant under rephasing of the quark fields. As a consequence, all observables must be proportional to Im A(4) = A2\er) = cic2c$s\s2szs8 ,
(3.1)
written in the notation of Sect. II—4. In particular, all CPviolating signals must vanish if any of the KM angles vanish. We shall now study the path whereby this phase is transferred from the lagrangian to experimental observables. For kaons, we have seen that the relevant amplitudes are those for AS = 2 K°K° mixing and for AS = 1 K—> TTTT decays. Tree level amplitudes in kaon decay can never be sensitive to the full rephasing invariant, so that one must consider loops. Typical diagrams are displayed in Fig. IX4. Experiment can help in simplifying the theoretical analysis. Note that e' is sensitive to AS = 1 physics through the penguin diagram [GiW 79], while e is sensitive to AS = 2 mass matrix physics as well as to AS = 1 effects. However, since experiment tells us that e » e r, it follows that the AS = 1 contributions to e must be small. Likewise, the longdistance contributions of Fig. IX2 and the contribution of Fig. IXl(d) must both be small because each also involves the AS = 1 interaction. This
(a)
(b)
Fig. IX4. (a) Penguin and (b) electroweakpenguin contributions to CP violation in AS = 1 transitions.
IX3 Kaon CP violation in the Standard Model
243
leaves the box diagrams of Fig. I X  l ( a ) , ( b ) as t h e dominant component of e. Moreover, since t h e KM phase 6 is associated with t h e heavy quark couplings, only t h e heavy quark parts of t h e box diagrams are needed. Hence e is very clearly shortdistance dominated.
Analysis of e The evaluation of e follows directly from the treatment of the box diagram in Sect. IX1, and we find (c/. Eq. (1.15)), G2 —^%
Im M12 = x
2
^^r)2m2H(xt)
c \r)iml
\ p )
 rjsm2cG(xc,
x c)
%
s\
(3.2)
The above relations depend on three principal quantities, viz. mt, the KM angles, and the B parameter. At present, we have only the bound mt > 89GeV, so that not much can be inferred about the influence of the tquark mass. The information on KM angles allows the range A2X6r] = sls2s3ss < 0.7 x 10~4 ,
(3.3)
where no firm value can yet be given because there is no independent measure of 6. However, by taking t h e representative values mt = 150 GeV, mc — 1.5 GeV, A = 1, p = 1/2, we can express e in t h e useful form
[G(xc, xt)~
(3 4)
'
In this way, the magnitude of e is naturally obtained. Penguin contribution to ef The analysis of ef is rather more involved because one must confront A/ = 1/2 amplitudes in K —• TTTT. These are not yet under theoretical control, so that the predictions of ef must necessarily be uncertain. However, we can review the main ingredients of the analysis, which by now have become clear. As can be seen from Eq. (2.8), the most natural scale for the magnitude of ef/e is just u = Re A2/Rje A$ ~ 1/22. Yet, the experimental value of e'/e is found to be suppressed by at least another order of magnitude. This suppression is a consequence of the CPviolating phase 6 which appears
244
IX Kaon mixing and CP violation
only in the heavy quark sector. Its contribution to the direct K —> TTTT process occurs only as a result of virtual oquark and tquark loops in the penguin diagram of Fig. IX4(a) and the electroweakpenguin diagram of Fig. IX4(b), from which we can extract some general features. The gluonic penguin diagram involves the s —• d transition and is purely A / = 1/2. It therefore generates a phase only in A$. The electroweak penguin would seem at first sight to be suppressed when compared to the usual penguin because of the factor of a/as. However, as will be described below, the electroweak penguin is relevant because it can generate a phase for A2. The full prediction emerges only when we have calculated the phases of both AQ and A*i. Let us first give estimates of various contributions to e' using QCD perturbation theory, but without renormalization group improvements. This serves to identify the important physics and can be adjusted afterward to incorporate the effects from shortdistance renormalization. The penguin and electroweakpenguin diagrams yield respectively the CPodd operators given in Eq. (VIII3.30). If one wishes to explicitly check the rephasing invariance, it is necessary to recognize that CPodd interference with other diagrams in general involves quantities like Im (V^VtsV^Vtid) which are invariant under rephasing the quark fields. The other feature to extract from the above operatorsis the symmetry structure. The gluonic penguin involves a lefthanded (ds) current which is a member of an SU(3) octet. The remaining current is a flavor singlet, so that overall the interaction transforms as an octet and hence carries A / = 1 / 2 . More precisely, under chiral SU(3) it transforms as (8^,1^). By contrast the electromagnetic coupling in the electroweak penguin is itself an octet, having both lefthanded and righthanded portions,
^
^
(U, sR),
(3.5)
and is a mixture of A / = 1 and A / = 0, __ fQq=
2uu — dd — ss uu — dd = ^ ~ + §
uu + dd — 2ss g •
(36)
Thus the symmetry structure of the electromagnetic penguin is more complicated, involving SU(3) octet and 27plet representations, isospin 1/2 and 3/2 portions, and the chiral properties ( 8 L , 1 # ) , (27^,1^) and
(0 In order to complete the estimate of e', one needs to compute hadronic matrix elements. To illustrate this procedure, we shall use the vacuum saturation method, with the understanding that this approximation could be seriously flawed. One interesting new ingredient, beyond those described in Sect. VIII4, is the presence of matrix elements of scalar and
^x
IX3 Kaon CP violation in the Standard Model
245
pseudoscalar operators which arise as a consequence of the Fierz relations, rfn)i(l + 75)sj<&7/i(l  75)% = 2dj(l  75)Mfc(l + 75)5j .
(3.7)
For matrix elements of such operators, we have
= Bo ( +
^ )
V2SF (l
,
§)
(3.8)
= iV2B0F U + J2 where So is a constant, F is F^ in the chiral limit and A defines an energy scale in the momentum dependent terms. To obtain these expressions, we have assumed chiral SU(3) symmetry and, for simplicity, have omitted loops. The momentum dependence is required to obtain a nonzero result, and the constant Bo can be identified from our previous chiral lagrangian studies, where the lowest order relations (ra = (mu + m^)/2), m% = (n\m{uu + rfrf)7r) ,
m2K — (K\m{uu + dd) + msss\K)
, (3.9)
reveal that Bo = $• = ~^r
•
(3.10)
m +m
2m
However, Bo and the quark masses are not separately obtained from chiral symmetry  only the product rhBo or the mass ratio rh/ms is welldefined. In Chap. XII, we shall obtain a more model dependent estimate of ms from hyperon mass splittings, where a first order treatment yields m\ — mp — (Aras5sA) — (p\msss\p) + O (m/ms) = msZm .
(3.11)
The matrix element Zm is estimated to lie in the range 0.5 < Zm < 0.8 within a variety of quark models, and so Bo *
m2 TTlA
*— Z ™
m
.
(3.12)
Note also that the momentum dependence is related to FK/FK, since ms + m
ms + m
(3.13) where the equations of motion have been used in the final relation. From Eq. (3.8), we find _
246
IX Kaon mixing and CP violation
With these ingredients now defined, we can piece together the structure of e'/e. Using the physical values of e and Re Ao, we find Im
(TV+Trf°
Inserting as ~ 0.2 (as is appropriate for heavy quarks), mt ~ 150 GeV, mc — 1.5 GeV and fixing A with the FR/F^ analysis, we obtain finally  = 0.0002 A2r) U + 3.9 Z2m
.
(3.16)
Remember that this result still lacks the renormalization group enhancement. However, it does indicate that one expects a small but nonzero value of e'/e in the Standard Model. Additional contributions to e' In the limit of isospin invariance, the gluonic penguin contributes only to Ao. Since e' is proportional to the difference between the phase of Ao and the phase of A%, we need to be certain also about the phase of Ai A little thought will convince one that A2 is capable of picking up a sizeable phase from small effects because Re A2 is itself a small number. That is, if there is a small CPviolating process which contributes to A2, the phase IDQLA2/R&A2 can be enhanced because of the smallness of Re^2. We can estimate such an effect. For example, if isospin is broken by quark mass effects, then the decomposition into Ao and A2 will not uniquely represent the A/ = 1/2 and A/ = 3/2 transitions. Some A/ = 1/2 effects will appear in the parameter A2. An example of this is given below. The size of such effects would be roughly yiiso—brk
A
*
A
A
m
d
~~ ™"u
~ °—^T~ '
/ o i v\
(3>17)
where the second factor is a measure of isospin breaking. In this case the phase of A2 is of order Im A2 Re A2
I m Ao Tiid — TRU Re A2 ms
Im Ao Re Ao vrtd — TUU Re A$ Re A2 ms A 22 = 22 • — ~ 00 7 30 Re Ao 30 Re Ao '
(318)
where we have used a first order chiral estimate of the quark mass ratio. In Fig. IX4(b) we display a second example, the electroweak penguin diagram. This can contribute directly to A2 with an estimated size
IX3 Kaon CP violation in the Standard Model
247
relative to the penguin of Im A™»
~ — Im ^ e n g ,
lmA2 Re ^2
aImAlengReA0 a s Re Ao Re A2
Im Ao n o ImA 0 Re Ao ~ ' Re ^4o (3.19) We see that because of the extra factor of 22 in Eqs. (3.18), (3.19), even suppressed effects can modify ef markedly if they contribute to A^ The above results are orderofmagnitude estimates only, not real calculations. However, they do indicate that a careful study of isospin breaking and the electroweak penguin is required to fully understand e1. It has become conventional to express e'/e in the form e
l
Re An
1/137 0.2
nn
 '
(3.20)
22
r^rT" Im AQ Re A
Im
To determine the correction factor ft, let us first estimate the isospin breaking correction due to the u — d quark mass difference. One such effect is clearly calculated from chiral perturbation theory. The u—d mass difference induces 7r°r/ mixing,and this in turn influences the transitions K » 7r°7r°and K+  • TT+TT0 through the diagram of Fig. IX5. The ingredients of the calculation are (i)the weak transition,
^K0^m
= ^=Af
,
(3.21)
which is found using the weak lagrangian Eq. (VIII2.4b) and where the superscript (0) has been added to indicate that one uses AQ before 7r°77 mixing, (ii) the amplitude for TTO778 mixing,
^ ^ ^ K  mD 
K+
Fig. IX5 Effect of 7r°rj mixing
(322)
248
IX Kaon mixing and CP violation
which is found from the chiral lagrangian of Eq. (IV5.10). Using these to calculate Fig. IX5 yields )
~ft > 2
\
2
<
(3.23) We see that, subsequent to mixing, this is equivalent to the modified isospin decomposition A  4<°)+ lmdmu Is A
_
4 (0)
1
(Q) m
m d  mu
(3.24)
(Q)
v
^
The change in A$ is small, but the effect in A% is relatively larger, with
^
= _ l ! ^ ^ 4 2 ^ _Q M
A2 3\/2 m s  m A2 For our purposes it is most important that a phase has been induced into A2, with magnitude O? = 0.14 .
(3.26)
There will also be a correction for the rf. Although this cannot be calculated in chiral perturbation theory, it can be estimated from quark model relations plus rj — r( mixing that ^isobrk ^ % +
(3.27)
We are also interested in the contribution of the electroweak penguin operator to Im A
(3.28)
whereas the (8/,, 1R) and (27^, 1#) operators each require two derivatives. Due to the presence of the electric charge operator Q, the matrix element vanishes for the neutral mode K° —> 7r°7r° but not for the charged mode K° —> 7r+7r~. From this, we can determine the relative contributions of the electroweak penguin to A2, AQ in terms of the K° —> TT+TT" amplitude, 22^2 3
Im (^\Hewp\K) Im (TT+TTiWlKO) '
(
]
IX3 Kaon CP violation in the Standard Model
249
Again, the need to calculate a hadronic matrix element hinders one's ability to make a firm prediction. However, in the approximation of vacuum saturation we estimate _
22x/2 a
A2
2B%  {m\  m2.)
Interestingly, the electroweak penguin operator is chirally enhanced. At the effective lagrangian level this follows because its operator is O(E°) in the energy expansion, while the penguin operator is O(E2). This is reflected in the above matrix element by the factor h? /{m2K — rnfy. Note also that in this approximation fiewp is negative so that it enhances e'. Inserting the numerical factors as ~ 0.2, A = 1 GeV, Zm ~ 3/4, we estimate that fiewp — —0.30. This is a surprisingly large correction for an electroweak contribution, and indicates that it can be an important part of the prediction for e'\ The final ingredient is the calculation of shortdistance effects in QCD using the renormalization group. This proceeds along the lines described in Sect. VIII3. The usual penguin can generate a phase primarily in Oi, O5 and OQ to yield Im Ao = ^ K d K s (TT+TTI £
Tma Oi \K°) ,
(3.31)
i=l,5,6
and the electroweak penguin contributes to A
ImC
* Oi \K°) .
(3.32)
Present estimates [BuBH 90] yield the values Im c7 ~ 0.004as2sss6 = 0.004aA4A2rj , Im c$ ~ 0.16as2Ssss = 0.16a\4A2r) ,
(333)
for mt  150 GeV, A ~ 0.2 GeV, and fi  1.0 GeV. In addition, the renormalization group analysis including the electroweak penguin induces a phase in the coefficient of the A/ — 3/2 operator O4. This arises from the purely lefthanded portion of the electroweak penguin, with transformation property (27^,1^). The result is a phase in A2, of magnitude /C4
Re c4
This result is independent of the QCD scale /i in the shortdistance analysis. Note that by use of Eq. (2.8) and the experimental value of e, this
250
IX Kaon mixing and CP violation
piece by itself generates
°0 1 *'/" =  0 . 3 X 1 0  ^ ,
U
(3.35)
C1
which can be significant given the size of the leading contribution in Eq. (3.16). The isospin breaking effect discussed above is unchanged by QCD shortdistance corrections. We see then that there are several important contributions to e'. Because of the possibility of cancelations and because we cannot yet calculate the matrix elements with acceptable precision, the only safe statement is that e1 is expected to be nonzero, and to occur in the range probed by present experiments. To summarize, we have outlined the main ingredients for the predictions of e and e'. As this is being written, the experimental results have not settled down, and important theoretical uncertainties lie in the KM angles, the tquark mass, and the hadronic matrix elements. There is cause for optimism that more will be known about all of these in the near future.
IX—4 Electric dipole moments The condition of T invariance forbids the existence of an electric dipole moment for any elementary particle. This can be seen heuristically in the following manner. The only vectorial quantum numbers associated with a particle are its momentum p and spin s. Thus, for a particle at rest the electric dipole moment must be proportional to the spin, dc = d e l ^ s
ll
.
(4.1)
The dipole interacts with an electric field as = deE
=de^
.
(4.2)
Under time reversal E is unchanged (i.e. think of the E field of capacitors or static charges, which do not change under T), but all angular momenta reverse sign. Hence, i^edm is °dd under T.* For a relativistic spin1/2 particle, the electric dipole moment contribution has the matrix element (p'\j;ia\p)\edm
= ideu(p')afiUql/l5u(p)
,
(4.3)
with qu = (p1 — p)u, which is equivalent to an interaction density ^
.
(4.4)
* One might be concerned that, since T is an antiunitary operator, this conclusion could be changed by adding a factor of i to He^m. However, hermiticity requires that no additional factors of i appear.
IX~4 Electric dipole moments Since FOi = E{
251
and
this interaction attains the hamiltonian form in the nonrelativistic limit, #edm = [cPxHedm "> ~de((r) • E .
(4.6)
Note that an electric dipole moment also violates parity. The best existing experimental limits on electric dipole moments are
{
1.2 x 10~25ecm
(neutrons [RPP 90]) ,
(2.7 ± 8.3) x 10"27ecm (electrons [Ab et al. 90]) , (4.7) 23 3.7 x 10" ecm (protons [RPP 90]) . The extreme sensitivity of these limits can be seen from the observation that if neutrons were expanded to the size of the earth, the limit quoted would correspond to a dipole with a unit electric charge separated by only a micron! In practice, the dipole moment is a sensitive probe of CP violation, especially for theories beyond the Standard Model. The Standard Model can produce an electric dipole moment either through the 0 parameter of QCD or through Wexchange. The discussion of 6 and the strong CP problem is given in the next section. Some of the possible diagrams involving the electroweak interactions are shown in Fig. IX6. There emerges the very important property that no electric dipole moment is produced at first order in the weak interaction (i. e.
O(Gp)) in the Standard Model This is because the process does not change flavor. Treelevel processes which do not change flavor always involve the combination of KM matrix elements V^V^ for some flavors i, j , and the loop diagrams of Fig. IX6 (a) involve ]T\ ^f^ij Hence these are real and cannot involve the CPviolating phase. Theories which do have first order CP violation tend to produce neutron dipole moments close to the experimental bounds. At second order in the weak interactions, there is enough structure to produce a dipole moment. Interestingly, the single quark line diagrams, of which one example appears in Fig. IX6(b), sum to zero net effect [Sh 78].
(a)
(b)
(c)
Fig. IX6 Contributions to an electric dipole moment
252
IX Kaon mixing and CP violation
However, with an extra gluon loop a dipole moment occurs at O{G2Fas). Since as ~ 1 at hadronic scales, this provides no extra suppression. The multiquark interactions of Fig. IX6(c) can also lead to a neutron dipole moment [GaLOPRP 82]. For the electron, CP violation vanishes if the neutrinos are massless. One can obtain a reasonable estimate of the neutron dipole moment from dimensional analysis alone. There is always a factor of Im A^4^ (cf. Eq. (3.1)) associated with any CPviolating process. In addition, the GIM mechanism would cancel the contributions of degenerate mass quarks, so a contribution of the form (mh — m\)/Myy is also expected. Altogether then, we have de(neutron)  e ^ f  ^  I m A^/i 3 ~ 10"31e  cm ,
(4.8)
where // is a typical hadronic scale (we use /i ~ 0.3 GeV) which has been included to make the dimensions correct, and we have incorporated factors of n as anticipated from loop diagrams. This estimate is in the middle of the range found in model calculations [BaM 89], which span from 10~33 to 10~30e — cm. It is well out of the reach of experiments in the foreseeable future. IX5 The strong CP problem The possibility of a 0term in the QCD lagrangian raises potential problems (see Sect. Ill—5). For 0 ^ 0, QCD will in general violate parity, and even worse, time reversal invariance. The strength of T violation (and hence by the CPT theorem, CP violation) is known to be small, even by the standards of the weak interaction. This knowledge comes from both the observed KL —• 2?r decay and bounds on electric dipole moments. From these it becomes clear that QCD must be T invariant to a very high degree. However, there is nothing within the Standard Model which would force the ^parameter to be small; indeed, it is a free parameter lying in the range 0 < 6 < 2TT. The puzzle of why 6 ~ 0 in Nature is called the strong CP problem. One is tempted to resolve the issue with an easy remedy first. If QCD were the only ingredient in our theory, we could remove the strong CP problem by imposing an additional discrete symmetry on the QCD lagrangian, the discrete symmetry being CP itself. This wouldn't really explain anything but would at least reduce a continuous problem to a discrete choice. In reality, this will not work for the full Standard Model since, as we have seen, the electroweak sector inherently violates CP. It would thus be improper to impose CP invariance upon the full lagrangian. Moreover, even if one could set #bare = 0 in QCD, electroweak radiative corrections would generate a nonzero value. These turn out to
IX5 The strong CP problem
253
occur only at high orders of perturbation theory, and are expected to be divergent by power counting arguments, although they have not been explicitly calculated. This divergence is not a fundamental problem because one could simply absorb #bare plus the divergence into a definition of a renormalized parameter 9ren, which could be inferred from experiment. However we are then back to an arbitrary value of 0ren and to the problem of why 9Ten is small. The parameter 0 The situation is actually worse than this in the full Standard Model, as the quark mass matrix can itself shift the value of 6 by an unknown amount. Recall that CP violation in the Standard Model arises from the Yukawa couplings between the Higgs doublet and the fermions. When the Higgs field picks up a vacuum expectation value, these couplings produce mass matrices for the quarks which are neither diagonal nor CP invariant. The mass matrices are diagonalized by separate lefthanded and righthanded transformations, and CP violation is shifted to the weak mixing matrix. However, because different lefthanded and righthanded rotations are generally required, one encounters an axial U(l) rotation in this transformation to the quark mass eigenstates, and as discussed in Sect. Ill—5, this produces a shift in the value of 0. Let us determine the magnitude of this shift. Denoting by primes the original quark basis, one has the transformation to mass eigenstates given by (cf. Eqs. 114.5,4.6) m = S* m'S* ,
tl>L = Slt//L,
1>R = S\tl/R .
(5.1)
Here, we have combined the u and d mass matrices into a single mass matrix. Expressing SL,R as products of U(l) and SU(N) factors, i
^
,
(5.2)
with SL, SR in SU(N), one obtains an axial U(l) transformation angle of (fR — (fL From the discussion of Sect. Ill—5, this is seen to lead to a change in the 6 parameter, e^0
= O + 2Nf(
(5.3)
where Nf = 6 for the threegeneration Standard Model. However, noting that the final mass matrix m is purely real, we have arg(det m) = 0 = arg (det 5^ det m' det SR)
= arg (det sty + arg (det m') + arg (det SR) = 2N (
arg (det m') ,
(54)
254
IX Kaon mixing and CP violation
where we have used the SU(N) property, detSn = det SL = 1. The resultant ^parameter is then 0 = 0 + arg(detm')
,
(5.5)
with m' being the original nondiagonal mass matrix. The real strong CP problem is to understand why 6 is small. One possible solution to the strong CP problem occurs if one of the quark masses vanishes. In this case, the ability to shift 6 by an axial transformation would allow one to remove the effect of 0 by performing an axial phase transformation on the massless quark. Equivalently stated, any effect of 8 must vanish if any quark mass vanishes. Unfortunately, phenomenology does not favor this solution. The u quark is the lightest, but a value mu ^ 0 is favored. Connections with the neutron electric dipole moment The 6 term is not the source of the observed CP violation in K decays. This can be seen because it occurs in a AS = 0 operator, and while this may ultimately generate effects in AS = 1 processes, its influence is stronger in the AS = 0 sector. In particular, it generates an electric dipole moment de for the neutron. Since no such dipole moment has been detected, one can obtain a bound on the magnitude of 6. To determine the effect of 0, it is most convenient to use a chiral rotation to shift the 6 dependence back into the quark mass matrix. A small axial transformation produces the modified mass matrix md
\ ^+ir)^T^
j
= ^LM^R+^RM^L
, (5.6)
where rj is a small parameter proportional to 0 having units of mass, and T is a 3 x 3 hermitian matrix. Consistency requires T to be proportional to the unit matrix. If this were not the case, and instead we wrote T = 1 + \{Ti/2, the effective lagrangian would start out with a term linear in the meson fields, £eff ~ iV IV (rt/T  t/Tt) = 2^ (T37ro + TsV8 + ...) ,
(5.7)
rather than the usual quadratic dependence. The vacuum would then be unstable because it could lower its energy by producing nonzero values of, say, the TTO field. Thus to incorporate ^dependence without disturbing vacuum stability, one chooses T = 1. The act of rotating away any depen
IX5 The strong CP problem
255
dence on 9 produces a nonzero value of argdet M, and also determines 77, 9 = arg det M = arg \{mu + irj) (rrid + ir)) (ms + irj)] , mumdms r/ ~ 0 (for small 77) Tnvrid + Tnums + rridm such that the mass terms become TUUUU 
msSS
mdmdms
,_

_
,
(5.8) v '
(5.9)
murrid + mums The last term is the CPviolating operator of the QCD sector. Note that, as expected, 6 vanishes if any quark is massless. A nonzero neutron electric dipole moment de requires both the action of the above CPodd operator and that of the electromagnetic current,
(5
*10)
where q — p'—p and we have inserted a complete set of intermediate states {/} in the neutrontoneutron matrix element. For intermediate baryon states, the matrix elements of Tp^tp are dimensionless numbers of order unity and magnetic moment effects are of order the nucleon magneton, \in. Thus we find for de, de * 9
^ ^  ^ , (5.11) murrid + mums + mdms AM where AM is some typical energy denominator. Using AM = 300 MeV, we obtain de~0x
10"15 ecm .
(5.12)
Far more sophisticated methods have been used to calculate this, with results that unfortunately have a spread of a factor of 50 [Pe 89]. Our simple estimate is near the average. In explicit calculations, some subtlety is required because one must be sure that the evaluation correctily represents the U(1)A behavior of the theory [AbGLOPR 91]. However, the precise value is not too important; the significant fact is that bounds on d e ^10~ 2 5 ecm requires 9 to be tiny, 0<1O~ 10. The strong CP problem does not have a good resolution within the Standard Model. It would appear that the abnormally small value of 0, and of the cosmological constant as well, are indications that more physics is required beyond that contained in the Standard Model.
256
IX Kaon mixing and CP violation Problems
1) CP violation and K> Sn The Standard Model makes definite predictions for the existence of CP violation within the K —• 3TT system. One such effect is the existence of the decay Ks —• 7r +7r~7r° which one would expect to be forbidden in the limit of CP conservation. a) Actually this expectation is not quite correct. Calculate the amplitude for Ks —• 7T+7r~7r° in the limit of CP conservation and show, using the results of Prob. VIII2, that
Thus there exists a CPeven piece of the amplitude, but it is small, arising from the A/ =  component of the hamiltonian, and vanishes at the center of the Dalitz plot. b) Now consider the CPviolating component of the amplitude. Using Prob. VIII2 together with the fact that in the Standard Model, CP violation is purely A/ = \, show that + 7T7T0
—
V2 c) Defining =
demonstrate that 77+0 = ^?oo = € — 2ef as first given in [LiW 80]. In this approximation, any deviation of r/+_o — e from zero is small due to the same mechanism which suppresses r/+_ — e (This result can be modified by electroweakpenguin contributions, which have a different chiral structure, and higher order energy dependence, but the result remains discouragingly small). d) Why is experimental measurement of 7/+_o much more difficult than that of r/ + _? 2) Strangeness gauge invariance a) Physics must be invariant under a global strangeness transformation \M) —• exp(iA5)M), where A is arbitrary. Explain why this is the case. b) Demonstrate that such a transformation has the effect ImMi2 ImMi2 ' ReMi 2 ~* ReM i 2 ~
Problems
257
as claimed in Eq. (2.9b), and that, while unphysical quantities such as e, £o are affected by such a change, physical parameters such as e, ef are.not. 3) Neutral kaon mass matrices and CPT invariance Some of the ideas discussed in this chapter can be addressed in terms of simple models of the neutral kaon mass matrix M which appears in Eq. (1.2). a) Consider the following CPconserving parameterization as defined in the (If0, If0) basis: A m0 where A is realvalued. Determine the basis states (if_,if+) in which Mo —> M± becomes diagonal and obtain numerical values for 7770,
A.
b) Working in the (if_,if + ) basis, extend the model of (a) to allow for CP violation by introducing a realvalued parameter <S, rri0
, , , _ / ra_ —id
0 \ 7774 /
y 2(5
7711
and assume there is no direct CP violation. This mass matrix corresponds to the superweak (SW) model. By expressing M± in the (if0, K°) basis, use the analysis of Sects. IX1,2 to predict (fe ' = phase e and determine 6 from the measured value of e c) Finally, extend the model in (b) to M±"=(m; * V X * m+ where Re x is a Tconserving, CPviolating and CPTviolating parameter. Show that the states which diagonalize M±" are *r5} ^
\K+)

K_)
,
\KL) ~ \K.) + £\K+) , where V = {TRL — rns)/2 + iYs/^> Then evaluate r?+_ and 7/00, allowing for the presence of direct CP violation (i.e. e' ^ 0), and derive the following relation between phases,
The result \m^o —m Ko\/mKo < 5 x 10~18 which follows from this relation is the best existing limit on CPT invariance.
X The 1/N C expansion
The Nc * expansion is an attempt to create a perturbative framework for QCD where none exists otherwise. One extrapolates from the physical value for the number of colors, Nc = 3, to the limit iVc —> oo while scaling the QCD coupling constant so that g%Nc is kept fixed ['t H 74]. The amplitudes in the theory are then analyzed in powers of N'1. The hope is that the Nc —* oo world bears sufficient resemblance to the real world to yield significant dynamical insights. There is no magical process which makes the Nc —• oo theory analytically trivial; nonlinearities of the nonabelian gauge interactions are present, and the theory is still not solvable. However any consistent approximation scheme for QCD is welcome, and the large Nc expansion is especially useful for organizing one's thoughts in the analysis of hadronic processes. X—1 The nature of the large N c limit In passing from SU(3) to SU(NC), the quark and gluon representations, originally 3 and 8, become N c and N^ — 1 respectively. The analysis of Feynman graphs at large Nc is simplified by modifying the notation used to describe gluons. As usual, quarks carry a color label j , with ,7 = 1,2,..., Nc. Gluons can be described by two such labels, i.e. A;
> A*
{A*, = 0) ,
(1.1)
where a = 1,..., N% — 1 and j , k = 1,..., Nc. In doing so, no approximation is being made. The new notation in simply an embodiment of the group product N c x N c —> (Nj? — 1) ^ 1. The quarkgluon coupling is then written
fi , 258
(1.2)
Xl
The nature of the large Nc limit
259
Fig. Xl. Double line notation: (a) quark and (b) gluon propagators, (c) quarkgluon, (d) threegluon and (e) fourgluon vertices.
and the gluon propagator is
Jd'x e** (0 r (A*j(x)Aj}(0)) I 0) = (1.3) The term proportional to Nc 1 must be present to ensure that the color singlet combination vanishes, A3 — 0. However, as long as we avoid the color singlet channel, this term will be suppressed in the large Nc limit and may be dropped when working to leading order. Using this new notation, the Feynman diagrams for propagators and vertices are displayed in Fig. Xl. A solid line is drawn for each color index, and each gluon is treated as if it were a quarkantiquark pair (as far as color is concerned). In this double line notation, certain rules which are obeyed by amplitudes to leading order in 1/NC emerge in an obvious manner. Although general topological arguments exist, we shall review these rules by examining the behavior of specific graphs. The power of Feynman diagrams to build intuition is rather compelling in this case. We consider first the familiar quark and gluon propagators. The quark propagator, unadorned by higher order corrections, is O(l) in the Nc —• oo limit. Fig. X2 depicts two radiative corrections. Fig. X2(a), the onegluon loop, is O(l) in powers of Nc because the suppression from the squared coupling g\ is compensated for by the single closed loop, which corresponds to a sum over a free color index and thus contributes a factor of Nc. The graph then is of order g\Nc which is taken to be constant. The graph Fig. X2(b) with overlapping gluon loops is O(N~2) because, with no free color loops, it is of order g% = (glNc) N~2 ~ N~2. The terms planar and nonplanar are used respectively to describe Figs. X2(a),2(b), because the latter cannot be drawn in the plane without at least some internal lines crossing each other.
Fig. X2. Radiative corrections to the quark propagator: (a) planar, (b) nonplanar.
260
X The 1/NC expansion
Four distinct contributions to the gluon propagator are exhibited in Fig. X3. Figs. X3(a),3(b) depict in double line notation the quarkant iquark and twogluon loop contributions. It should be obvious from the above discussion that these are respectively O(N~1) and 0(1). A new diagram, involving the threegluon coupling, appears in Fig. X3(c). With three colorloops and six vertices, it is of order (glNc) = O(l). Figure X3(d) is a nonplanar process with six vertices and one color sum, and is thus O{N~2). The discussion of the gluon propagator indicates why we constrain . to be fixed when taking the large Nc limit. The beta function of QCD is determined to leading order by Figs. X3(a),3(b). If g\ were held fixed, the beta function would become infinite in the large Nc limit, leading to the immediate onset of asymptotic freedom. The choice g\Nc ~ constant leads to a running coupling constant and is compatible with the behavior for the realistic case of Nc = 3. To summarize, there are several rules which can be abstracted from examples such as these: (i) the leading order contributions are planar diagrams containing the minimum number of quark loops; (ii) each internal quark loop is suppressed by a factor of A^T1; and (iii) nonplanar diagrams are suppressed by factors of N~2. The suppressions in rules (ii), (iii) are combinatorial in origin. Quark loops and nonplanarities each limit the number of colorbearing intermediate states, and consequently cost factors of N~l. X—2 Spectroscopy in the large N
c
limit
In order for the large Nc limit to be relevant to the real world, it must be assumed that confinement of color singlet states continues to hold. In this case, we expect the particle spectrum to continue to be divided into mesons and baryons. Let us treat the mesons first, as the baryons are more problematic. One can form color singlet meson contributions from QQ pairs. To form a color singlet, one must sum over the quark colors. In order to produce a properly normalized QQ state one must therefore include a —
I/O
normalization factor of Nc ' into each QQ meson wavefunction, such
(a)
(b)
(c)
Fig. X3 Various radiative corrections to the gluon propagator.
X2 Spectroscopy in the large Nc limit
(a)
261
(b)
Fig. X4 Mesons in the double line notation. that Q(*)Q(0)) color
singlet
i^6( Q )t^)to) y/Nc
5
(2.1)
where a, /3 are flavor labels, i = 1,..., Nc is the color label, and tf (rf) are the quark (antiquark) creation operators. Meson propagators, as represented in Fig. X4(a), are then 0(1) in Nc since the factors of (JV«T ' )2 from the normalization of the wavefunction are compensated by a factor of Nc from the quark loop. This leads to the prediction that meson masses are of (9(1) in the large Nc limit, i.e. they remain close to their physical values. Multiquark intermediate states, as in Fig. X4(b), are suppressed by 1/NC, indicating a suppression of mixing between QQ and Q2Q2 sectors. That is, large Nc plus confinement implies the existence of QQ mesons which contain an arbitrary amount of glue in their wavefunction, but which do not mix with Q2Q2 states. What about the decay widths of QQ mesons? The decay amplitude is pictured in Fig. X5 (other possibilities involve the suppressed quark loops). This diagram contains three meson wavefunctions and one quark loop and hence is of order (iVc~ ' )3/Vc = Nc in amplitude or N~l in rate. The large Nc limit thus involves narrow resonances, i.e. T/M —> 0, where F is the meson decay width and M is the meson mass. This is reasonably similar to the real world, where most of the observed resonances have T/M ~ 0.10.2 [RPP 90]. Color singlet gluonic states, called glueballs, may also exist. The normalization of a glueball state can be fixed by means of the following argument. Suppose, as will be defined in a gauge invariant manner in Sect. XIII4, that a neutral meson can be created from two gluons. Then in normalizing this configuration, one must sum over the Nc2 gluon color labels. As a consequence, a normalization factor A^"1 is associated with
Fig. X5 Strong interaction decay of a QQ meson.
262
X The 1/NC expansion
(a)
(b)
Fig. X6 Mesonmeson scattering. each glueball state. Glueball propagators also emerge as being O(l). There is no physical distinction between twogluon states, threegluon states, etc., because all are mixed with each other by the strong interaction. As a result, there need not be any simple association between a specific physical state and gluon number, and thus the concept of a 'constituent gluon' need not be inferred. In glueball decays, however, one must distinguish between glueballs decaying to other glueballs, and those decaying to QQ mesons. Where kinematically allowed, the decay of glueballs to glueballs is 0(1), while that to QQ states is O (1/NC). The lowest lying glueball(s) will then be narrow, while those above the threshold for decay into two glueballs will be of standard, nonsuppressed width. Mesonmeson scattering amplitudes are also restricted by large Nc counting rules. Consider the diagrams of Fig. X6. That of Fig. X6(a) is of order (N~1/2)4NC ~ TV"1, whereas Fig. X6(b) is O(N~2) because of the extra quark loop. The scattering amplitudes thus vanish in the large Nc limit, and the leading contributions are connected, planar diagrams. The large Nc limit also predicts that neutral mesons (i.e. Q^Q^ composites with a = j3) do not mix with each other. The possible mixing diagram is given in Fig. X7, and includes any number of gluons. However, because of the extra quark loop, it is of order N~l, and thus vanishes in the infinite color limit. This means that uu states do not mix with dd or 55, nor do the latter two mix. The large Nc spectrum thus displays a nonet structure with the uu and dd states degenerate (to the extent that electromagnetism and the mumd mass difference are neglected) and the ss states somewhat heavier. This pattern is reflected in Nature, except that the uu and dd configurations now appear as states of definite isospin, uu ± dd. For example, let us consider the JPC — 1 L 2 + + mesons. For the former, p(770) and a;(783) are interpreted as uu, dd isospin 1 = 1 and
Fig. X7 Mesonmeson mixing.
X3 Goldstone bosons and the axial anomaly
263
7 = 0 combinations, while y>(1020) is the ss member of the nonet. Including the if*(892) doublet as the us, ds combinations, a simple additivity in the quark mass would imply m
v?(i020) 
m
p(770) = 2 (m#*(892)  m p ( 7 7 0 )) ,
(2.2)
+
which works well. A similar treatment of the 2 + mesons, identifying a2(1320) and /2(1270) as the corresponding uu, dd states and /2(1525) as an ss composite, predicts 
m
a2a(1320) 2(1320) =
22
( mm/q(1430) ~
mm
a2(1320)
which is also approximately satisfied. The fact that p(770), a;(783), /2(1270) and a2(1320) decay primarily to pions, and
(Pj(q)\dllAU0)\0) = Fjmfak .
(3.2)
For the octet of currents, the divergence vanishes for zero quark mass, and as usual leads to the identification of TT, K, rjs as Goldstone bosons. However, for the singlet current the anomaly is present. Even in the
X The 1/NC expansion
264
limit of vanishing quark mass, the current divergence has nonzero matrix elements, in particular, (3.3)
If one repeats the calculation of the anomalous triangle diagram as in Sect. Ill—3 but now allows N c to be arbitrary, one sees that it is proportional to Tr (AaAfe) = 26ab and is therefore independent of Nc. However, by using large Nc counting rules, the matrix element in Eq. (3.3) is seen to be of order g\Nc ~ Nc .* This implies that the gluonic contribution to the axial anomaly vanishes in the large Nc limit. When we take into account the behavior of jfy, we conclude that m2, ~ 1/NC —> 0. The rf is thus massless in the large Nc limit, and we end up with a nonet of Goldstone bosons. To illustrate what happens when the number of colors is treated perturbatively, let us consider the 1/NC corrections to the meson spectrum together with the effects of quark masses. If we first add quark masses, we have, in analogy with the results of Sect. VII1, the mass matrix \m(uu + dd) + msss\
(3.4)
where we have taken mu = rrid =TO.This leads to a squaredmass matrix (2m ins
0
0
0
0
0
+m
0
0
0
\
(3.5)
§(2m8 \{ rns
0
in the basis (TT, if, r)g,r)o). If this were diagonalized, one would find an isoscalar state degenerate with the pion. This is a manifestation of the U(l) problem which arises when there is no anomaly. However, at the next order in large Nc, the matrix picks up an extra contribution in the SU(3) singlet channel due to the anomaly, yielding /2m 2
m = B0
0
0 ms
i
I
m
0
0
0
0
0
0
 (2ms + m)
\ 0
0
mmms)
{m
\(
\
(3.6)
2rh)
This result depends on the assumption that topologically nontrivial aspects of vacuum structure are smooth in the 7VC —> oo limit.
X4
The OZI rule
265
where e = O(N®). This mass matrix yields an interesting prediction. The quantities Born and Boms are fixed as usual by using the n and K masses. Also the trace of the full matrix must yield rn^+rn\+mi+m?,, which fixes e = 2.16GeV2. The remaining diagonalization then predicts m^ = 0.98 GeV, 77?^ = 0.50 GeV with a mixing angle of 18°. This is a remarkably accurate representation of the situation in the real world. Although e/N c is suppressed in a technical sense, note how sizeable it actually is. One is hard pressed to imagine any sense in which the physical rj mass can be taken as a small parameter. X  4 The OZI rule In the 1960's, an empirical property, called the OkuboZweigIizuka (OZI) rule [Ok 63, Zw 65, Ii 66], was developed for mesonic coupling constants. Its usual statement is that flavor disconnected processes are suppressed compared to those in which quark lines are connected. In the language which we are using here, flavor disconnected processes are those with an extra quark loop. Unfortunately, the phenomenological and theoretical status of this socalled rule is ambiguous. We briefly describe it here because it is part of the common lore of particle physics. The empirical motivation for the OZI rule is best formulated in the decays of mesons. Let us accept that <^(1020) and /2(1525) are primarily states with content ss whereas a;(783) and /2(1270) have content (uu + dd) / \ / 2 . Mixing between the ss and nonstrange components can take place with a small mixing angle, such that
Amp (ss) Amp (\uu + dd)/\/2)
EE
tan 0 ,
(4.1)
with 9 = 9y for the vector mesons and 0 = 6T for the tensor mesons. In both cases, 9 is small. Experimentally, the
= 0.012 ± 0.002 ~ 0.003 x p.s. ,
E^r+p.^
Y ^ = °004 ± °001
(42)
 0002 x p.s. .
This suggests the hypothesis 'ss states do not decay into final states not containing strange quarks'. Diagrammatically this leads to a pictorial representation of the OZI rule, viz., the dominance of Fig. X8(a) over Fig. X8(b). Some scattering processes also show such a suppression. For
266
X The 1/NC expansion
example, we have ***<"
~ 0.03 ,
(4.3)
which can be interpreted as an OZI suppression. A stronger version of the OZI rule would have the ip/u and /2//2 ratios equal to a universal factor of tan 2 0 (cf. Eq. (4.1)) once kinematic phase space factors are extracted. The narrow widths of the J/ip and T states are also cited as evidence for the OZI rule, since these hadronic decays involve the annihilation of the cc or bb constituents. This can be correct almost as a matter of definition, but it is not very enlightening. Indeed, the small widths of heavyquark states can be understood within the framework of perturbative QCD without invoking any extra dynamical assumptions. However, perturbative QCD certainly cannot explain the OZI rule in light mesons. It must have a different explanation for these states. There actually exist several empirical indications counter to the OZI rule [Li 84, E1GK 89, RPP 90]. Among the more dramatic examples of OZ/forbidden reactions, expressed as ratios, are =1.2 ± 0 . 5 , 0 2 3
+0.14
a
i™*+*
=
2.0 ± 0 . 7 ,
0TP+PVK+*
The universalmixing model is incorrect more often than not, with counterexamples being = o. 1O ± 0.02 ,
(4.5) _ +0.011 — u.uzy instead of the values 0.03, 0.03 and 0.006 expected from the previous ratios. The empirical 777/ mixing angle 6^^ ~ —20° also violates the OZI rule, which would require a mixing angle of —35°. l v
(a)
^
(b)
Fig. X8 OZI (a) allowed, (b) forbidden amplitudes.
X5 Chiral lagrangians
267
There is also an intrinsic logical flaw with the simplest formulation of the OZI rule. This is because OZJforbidden processes can take place as the product of two OZ/allowed processes. For example, each of the following transitions is OZ/allowed: (4 6)
fi  * VV J
777?  • TTTT .
Hence the OZ/forbidden reaction f2 —• TTTT can take place by the chains f2 —• KK
 > 7T7T ,
f2  > 7?77  > 7T7T .
(4.7)
These twostep processes are in fact required by unitarity to the extent that the individual scattering amplitudes are nonzero. The large Nc limit provides the only known dynamical explanation of the OZI rule at low energies. Although the gluonic coupling constant is not small at these scales and suppressed diagrams have ample energy to proceed, they are predicted to be of order 1/N% in rate because of the extra quark loop. Yet large Nc arguments need not suggest a universal suppression factor of tan 2 0, because there is no need for the 1/NC corrections to be universal. Note that the large Nc framework also forbids the mixing of 77 and 7/ and more generally, the scattering of mesons. Thus, the OZI rule in lightmeson systems remains somewhat heuristic. It has a partial justification in large Nc counting rules, but it also has known violations. It is not possible to predict with certainty whether it will work in any given new application. X—5 Chiral lagrangians The large Nc limit places restrictions on the structure of chiral lagrangians [GaL 85a]. To describe these, we must first allow for an enlarged number Nf > 3 of quark flavors. The threeflavor O(E4) lagrangian is expanded as 10
A = $>A
,
(5.1)
where the {Oi} can be read off from Eq. (VI2.7). Recall that in constructing £4, we removed the O(E4) operator Oo = Tr (D^UD^D^UD^U^
,
(5.2)
because for Nf — 3 it is expressible (cf. Eq. (VI2.3)) as a linear combination of Oi52,3 However, if the number of flavors exceeds three, one must append Oo to the lagrangian of Eq. (5.1), 10
3
10
aiOi .
(5.3)
268
X The 1/NC expansion
In view of the linear dependence of Oo on Oi,2,3, note that we have needed to modify the coefficients 0:1,2,3 —• /?i,2,3 Upon returning to three flavors, we regain the original coefficients, ai = y + A ,
<*2 = /% + /%,
a 3 =  2 # ) + /%.
(5.4)
We can now study the large Nc behavior of the extended O(E4) chiral lagrangian. The distinguishing feature is the number of traces in a given O(E4) operator. Each such trace is taken over flavor indices and amounts to a sum over the quark flavors, which in turn can arise only in a quark loop. In particular, those operators with two flavor traces (Oi,2,4,6,7) will require at least two quark loops, while those with one flavor trace need only one quark loop. However, our study of the large Nc limit has taught us that every quark loop leads to a 1/NC suppression. Thus the O(E*) chiral contributions having two traces will be suppressed relative to those with one trace by a power of 1/NC, and provided Ps ^ 0 we can write* Pi
02
OL±
C*6
(xti\
n
^
(
a)
Alternatively, this 7Vccounting rule implies (provided Po/Ps 7^ 1/2) for the {cti} coefficients of flavor 51/(3), () . (5.5b) as as as The overall power of Nc for the remaining terms can be found by noting that the TTTT scattering amplitude should be of order A^T1, implying ai,2,3 = O(NC). The only exception to the above counting behavior is the operator with coefficient 0:7. This exception occurs because the operator can be generated by an 7/pole, and the 7/ masssquared is O(l/Nc). In particular, the coefficient of this term is absolutely predicted in the large iVc limit. This follows if we include the large Nc result for ?TIQ8 from Eq. (3.6) in the 777/ mixing analysis of Prob. VII5. The result of the mixing is
ifif
(5 6)

It is the factor of ra~,2 which overcomes the counting rules. Although the double trace suggests that this operator is suppressed in the large iVc limit, we have m~? oc Nc. Thus, at least formally, an extra enhancement would be predicted. The operator Oj presents a special case and is discussed below.
X6
Weak nonleptonic decays
269
The large Nc limit then predicts the following ordering of the chiral coefficients in £4: a7 = O(N2C) ,
ai, a 2 , a 3 , a 5 , a 8 , a 9 , ai 0 = O{NC) ,
(5.7)
 a 2 , ^4,^6 = C?(1) • We have built these properties into the coefficients appearing in Sect. VI2. The only existing empirical test involves the occurrence of 2OL\ — OLj Nc—>OO yj I
,
(6.1)
where each current in the above is a color singlet. Some added insight can be obtained by considering directly the K —• 2?r nonpenguin diagrams, such as those in Fig. X9. There is a modification in the large Nc rules when two weak currents interacting via VK±exchange are involved. Because the currents and the W± are color singlets whose couplings do not involve #3, we see that an extra quark loop connected
n d
(a) Fig. X9
K —• 7T7T amplitudes to leading N~l order.
270
X The 1/NC expansion
by W± gains an extra factor of Nc from the color sum. Thus Fig. X9(a) is a factor of Nc larger than Fig. X9(b). Next consider the case where gluons are exchanged across the interaction vertex as in Fig. X10(a). The singlegluon exchange amplitude vanishes since a gluon belongs to a color octet whereas both weak currents and mesons are color singlets. Twogluon amplitudes are expressed in double line notation in Fig. X10(b). Application of the counting rules to these diagrams indicates they are of O(N~1) compared to those in Fig. X9(a). Thus in the Nc —• oo limit we are left with only the vacuum saturation diagram of Fig. X9(a). This diagrammatic exercise explains the nonrenormalization of HwTo leading order, the only gluonic diagrams are those which act on a single current and do not act across a vertex. In QED, the single current vertex is not changed in normalization as a consequence of the Ward identity. Here, one gets more in that the absolute normalization of the kaon amplitudes is also calculable. One finds 76)« 0)
 ml) ~ 9.8 x l^mK
+ 75)*
,
=0, 75)« 0)
^
K°)
(6.2) (6.3) K~)
(6.4)
These results bear no relation to reality either in absolute magnitude or in relative rates (e.g., see Eq. (VIII4.2)). There is certainly no A/ = 1/2 rule here, because the ratio A2/A0 = l/\/2 implied by the above is far larger than the measured value, A2/A0 ~ 1/22. Since the large Nc limit fails, the origin of the A/ = 1/2 rule must lie beyond leading order in Nc. Unfortunately there is no unique prediction for the nexttoleading corrections, although some nonleading aspects which enhance A/ = 1/2 have been identified. For example, the QCD short distance corrections provide a partial enhancement, and the purely
(a) Fig. X10
(b) Nonleading K —• TTTT contributions.
Problems
271
A / = 1/2 penguin graphs enter at the first nonleading order in Nc. There has been an interesting attempt [BaBG 87] to identify a set of nonleading corrections in 1/NC and to combine them to obtain the A / = 1/2 rule for kaons. This method involves the use of the large Nc chiral lagrangian at low energy and QCD perturbation theory at high energy. Nonleading corrections enter when one considers oneloop corrections, employing meson loops or quark/gluon loops in the respective domains. An interesting innovation is a serious attempt to match the scale dependence of the QCD coefficients in the weak nonleptonic hamiltonian with the scale dependence of the low energy matrix elements. If matrix elements and coefficient functions are computed with reference to an energy scale //, the key requirement is that the physical nonleptonic amplitude,
Aonlept. = I ^ K d K s J2 Ci(l*)(™\Oi\K) = J2 *(»)Mi(n)
, (6.5)
be /i independent. In [BaBG 87], the scale dependence in the matrix elements Mi(fi) arises when one calculates chiral loops with a high energy cutoff A. The identification \x — A allows a partial cancelation of the scale dependence. Although the results are promising, a full resolution will require a more complete understanding of the relevant hadronic physics at low energy. Problems 1) The large Nc weak hamiltonian Retrace the calculation of the QCD renormalization of the weak nonleptonic hamiltonian described in Sect. VIII3, but now in the limit Nc —• oo with g\Nc fixed. Show that the penguin operators do not enter and that all short distance effects are of order A^T1, with the operator product coefficients c\ = 1,C2 = 1/5, C3 = 2/15, C4 = 2/3, C5 = ce = O. 2) The strong CP problem in the large N c limit In the large Nc limit, the 770 can be united with the Goldstone octet in the effective lagrangian. Generalizing the chiral matrix to nine fields we write C = Co + CNi, where Co = ^ Tr (dpUdfU^ + ^Bo Tr (m(U +
U = exp (iX • tp/F) exp \i\
—
272
X The 1/NC expansion
a) Confirm that this reproduces the mixing matrix of Eq. (3.6). b) Another way to obtain this result is to employ an auxiliary pseudoscalar field q(x) (with no kinetic energy term) to rewrite £Ni as
Identify the SU(3) singlet axial current and calculate its divergence to show that q(x) plays the same role as FF, i.e. q(x) ~ aFF/87r. Integrate out q(x) to show that this is equivalent to the form of part (a). c) Several authors [RoST 80, DiV 80] suggest adding the 0term through C = Co + CNi  6q{x) . From this starting point, integrate out q(x) and show that a chiral rotation can transfer 9 to argdet m. However, in the sense described in Sect. IX5, this theory is unstable about U = 1. The stable vacuum corresponds to U^ = 8^exp(i(3j). For small 0, solve for fij in terms of 6. d) Using U — elP/2Ue%P/2, define the fields about the correct vacuum to find the CPviolating terms of the form
identifying a and b and showing they vanish if any quark mass vanishes. Calculate the CPviolating amplitude for rj —> TT+TT".
XI Phenomenological models
QCD has turned out to be a theory of such subtlety and difficulty that a concerted effort over an extended period has not yielded a practical procedure for obtaining analytic solutions. At the same time, vast amounts of hadronic data which require theoretical analysis and interpretation have been collected. This has spurred the development of accessible phenomenological methods. We devote this chapter to a discussion of three dynamical models (potential, bag, and Skyrme) along with a methodology based on sum rules. Although the dynamical models are constructed to mimic aspects of QCD, none of them is QCD. That is, none contains a rigorous program of successive approximations which, for arbitrary quark mass, can be carried out to arbitrary accuracy. Therefore, our treatment will emphasize issues of basic structure rather than details of numerical fits. By using all of these methods, one hopes to gain physical insight into the nature of hadron dynamics. Despite its inherent limitations the program of model building, fortified by the use of sum rules, has been generally successful, and there is now a reasonable understanding of many aspects of hadron spectroscopy. XI—1 Quantum numbers
of QQ and Q3 states
Among the states conjectured to lie in the spectrum of the QCD hamiltonian are mesons, baryons, glueballs, hybrids, dibaryons, etc. However, since practically all currently known hadrons can be classified as either QQ states (mesons) or Q3 states (baryons), it makes sense to focus on just these systems. We shall begin by determining the quark model construction of the light hadron ground states. Much of the material will be valid for heavyquark systems as well. 273
274
XI Phenomenological models Hadronic flavorspin state vectors
In many respects, the language of quantum field theory provides a simple andflexibleformat for implementing the quark model. Let us assume that for any given dynamical model, it is possible to solve the field equations of motion and obtain a complete set of spatial wavefunctions, {*/ja(x)} for quarks and {^{x)} for antiquarks, where the labels a and a refer to a complete set of observables. A quark field operator can then be expanded in terms of these wavefunctions, (a)] ,
(1.1)
where a;Q, uJa are the energy eigenvalues, b(a) destroys a quark and eft (a) creates the corresponding antiquark. The quark creation and annihilation operators obey
{b(a),b(a')} = 0 ,
{&(<*),
{d(a),d(ct)} = 0 ,
(1.2)
=0 ,
which are the usual anticommutation relations for fermions. In all practical quark models, an assumption is made which greatly simplifies subsequent steps in the analysis, that the spatial, spin, and color degrees of freedom factorize, at least in lowest order approximation. This is true provided the zeroth order hamiltonian is spinindependent and colorindependent. Spin dependent interactions are then taken into account as perturbations. This assumption allows us to write the sets {a} and {a} in terms of the spatial (n), spin (s, ras), flavor (g), and color (fc) degrees of freedom respectively, i.e. a = (n, 5,m s ,g, k). If we are concerned with just the ground state, we can suppress the quantum number n, and for simplicity replace the symbolsfe,d)', etc. for annihilation and creation operators with the flavor symbol q (q = u,d,s for the light hadrons),
Hadrons are constructed in the Fock space defined by the creation operators for quarks and antiquarks. Light hadrons are labeled by the spin (S2,53), isospin (T2,T3), and hypercharge (Y) operators as well as by the baryon number (B). Other observables like the electric charge Qe\
XI1 Quantum numbers of QQ and Q3 states
275
•T,
3
8
3*
10
Fig. XI1 Some SU(3) flavor representations.
and strangeness S are related to these, Qel = T3 + Y/2 ,
S=
YB
(1.4)
Since quarks have spin onehalf, the baryon (Q3) and meson (QQ) configurations can carry the spin quantum numbers S = 1/2,3/2 and S = 0,1 respectively. If we neglect the mass difference between strange and nonstrange quarks, then flavor SU(3) is a symmetry of the theory, and both quarks and hadrons occupy SU(3) multiplets. The quarks are assigned to the triplet representation 3 and the antiquarks to 3*. The QQ and Q 3 constructions then involve the group products 3 x 3* = 8 0 1 , (3 x 3) x 3 = (6 0 3*) x 3 = 10 © 8 0 8 0 1
(1.5)
so that baryons appear as decuplets, octets, and singlets whereas mesons appear as octets and singlets. The 577(3) flavor representations 3, 3*, 8, 10 are depicted in Y vs. T3 plots in Fig. XI1. The circle around the origin for the eightdimensional representation denotes the presence of two states with identical Y, /3 values. Finally, quarks and antiquarks transform as triplets and antitriplets of the color SU(3) gauge group, and all baryons and mesons are color singlets. Two simple states to construct are the p\ meson and the ££% baryon, lA3+/2> = «' where the superscript and subscript on the hadrons denote electric charge and spincomponent, and a summation over color indices for the creation operators is implied. The normalization constants are fixed by requiring that the hadrons {Hn} form an orthonormal set, (Hm\Hn) = 8mn. The other ground state hadrons can be reached from those in Eq. (1.6) by means of ladder operations in the spin and flavor variables. In this manner, one can construct the flavorspincolor state vectors of the 0~ octet
276
XI Phenomenological models Table XI1. State vectors of the pseudoscalar octet and singlet mesons
^6 [ «  « ]
«
7
10)
+ 44 + 44  44i
_
7b W  44 + 44  44
44 4  44
and singlet mesons and the \ octet baryons displayed in Tables XI1 and XI2. A convenient notation for fields which transform as SU(3) octets involves the use of a cartesian basis rather than the 'spherical' basis of Tables XI1,2. In fact, we have already encountered this description in Sect. VII1 during our discussion of SU(3) Goldstone bosons where the quantity U = exp(iip • A) played a central role. The eight cartesian fields
Table XI2. State vectors of baryon spin1/2 octet
1°)
XI1 Quantum numbers of QQ and Q3 states
277
{(fa} are related to the usual pseudoscalar fields by 4
1 /
\
n
(1.7) 1
/
•
N
={tpavpi),
T70
K
which is an alternative way for stating the content of Eq. (VII2.4). The physical spin onehalf baryons p, n , . . . can likewise be expressed in terms of an octet of states {Bi} (i — 1,..., 8) in cartesian basis as ± =  ^ ( B i =F iB2) , ^
E° = Bz , ^
~°= j=(B6 + iB7),
A= 58 , ,
(1.8)
~ = L(
In the quark model, hadron observables have simple interpretations, e.g. the baryon number is simply onethird the difference in the number of quarks and antiquarks, etc. Thus, writing quark and antiquark number operators as N(q) and N(q) for a quark flavor g, we have B = [N(u) + N(d) + N(s)  N(u)  N(d)  N(s)}/3 , T3 = [N(u)N(d)N(u) Y = [N(u) + N(d)  2N(s)  N(u)  N(d) + 2N(s)]/3 , Qei = [2N(u)  N(d)  N{s)  2N(u) + N(d) + N(s)]/3 , and the hadronic spin operator is
Quark spatial wavefunctions Many applications of the quark model require the knowledge of the quark spatial wavefunctions within hadrons. It is here that the greatest variation in the different models can occur, but several general features still remain. Indeed, in many instances it is the general features that are primarily tested. For example, the ground state in all models is a spatially symmetric Sstate in which the wavefunction peaks at r = 0. The normalization
278
XI Phenomenological models
0.020

0.015

0.010
~
0.005

0.0
0.2
0.4
0.6
0.8
1.0
1.2
r(fm)
Fig. XI2 Quark probability density in the bag and oscillator models condition of the quark spatial wavefunction, /
dsx
ensures that the magnitude of t/> will be similar in those models having wavefunctions of comparable spatial extent. This accounts for the agreement which can be found among diverse quark models in specific applications. How does one fix the spatial extent? One approach is to use an observable like the hadronic electromagnetic charge radius, e.g., = 087 ± 0.02 fin
= 066 ± 0.02 fin
(1.12)
Viewed this way, the bound states are seen to define a scale of order 1 fm. For example, we display two models in Fig. XI2, the oscillator result with a2 = 0.17 GeV2 and the bag profile, which are each obtained by fitting to ground state baryon observables like the charge radius. Not surprisingly their behaviors are quite similar. Also shown in Fig. XI2 is an oscillator model wavefunction whose parameter (a2 = 0.049 GeV2) was determined by using data from decays of excited hadrons. The difference is rather striking, and serves to demonstrate that the most important general feature in setting the scale in quark model predictions of dimensional matrix elements is the spatial extent of the wavefunction.* Another aspect of quark wavefunctions involves the issue of relativistic motion. A relativistic quark moving in a spinindependent central * We could obtain a bag result which behaves similarly by employing a charge radius of 0.5 fm rather than the 1 fm value shown.
XI1 Quantum numbers of QQ and Q 3 states
279
potential has a ground state wavefunction of the form i u(r)x where u, £ signify 'upper' and 'lower' components. As we shall see, in the bag model these radial wavefunctions are just spherical Bessel functions. The above form also appears in some relativized harmonic oscillator models which use a central potential. If we allow for relativistic motion, then the major remaining difference in the quark wavefunctions concerns the lower two components of the Dirac wavefunction. Nonrelativistic models automatically set these equal to zero, while relativistic models can have sizeable lower components. Which description is the correct one? Quark motion in light hadrons must be at least somewhat relativistic since quarks confined to a region of radius R have a momentum given by the uncertainty principle,* p > V3R1 ~ 342 MeV
(for R ~ 1 fin) .
(1.14)
Since this momentum is comparable to or larger than all the lightquark masses, relativistic effects are unavoidable. A more direct indication of the relativistic nature of quark motion comes from the hadron spectrum. Nonrelativistic systems are characterized by excitation energies which are small compared to the constituent masses. In the hadron spectrum, typical excitation energies lie in the range 300500 MeV, again comparable to or larger than lightquark masses. Such considerations have motivated relativistic formulations of the quark model. Interpolating fields In the LSZ procedure for analyzing scattering amplitudes the central role is played by interpolating fields. These are the quantities which experience the dynamics of the theory in the course of evolving between the asymptotic instates and outstates. They turn out to be also useful as a kind of bookkeeping device. That is, one way to characterize the spectrum of observed states is to use operators made of appropriate combinations of quark fields ip(x). For example, corresponding to the meson sector of QQ states, one could employ a sequence of quark bilinears, the simplest of which are r/;ip, ^75^, ^7^> 7/i75V>, ^
(1.15)
Any of these operators acting on the vacuum creates states with its own quantum numbers. The lightest states in the quark spectrum will be The <s/3 factor is associated with the fact that there are three dimensions.
280
XI Phenomenological models
associated with those operators which remain nonzero for static quarks, i. e. with creation operators and Dirac spinors of the form
Only the pseudoscalar operators ^75 T/>, ^7075 ^ a n d the vector operators ifr'jiipiipcroiip a r e nonvanishing in this limit. All the other operators have a nonrelativistic reduction proportional to spatial momentum, indicating the need for a unit of orbital angular momentum in forming a state. The interpolating field approach is particularly useful in situations where the imposition of gauge invariance determines whether a given field configuration can occur in the physical spectrum. We shall return to this point in Sect. XIII4 in the course of discussing glueball states. We now turn to a summary, carried throughout the next three sections, of various attempts to model the dynamics of light hadronic states. XI—2 Potential model The potential model posits that there is a relatively simple effective theory in which the quarks move nonrelativistically within hadrons. In the light of our previous comments on relativistic motion, this would seem to be acceptable only for truly massive quarks like the frquark and certainly questionable for the light quarks ix, d, s. However, in the potential model it is assumed that QCD interactions dress each quark with a cloud of virtual gluons and quarkantiquark pairs, and that the resulting dynamical mass contribution is so large that quarks move nonrelativistically. These 'dressed' degrees of freedom are called constituent quarks, and their masses are called 'constituent masses'. Constituent masses are not to be directly identified with the mass parameters occurring in the QCD lagrangian.* Energy levels and wavefunctions are then obtained by solving the nonrelativistic Schrodinger equation in terms of the constituent masses and some assumed potential energy function. The potential model is not without flaws. For lightquark dynamics, it is far from clear that a static potential can adequately describe the QCD interaction. Even with the use of constituent masses, one finds from fits to the mass spectrum and/or the charge radius that quark velocity is nevertheless near the speed of light (c/. Prob. XI1). Also, although it is possible [LeOPR 85] to make a connection between the lightest pseudoscalar mesons as Goldstone bosons on the one hand and QQ composites on the other, this is not ordinarily done. Such criticisms notwithstanding, * We shall continue to denote the QCD mass parameter of quark qi as rrii, and shall write the corresponding constituent mass as Mi.
XI2 Potential model
281
the nonrelativistic quark model does provide a framework for describing both ground and excited hadronic states, and brings a measure of order to a spectrum containing hundreds of observed levels. Besides, virtually all physicists are familiar with the Schrodinger equation, and find the potential model to be an understandable and intuitive language.
Basic ingredients One begins by expressing the mass Ma of a hadronic state a as Ma = ^
Mi + Ea ,
(2.1)
where the sum is over the constituent quarks and antiquarks in a. The internal energy Ea is an eigenvalue of the Schrodinger equation = EQi>a
,
(2.2)
with hamiltonian H
= E ^ P * 2 + E ^colorM ,
(2.3)
where r^ = r^ — r^, and the subscript 'color' on the potential energy indicates that the dynamics of quarks necessarily involves the color degree of freedom in some manner. It is standard to assume that the potential energy is a sum of twobody interactions. Although there exists no unique specification of the interquark potential V^oior from QCD, the following features are often adopted: 1) 2) 3) 4)
a spin and flavor independent long range confining potential, a spin and flavor dependent short range potential, basis mixing in the baryon and meson sectors, and relativistic corrections.
We shall discuss specific models of the potential energy function in Sect. XIII1. They all have in common the color dependence in which the twoparticle potential is twice as strong in mesons as it is in baryons,
{
V{rij) 2 V\Tij)
(mesons) , (baryons) .
We shall describe a simple empirical test for such behavior at the end of this section. To appreciate its theoretical basis, note that the quarkantiquark pair in a meson must occur in the 1 representation of color, whereas any two quarks in a baryon must be in a 3* representation (in
282
XI Phenomenological models
order that the threequark composite be a color singlet), (F(3).F(r) (mesons), oc j F ( 3 ) F ( 3 ) (baryong)
(2.5)
(F 2 (l)  F 2 (3)  F 2 (3*))/2 =  4 / 3 (mesons) , 2 2 (F (3*)  2F (3))/2 =  2 / 3 (baryons) , a where F (R) is a color generator for £J7(3) representation R. Thus, the color dependence in Eq. (2.4) is that which one would naturally associate with the interaction between two quarks or a quarkantiquark pair. Mesons For the twoparticle QQ system, it is straightforward to remove the centerofmass dependence. In the centerofmass frame the Schrodinger equation becomes /p 2 \ [ _£_ _i_ V(r) I tb (r) — E ib (r)
(2 6)
where r = TQ — TQ and M" 1 = MQ1 + MQ1 is the inverse reduced mass. The LS coupling scheme is typically employed to classify the eigenfunctions of this problem. One constructs the total QQ spin, S = SQ + SQ, and adds the orbital angular momentum L to form the total angular momentum J = S + L. There is an infinite tower of eigenstates, each labeled by the radial quantum number n and the angular momentum quantum numbers J, J 2 , L, S. The QQ states are sometimes described in terms of spectroscopic notation 2S+lLj (JPC), where P is the parity and C is the charge conjugation, P = (_)£+i ,
C = ()L+5 .
(2.7)
Strictly speaking, although only electrically neutral particles like TT°can be eigenstates of the charge conjugation operation, C is often employed as a label for an entire isomultiplet, like n = (TT+, TT0, TT~). The lowest QQ Table XI3. Quantum numbers of QQ composites. Singlet
Triplet
0 2 3
1
ZJ 2 (2+) ^3(3+)
3 3
r>12f3(l—,2—,3—) F 2 , 3 ,4(2++,3++,4++)
XI2 Potential model
283
orbital configurations, expressed in 2S+1Lj (JPC) notation, are displayed in Table XI3. The 0 + , 1~, 2 + , . . . series of Jp states is called natural, and has the same quantum numbers as would occur for two spinless mesons of a common intrinsic parity. The alternate sequence, 0~, 1 + , 2~,... is referred to as unnatural. There are a number of Jpc configurations, called exotic whichstates, cannot be accommodated within the QQ construction. For example, the 0 state is exoticbecause any state with J = 0 must have L = 5, and according to the QQ constraint of Eq. (2.7) must therefore carry C = +. Likewise, the CP = — 1 sequence 0 + ~, 1~+, 2 + ~ , . . . is forbidden because the QQ model requires CP = (—)^ +1, implying 5 = 0 and hence J = L. Thus one would obtain P = (—) J+1 in the QQ model and not P = (  ) J . Baryons Most applications of the quark model for Q 3 baryons involve the light quarks. If, for simplicity, we assume degenerate constituent mass M, the Schrodinger equation is
where the prefactor of 1/2 in the potential energy term follows from Eq. (2.4). It is convenient to define a centerofmass coordinate R and internal coordinates A and Q by R = (n + r 2 + r 3 )/3 , Q = (n  v2)/V2 ,
(2.9)
A = (n + r2  2r3)/V6 . Because it is not possible to remove the threeparticle centerofmass dependence for an arbitrary potential, the following approach is often followed [GrS 76, IsK 78]. The potential V(r^) is rewritten as V(rlj) = Vosc(rij) + U(rij)
,
(2.10)
where VOBC = ^ r i ] ,
U = VVOSC
.
(2.11)
The Schrodinger equation is solved in terms of the oscillator potential and U is evaluated perturbatively in the oscillator basis. Having removed the centerofmass coordinate, we are left with the following hamiltonian for the internal energy: Hint
{
+
Q)
+
(
+
A )
(212)
284
XI Phenomenological models
which is just that of two independent quantum oscillators each with spring constant 3k. For later purposes, we write the number of excitation quanta for the two oscillators as Np and N\ (Np,\ = 0,1,2,...) and let TV = Np + N\. The angular momentum for the threequark system is found in a similar manner as for the QQ mesons, J = L + S. The total quark spin is S = Yl Sj, the orbital angular momentum is given by L = L p + Lj\, and the parity is P = (—)^>+^. The ground state wavefunction has the form v2\3/2
e i P R e x p [a\Q2
+ A2)/2]
,
(2.13)
where a2 = (3km)1/2. A cautionary remark is in order. One should not misinterpret the use of an oscillator potential  it is not the intent to model the observed baryon spectrum as that of a system of quantum oscillators because such a picture would fail. For example, the oscillator spectrum has EN ~ TV, whereas the baryon spectrum obeys the law of linear Regge trajectories (cf. Sect. XIII2), E^ ~ N. The oscillator potential provides a convenient basis for structuring the calculation and nothing more.
Color dependence of the interquark potential Short of doing a complete spectroscopic analysis, we can find experimental support in the following simple example for the assertion that the twoparticle interquark potential is twice as strong in mesons as it is in baryons. A potential model description for the meson and baryon mass splittings p(770)  TT(138) and A(1232)  N(939) is given by a QCD hyperfine interaction, i?hyp> akin to the delta function contribution in the QED hyperfine potential of Eq. (V1.16),
Yl
(a = M,B) ,
(2.14)
where the {Wij} are constants and, assuming the color dependence is that given by Eqs. (2.4), (2.5), &M = 1 for mesons and fc# = 1/2 for baryons. We shall discuss in Sect. XIII2 how this effect could arise from gluon exchange. Although there is ordinarily dependence on quark mass in the {Hij}, it suffices to treat the {Wij} as an overall constant since the hadrons in this example contain only light nonstrange quarks. The point is then to see whether the condition &M — 2fc# is in accord with phenomenology. Noting that for mesons the spin factors yield 2S23
f 1/4 = 
(5 = 1 ) , {s
o h
(2.15a)
XI3 Bag model
285
whereas for baryons one has 4S29
/ 3/4 .
(S = 3 / 2 ) ,
,
,=1/2),
we find after taking expectation values that nip — mn
2UM V ; M(0) 2
2&M (Volume)^
(Volume) M MI
V)B1"'" i3/2 2v
(2!6) V
/
2
.{T )M\
The measured values (c/. Eq. (1.13)) of the proton and pion charge radii imply that /CM/^B — 2. This example, along with others, lends credence to the assumed color dependence of Eq. (2.4). At this point we shall temporarily leave our discussion of the potential model to consider other descriptions of hadronic structure. We shall return to the potential model for the discussion of hadron spectroscopy in Chaps. XIIXIII. XI3 Bag model A superconductor has an ordered quantum mechanical ground state which does not support a magnetic field (Meissner effect) and which is brought about by a condensation of dynamically paired electrons (Cooper pairs). An order parameter for this medium is provided by the LandauGinzburg wavefunction of a Cooper pair. Even at zero temperature, a sufficiently strong magnetic field, S c r , can induce a transition from the superconducting phase to the normal phase. For example, in tin the critical field is 5 c r (tin)~ 3.06 x 10~2 tesla, and the energy density of superconducting pairing (condensation energy ) is Usnper/V ~ 373 J/m 3 . Chromodynamics exhibits similar behavior, and this is the basis for the bag model [ChJJTW 74, Jo 78]. The QCD ground state evidently does not support a chromoelectric field, and is thus analogous to the superconducting state, although a compelling description of the QCD pairing mechanism has not yet been provided. In the bag model, the analog of the normal conducting ground state is called the perturbative vacuum. The vacuum expectation value of the quark bilinear qq (q = u, d, s) plays the role of an order parameter by distinguishing between the two vacua, QCD (OWO)QCD < 0 ,
pert (0w0) pert = 0 .
(3.1)
Hadrons are represented as color singlet 'bags' of perturbative vacuum occupied by quarks and gluons. The bag model employs as its starting
286
XI Phenomenological models
point the lagrange density [Jo 78] Aag = (£QCD  B) 0(qq)
,
(3.2)
where the ^function (which vanishes for negative argument) defines the spatial volume encompassed by the perturbative vacuum. B is called the bag constant, and is often expressed in units of (MeV) 4 . Physically, it represents the difference in energy density between the QCD and perturbative vacua. Phenomenological determinations of B yield jgi/4 ^ 150 MeV, which translates to a QCD condensation energy of 34 3 UQCD/V ~ 1.0 x 10 J/m . Although huge on the scale of the condensation energy for superconductivity, this value appears less remarkable in more natural units, B ~ 66 MeV/fm 3 .
Static cavity To obtain the equations of motion and boundary conditions for the bag model, we must minimize the action functional of the theory. We shall consider at first a simplified model consisting of a bag which contains only quarks of a given flavor q and mass m. The equations of motion which follow from the lagrangian of Eq. (3.2) are (ifi  m)q = 0 ,
(3.3)
within the bag volume V and = Q > = 2B
(3.4a) (3.4b)
on the bag surface *S, where n^ is the covariant inward normal to S. Eq. (3.3) describes a Dirac particle of mass m moving freely within the cavity defined by volume V. Since the order parameter qq vanishes at the surface of the bag, the linear boundary condition in Eq. (3.4a) amounts to requiring that the normal component of the quark vector current also vanish at the surface. Thus quarks are confined within the bag. The nonlinear boundary condition represents a balance between the outward pressure of the quark field and the inward pressure of B.
Spherical cavity approximation In principle, the bag surface should be determined dynamically. However, the only manageable approximation for light quark dynamics is one in which the shape of the bag is taken as spherical with some radius R. For such a static configuration, the nonlinear boundary condition becomes equivalent to requiring that the energy be minimum as a function of R. The static cavity hamiltonian is H = f d3x [q\iot Jv
 V)q + q](3mq + B] .
(3.5)
XI3 Bag model
287
Observe that B plays the role of a constant energy density at all points within the bag. As in Eq. (1.1), the normal modes of the cavityconfined quarks and antiquarks provide a basis for expanding quantum fields. They are determined by solving the Dirac equation Eq. (3.3) in a spherical cavity. We characterize each mode in terms of a radial quantum number n, an orbital angular momentum quantum number £ (as would appear in the nonrelativistic limit), and a total angular momentum, j . Only j = 1/2 modes are consistent with the nonlinear boundary condition since the rigid spherical cavity cannot accommodate the angular variation of j > 1/2 modes. Such nonspherical orbitals can be treated only approximately, by implementing the nonlinear boundary condition as an angular average or by minimizing the solution with respect to the energy. In addition, since neither pi/% modes nor radially excited sly/2 modes are orthogonal to a translation of the ground state, they must be admixed with some of the j = 3/2 modes to construct physically acceptable excitations. For these reasons, the bag model has been most widely applied in modeling properties of the ground state hadrons rather than their excited states. Let us consider the S\/2 case in some detail. Even with the restriction to a single spinparity state, there are still an infinity of eigenfrequencies u)n. Each un is fixed by the linear boundary condition, expressible as the transcendental equation n
=
Pn
(n = l , 2 , . . . ) ,
LUji  r TYlrt — 1
(3.6)
where pn = y/ool — m2R2. For zero quark mass, the lowest eigenfrequencies are u = 2.043, 4.611, For light quark mass (mR < 1) the lowest mode frequency is approximated by UJ\ ~ 2.043 + 0.493rai?, and in the limit of heavy quark mass (mR ^> 1) becomes OJ\ —• \/m2R2 + n2. The spatial wavefunction which accompanies destruction of an S1/2 quark with spin alignment A and mode n is
iJ Pr
f /*)X\ ) ,
(3.7)
while for creation of an sy2 antiquark we have
where e = ((un — mR)/(un + mR))1/2, x\ ls a twocomponent spinor, and x\ = ^a2X\' The full quark field q(x), expanded in terms of the modes, is given by
q(x) = J2 N(un) [il>n(*)eiUnt/RKn) + ^ ( ^ e ^ ^ d ^ n ) ] ,
(3.9)
288
XI Phenomenological models
where
is a normalization factor which is fixed by demanding that the number operator Nq = Jh d3x q\x)q{x) for quark flavor q have integer eigenvalues. By computing the expectation value of the hamiltonian in a state of N quarks and/or antiquarks of a given flavor, one obtains (H) = NUJ/R + ATTBRS/3  Zo/R .
(3.11)
In the final term, Zo is a phenomenological constant which has been used in the literature to summarize effects having a 1/R dimension, most notably the effect of zeropoint energies, which for an infinite volume system would be unobservable. However, just as the Casimir effect is present for a finite volume system with fixed boundaries, such a term must be present in the static cavity bag model [DeJJK 75]. Unfortunately, a precise calculation of this effect has proven to be rather formidable, and so one treats Zo as a phenomenological parameter. Upon solving the condition d (H) /OR = 0, we obtain expressions for the bag radius R4 = ^(Nu
 Zo) ,
(3.12)
and the bag energy E=
\{AKB)1/A(NUJ
 Z 0 ) 3/4 .
(3.13)
The bag energy E is not precisely the hadron mass. Although the bag surface remains fixed in the cavity approximation, the quarks within move freely as independent particles. Thus at one instant, the configuration of quarks might appear as in Fig. XI3(a), whereas at another time, the quarks occupy the positions of Fig. XI3(b). As a result, there are unavoidably fluctuations in the bag centerofmass position. The bag energy is thus E = I ^/p 2 + M2 \, where M is the hadron mass and p
(a)
(b)
Fig. XI3 Quarks in a bag
XI3 Bag model
289
represents the instantaneous hadron momentum. Although the average momentum vanishes ((p) = 0), the fluctuations do not, ((p 2 ) 7^ 0). For all hadrons but the pion, it is reasonable to expand the bag energy in inverse powers of the hadron mass, (3.14) E = M + (p2)/2M + ... . For the pion, one should instead expand as £;=(p) + M 2 <  P r 1 ) / 2 + ... . (3.15) One can employ the method of wave packets, to be explained in Sect. XII1, to estimate that (p) ~ 2.Si?"1, ( I P I " 1 ) — 0.7R for the pion bag, and ^p2) ~ N(JiR~2 for a bag containing N quarks and/or antiquarks in the Si/2 mode. Gluons in a bag Any detailed phenomenological fit of the bag model to hadrons must include the spinspin interaction between quarks. One way to incorporate this effect is to posit that gluons, as well as quarks, can exist within a bag. With only gluons present, the lagrangian is taken to be [Jo 78]
 B] # (  W / 4  B) ,
(3.16)
and the EulerLagrange equations are &F%, = 0
(3.17)
in the bag volume V, and = 0 (3.18a) ^ = 45 (3.18b) on the bag surface S. In the limit of zero coupling, the gluon field strength becomes F*v = d^A^ — dvAa^. The field equations in V are sourceless Maxwell equations with boundary conditions xE a = 0 and x x B a = 0 on 5, where E a and B a are the color electric and magnetic fields respectively. It is convenient to work directly with the gluon field Aa(x), and with a gauge choice to restrict the dynamic degrees of freedom to the spatial components. In mode n, these obey [V2 + (kn/R)2}Aan = 0 ,
(3.19)
and V • A£ = 0 (3.20) within the bag. The gluon eigenfrequencies kn are determined by the linear boundary condition rx(VxA;)=0.
(3.21)
290
XI Phenomenological models
Restricting our attention to modes of positive parity, we have for the gluon field operator
Aa(x) = J2NG(kn)(ji(knr/R)X1(T(n)aan^
+ H.c.) ,
(3.22)
n,a
where X ^ is a vector spherical harmonic. The gluon normalization factor is obtained, analogously to N(ujn) for quarks, by constraining the gluon number operator to be integer valued and we find [NG(kn)}2 = [3(1  sin(2kn)/2kn)  2(1 + k2n) sin2(kn)]R2 .
(3.23)
The quarkgluon interaction In the following, we shall work with the lowest positive parity mode, for which k\ = 2.744. The quark hyperfine interaction in hadron H can be computed from the second order perturbation theory formula, E"hyp — \H\Hqg(Eo — HQ + ie)~ Hqg\H)
(3.24)
where the unperturbed hamiltonian HQ is given in Eq. (3.5) and Hqg is the quarkgluon interaction
# q _ g = gs f Sx 3a(x) • Aa(x) , Jv
(3.25)
defined in terms of the quark color current density J
a
(rr\
n .(rp\^\a
n
.(rp\
^9^
Implicit in Eq. (3.24) is an infinite sum over all intermediate states. In practice the sum can be well approximated by the lowest energy intermediate state, and we find for hadron H ,
(3.27)
where hH = O.177(H\ Y, <*i ' *j Fz • Fj \H) .
(3.28)
The numerical factor arises from an overlap integral of quark and gluon spatial wavefunctions, and F^,cr^ are respectively the color and spin operators for quark i. It is straightforward to demonstrate that hn — 0.708, IIN — —h/s. = hn/2, and hp = —h n/3.
XIS Bag model
291
Table XI4. Results of bag model fit £1/4
Zo
Ro
RN
RA
135.
1.01
2.13
5.5
5.6
m
3.5
33.
Notes: Rl/4 and rh are given in MeV and R0,RA I I ^ A i, Rn are given in GeV l
A sample fit An example of how to determine parameters of the bag model is provided by quoting a simultaneous fit to JV, A, and n states [DoJ 80]. The SU(2) limit with mu = m^ = m is employed and the constraint of chiral symmetry is implemented by arranging the pion mass to vanish for zero quark mass,TO= 0. Since the fit entails working at different momentum scales, a running effective coupling as(R) is introduced in place of the fixed one. Although the precise form of as(R) is not known for small i?, let us assume its dependence to be logarithmic. A naive identification of the lowest order QCD formula with the bag radius, as(R) = 27r/[91n(jRo/i2)] for three flavors (Ro is the spatial counterpart of the confinement scale A in momentum), is untenable, as the coupling would diverge for i? — Ro. Hadrons, such as the pion and the nucleon, which have hyperfine contributions proportional to — as(R)R~1 would suffer mass instability as a result. This problem is avoided by employing a modified form, as(R) = 2TT/[9 ln(l + Ro/R)]. The bag model then implies the system of equations 2
+ ml)1'2)
= Rl[2un Zo
4
3  hNas(RN)] (p2 + ™i)
1/2
) = R^fivA
,
(3.29)
Zo + 4nBRi/3  hAas(RA)} ,
which contain a total of seven unknowns, i?, Zo, i?o, RN,A,KI a n d TO. The lefthand side is calculated using the wavepacket method of [DoJ 80], which will be described in Sect. XII1. Employing Eq. (3.29) in a fit to the TV and A masses and using drriN^/dR \R=RN A = 0 yields four relations. For the pion bag, the energy is expanded in powers of the light quark mass,
En(m) = En(0) + mdE^/drhiO) + . . . .
(3.30)
In addition to fitting to the pion mass and minimizing with respect to Rn, a further constraint follows fromTO2(TO = 0) = 0. Results of the fit are given in Table XI4. With the bag parameters now determined, the quark wavefunctions are known and various observable matrix elements
292
XI Phenomenological models
can be evaluated. Also, the p meson mass can be predicted, resulting in mp = 704 MeV. XI—4 Skyrme model In Chap. X, we explored the Nc —• oo limit of QCD. In some respects the world thus defined is not unlike our own. Mesons and glueballs exist with masses which are O(l) as Nc —• oo. To lowest order, these particles are noninteracting because their coupling strength is O(N~l). What becomes of baryons in this world? It takes Nc quarks to form a totally antisymmetric color singlet composite, so baryon mass is expected to be O(NC). Note the inverse correlation between interparticle coupling O(N~1) and baryon mass O(NC). This is reminiscent of soliton behavior in theories with nonlinear dynamics. SineGordon soliton An example is afforded by the SineGordon model, defined in one space and one time dimension by the lagrange density, £SG = \{d^)2
 ^ ( 1  cos 0
(4.1)
where a and j3 are constants. For small amplitude field excitations, an expansion in powers of
£SG = \{dM2  f v>2 + ^ V + O(PV),
(4.2)
identifies the parameter a as the boson squared mass and (3 as a coupling strength. For j3 —> 0 we recover the free field theory. The SineGordon lagrange density has also a nonperturbative static solution, <po(x) =  tan" 1 (exp( v / ax)) ,
(4.3a)
with energy EQ  8V^//3 2 •
(4.3b)
This solution is a SineGordon soliton. The natural unit of length for the soliton is a" 1 / 2 , and the energy E$ diverges as the coupling is turned off (/?—•()). The potential energy in this theory has an infinity of equally spaced minima, with (pW = 2im/(3 (n = 0, ±1, ±2,...). As the coordinate x is varied continuously from —oo to +oo, the soliton amplitude (fo(x), starting from the minimum ip^ — 0, moves to the adjoining minimum
XI4 Skyrme model
293
current density,
J" = i
(4.4)
such that AN =
dx J°{x) = — [
(4.5)
^
For
Tr (d^Ud^) ,
(4.6)
where U is an 5/7(2) matrix which transforms as U —> LUR~l under a chiral transformation for L E SU(2)L and R G SU(2)R. Unfortunately £2 cannot support an acceptable soliton, as the soliton would have zero size and zero energy. To see why, recall that the SineGordon soliton has a natural unit of length a" 1 / 2 . Suppose there is an analogous quantity, i?, for the chiral soliton. Then we can write the radial variable as r = fi?, where f is dimensionless. For a static solution, the energy becomes 3
3
E = f d x H =  I d x £=^f
I d3x Tr ( W • W f ) .
(4.7)
Upon expressing the integral in terms of the dimensionless variable r, we find E = aR where a is a nonnegative number. The energy is minimized at R = 0 to the value E = 0. This trivial solution is unacceptable, and thus the model must be extended. The Skyrme model [Sk 61] employs, in addition to £2, a quartic interaction of a certain structure, ^
^
Tr [d^U U\dvU U]}2 ,
(4.8)
where e (not to be confused with the electric charge!) is a dimensionless realvalued parameter. The above chiral lagrangian should look familiar, since it is part of the general fourth order chiral lagrangian used in Chap. VI. In particular, Eq. (4.8) is reproduced if 2oc\ + 2^2 + a% — 0, in
294
XI Phenomenological models
which case (32c 2)"1 = (a2 — 2ai — 0:3 )/4. The comparison with the phenomenology of Chap. VI is not completely straightforward, as the pion physics was treated to oneloop order while the Skyrme lagrangian is used at treelevel. We note, however, that the coefficients in Table VI1 give 2 Q 1 + 2a2
+
°3 = 0.65 ,
a 2  2a!  03 = 0.0036 .
(4.9)
The latter combination, which is independent of renormalization scale, numerically gives e ~ 5.9. In the following development, we shall follow standard practice by taking the parameter e as arbitrary. We seek a static solution of the Skyrme model. Our strategy shall be to first determine the energy functional of the theory, and then minimize it. Following the procedure leading to Eq. (4.7), we can write the energy as (4.10)
 / •
where X^ = Ud^U^ and X^ = —Xjj,. It is necessary that Xi —> 0 as x —• oo in order that the energy be finite. Thus U must approach a constant element of 5C/(2), which we are free to choose as the identity / . For the mesonic sector of the theory, the vacuum state corresponds to C/(x) = / for allx. In this state, both the field variable X{ and the energy E vanish. The form U ~ I+iTZr/F^, used extensively in earlier chapters, corresponds to small amplitude pionic excitations of the vacuum. To see that the Skyrme model does support a nontrivial soliton, we cast the energy integrals of Eq. (4.7) in terms of a natural length scale R and find E = aR + bR~1 ,
(4.11)
where a, b are nonnegative. For a, b ^ 0, the energy is minimized at nonzero R and nonzero E. Thus, the quartic term of Eq. (4.8) is seen to have the desired effect of inducing soliton stability. Moreover, for arbitrary U a lower bound on the energy is provided by applying the Schwartz inequality to Eq. (4.11), E > ^
fd3x Tr EijkXiXjXkl .
(4.12)
It is not hard to show that the integrand of Eq. (4.12) is proportional to the zeroth component of a fourvector current,
XI~4 Skyrme model
295
which is divergenceless, d^B^ = 0, and thus has conserved charge dsx JB°(X) .
(4.14)
• / •
It turns out that the current B^ can be identified as the baryon current density and B is the baryon number of the theory. Note that this is consistent with our prescription U(x) = I for the meson vacuum, where we see from Eq. (4.13) that B = 0. Interestingly, B turns out to have an additional significance. It is the topological winding number for the Skyrme model, analogous to AN for the SineGordon model. The point is, by having associated spatial infinity with a group element of £77(2) to ensure that the field energy is finite, we have placed the elements of physical space into a correspondence with the elements of the compact group SU(2). The parameter space of each set is £ 3 , the unit sphere in four dimensions, and it is precisely the field U which implements the mapping. The mappings from 5 3 to 5 3 are known to fall into classes, each labeled by an integervalued winding number. In this context, B serves to measure the number of times that the set of space points covers the group parameters of SU(2) for some solution U of the theory.
The
Skyrme soliton
The Skyrme ansatz for a chiral soliton (skyrmion) has the functional form [BaNRS 83, AdNW 83] tfo(x) = exp [iF(r)r • x] .
(4.15)
The unknown quantity is the skyrmion profile function F(r). To specify it, we first determine the energy functional by substituting Uo into Eq. (4.10),
E[F) =
4TT
f Jo
(4.16)
where a prime signifies differentiation with respect to the argument. For a static solution, the minimization of the energy generates an extremum of the action, and is hence equivalent to the equations of motion. The variation 6E/6F = 0 generates a differential equation for F, 4 +
2 s i n ^ V /
2
+
V
+
^
s
i
4
n
2
F
rl
!
0
,
(4.17) as expressed in terms of a dimensionless variable f = r/R, with R 1 = 2eFn. This nonlinear equation must be solved numerically, subject to
XI Phenomenological models
296
certain boundary conditions. The condition U = I at spatial infinity implies F(oo) = 0. The boundary condition at r = 0 is fixed by requiring that the soliton correspond to baryon number 1. For the Skyrme ansatz, the baryon number charge density is
B°(r)= 
(4.18)
2TT 2
and corresponds to a baryon number B = ! [2F(0)  2F(oo)  sin2F(0) •sin2F(oo)] . Z7T
(4.19)
This leads to the choice F(0) = n. Although the profile F(r) cannot be determined analytically over its entire range, it is straightforward to show that f 7T — const, r \ const. r~ 2
F(r)
(r —• 0) , (r —• oo) .
(4.20)
We display F(r) in Fig. XI4. Insertion of the solution to Eq. (4.17) into the energy functional E[F] yields the mass M of the skyrmion, and from a numerical integration we obtain M ~ 73 F^/e. There is an important point to be realized about the skyrmion  it represents a use of chiral lagrangians outside the region of validity of the energy expansion. Recall that the full chiral lagrangian is written as a power series, C = £2 + £4 + • • • in the number of derivatives. When matrix elements of pions are taken, terms with n derivatives produce n powers of the energy. Hence at low energy, one may consistently ignore operators with large n, as their contributions to matrix elements are highly suppressed. However, in forming the skyrmion one employs only £2 ami a subset of £4. The relative effects of the two are balanced in the minimization of the energy functional, and as a result both contribute
1
I
r
1
I
r
F(r)
Fig. XI4 Radial profile of the skyrmion
XI4 Skyrme model
297
equally. In an extended model containing CQ, one would expect the import of CQ to be analogously comparable to £4, etc. Higherderivative lagrangians thus will contribute to skyrmion matrix elements, and the result cannot be considered a controlled approximation. However, this is not sufficient cause for abandoning the skyrmion approach. It simply becomes a phenomenological model rather than a rigorous method, and thus has a status similar to potential or bag models.
Quantization and wavefunctions The analysis done thus far is at the classical level, and merely shows that the chiral soliton satisfies the equations of motion. To determine the quantum version of the theory, we shall follow a canonical procedure. An analogy with quantization of the rigid rotator may help in understanding the process. A classical solution consists of the rotator being at any fixed angular configuration {0, (p}. To obtain the quantum theory, one allows the rotator to move among these solutions, and describes its motion in terms of the angular coordinates and their conjugate momenta {pe,Pip} The quantum states are those with definite angular momentum quantum numbers {£,ra},and have wavefunctions given by the spherical harmonics, {9,(p\t,"i)=Ye,m(6,
= = ~
[x(t)] over trajectories one has instead a sum J [d(p(x)] over all possible field configurations. Nevertheless the analogy is rather direct.
d^{x)] f[
W , (3.9)
.
(4.21)
The classical skyrmion solutions consist not only of U$ (cf. Eq. (4.15)), but also of any constant SU(2) rotation thereof, UQ — AU$A~l with AeSU(2). A particularly simple approach to quantization is then to allow the soliton to rotate rigidly in the space of these solutions, U = A(t)U0A\t)
,
(4.22)
where now A(i) is an arbitrary timedependent SU(2) matrix. One proceeds to define a set of coordinates {a^}, their conjugate momenta {?!"£ = dC/ddk}, and a hamiltonian constructed via Legendre transformation H = 7rkdkL
.
(4.23)
We shall presently describe how to choose quantum numbers and determine the associated wavefunctions. Note that this approach is approximate in that it neglects the possibility of spacetime dependent excitations such as pion emission. As such, it would be most appropriate for a weakly coupled theory (as occurs for Nc —> oo) where the soliton rotates slowly, but is only approximate in the real world. In general, an SU(2) matrix like A can be written in terms of three unconstrained parameters {9k} as A(t) = exp(ir • 0) = I cos (9 + ir • 9 sin0 .
(4.24)
298
XI Phenomenological models
However, we can equivalently employ the four constrained parameters, ao = cos# ,
a = 0 sin0 ,
(4.25a)
where 3
l=l •
( 4  25b )
Substitution of the rotated quantity U into Eq. (4.7) and evaluation of the spatial integration yields 3
L = M + \Tr(d0A^d0A)
= M+
2A^~] a\ ,
(4.26)
k=0
where A = TrA/3esF7r, with A = f drr2
sin2 F [1 + 4(F^ + sin2 F/r2)} ~ 50.9 .
(4.27)
As written in terms of the conjugate momenta 7T& = 4Ad&, the hamiltonian is 1
3
Adopting the canonical quantization conditions (4.29) we see that the canonical momenta can be expressed as differential operators, TTfc = —id/dak Thus the hamiltonian has the form
H = M±V24,
(4.30)
where V 2 is the fourdimensional laplacian restricted to act on the threesphere by the constraint of Eq. (4.25b). We can determine the eigenvalues and eigenvectors of H by working in analogy with the more familiar threedimensional laplacian,  ^
^    1 L
2
2
.
(4.31)
r or r If constrained to the unit twosphere by the condition Ylk=i x\ ~ r^ ~ •'» the threelaplacian V3 reduces to —L 2. As is well known, the three components of L are operators Li, L2, £3 which satisfy [Lj, Lk] = i6jkeLe ,
(4.32)
XI4 Skyrme model
299
and generate rotations in the 23, 31, 12 planes respectively. The underlying symmetry group is SO (3), and the eigenfunctions are the spherical harmonics. The fourdimensional problem is treated by analogy. Upon adding an extra dimension labeled by the index 0, we encounter the additional operators K\, K2, K% which generate rotations in the 01, 02, 03 planes. The full set of six rotational generators can be represented as Kk = aoTTfc — afcTro .
(4.33)
The extended symmetry group is SO (A) and the commutator algebra of the rotation generators is [Lj, Lk) = itjkeLe ,
[Lj, Kk] = iejMKt
,
[Kj, Kk] = iejkeLe . (4.34)
The mathematics of this algebra is well known, underlying for example the symmetry of the Coulomb hamiltonian in nonrelativistic quantum mechanics. By the substitutions T = (L  K)/2 ,
J = (L + K)/2 ,
(4.35)
we arrive at operators T and J which generate commuting 5(7(2) algebras. We associate T with the isospin and J with the angular momentum. The explicit operator representations, Tk = i(eijkaidj
+ aodk  akdo) ,
Jk — i(—€ij kaiOj — aoO
k
+ akOo) ,
follow immediately from Eq. (4.33), and the Skyrme hamiltonian becomes H = M + (T 2 + J2)/4A .
(4.37) 2
2
It follows from the commutator algebra of Eq. (4.34) that T = J . Thus the quantum spectrum consists of states with equal isospin and angular momentum quantum numbers, T — J. This is no surprise. After all, in the Skyrme ansatz of Eq. (4.15), the isospin and spatial coordinates appear symmetrically, and we expect the quantum spectrum to respect this reciprocity. Our final form for the hamiltonian, H = M + J2/2A ,
(4.38)
has the eigenvalue spectrum E = M + J ( J + 1)/2A ,
(4.39)
where in general J = 0,1/2,1,3/2,.... By analogy with the usual spherical harmonics, the eigenfunctions of H are seen to be traceless symmetric polynomials in the {ak}. However, both {ak} and {—a k} describe the same solution U (cf. Eq. (4.22)). In the quantum theory, eigenfunctions thus fall into either of two classes,
300
XI Phenomenological models
— ak}) = =tV>({afc}) Since fermions correspond to the antisymmetric choice, we select only the halfinteger values in Eq. (4.39). In the Skyrme model, the N and A baryons will have wavefunctions which are respectively linear and cubic in the {a^}. To construct such states, it is convenient to employ the differential representations of Eq. (4.36) to prove L3(ai ± za2) = ±(ai db ia2) , ± za3) = ±(a 0 ± za3) ,
^3«o,3 = 0 , #3^1,2 = 0 .
From these and Eq. (4.36), the T3 = J3 = 1/2 eigenstate of a proton with spin up is found to be CAPT) = ±
(a1+ia2) .
(4.41)
The normalization of this state is obtained from the angular integral over the threesphere
=
= Jdn3
{a\ + 4) ,
(4.42)
where the angular measure is / » 22 T T
/
dft3 = /
Jo
PIT
P7T
d(p / d6 sin(9 / dx sin 2 * , Jo
(4.43)
Jo
and spherical coordinates in four dimensions are defined by (4.44) a2 == cos sinx\ • aa\3 = = sin sin xx sin cos 66 cos ,
(4.45) where dk = d/dak The T = J = 3/2 A states are formed by employing analogous ladder operations on is/2 (A\ A++) =  ^ (01 +ia 2 ) 3 .
(4.46)
It is remarkable that fermions can be constructed from a chiral lagrangian which contains nominally bosonic degrees of freedom. However, the presence of a nonzero fermion quantum number can be easily verified by direct calculation. The wavefunctions for the eigenstates (the equivalents of Y^m(0, if) for the rigid rotator) are given by 5/7(2) rotation matrices with halfinteger
XI4 Skyrme model
301
values. These are defined by the transformation properties of states under an SU(2) rotation A [Ed 60], ^
3
)
3
.
(4.47)
The simplest case is then just the T =1/2 representation, which we know is rotated by the matrix A, a
i(ai °+ iaz) " ia2) \ (4 48) a1+ia2) (aoia3))&h ' ^AS) Comparison with Eq. (4.41) and with the results of Eq. (4.45) shows that the properly normalized nucleon wavefunctions are
(A\ NT3,Ss) = I (  f 3 + i / 2 V^]Sa(A)
.
(4.49)
The general case for a nonstrange baryon B of isospin T and spin S (S = T) is given by
[^]
V2
(
X
,
(4.50)
of which the A states are specific examples. Finally, the N and A masses are MN = M + 3/8A = 73Fn/e + If the measured JV, A masses are used as input, one obtains e = 5.44 and Fn = 65 MeV. Alternatively, from the empirical value for Fn and the determination e ~ 5.9 from pionpion scattering data, the model implies Mjy ~ 1.27 GeV, MA ^ 1.80 GeV. In either case, agreement between theory and experiment is at about the thirty percent level. The next state in the spectrum would have quantum numbers T = J = 5/2 and is predicted by the first of the above fitting procedures to have mass M5/2 = M + 35/8A ~ 1.72 GeV. There is no experimental evidence for such a baryon. Although our discussion has been based on the SU(2) flavor symmetry of isospin, it is possible to extend the analysis to flavor SU(3) [Gu 84]. There, the action consists of the usual quadratic and quartic Skyrme forms, the quark mass term, and also the WessZuminoWitten action of Eq. (VII3.21). This final contribution was not considered in our previous analysis because it vanishes identically for SU(2). It turns out not to affect the equation for the static classical soliton, but does enter into the quantization procedure and thus contributes to the various quantum currents, etc. Two distinct approaches have been adopted for analyzing
302
XI Phenomenological models
the SU(S) model. Either one expands about zero quark mass and treats the quark mass term perturbatively [Ch 85], or one works in the limit of a large squark mass [CaHK 88] and thereby generates equations which describe the motion of kaons in a classical background of the 5(7(2) soliton. We shall not pursue this point further, but shall return to the Skyrme model in Sect. XII1, where the problem of computing matrix elements is addressed. Although the development of the skyrmion and its quantization have been motivated by largeNc ideas, we know of no proof that requires the skyrmion to come arbitrarily close to the baryons of QCD in the Nc —> oo limit. An oftcited counterexample is the existence of a oneflavor version of QCD. Such a theory still contains baryons, such as the A + + . However, it makes no sense to speak of a oneflavor Skyrme model, as an SU(2) group is required for the underlying soliton C/o The Skyrme model remains an interesting picture for nucleon structure because it is in many ways orthogonal to the quark model, and thus offers opportunities for new insights.
XI5 QCD sum rules Low energy QCD involves a regime where the degrees of freedom are the hadrons and where it is futile to attempt perturbative calculations of hadronic masses and decay widths. Contrasted with this is the shortdistance asymptotically free limit in which quarks and gluons are the appropriate degrees of freedom, and in which perturbative calculations make sense. The method of QCD sum rules represents an attempt to bridge the gap between the perturbative and nonperturbative sectors by employing the language of dispersion relations [ShVZ 79a, ReRY 85, Na 89]. The existence of sum rules in QCD is quite general, and some might dispute the classification on these sum rules as a phenomenological method. However in practice, to utilize the sum rules involves the introduction of various approximations and heuristic procedures. Like quark model methods, these are motivated by an intuition of the most important physics involved, but are not always rigorous consequences of QCD. As a result, there remains considerable art in their use.
Correlators It is convenient to approach the subject by considering the relatively simple twopoint functions. Thus we consider the quark bilinear, Mx)=q1(x)Tq2(x)
,
(5.1)
where F is a Dirac matrix, and analyze the correlator,
i f dAxeiqx(0\T(JT(x)4(0))\0)
.
(5.2)
XI5 QCD sum rules
303
Such quantities can be expressed in terms of invariant functions H*r(q2) and attendant kinematical factors, e.g. as for the correlators of pseudoscalar currents {Jp) and of conserved vector currents (Jy), UP(q2) = i ( dAx eiqx(0\T(Jp(x)JP(0))
0)
(5.3a)
e*x(0\T (J^(x)J^(O)) 0). (5.3b) Analogous structures occur for other currents. There are several means for analyzing a quantity like IIr(
 g)(OJF(O)n)^ ,
(5.4)
n
where so is the threshold for the physical intermediate states. Such considerations, together with the application of Cauchy's theorem in the complex q2 plane, imply a dispersion relation for IIr(<72),
where the {an} are N subtraction constants.* One attempts to introduce a phenomenological component to the dispersion relation by expressing Im Ilr(s) in terms of measureable quantities, e.g. with cross section data as in the case of the charm contribution ey^c to the vector current, 1
CTe+e—•charm
$s
where ec is the cquark electric charge and s is the squared centerofmass energy. If, as is usually the case, such data is not available, another means must be found for expressing Im Ilr(s) in the range so < s < oc. To approximate the low5 part of Im lip (5), one usually employs one or more singleparticle states. As an illustration, let us determine the contribution to Hp(q2) of a flavored pseudoscalar meson M which is a bound state or a narrowwidth resonance of the quarkantiquark pair * The number of subtraction constants needed depends on the behavior of Im Ilr (s) in the s • limit, with Ur{q2) ~ q2N Ing 2 requiring TV subtractions.
304
XI Phenomenological models
i2 In this instance, we take the pseudoscalar current in the form of an axialvector divergence, Jp —> d^A^L, with 2 ,
= y/2FMm2M
,
where TRM and FM are the meson's mass and decay constant. Eq. (5.4) implies
= 2F2MmlM6(q'1
which yields pp(q2) = 2F^m\I6(q2
Then

— m2M) for the spectral function or
Im n P  m eson = 2FlIm\I7r6(s
 m2M)
(5.9)
for the dispersion kernel. Thus, bound state or narrowresonance contributions give rise to deltafunction contributions. It is not difficult to take resonant finitewidth effects into account if desired. One or more of these singleparticle contributions are then used to represent the lows part of the dispersion integral. Proceeding to higher svalues in the dispersion integral, one enters the continuum region, where multiparticle intermediate states become significant and the boundstate (or resonance) approximation breaks down. Although, as described below, one ordinarily attempts to suppress the large5 part of Im Ilr(s) by taking moments or transforms of the dispersion integral, it is common to add to the lows contribution a 'QCD continuum' approximation, Imllr(s) — • e(ss large—s
c)ImUcontXs)
,
(5.10)
taken from discontinuities of QCD loop amplitudes and their O(as) corrections. In Eq. (5.10), sc parameterizes the point where the continuum description begins and the form of Im Ilc^t. depends on the specific correlator. Experience has shown that this 'parton' description can yield reasonable agreement of scattering data even down into the resonance region, provided the resonances are averaged over (duality).
Table XI5. Local operators of low dimension. d:
0
4
4
On'. 1 mqqq G^ G T
6 qTqqVq
6
6
XI5 QCD sum rules
305
Operator product expansion A representation for correlators which is distinct from the above phenomenological approach can be obtained by employing an operator product expansion for the product of currents,
i fd4x e*xT (Jr(z)jf(O)) = £c£(92)G>n • **
(5.11)
n
The {On} are local operators and the {C£(g 2 )}, called Wilson coefficients, are cnumbers. The {On} are organized according to their dimension, and aside from the unit operator / , are constructed from quark and gluon fields. Table XI5 exhibits the operators up to dimension 6 which might contribute to the correlator of Eq. (5.2). Although one may naively expect all the operators but the identity to have vanishing vacuum expectation values (as is the case for normalordered local operators in perturbation theory), nonperturbative longdistance effects like those discussed in Sect. Ill—5 generally lead to nonzero values. Most often, the operator product approach contains vacuum expectation values like (^G^Ga^o = (^fG2)o and (mqqq)o as universal parameters, 'universal' in the sense that the same few parameters appear repeatedly in applications. Calculation reveals that the quantity (^G2)o is divergent in perturbation theory, so the perturbative infinities must be subtracted off if one is working beyond treelevel. In principle, all the vacuum expectation values should be computable from lattice gauge theory once the renormalization prescriptions are specified. At present, the only theoretically determined combinations are the products (msss)0 ^ Flm\
,
(m(uu + dd))0 ~ 2i^2ra2 ,
(5.12)
which follow from the lowest order chiral analysis in Chaps. VI,VII. Although only the product rrnpt/j is renormalization group invariant and it is difficult to separate out quark masses uniquely, sum rule calculations often adopt the value (qq)0 = (225±25MeV) 3
(q = u,d,s) ,
(5.13)
corresponding to ms = 180 MeV. In any given application, one must cope with the other local operators as best one can, e.g. the value of (^G 2)o has been estimated from charmonium data. Of course, use of the shortdistance expansion must be justified. We have seen in previous chapters how a given hadronic system is characterized in terms of the energy scales of confinement (A) and quark mass ({mq}). Given these, it is indeed often possible to choose the variable q such that shortdistance, asymptotically free kinematics obtain. Two situations which have received the most attention are the heavy quark limit
306
XI Phenomenological models
x (a)
x
>g x
x
(b)
xx (c)
(d)
Fig. XI5 Contributions to coefficient functions. (m2 » A2,g2) and the light quark limit (q2 » A2 » m2q). Once in the asymptotically free domain, it is legitimate to apply QCD perturbation theory to the C^(q2), with the expansion being typically carried out to one or two powers of as, CTn(q2) = ATn(q2) + BTn(q2)as
+ ... .
(5.14)
Rather extensive lists of Wilson coefficients already appear in the literature. Fig. XI5 depicts contributions to a few of the Wilson coefficients. Denoting there the action of a current by the symbol c x', we display in (a)(b) the lowest order and an O(as) correction to operator / and in (c)(d), the lowest order contributions to (^fG2)o and to (mqq)o respectively. Finally, besides the vacuum expectation values mentioned above, additional parameters which generally occur in the operator product representation are the quark mass mq and the strong coupling as. Since these quantities will depend on the momentum q, one must be careful to interpret them as running quantities whose renormalization is to be specified. Due to asymptotic freedom, they too can be treated perturbatively, e.g. as in the familiar expression Eq. (II2.78) for as.
Master equation The essence of the QCD sum rule approach is to equate the dispersion and the operator product expressions to obtain a 'master equation', W )
I
Jn
"""U"/
,
_ \
^/nf!7«2\/^i \
(5 jg\
It is important to restrict use of this equation to a range of q2 for which both the shortdistance expansion and also any 'resonance + continuum' approximation to Im Ilr are jointly valid. To satisfy these twin constraints, it is common practice to not analyze Eq. (5.15) directly, but rather to first perform certain differential operations leading to either moment or transform representations. The nth moment M^(QQ) is de
XI5 QCD sum rules
307
fined as 1
s
f°
,
Imnr(s)
(7TW
(5.16) where, in the spacelike region #2 < 0, one usually works with the variable Q2 = —q 2. By taking sufficiently many derivatives, one can remove unknown subtraction constants from the analysis and at the same time, enhance the contribution of a singleparticle state at low s in the dispersion integral. Alternatively, one can express the dispersion integral as a kind of transform. The Borel transform is constructed from the moment M^(Q2) as 2n
2
n Q Ml(Q )
1
—> —
C°°
ds e ^ I m U
n,Q 2 ^oo 7TT JSQ
T(s)
,
(5.17)
where Q2/n = r remains fixed in the limiting process and defines the transform variable r. To obtain the factor e~~s/r in the above dispersion integral, we note
A slightly different version of exponential transform which has appeared in the literature is defined analogously, 1 r° 7T J8o 8o
where the transform variable is now a = n/Q2. The transform method serves to remove the subtraction constants and to suppress the contributions from operators of higher dimension in the operator product expansion. Examples Applications of the QCD sum rule approach generally proceed according to the following steps. 1) Choose the currents and write a dispersion relation for the correlator. 2) Model the dispersion integrals with phenomenological input, usually some combination of singleparticle states and continuum. 3) Employ the operator product expansion, including all appropriate operators up to some dimension at which one truncates the series. 4) Obtain the Wilson coefficients as an expansion in as. 5) Use the moment or transform technique to extract information from the master equation. 6) Vary the underlying parameters until stability of output is achieved.
308
XI Phenomenological models
Let us consider three examples, keeping the treatment on an elementary footing to better emphasize the kinds of relationships which QCD sum rules entail. (i) Charmonium spectrum: The correlator for the charm quark vector current J^c m ' = cj^c has the form
?m)(92) = iJdAx e* (5.20)
and the corresponding dispersion relation is
In the following we use the dispersive kernel of Eq. (5.6). Following the original treatment of this system [ShVZ 79a], we work at Q2=0 and employ a moment analysis of the shortdistance expansion containing just the identity and gluon contributions. This yields for the n th moment, expanded to first order in as, 1 Q2=0
{
3 ( n  l ) ! ( n + l)
c)
v
^v
s
' 7r2n(n + l)(n + 2)(rc + 3) >n^°A 36 (2n + 5) m^
n
(5 22)
where {a%} = {0.73, 0.71, 0.51, 0.22, 0.14, etc.} for n = 0,1,.... In practice, one analyzes the ratios rn = Mn/Mn\, expanded in powers of a s , rather than the individual moments. Ratios should be less sensitive to experimental errors and tend to yield a more highly convergent expansion in as. Theory and experiment can then be compared in a plot of rn vs. n. For simplicity, suppose that the strong coupling is fixed from J/i/j decay data, as(s = Am2) ~ 0.2, leaving just the two parameters mc and (^G2)o to be fitted by sum rules. Since the gluon contribution increases as n 3 relative to the identity operator, the mass parameter mc is fixed from lown ratios (n < 4), yielding a value mc ~ 1.26 GeV. Proceeding to higher n (n < 10), but keeping only the identity contribution, one notices the agreement between theory and experiment begins to break down. Attributing this to the need for a gluon condensate term, one extracts the value (^G2)o — (350 MeV)4. Continuing on to even higher n would not be justified in the context of this simple model, because other condensate terms grow even more rapidly with n. Were we to extend the above procedure from cquarks to frquarks, we see from Eq. (5.22) that the relative strength of the gluon contribution would be heavily suppressed by the factor (ra c/rab)4. That is, nonper
XI5 QCD sum rules
309
turbative effects are able to compete with radiative corrections on a relatively equal footing for cquark physics, whereas they are overwhelmed in 6quark systems. In this sense, charm represents a fertile testing ground for the QCD sum rules. However, care must be exercised in interpreting numerical values of mc and (^G2)o due to gauge and renormalization dependence. In particular, the above analysis was carried out in Landau gauge. Also, the mass parameter mc involves a determination at spacelike momentum, q2 = —m 2, and is not the same as would be inferred from a timelike e+e~ scattering experiment (for which q2 = +m2).* It is possible to extend and improve upon the above simple development in a number of ways. By taking Q2 ^ 0, one increases the range of n values for which the 'identity plus gluoncondensate' approximation is valid. Instead of using cross section data, one can saturate the dispersive kernel in terms of individual J/ij) states, and by considering different correlators, it is possible to study a variety of partial waves. Finally of course, additional condensates of higher dimension should be included, and alternative transform methods studied. For example, a study using the exponential transform and including all operators of dimension up to and including d = 8 obtains a rather larger gluon condensate, in the range (^G2)0 ^ (450 > 510 MeV)4 [Pa 90]. (ii) Decay constant of a heavy meson: We consider the axialcurrent divergence and correlator associated with a heavy quark Q and a light antiquark q of mass TRQ and m ~ 0 respectively,
UP(q2) =i f
d4x eiq
A dispersion representation free of subtraction constants is 2
°° , Im Up(s) ds
(JTW '
,
A,
(5 24)
'
Recalling Eq. (5.9), we have in the 'particle plus continuum' approximation, Im UP(s) = 2F2MMlI6{s  M2M) + 6{s  sc)lmllcont.(s)
(5.25)
7T
for a heavy flavored meson of mass MM and decay constant FM Upon performing an exponential transform of the type in Eq. (5.19), the sum The relation between the two is m*.\qi=rn2
= (1 + 4In2 as//K)'m%\q2 =
_rri2.
310
XI Phenomenological models
rule can be cast in the form
2F2MMiIeMM°=  r
ds ds e^ImnpertW (5.26)
where Imnpert. (s) denotes perturbative contributions to the identity operator in the shortdistance expansion and we suppress writing the explicit form of the operators O^Q. Regarding the stability aspect of the fit, one seeks (for a fixed value of the parameter sc in Eq. (5.26)) a minimum F]^m in the value of FM as a is varied. Thus, a stable prediction for FM is found provided one can find a 'plateau of stability' in the response of Fjtfin as sc is varied. The estimates FB — Fp ~ lAFn have been obtained in this manner [Na 87]. Nucleon mass: It is not necessary to restrict oneself to mesonic currents as in Eq. (5.1). Here, we consider a current r)w (and its correlator) which carries the quantum numbers of the nucleon, W =
U(q2) = ni(g 2 ) + #I2(<,2) = i Jd4x
e**(0\T(r,N(x)rjN(0))
0> ,
(5.27) where C is the charge conjugation matrix. The simplest approximation to the dispersion integral comes from the nucleon pole, n
(92)U = A ^ J ^ , q — ivi
(5.28)
N
where the coupling A^ is proportional to the 'nucleon decay constant', i.e., the probability of finding all three quarks within the nucleon at one point. Upon making a simple approximation to the operator product expansion, ^
^
ln(q2)
,
(5.29)
and employing a Borel transform, one obtains an amusing relation between nucleon mass and quark condensate [Io 81], MN = (87T2(qq)0)1/3
+ ... 
1 GeV ,
(5.30)
and implies the vanishing of the former with the latter. However, it should be realized that this result is subject to important corrections in a more careful treatment. Each of the above examples has involved twopoint functions. It is possible to apply the method to threepoint functions as well, where one
Problems
311
can obtain coupling constant relations. The underlying principles are the same, but some technical details are modified due to the larger number of variables, e.g. one encounters doublemoments or doubletransforms. QCD sum rules work best when there is a reliable way to estimate the dispersion integral, most often with ground state singleparticle contributions. However, the method has its limitations. It is not at its best in probing radial excitations since their dispersion effects are generally rather small. Even having a good approximation to the dispersion integral is not sufficient to guarantee success. For example, the method has trouble in dealing with highspin (J > 3) mesons because, even with dispersion integrals which are dominated by ground state contributions, power corrections in the operator product expansion become unmanageable. Problems 1) Velocity in potential models Truly nonrelativistic systems have excitation energies small compared to the masses of their constituents. However, fitting the observed spectrum of light hadrons requires excitation energies comparable to or larger than the constituent masses. Assuming nonrelativistic kinematics, consider a particle of reduced mass m moving in a harmonic oscillator potential of angular frequency u. Expressing u in terms of the energy splitting E\ — Eo between the firstexcited state and the ground state, use the virial theorem to determine the 'rms' velocities of the ground state (vnns) and of the firstexcited state (tins) in terms of E\ — EQ. Compute the magnitude of Vrms/c and Vrms/c using as inputs (i) m/2 — mp ~ 500 MeV for light hadrons and (ii) m^S) ~ mJ/tp — 590 MeV for charmed quarks. Your results should demonstrate that the kinematics of quarks in light hadrons is not truly nonrelativistic. However, one tends to overlook this flaw given the potential model's overall utility. 2) Nucleon mass and the Skyrme model a) Use the Skyrme ansatz of Eq. (4.15) to derive the expression Eq. (4.16) for the nucleon energy E[F]. b) Using the simple trial function F(r) = 7rexp(—r/R), scale out the range factor R to put E[F] in the form of Eq. (4.11), where a ~ 30.8F^ and b ~ 44.7/e2 are determined via numerical integration. c) Minimize E[F] by varying R and compare your result with the value 73Fn/e determined with a more complex variational function.
312
XI Phenomenological models d) Using the numerical value of the nucleon mass, determine e and compare with the value 1 4 1
expected from chiral scaling arguments. 3) A 4QCD sum rule' for the isotropic harmonic oscillator Consider threedimensional isotropic harmonic motion with angular frequency u) of a particle of mass ra. a) Using ordinary quantum mechanics or more formal path integral methods, determine the exact Green's function G(T) for propagation from time t = 0 to imaginary time t = —IT at fixed spatial point x = 0. G(T) is the analog of the 'correlator' for our quantum mechanical system. b) Prom the representation G{r) = (0, — ir0,0), use completeness to express G{r) in terms of the Swave radial wavefunctions {Rn(0)} evaluated at the origin and the energy eigenvalues {En}. What values of n contribute? This representation is the analog of the dispersion relation expression for a correlator in which one takes into account an infinity of resonances. c) Plot the negative logarithmic derivative —d[ln G(r)]/dr for the range 0 < uor < 5 and interpret the large uor behavior in terms of your result in part (b). d) Obtain the first three terms in a power series for —d[lnG(r)]/dr, expanded about r = 0. This is the analog of the series of operator product 'power corrections' to — d[lnG(r)}/dr. Assume, as is the case in QCD, that you know only a limited number of terms in this series, first two terms and then four terms. Is there a common range of UJT for which (i) your truncated series reasonably approximates the exact behavior, and (ii) the approximation for keeping just the lowest bound state in part (b) is likewise reasonable? It is this compromise between competing demands of the resonance and operator product approximations which must be satisfied in sucessfully applying the QCD sum rules to physical systems.
XII Baryon properties
An important sector of hadron phenomenology is associated with the electroweak interactions. Baryons provide a particularly rich source of information, with data on vector and axialvector couplings, magnetic moments, and charge radii. In Sect. XII1, we describe the procedure for computing matrix elements in the quark model, and then turn to a variety of applications in the succeeding sections. XII—1 Matrix element computations Much of the application of the quark model to physical systems involves the calculation of matrix elements. The subject divides naturally into two parts. On the one hand, many quantities of interest follow from just the flavor and spin content of the hadronic states. On the other, it is often necessary to have a detailed picture of the quark spatial wavefunction. Flavor and spin matrix elements For the first of these, the quark model is particularly appealing because of the intuitive physical picture which it provides. For example, consider the quark content of the proton state vector, which we reproduce here from Table XI2,
= ^e^Kul^
 uX>!T] 1°) •
The first two quarks form a spinzero, isospinzero pair with the net spin and isospin of the proton being given by the final quark. The prefactor of I / A / 1 8 ensures that the state vector has unit normalization. Calculation reveals that onethird of the magnitude of this normalization factor comes from the u^u^ term and twothirds from the u^u^di term, i.e. one concludes that 4the proton is twice as likely to be found in the configuration 313
314
XII Baryon properties
with the ?/quark spins aligned than antialigned', (1.2) The 'six parts in eighteen' of the u^u^ configuration arises entirely from the six ways that color can be distributed among three distinct entities. The configuration u^u^dy is twice as large due to the presence of two u^ states. Similar kinds of inferences can be drawn for the remaining baryon state vectors in Table XI2. We can proceed analogously in deriving and interpreting various matrix element relationships. It is instructive to work at first in the limit of 5/7(3) invariance because more predictions become available. The effect of symmetry breaking is addressed in Sect. XII2. Let us consider matrix elements, taken between members of the spin1/2 baryon octet, of the operators Squared chargeradius : / d3x r2ift^Qi/j
oc (Q) ,
Axialvector current : / d3x ^7375X31P
oc (Xsaz) ,
Magnetic moment : / d3x  ( r x ip^cxQip)^
(1.3)
oc (Qcrz) .
Along with the definition of each operator is indicated the flavor/spin attribute of an individual quark which is being averaged over. For example, a magnetic moment is sensitive to the combination Qaz of each quark within the baryon. Matrix elements will then be products of such averages times quark wavefunction overlap integrals. The flavor/spin averages for the baryon octet are displayed in Table XII1. To see how these values are arrived at, let us compute the value 5/3 obtained for the proton axialvector matrix element. For the configuration u^u^di, which occurs with a probability of 2/3, the average value of XzCFz equals (1 + 1 + 1) x 2/3 = 2, whereas for the configuration u^u^ one finds (1  1  1) x 1/3 =  1 / 3 . Together they sum to the value 5/3. Table X I I  1 . Some baryon octet expectation values P
n
0 } 1 2/3 (A3
a
1
A
£+
0 1/3
1 1
2/V6 4/3
E~
0 l/3 a 0
1/3 4/3
0 2/3 1/3
The offdiagonal transition E° • A has \(Qaz)\ = l/>/3.
1 1/3 1/3
XII1 Matrix element computations
315
Overlaps of spatial wavefunctions The spatial description of quark wavefunctions is less well understood than the spin/flavor aspect of the phenomenology. The most extensive studies of the spatial wavefunctions are associated with matrix elements of currents. Because these are bilinear in quark fields and because of the wavefunction normalization condition, the magnitudes of these amplitudes are constrained to be nearly correct. Dimensional matrix elements are primarily governed by the radius of the bound state. As long as the proper value is fed into the calculation, the scale should come out right. As noted in Sect. XI1, a relativistic quark moving in a spin independent central potential has a ground state wavefunction of the form eiEt
,
(1.4)
gnd
where u, £ signify 'upper' and 'lower' components. For the bag model, these radial wavefunctions are just spherical Bessel functions. This form also appears in some relativistic harmonic oscillator models which use a central potential. To characterize different types of relativistic behavior, it is worthwhile to express matrix elements in terms of u and £ without specifying them in detail. The normalization condition for the spatial wavefunction is then ! dsx Vf(x)^(x) = f dsx (u2(r)+£2(r)) = 1 .
(1.5)
In the nonrelativistic regime, the lower component vanishes {£ = 0). Let us consider the size of the lower components which occur in various approaches. In the bag model one obtains for massless quarks the integrated value [ dsx£2(r)~0.26
.
(1.6)
Relativistic effects are often included in potential models by working in momentum space and employing the spinor appropriate for a quark q in momentum eigenstate p,
«(p) = yjE + mA
 .
a p
E + mq
(1.7)
X
In this case the relevant prescription is
I*'
(I 8)

316
XII Baryon properties
where the averaging is taken over the momentum space wavefunction of the particular model. Using the uncertainty principle relation of Eq. (XI1.14) to estimate (p 2), we find typical values P
2
x
~ 0.13 > 0.20
(1.9)
for a confinement scale of 1 fm. Larger effects are found in the harmonic oscillator model if one uses the value a2 = 0.17 GeV2 (see Fig. XI2). Generally, the lower component is found to be significant but not dominant in quark wavefunctions. Connection to momentum eigenstates In all cases except for the nonrelativistic version of the harmonic oscillator model, one cannot explicitly separate out the centerofmass motion. The result of a quark model description of a bound state is a configuration localized in coordinate space, i.e., a position eigenstate. However, the analysis of scattering and decay deals with the plane waves of momentum eigenstates. The basic assumption made in all quark models is that the bound state with a given set of quantum numbers is related to only those momentum eigenstates of the same type. If we denote iJ(x)} as a unit normalized hadron state centered about point x and \H(p)) as a plane wave state, then we have
)) = J dsp ¥>(p)e^ x \H(p)) .
(1.10)
We shall give a prescription for obtaining a functional form for (p(p) shortly. Let us normalize the plane wave states for both mesons and baryons as (H(p')\H(p)) = 2cup(27r)3^3)(p'  p) .
(1.11)
The constraint of unit normalization then implies
We can employ the above wavepacket description to derive a general procedure within the quark model for calculating matrix elements [DoJ 80]. Many matrix elements of interest involve a local operator O evaluated between initial and final singlehadron states. Let us characterize the magnitude of the matrix element in terms of a constant g. Then, for baryons in the momentum basis, the spatial dependence is given by (B'(p') OCr) £(p)> = g u(p')r O n(p) * ,
(1.13)
XII1 Matrix element computations
317
where To is a Dirac matrix appropriate for the operator O. By comparison, one obtains in any bound state quark model (QM) calculation a spatial dependence whose specific form is model dependent, f QM(B \O(x)\B)QM
= f(x) .
(1.14)
Hereafter, let us center all quark model states at the origin. The method of wavepackets then implies QU{B'\Jd
3
x O{x)\B)m=g J dzx J dzp'Sp
(2TT)3
(1.15)
^(p) 2 tZ(p)r o «(p) .
For sufficiently heavy bound states the fluctuation in squared momentum (p2) is small, and one may expand about p — 0, (1.16) n(p)r o u(p) = «(0)r o «(0) + O (( P 2 )/m) . A common approach consists of keeping only the leading term to obtain
u(0)rou(0) 
QM (£'
J dzx O{x)\B)m .
(1.17)
It is interesting to note that this relation, often thought of as fundamental, is in fact only an approximation. As an example, let us perform the complete quark model procedure for the neutronproton axialvector current matrix element. We begin by defining as usual
^
(1.18)
For spinup nucleons the choice /i = 3 gives u(0,1)7375^(0, T) = 2mN ,
(1.19)
yielding for Eq. (1.17) the basic formula, 9A =
QM(PI
/ d*x u(x)wsd(x)
n T ) QM .
(1.20)
The field operator for any quark q is expanded as in Eq. (XI1.1), Qa(x) =
Substituting, we have 9A =
/
d?>x
V>o,s'(x) 7375^o,s(^) ui(s)da(sf)\n^)QM
,
(1.22)
318
XII Baryon properties
where only the n = 0 ground state mode contributes. At this stage, one can factorize the spin and space components by using the general ground state wavefunction of Eq. (1.4). This leads to /
r I d x Xs(^ ^3 — ^ **3
=
d x ipQ s7375'0o s' —
CTo of
I
CL X
I ix v
Ks )
«
Q
and thus dsx (u2  i2) QM(PT ^(s.a)^ d(s',a) U^)QM (124) J o Finally, upon dealing with the spin dependence in Eq. (1.24), we obtain gA =
Any nonrelativistic quark model, having zero lower components, would simply yield QA — 5/3. If one desires to make relativistic corrections to such a model, the result can be inferred from the above general formula with the appropriate substitution of Eq. (1.8). Clearly, the procedure just given can be extended to matrix elements of any physical observable. The wavepacket formalism also allows for the estimation of the 'centerofmass' correction. This arises from the (p2) modifications to Eq. (1.16). For the axial current, the zero momentum relation in Eq. (1.19) is extended for nonzero momentum to
where an average over the direction of p has been performed. This expression generalizes Eq. (1.25) to 1
9A\1
3mnmp 2
3 rap
4
8mn 2
3mn\l
8mpJ\
=
5 f
SJ
3
/
2
\
_ 1
3 (1.27)
where (p )np — 0.5 GeV is a typical bag model value. It is possible to argue that in the transition from the current quarks of the QCD lagrangian to the constituent quarks of the quark model, the couplings to currents should be modified. For example, one might suspect that the coupling of a constituent quark to the axial current occurs not with strength unity, but with a strength g^ such that the nonrelativistic expectation is not g\ = 5/3 but rather g\ = 5gJ[ /3. The choice g£ ~ 3/4 would then yield the experimental value. This is not unreasonable, but if fully adopted, leads to a lack of predictivity. In such a picture, not only can the magnetic moments and weak couplings be
XII1 Matrix element computations
319
renormalized, but also the spin and flavor structures. That is, in the 'dressing' process which a constituent quark undergoes, there could be 'sea'quarks, such that the constituent ixquark could have gluonic, dquark, or squark content. Likewise, some of the spin of the constituent quarks could be carried by gluons. One is then at a loss to know how to calculate matrix elements of currents. In practice, however, the naive quark model, with no rescaling of g& or of the magnetic moment, does a reasonable job of describing current matrix elements. It is then of interest to study both the structure and limitations of this simple approach. Calculations in the Skyrme model There are several differences between taking matrix elements in the quark model and in the Skyrme model. To begin, in the quark model a current is expressed as a bilinear covariant in the quark fields (c/. Eq. (1.3)), whereas in the Skyrme model the representation of a current is rather different. As an example, application of either Noether's theorem or the external source method of Sect. IV6 identifies the SU(2) vector and axialvector currents to be
±Tr where U = A(t)U$A~l(t) is the quantized skyrmion form and A(t) is an SU(2) matrix. We shall neglect derivatives of A(t), as the quantization hypothesis corresponds to slow rotations. This leads to a result similar in form to Eq. (1.28), but with U * Uo and ra » A' 1 (t)raA(t). The answer may be simplified by use of the explicit form of Uo appearing in Eq. (XI4.15). Let us use Eq. (1.28) to compute the spatial integral of the axial current. After some algebra, we obtain a product of spatial and internal factors,
Jd sin2F 8sin 2 F ,, 8s
4sin 2 F sin2F
320
XII Baryon properties
where a is the isospin component and j is the Lorentz component. This is now suitable for taking matrix elements, such as )aj IPT) = I d°x I dtl3 (p T A) (JA)aj a iQ j_y
i
i I J~\. J _L1 I /
./a/
X\
) LJ
~2'2
( L3 °)
1 i I / I J ,
"2'2
where we have used the completeness relation of Eq. (XI4.42). Upon expressing the trace in Eq. (1.30) as a rotation matrix, Tr (rkAr^A~1)/2 = D\J i we can determine the group integration in Eq. (1.30) in terms of SU{2) ClebschGordan coefficients [Ed 60],
T>
— ( — \^{
V /
T+m)
9
Z7r
o
siT'TT"
(131)
fiT'TT"
27"1" I 1 ^ m
^Jn
Alternatively, one can work directly with the collective coordinates, e.g. with the aid of Eqs. (XI4.4144) we obtain for a = j = 3 r
I
K
2
i
2
2
CLr\ ~r tto — Qi —
2\/
i
•
\
CLn)[CL\ \ 1CL2) ^
9
*r^ —^5
(A
•
Qo^
[l.oZj
Before Jone can infer a Skyrme model prediction for gA from this calculation, there is a subtlety not present for the quark calculation which must be addressed. Due to the original chirally invariant lagrangian, the Skyrme model is unique among phenomenological models in being completely compatible with the constraints of chiral symmetry. As a consequence, the nearstatic axialvector matrix element is constrained to obey ?,(p(p')IM?p(p)) = 0 , and hence must be of the form [AdNW 83],
(q = pp')
(p(p')\(JA)3MP)) = 2mp9A (Sjk ~ H * ) K ) •
(1.33a)
(1.33b)
The term containing q~2 arises from the pion pole, as will be discussed in Sect. XII3 in connection with the GoldbergerTreiman relation. An angular average of Eq. (1.33b) then yields 2 ^ / 3 , which from comparison with Eq. (1.32) implies gA — G^. Thus in the Skyrme model, the axialvector coupling constant equals the radial integral in Eq. (1.29) which defines G5. Use of the profile given in Sect. XI4 leads to the prediction gA — 0.61, which is about only onehalf the experimental value and constitutes a wellknown deficiency of skyrmion phenomenology. Presumably, consideration of a more general chiral lagrangian could modify this result by including higher derivative components in the weak current.
XII2 Electroweak matrix elements
321
Pions may be added to the Skyrme description through introduction of the matrix £ previously seen in Sects. IV7,VII3 [Sc 84], U = £A(t)U0A\t)ti,
£ = exp [ir • w/(2Fff)]
.
(1.34)
If currents are formed using this ansatz, some terms occur without derivatives on the pion field, while others contain one or more factors of d^n. Since d^n gives rise to a momentum factor q% when matrix elements are taken and softpion theorems deal with the limit <$—•(), the lowest order softpion contribution will consist of keeping only terms without derivatives. Thus in the process v^ + N —> N + ir + /i the finalstate pion is produced by a hadronic weak current and the softpion theorem relates the N —> NTT matrix element to the N —> N current form factors. Expanding the currents to first order in the pion field yields
(J
x ^ = b^ [* ( rVrl M u°±w* uo)A) •
b
1
(135)
where for notational simplicity we have displayed only the first term in the current. Note the sign flip in the second line. This form is in accord with the softpion theorem Urn ( fabc e
(1.36) where the current commutation rules of Eq. (IV5.14) have been used. XII—2 Electroweak matrix elements The static properties of baryons can be determined from their coupling to the weak and electromagnetic currents. In this section, we shall describe these features in terms of the quark model.
Magnetic moments The generic quark model assumption for the magnetic moment is that the individual quarks couple independently to a photon probe. For ground state baryons where all the quarks move in relative Swaves, the magnetic moment is thus the vector sum of the quark magnetic moments, (21)
322
XII Baryon properties
where (T{ is the Pauli matrix representing the spin state of the ith quark and fii is the magnitude of the quark magnetic moment.* Since the light hadrons contain three quark flavors, the most general fitting procedure to the moments of the baryon octet will involve the magnetic moments V>u,V>d, Ms
It is straightforward to infer baryon magnetic moment predictions in the quark model directly from the state vectors of Table XI2. For example, we have seen that the proton occurs in the two configurations u^u^d^ and iXi/d with probabilities 2/3 and 1/3 respectively. This can be used to carry out the construction defined by Eq. (2.1) as follows: 2 1 3 v(uXu\dl) + 3 2 1 4 1 g g 3 3 (2.2) and similarly for the other baryons. Experimental and quark model values are displayed in Table XII2. It is of interest to see how well the assumption of SU(3) symmetry fares. In the limit of degenerate quark mass (denoted by a superbar), the quark magnetic moments are proportional to the quark electric charges, fid = p8 = —zp*
(SU(3) limit) ,
(2.3a)
Table XII2. Baryon magnetic moments Mode
ME+ I^SOAI A*E
Ms
Experiment 0 2.792847386(63) 1.91304275(45) 0.613(4) 2.42(5) 1.61(8) 1.16(2) 1.25(1) 0.69(4)
Quark Model
Fit A 6
Fit B c
V
2.79 1.86 0.93 2.79 1.61
2.79 1.91 0.61 2.67 1.63
0.93 1.86 0.93
1.09 1.44 0.49
4fJ,d^JLu
3 3
3 S
3
S
3
a
Expressed in units of the nucleon magneton fij\f = eh/2Mp. SU(3) symmetric fit. c // u ,//^,/i s taken as independent parameters. b
When referring to the 'magnetic moment' of a quantum system, one means the maximum component along a quantization axis (often chosen as the 3axis). Thus the magnetic moment is sensitive to the third component of quark spin as weighted by the quark magnetic moment.
XII2 Electroweak matrix elements
323
while isospin symmetry would imply pd = Ifa
(SU(2) limit) .
(2.3b)
If we determine the one free parameter by fitting to the very precisely known proton moment, we obtain the SU(3) symmetric Fit A shown in Table XII2. More generally, allowing /iw, /i^, /i s to differ and determining them from the proton, neutron, and lambda moments yields jiu = 1.85 /i^v 5
ftd
— —0.972 /ijv
5
l^s — —0.613 fij\f ,
(24)
and leads to the improved (but not perfect) agreement of Fit B in Table XII2. We see from Eq. (2.4) that the main effect of SU(3) breaking is to substantially reduce the magnetic moment of the strange quark relative to that of the down quark. The deviation of ^d/Hn from the isospin expectation of fid/l^u — —1/2 is smaller and perhaps not significant. Observe that the famous prediction of the SU(2) limit, fin/{\ip = —2/3, is very nearly satisfied. The magnetic moment as derived from the multipole expansion of the electric current is defined by r
3
xrxJem(x)
.
(2.5)
It follows from this expression that the contribution of a nonrelativistic quark V to the hadronic magnetic moment is just the Dirac result,
"• = 55 •
(2 6)
'
where Mq is the quark's constituent mass and Q is its charge. We can use this together with Eq. (2.4) to determine the constituent quark masses, with the result Mu~Md~
320 MeV ,
Ms ~ 510 MeV .
(2.7)
As we shall see in Sect. XIII1, these masses are comparable to those extracted from mass spectra of the light hadrons. One can also construct models involving relativistic quarks. For these, the magnetic moment contribution of an individual quark becomes
/ / = Q
(2.8)
Note the absence of an explicit dependence on quark mass. This is compensated by some appropriate dimensional quantity. The inverse radius i?" 1 plays this role in the bag model, and other determinations of R allow for a prediction of the hadronic magnetic moment. For example, the bag model defined by taking zero quark mass (corresponding to the ultrarelativistic limit) and R = 1 fm yields the value fip ~ 2.5 in a treatment
324
XII Baryon properties
which takes centerofmass corrections into account [DoJ 80]. Although this specific value is somewhat too small, it is fair to say that quark models give a reasonable first approximation to baryon magnetic moments. Semileptonic matrix elements The most general form for the hadronic weak current in the transition Bi »• B 2£ve is
where the {fi} and {gi} form factors correspond respectively to the vector and axialvector current matrix elements, and q = p\ — pi is the momentum transfer.* The form factors are all functions of q2 and the phases are chosen so that each form factor is realvalued if time reversal invariance is respected. In practice, the form factors accompanying the two terms with the kinematical factor q^ are difficult to observe because each such contribution is multiplied by a (small) lepton mass upon being contracted with a leptonic weak current. Thus we shall drop these until Sect. XII4. As regards the remaining form factors, we have already presented the ingredients for performing a quark model analysis. Using the n —+ p transition as a prototype, we have ft*> = (p T  f dsx Wyod raT) = f dsx (uuud + ejd) = 1 , (2.10a) TTlp \ fJlfi
J
Z
= lj^r{ujd 97 = (Ptl ff mn + m p
+( \2mn
d3x
+ udeu)
u^lsd
2mpJ
=l^
u +
^ ,
nT) =  // d3x (uuud  ^^£Jd),
"P +2 g3"P = vni (m  iJf
(2.10b) (2.10c)
d3x zuyOKd n T
i ( i i ) .
(2.10d)
Given the context of application, there should be no confusion between the QCD strong coupling 2 constant g% and the axialvector form factor
XII3 Symmetry properties and masses
325
In each case, we first give the defining relation, then the general Dirac wavefunction (c/. Eq. (1.4)), and finally the nonrelativistic quark model limit. The vanishing of g^p in the limit of exact isospin symmetry is a consequence of Gparity (cf. Eq. V3.22)). Predictions for the other baryoiiic transitions are governed by 5/7(3) invariance, amended by small departures from SU(3) invariance as suggested by the quark model, i.e. s —> u transitions are similar to those of d —• u as given above, but with the down quark mass and wavefunction replaced by those of the strange quark. 5/7(3) breaking in the form factors arises from this difference in the wavefunction. As a quark gets heavier, its wavefunction is more concentrated near the origin and the lower component becomes less important. The form factors of the matrix element (JB& Jc\Ba) evaluated in the 5/7(3) limit at q2 = 0 give for the vector current, /l(0) = ifabc , 1
/2(0) = ifabcf + dahcd , 3
(2.11a)
with f/d = 0.29, and for the axialvector current, fli(0) = ifabcF + dabcD ,
(2.11b)
9
with F + D = g^ = QA = 1.26. In the above, the indices a,b,c— 1,...,8 label the 5/7(3) of flavor, with c = (1 + i2) for AS = 0 and c = 4 + i5 for AS = 1. There is no SU(3) parameterization for the g u transition. The wavefunction overlaps in g\ lead to a slight increase in the strength of the s —• u transition compared to d —> u because of the reduced lower component of the s quark. For #2, a nonzero but highly model dependent value is generated, typically of order 132/511 — 03.
XII3 Symmetry properties and masses In our discussion of baryon properties, we have given priority to quark models because they are generally simple and have predictive power. However, symmetry methods are also useful, particularly when expressed in the language of chiral lagrangians. We shall combine the two descriptions in this section.
XII Baryon properties
326
Effective lagrangian for baryons We begin by writing effective lagrangians which include baryon fields, using the procedure described in Eqs. (IV7.3), (VII3.2). The lowestorder SU(2) invariant lagrangian describing the nucleon and its pionic couplings has the form  m0) N N
—N
~Y U=
= exp [ir •
(31)
where N — (^) is the nucleon field, in is the mass matrix for current quarks (with mu — rrid = m), Zo and Z\ are arbitrary constants which parameterize terms proportional to the quark mass matrix, and the constant QA is the nucleon axialvector coupling constant QA — 1.26 (c/. Prob. XII1). The mass parameter mo represents the nucleon mass in the SU{2) chiral limit. For the full SU(3) octet of baryons, the analog of W is A
E+ A
E
B = \
1—l
p n
(3.2)
2A
k
"V6
—'
where the phases have been adjusted to match our quark model phase convention of Eq. (XI1.8). The SU(3) version of Eq. (3.1) becomes  F
 D
, B]))
(33)
where the covariant derivative is now T>^B = d^B + ifV^ji?], ^ is the 5C/(3) generalization of the quantity in Eq. (3.1) with r replaced by A, m is the diagonal SU(3) quark mass matrix, m
— 7=(mms)\$
rh m.
v3
(3.4)
XII3 Symmetry properties and masses
327
and mo is the degenerate baryon mass in the SU(3) chiral limit. Consistency of the SU(2) and SU(3) lagrangians requires D + F = gA , dm + f m = 1 , mo = fho + Z\ms  Zoms(fm  dm) . The description thus far is based on symmetry. It includes quark mass, but not higher powers of derivatives. Baryon mass splittings and quark masses The various parameters (m, m s , ZQ etc.) appearing in the chiral lagrangians of Eqs. (3.1), (3.3) can be determined from baryon mass and scattering data. In the nonstrange sector, the nucleon mass is given in the notation of Eq. (3.1) as mN = m0 + (Zo + 2Zi)m .
(3.6)
To isolate the effect of the nonstrange quark mass rh and of the constants Zo, Zi, it will prove useful to define a quantity a, rh(Zo
+
2Z1) .
(3.7)
Shortly, we shall see how this quantity can be determined from pionnucleon scattering data. However, let us first consider the baryonic mass splittings generated by the mass difference ms — rh. Upon using Eq. (3.3) to obtain expressions for the baryon masses and working with isospinaveraged masses, it is possible by adopting the numerical values Z0(ms  rh) = 132 MeV ,
dm/fm = 0.31 ,
(3.8)
to obtain the following good fit: mxmN
= (fm  dm)Z0(ms  rh) = 251 MeV (Expt. : 254.2 MeV), 4 mx  mA = —d m Zo(m s  rh) = 79MeV (Expt. : 77.5Mev),
mr=mN = 2fmZ0(ms  rh) = 383 MeV
(Expt. : 379.2MeV). (3.9) Observe that these masssplittings depend on Zo but not on Z\. The three relations of Eq. (3.9) imply the GellMannOkubo formula, m s mN
= (mE
mN) + (m E  mA)
(Expt. : 254 MeV = 248 MeV ) . which displays an impressive level of agreement (c± 3%) with experimental values.
328
XII Baryon properties
The above analysis, based on a chiral lagrangian, can be enhanced by using ideas taken from the quark model. In the limit of noninteracting quarks, the quark model yields for a general spatial wavefunction,* m\ — rriN = m^ — m^ = THE — ra^ — (m
s
— rh) I Sx (u2 — i2) . (3.11)
However, observe that ra^ = m\ (corresponding in the chiral lagrangian description to dm = 0) for noninteracting quarks. Of course, the actual A and S baryons are not degenerate, so additional physics is required. A quark model source of the A — S mass splitting lies in the hyperfine interaction of Eq. (XI2.14),
where the prefactor of 1/2 is associated with the color dependence of Eq. (XI2.4). Matrix elements of this operator give rise to the additive mass contributions, m
mjsf = . . . —  W m i ?
1 8
A = . . . — Q^nn ,
1 2
"
1 2
(3 13)
1
'
8 2
where H^ and TYy are related by Hy = 7^ij^(O) and the subscripts 'n', V denote an interaction involving a nonstrange quark and a strange quark respectively. For TLnn ^ 7Yns, the S and A will not be degenerate. Treating both quark mass splittings and hyperfine effects as firstorder perturbations (e.g. Hss — Hns = Hns — ^nn), one obtains quark model mass relations = (ms rh) I dsx (u2  £2) , = l(HnnHns)
,
(3.14)
m*  mN = (Wnn  Wns) + 2(ms rh) d3x (u2  f) " 4 J in accord with the sum rule of Eq. (3.10). These formulae can provide an estimate of quark mass. For the usual range of quark model wavefunctions (encompassing both bag and potential descriptions), the overlap integral One could equivalently use the language of the potential model, where these baryon mass splittings arise from the constituent quark mass difference Ms — M.
XII3 Symmetry properties and masses
329
has magnitude r
*x{u2l2)~\^A
.
(3.15)
To the extent that this estimate is valid, it produces the values ms  m ~ 230 > 350 MeV ,
m ~ 11 > 14MeV ,
(3.16)
where the chiral symmetry mass ratio of Eq. (VII1.6a) has been used to obtain m. In general, quoting absolute values of quark masses is dangerous as one must specify how the operator qq which occurs in the mass term mqqq has been renormalized. It is all too common in the literature to ignore this point by using ms — m = m\ — TUN The values quoted here are actually currentquark mass differences, renormalized at a hadronic scale using quark model matrix elements. The parameter Z\ which appears in the SU(3) lagrangian of Eq. (3.3) is difficult to constrain in a quark model. For example, one might consider the matrix element (N\msss\N)
= ms \Z\ — Zo(jm — dm)) .
(^17)
The most naive assumption, that (N \ms'ss\ N) vanishes, would imply Z\ = Zo(fm — dm) ~ 1.9 ZQ. However, one may legitimately question whether such an assumption is reasonable. We shall return to the issue of the 'strangeness content' of the nucleon later in this section. GoldbergerTreiman relation Moving from the study of baryon masses to the topic of interactions, let us consider the coupling of pions and nucleons. The SU(2) lagrangian of Eq. (3.1), expanded to order TT2, becomes CN = N(ip — rriN)N + —N^j^—N 1
~" "
• C^TT
7T x d^K N + ^7T2NNa
+ ... ,
where a is defined in Eq. (3.7). The second term describes the NNn vertex. Upon using Eq. (3.18) to compute the pion emission amplitude N —> NTT1 and comparing with the Lorentz invariant form MN^N7ri
= ignNNuip'hsr'uip)
,
(3.19)
one immediately obtains the GoldbergerTreiman relation [GoT 58], 9AmN —^— • (3.20)
330
XII Baryon properties
Inserting the experimental value, ff^v/^71" ~ 13.5, for the nNN coupling constant, one finds the GoldbergerTreiman relation to be satisfied to about 5%. There also exist important implications for the g% term in the general expression given in Eq. (2.9) for the axialcurrent matrix element. In forming the n —• p axial matrix element, one encounters a direct 7^75 contribution and also a pionpole term which corresponds to pion propagation from the n —> pn~ emission vertex to the axialcurrent. Making use of Eq. (3.20) and Prob. XII1, we have gA
«(p)
2™>N9A
= u(pf)
!li
H
(3.21)
1 'K
J
where q = p — p1'. It is this induced pseudoscalar modification which allows the axialcurrent to be conserved in the chiral limit ra^ —» 0, = 2mN9A \l L /
2
^
J u(p')75«(p)
mA
(3.22)
,2 _
Note that for nonzero pion mass, the above is consistent with the PC AC relation of Eq. (IV5.7), ; , as both sides have the same matrix element, t
(3.23) i
q
^ m^
2mNgAml_ , g 2 _ m 2 "(P)75«(P) •
(3.24) The pionpole contribution of the axialvector current matrix element has been probed in nuclear muon capture, as will be described in Sect. XII4. The nucleon sigma term One of the features immediately apparent from the effective lagrangian of Eq. (3.1) is that all the couplings of pions to nucleons, with the exception of the quark mass terms, are derivative couplings. Before turning to the sigma term, which appears in the nonderivative sector, let us briefly consider the expansion in powers of the number of derivatives for pionnucleon scattering. Recall for pionpion scattering (c/. Sect. VI4), there
XII3 Symmetry properties and masses
331
were no large masses and the chiral expansion was expressed in terms of m\ or E\. However, correction terms in the chiral expansion for nucleons will enter at relatively low energies since a term like 2p  q ~ 2mpE7r can get large quickly (it is linear in the energy and has a large coefficient, e.g. En = 250 MeV gives 2mpEn = (700 MeV)2). To combat this difficulty, additional (but still general) inputs such as analyticity and crossing symmetry are often invoked. Fortified with these theoretical constraints, one then matches intermediate energy data to the low energy chiral parameterizations. The low energy chiral results thereby obtained appear to be well satisfied [Ho 83, GaSS 91]. The nonderivative pionnucleon coupling coming from the quark mass terms in Eq. (3.1) is of particular interest. To determine this contribution from experiment, one must be able to suppress the various derivative couplings. Thus, if one extrapolated in the chiral limit to zero fourmomentum, the derivative couplings would vanish. Not surprisingly then, a softpion analysis reveals that the nonderivative coupling can be isolated by extrapolating the isospineven nN scattering amplitude with Born term subtracted (called D+ in the literature) to the socalled 'ChengDashen point' t = ra2, s — m2N [ChD 71]. It is conventional to multiply the extrapolated amplitude by F% and thus define a quantity S, X = F%D+D
.
(3.25)
To lowest order in the chiral expansion, the measured quantity £ is just the matrix element a defined in Eq. (3.7), 2mN
v
}
It is this isospineven scattering amplitude D+ which provides a unique window on the nonstrange quark mass ra. Because £ is proportional to the small mass m, it is difficult to determine this quantity precisely, and considerable effort has gone into its extraction. The ChengDashen point lies outside the physical kinematic region, and extrapolation from the experimental region must be done carefully with dispersion relations. Present estimates are f 64 ± 8 MeV S = I [60 MeV
[Ko 82] ,
[
[GaLS90] .
(3.27)
The result a = £ — 15 MeV has been obtained from studies of higher order chiral corrections, implying
(3.28)
332
XII Baryon properties Strangeness in the nucleon
In light of the above discussion, it is tempting to interpret various contributions to the nucleon mass by making use of the energymomentum trace. Recall the trace anomaly of Eq. (Ill—4.16), muuu + mddd + msss .
(3.29)
Taking the nucleon matrix element gives = TUQ + (7 ,
2
N)  8 9 4 ± 8 MeV , (3.30)
'
. (N\uu + dd\N) ! a — m— ! — ~ 45 MeV This result is already quite interesting in that the largest contributions, the gluon and strange quark terms in mo, appear to be cnonvalence'. At this stage, the separation is essentially model independent. One can learn something about the 'strangeness content of the nucleon' by using an SU(3) analysis of hyperon masses. Thus, we introduce a masssplitting operator which transforms as the eighth component of an octet, CmS —(m — ms)(uu\dd— 2ss) . (3.31) o Since the hyperon mass splittings are governed by this octet operator, we find (p\(msm)(uu + dd2ss)\p) 3 ^ M V ! 6S = — ! — = (m E  mN) = 574MeV . Zrrip
1
(3.32a) When scaled by the quark mass ratio m/ms, Eq. (3.32a) becomes ^ (N\uu + dd2ss\N) ! ! o=m 2mN , ( (6.62b) o = o l 9 * 0(m'=mA)~25MeV (35MeV) , 2m Km\ where the figure in parenthese includes higher order chiral corrections [Ga 87]. Comparison of 6 and a immediately indicates that they are compatible only if the strange quark matrix element does not vanish. Indeed, one requires (N\ss\N) 0.18 (0.09) . {N\uu + dd + ss\N)
(3.33)
XII3 Symmetry properties and masses
333
This gives for the constant Z\ of Eq. (3.1) the value Z\ ~ 3.9Z0 (2.9 Zo) to be contrasted with the estimate which follows Eq. (3.17). At the same time, one can separate out the following matrix elements (2mN)l(N
PQCD pg
N)  6 3 4 MeV (764 MeV) ,
(3.34)
1
(2mN) (N \msss\ N) ~ 260 MeV (130 MeV) , where figures in brackets use the corresponding bracketed quantity in Eq. (3.32b). Note the surprisingly large effect of the strange quarks. These results are not without controversy as they rely on the use of SU(3) symmetry. However, the difference between a and the SU(3) value of 6 is large enough that some ss contribution is likely to be required in any case. This analysis does not go well with the naive interpretation of the quark model as embodied, for example, in the proton state vector formula which began this chapter. However, it is not incompatible with a more sophisticated interpretation of the constituent quarks which enter into quark models. In the process of forming a constituent quark, the quark is 'dressed' by gluonic and even ~ss quark fields. It is no longer the naive object that occurs in the QCD lagrangian. It is this dressed object which then may easily generate gluonic and perhaps strange quark matrix elements. Recall that even the vacuum state has gluonic and quark matrix elements. Similar explanations exist in bag and Skyrme models [DoN 86]. Further work is clearly needed to decide on the proper decomposition of the nucleon mass and to understand its significance. Quarks and their spins in baryons The quark model provides a most simple picture of the contents of baryons as systems composed of three constituent quarks and nothing else. Although a description using the quark and gluon degrees of freedom which appear in the fundmental lagrangian may be more complicated, it is nevertheless instructive to explore the constituent picture of the spin structure of the nucleons. For any Lorentz invariant theory, Noether's theorem requires that there be an angular momentum tensor M^a^ which is conserved (d^M^P = 0) and which gives rise to three angular momentum charges associated with rotational invariance, = f dsxMOaP(x)
.
(3.35)
334
XII Baryon properties
In the rest frame of a particle, the {J Q ^} are related to the three components of angular momenta as j
For the example of a free fermion, the above quantities take the form up to total derivatives which do not contribute to the charges, and
J = I dsx \i^
(x x d) $  ^ 7 5 J = L + S .
(3.38)
The two contributions in Eq. (3.38) may be labeled the orbital and spin components of the angular momentum. The quarks in the Noether current are lagrangian (current) quarks, not constituent quarks. Nevertheless, in the spirit of the quark model let us apply Eq. (3.38) to the quarks in a spinup proton. As expressed in terms of upper (u) and lower (£) components (cf. Eq. (XI1.13)), the orbital and spin contributions are found to be
(L) = 1 1 d*x /V> ,
(S) = / d*x (u2  ^ 2 ) M .
(3.39)
Aside from the factor 1/2 occurring in a/2, the quark spin contribution to S is just the axialvector matrix element of Eq. (1.24), whereas the orbital angular momentum contains just the lower component £ because the x x d operator has a nonzero effect only when acting on the a • x factor in the lower component of Eq. (XI1.13). Observe that the orbital angular momentum is nonvanishing and proportional to the quark spin. The spin and orbital portions for the individual u, d flavors are easily computed to yield
=\ f
(3.40) A first lesson is that, despite the spin wavefunction of the protons being written entirely in terms of quarks as in Table XI2, the quark spin averages of Eq. (3.40) do not add up to yield the proton spin. The sum is reduced from the anticipated value of 1/2 by the lower component £ in the Dirac spinor. It is the total angular momentum J which has the expected result, <J> = ^<0> ,
(341)
XII3 Symmetry properties and masses
335
but the total is split up between the orbital and spin components. Because of the identification of the spin with the axial current, there is data on spin contributions of the individual quark flavors. Thus from nucleon beta decay we have a combination which transforms as the third component of an octet,
With the use of SU(3) symmetry, the eighth component of the same octet can be inferred from hyperon decay data, <5 W
+ sW  2fl<')> = ^ • ^
§
* \ (0.57 ± 0.05) ,
(3.43)
where we use (3F — D)/(F + D) ~ 0.45 ± 0.04 as implied by empirical fits.* Some caution is required because SU(3) breaking is both expected and observed in fits to hyperon data. The quoted error bar includes our estimate of the uncertainty due to SU(3) breaking, where the purely experimental uncertainty is about half the quoted error bar. If the strange quark matrix element were zero, Eq. (3.43) would be the total spin component and one could extract the content of each separate quark spin. However, other data appear to contradict this assumption. The weak neutral current contains an isoscalar axial component [CoWZ 78] associated with nonvalence quarks, f
(3.44)
Neutral current neutrino elastic scattering experiments give an indication of a possible signal of this operator [Ah et al 87],
(p(P,A')l4/=0)p(p,A)) =gou(pA'h^5u(p,X)
,
(3.45)
with 50 = 0.15 ±0.09. There are also results from polarized deep inelastic experiments [As et al. 88] which are sensitive to the matrix element*
and which have thus been interpreted as probing quark spin,
(SM + S^ + Si\2
= \x(g2) .
(3.46)
(3.47)
While the experimental error bar is such that the result S(g 2) = 0.01 ± 0.29 at the q2 range of the experiment* is only two standard devia* The quark model predicts 0.50 for this ratio in the SU(S) limit. t Here T,(q2) should not be confused with the use of E in the crterm discussion, Eq. (3.25). * Because of the axial anomaly, the current in Eq. (3.46) is not conserved, which implies a QCDinduced dependence on q2. Some of the issues in this extraction are discussed in [JaM 90].
336
XII Baryon properties
tions from the result of Eq. (3.43) together with the simplest assumption (Sz ) ^ 0, the central value of this measurement has aroused substantial interest since it would imply that the spin component of light quarks is nearly vanishing. This latter result along with aspects of the analysis remains controversial, and hopefully will be clarified in the near future.
XII4 Nuclear weak processes One area in which the structure of the weak hadronic current has received a great deal of attention is that of nuclear beta decay and muon capture. Although in some sense this represents simply a nuclear modification of the basic weak transitions n —> p e~~ ve , p —> n e+ ve, the use of nuclei allows specific features to be accented by the choice of levels possessing particular spins and/or parities [Ho 89]. Here, we shall confine our attention to allowed decays (A J = 0, ±1, no parity change) and will emphasize those aspects which stress the structure of the weak current rather than that of the nucleus itself. In particular, nuclear beta decay provides the best determination of Vu^ while muon capture provides the only measurement of the pseudoscalar axial weak form factor predicted by chiral symmetry. Measurement of Vud There are many occurrences in nuclei of an isotriplet of Jp — 0 + states. Examples are found with A = 10,14,26,34,42, Because coulombic effects raise the mass of the protonrich /3=1 state with respect to that with /3=0, the positron emission process N\(IZ = 1) —• N2(IZ — 0) + e + + ve can occur. These transitions are particularly clean theoretically, and this is the reason why they are important. Since the transition is 0+ —> 0 + , only the vector current is involved, and because of the lack of spin there can be no weak magnetic form factor. The vector current matrix element involves but a single form factor a(g 2), (N2(P2)\Vli\N1(Pl))
= a(
•
(4.1)
This form factor is known at q2 = 0 because the charged vector weak current V^ is just the isospin rotation of the electromagnetic current, [I,JU = dfu
•
(4.2)
This relation is often called the conserved vector current hypothesis or CVC, and requires for each of the 0 + —• 0 + transitions, a(0) = y/2 .
(4.3)
XII4 Nuclear weak processes
337
Table XII3. Energy release and Tt\j2 values for 0 + •+ 0+ Fermi decays. Nucleus
£ 0 (KeV)
.Fix/2 (sec)
14Q
1868.44(27) 3209.95(25) 4470.27(18) 5020.49(56) 5403.02(28) 6028.62(69) 6610.01(41) 7220.14(32)
3078.7(3.6) 3071.7(3.4) 3081.4(4.4) 3073.7(5.1) 3081.1(7.1) 3081.6(4.0) 3075.6(5.3) 3077.1(4.2)
26m^ 34
C1
38m j^" 42
Sc
46y 50
Mn Co
54
What is generally quoted for such decays is the Tt\j2 value, essentially the halflife ti/2 multiplied by the (kinematic) phase space factor T [WiM 72]. Theoretically, one expects a universal form
which should be identical for each isotriplet transition. G^ is the weak decay constant measured in muon decay while the logarithmic correction arises from 'hard' photon corrections, as discussed in Chap. VI. The 'soft' photon piece as well as finitesize and Coulombic corrections are contained in the phase space factor T. Much careful experimental and theoretical study has been given to this problem, and the current situation is summarized in Table XII3 where the experimental Tt\j2 values are tabulated. A fit produces the value Tt\j2 — 3077.3 ± 1 . 5 sec with chisquared per degree of freedom %2/u = 0.8. This excellent agreement over a wide range of Zvalues is evidence that soft photon corrections are under control. Comparison of the experimental Tt\j2 value with the theoretical expression given in Eq. (4.4) yields the determination Vud = 0.9744 ± 0.0010 ,
(4.5)
which makes Vud the most precisely measured component of the KM matrix. The pseudoscalar axial form factor Chiral symmetry predicts a rather striking result for the form factor gs(q2) of Eq. (2.9), namely that it is determined by the pion pole with a coupling
338
XII Baryon properties
fixed by the PC AC condition. One cannot detect this term in either neutron or nuclear beta decay because when the full matrix element is taken, one obtains 53
(4.6)
v(P)
2mN
which is proportional to the electron mass and is thus too small to be seen (effects in the spectra are O(m2/mjsfEe) « 1). However, in the muon capture process /i~p —• i^n, the corresponding effect is 0(171^/171^) ~ 10%. Thus muon capture is a feasible arena in which to study the chiral symmetry prediction. The drawback in this case is that typically one has available from experiment only a single number, the capture rate. In order to interpret such experiments, one needs to know the value of each nuclear form factor at q2 ~ — 0.9 raj;, which introduces some uncertainty since these quantities are determined in beta decay only at q 2 ~ 0. Nevertheless, predicted and experimental capture rates are generally in good agreement provided one assumes (i) the q2 ~ 0 value of form factors from the analogous /3decay, (ii) q2 dependence of form factors from CVC and electron scattering results, (iii) the CVC value for the weak magnetic term /2, and iv) the PC AC value of Eq. (3.14) for #3. The results are summarized in Table XII4. Obviously, agreement is good except for 6 Li, for which the origin of the discrepancy is unknown, although it has been speculated that perhaps the spin mixture is not statistical. Table XII4. Muon capture rates. Reaction
\T
3
He >
J/M
3
H
Theory (103 sec"1)
Experiment (103 sec"1)
0. 664 ± 0.020a 0. 506 ± 0.0206
0.651 ± 0.057 0.515 ± 0.085 0.464 ± 0.042
1.510 ± 0.040
1.410 ± 0.140 1.505 ± 0.048 1.465 ± 0.067
/* 6 Li + i/M 6 He
0.98 ±0.15
u i a C _
7.01 ±0.16
w
i2B
012
6.2 ± 0.3 6.7 ± 0.9 0.75
XII5 Hyperon semileptonic decay
339
Before proceeding, we should emphasize one relevant point. When PC AC is applied, it is for the nucleon 2mNgi(q2) 
2
^9S(Q2)
= 2F^NN{q2)
2 \ ~1
/
[l  ^J
.
(4.7)
Then at q2 = 0, we have 1.26 = gi(0) ~ F^NN^> = 1.33 (4.8) m which is the GoldbergerTreiman relation. On the other hand, taking similar q2 dependence for gi(q2) and gKNN{(l2), we find mM <73(0.9m2) m2 + 0.9m2 "
'
v
J
PC AC is generally applied in nuclei in the context of a simple impulse approximation, and it is this version of PC AC which is tested by the muon capture rates listed in Table XII4. The direct application of PC AC in nuclei cannot generally be utilized since the pion couplings are unknown. In the case of muon capture on 12C, additional experimental data is available. One class of experiment involves measurement of the polarization of the recoiling 12B nucleus. Combining this measurement with that of the total capture rate yields a separate test of CVC as well as of PC AC. The results ™
^ ~
™ = 8.0 ±3.0 ,
(4.10)
are in good agreement with both symmetry assumptions. In addition, one can measure the average and longitudinal recoil polarizarions in the 12C muon capture, yielding a value for the induced pseudoscalar coupling (0.9m2) — " which is again in good agreement with PC AC.
'
v~">
XII5 Hyperon semileptonic decay The goals in studying hyperon semileptonic processes are to confirm the value of V^ obtained in kaon decay and to use the form factors to better understand hadronic structure. These two goals are interconnected. In earlier days when data was not very precise, fits to hyperon decays were made under the assumption of perfect SU{3) invariance in order to extract V^. Presently, the experiments are precise enough that exact
340
XII Baryon properties
SU(3) no longer provides an acceptable fit. The desire to learn about Vus is thus impacted by the need to understand the SU(3) breaking. We have already described in Sect. XII1 the physics ingredients which lead to SU(3) breaking within a simple quark description. These include recoil or centerofmass corrections, wavefunction mismatch (in which a normalization condition realized in the symmetry limit no longer holds), and generation of the axial form factor #2 For hyperons, because of the presence of the axial current, SU(3) breaking can occur in first order. This means that hyperon decays are more difficult to use for determining V^s than are kaon decays, where the AdemolloGatto theorem reduces the amount of symmetry breaking. Thus at the moment it is probably best to use the value of Vus determined from kaon decay, and require that hyperon decays yield a consistent value. The clearest evidence on SU(3) breaking comes from the S~ —• Ae~i>e rate. Since this is a AS — 0 process, V^ does not enter, and in addition, the vector current matrix element must vanish. Thus the rate is determined by the axialcurrent contribution alone, for which the theoretical prediction is
where p is a 517(3) breaking factor due to the centerofmass effect. A bag model estimate yields p — 0.939. Taking p = 1, the best SU(3) symmetric fit to all the data [Ro 90] would require D/(D + F) = 0.629 ± 0.001, and hence gf ~A = 0.647. On the other hand, the data on E~ —> Aez/e requires A = 0.591 ± 0.014, which implies the correction p = 0.914 ± 0.022. gf~ There seems to be no way to avoid this need for 517(3) breaking. The full pattern of SU(3) breaking is more difficult to uncover. One problem is experimental. When the g\ values are extracted from the data, they have generally been analyzed under the assumptions that the /i and J2 form factors have exactly their SU(3) values and that g2 = 0. If these assumptions are not correct, then the values cited in [RPP 90] do not reflect the true g\ but rather some combination of g\, j \ , fa and #2 The correlation with g Ae ^ e , seems likely incorrect as it fits so badly with the remaining patterns. Discarding it, the remaining data can be fit well either by the centerofmass correction described above, with g2 = 0, or by the full corrections including wavefunction mismatch, with 92/gi = 0.20 d= 0.07 in A —• peve. Without an independent measurement of #2 one cannot decide between these. We note however, that either
XII6 Nonleptonic decay
341
option yields a value of Vus consistent with that found in kaon decays, Vus = 0.220 ± 0.004 .
(5.2)
XII6 Nonleptonic decay The dominant decays of hyperons are the nonleptonic B —• Bfn modes. Because of the spin of the baryons and the many decay modes available, the nonleptonic hyperon decays present a richer opportunity for study than do the nonleptonic kaon decays. Phenomenology The B —• B'n matrix elements can be written in the form MB^B'* = u(p')[A + B>K]u{p) ,
(6.1)
with parityviolating (A) and parityconserving (B) amplitudes. Watson's theorem implies that if CP is conserved, the phase of these amplitudes is given by the strong B'TT scattering phase shifts in the final state Swave (for A) or Pwave (for B), i.e. A = A0 exp (itff/?r) ,
B = Bo exp (i6%,n) ,
(6.2)
with Ao, Bo real (if CP is conserved). Aside from the nN system, these phase shifts are not known precisely, but are estimated to be ~ 10° in magnitude. The decay rate is expressed in terms of the partial wave amplitudes by \g\(E' where q is the pion momentum in the parent rest frame and we define B = (E1 — TUB'/Ef + rriBf)l^B. Additional observables are the decay distribution W(0), W(9) = 1 + aPB • pB, ,
a= " j ; v ^
~'
,
(6.4)
and the polarization (P#') of the finalstate baryon, (a + PB • i>B')PB' + 0 (PB x pBf) + 7 [PB' X (Pfi x PB')] W(9) B) =
V
p
'
(
}
where P # is the polarization of B and p#/ is a unit vector in the direction of motion of Bf. Experimental studies of these distributions lead to the amplitudes listed in Table XII5.
342
XII Baryon properties Table XII5. Hyperon decay amplitudes" A amplitudes Thy6
Mode
Expt
A —•
pn~
A —•
UTT0
3.38 3.25 2.37 2.39 0.00 0.13 3.27 3.18 4.27 4.50 3.14 3.43 4.51 4.45
£ +  > 717T+ E + ^j97T 0 S ~ —• 7i7T~ ^ 0  > ATT° S ~ —• A?r~
B amplitudes Expt
Thy
22.1 23.0 15.8 16.0 42.2 4.3 26.6 10.0 1.44 10.0 12.3 3.3 16.6 4.7
°In unitsof 10 7 b Lowestorder chiral fit.
The nonleptonic amplitudes may be decomposed into isospin components in a notation where superscripts refer to A / = 1/2, 3/2,
AA_^ = V2A?  Af ,
^ — = f + 43)
(6.6) and X% is of mixed symmetry. Similar relations hold for the B amplitudes. Prom the entries in Table XII5, it is not hard to see that the A / = 1/2 rule, described previously for kaon decays, is also present here. Table XII6 illustrates that the dominance of A / = 1/2 amplitudes compared to those with A / = 3/2 holds in the six possible tests in 5wave and Pwave hyperon decay, at about the same level (several per cent) as occurs in kaon decay.* Thus the A / = 1/2 rule is not an accident of kaon physics, but is rather a universal feature of nonleptonic decays. This makes the failure to clearly understand it all the more frustrating. The assumption that the dominant A / = 1 / 2 hamiltonian is a member of an SU(3) octet leads to an additional formula, called the LeeSugawara relation, ^Imo=2AE^+AA^FK
,
(6.7)
* For Pwaves, the observed smallness of # £  _ > n 7 r  indicates that 2?^ is small, presumably accidentally so. In this case the measure of A/ = 3/2 to A/ = 1/2 effects is given by By, ' /Xy^ •
XII6 Nonleptonic decay
343
which also is well satisfied by the data. In this case, the corresponding formula for the B amplitudes is not a symmetry prediction [MaRR 69], although for unknown reasons it is in qualitative accord there also. Lowestorder chiral analysis Chiral symmetry provides a description of hyperon nonleptonic decay which is of mixed success when truncated at lowestorder in the energy expansion. Given our comiii@nts on the convergence of the energy expansion for baryons made in Sect. XII3, the need for corrections to the lowestorder results is not surprising. We shall present the lowestorder analysis here, as it forms the starting point for most theoretical analyses. Recalling from Sect. IV7 the procedure for adding baryons to the chiral analysis, one finds that the two following nonderivative lagrangians have the chiral (8^, 1R) transformation property:
=DbTr ( B 7 5 {&A6£,B}) + FbTV ( [ % ] ) where £,i? are defined in Eqs. (3.1), (3.2) respectively. However, the operator C(P) w must vanish, as it has the wrong transformation property under CP [LeS 61]. That is, a CP transformation implies B • (i7 2B)
,
£ * UTJ
,
(6.9)
and including the anticommutation of B and B, Cw is seen to return to itself, but C\y changes sign and hence must vanish. This leaves C\^ as (S)
the only allowed chiral lagrangian at lowest order. Observe that CKW} lacks a 75 factor. Thus its B —> B'TT matrix elements will be parityviolating, leading to only A amplitudes. The parityconserving B amplitudes in Table XII6. Ratio of A/ = 3/2,1/2 amplitudes 5wave Pwave A 0.014 0.006 E 0.017 0.047 S 0.034 0.023
344
XII Baryon properties TT/
Hw
T7 /
gfl Hv
Fig. XII1 Pwave hyperon decay amplitudes
B —• B'TT are produced through pole diagrams as in Fig. XII1, and are proportional to the parityconserving B —> B1 matrix elements of Dw'. The counting of powers of energy (momentum transfer) in the energy expansion goes as follows. Both the B —> Bf transition and the A amplitudes in B —• B'n are obtained as matrix elements of C\^, which is zeroth order in the energy. The pole diagrams are likewise of zeroth order in the energy, being the product of the C\y vertex (0(1)), a baryon propagator (O(q~1)) and an NNn vertex (O(q)). Since the kinematic part of the pole diagrams, u'^u ~ cr • q, is of first order in g, the B amplitudes themselves are of order B ~ q~l ~ I/Am for the baryon pole. Kaon poles and higherorder chiral lagrangians enter at next order, i.e. having one power of the momentum transfer. The lowestorder chiral SU(3) analysis provides a fit to the data in terms of two parameters, called F and D,
with other amplitudes being predicted by the A/ = 1/2 rule. Use of the numerical values ^ = 0.42,
^  = 0.92xl(T 7 ,
(6.11)
leads to the excellent fit of the Swave amplitudes seen in Table XII5. Note that this form has one less free parameter than the general SU(3) structure [MaRR 69]. Thus the prediction of chiral symmetry that ^4x;+^n7r+ — 0 is independent of the D/F ratio (up to A/ = 3/2 effects), and represents a successful explanation of the smallness of this amplitude. In principle, the A amplitudes, together with the strong BB'n vertices, determine the baryon pole contributions to the B amplitudes. There should be no ambiguity since the strong vertices follow from the effective lagrangian of Eq. (2.1) already given for the axialcurrent matrix elements. These are then parametrized by the same d/f ratio as in the
I* 6 Nonleptonic decay
345
axial current,* e.g. 2/ J

Using this parameterization for the pole diagrams, one finds contributions such as w
2mNF7r
Taking d + f = 1.26, d/f — 1.8, one obtains from relations like this the disappointing Pwave predictions quoted in Table XII6. This failure to simultaneously fit the 5waves and Pwaves is a deficiency of the lowest order chiral analysis. To correct this problem one has various options [DoGH 86]. One must introduce momentum dependent vertices, but this may be done in the 5wave, the Pwave, or both. A nextorder effective lagrangian analysis is not very useful, because although the weak amplitude may be easily fit, there are too many possible effective lagrangians to decide whether the corrections are in the Swave or Pwave. Although various solutions have been proposed in quark models, such as inclusion of 1/2" pole contributions [LeOPR 79], there is no general agreement among the models. A fair summary of the situation is that, while solutions to the 5wave/Pwave puzzle do seem possible through nextorder corrections, a consensus on the unique solution chosen by Nature has not yet emerged. Quark model predictions For hyperon nonleptonic decay, there is a remarkably simple starting point which yields the A/ = 1/2 rule for at least a portion of the amplitude. In the B —• B' matrix elements, the A/ = 1/2 rule is automatic for ground state baryon state vectors of three quarks which are antisymmetric in color. This in turn implies that the portion of the decay amplitudes described by the lowestorder chiral analysis above will satisfy the A/ = 1/2 rule. The proof makes use of the Fierz rearrangement property (cf. Eq. (C2.11)) of the product of chiral currents to rewrite the A/ = 3/2 operator O4 as _ 1 O 4 = 4 (<5ij<5ki + <5ii<5kj) diLlffMUjLUk^siL  zdiL^^djLdk^siL
L
i
,
(6.14)
* This statement is the SU(S) generalization of the GoldbergerTreiman relation, Eq. (3.24).
346
XII Baryon properties
where we work with lefthanded fields as in Eq. (12.3) and i,j,k,l are color indices. Note that O4 is symmetric under the interchange of the colors of u and s (or d and u) in the first term and of d and s in the second. When a matrix element such as
Mfn = (nO4A) (6.15) is taken, within the approximation that baryons only contain three quarks the fields u and s must annihilate the quarks in the A. However, in this picture the A is antisymmetric under the colorinterchange of any two quarks. Since O4 is symmetric and A is antisymmetric under this interchange, the matrix element MAN must vanish when all quark colors are summed over. As pleasing as this simple result is, it cannot be regarded as completely satisfactory. The B —> B' matrix elements are not responsible for all the B —> B'TT amplitudes, so the A/ = 1/2 rule must be also shown to hold for the remaining contributions. In addition, the 'threequark' model of baryons is an oversimplified approximation, and we must hope that the 'sea' of quarks and antiquarks does not upset this result. A second prediction which can be derived simply in the threequark approximation is that, in the absence of the penguin diagram, one must have D/F = — 1. Recall from Sect. VIII3 that with out including the penguin diagram the hamiltonian can be written entirely in terms of the operators O± = d7M(l + 75)imYi(l + 75)5 ± ^ ( l + 7 5 ) ^ ( 1 + 75)5 . (6.16) Among the B —> Bf matrix elements parameterized by D and F, one finds F) . (6.17) However S~ and S~ contain no i^quarks, while matrix elements of O+ vanish unless the states contain ^quarks. Therefore we find . M ^  ^ s  = 0, implying D/F = — 1. Modification of this result would require sea quarks and/or the penguin diagram, interpreting the latter as a perturbative way to include the effects of nonvalence quarks. As regards the more model dependent issue of the overall magnitude for nonleptonic decay amplitudes, quark models are seen to give approximately the right size for the matrix elements. Consider, for example, the contribution of operator O\. Using the general Swave quark wavefunctions of Table XI2, it is a relatively simple calculation to show that = ^V:dVusd
Jd3x
(u2 + £2)2 .
(6.18)
Insertion of bag model wavefunctions with bag radius R ~ 1 fm yields M^+p — —5.5 x 10~ 8 GeV, while the parameterization of Eq. (6.11)
Problems
347
would require M^p = V6(DF) = 6.0 x 10"8 GeV. This calculation is essentially universal for all quark models, aside from the magnitude of the wavefunction overlap. Any variation of the magnitude may be understood by noting that due to the normalization of the quark wavefunction ipty ~ 1/V (where V is a typical baryon volume), a fourfermion matrix element scales like V~l, /
Sx (VV) ~V~l
.
(6.19)
Those quark models in which the quarks are tightly bound (small V) yield larger matrix elements than those with loosely bound quarks. In summary, the theoretical status of hyperon nonleptonic decays, while far from perfect, is better than that for kaons. Here, we have a simple starting point which comes within factors of two of the desired results. However, additional ingredients are clearly required for a more believable description. Problems 1) The axialvector coupling Consider the effective lagrangian in Eq. (3.1) for nucleons and pions. For combined lefthanded and righthanded transformations of the fields, we have
U > LUR] , £ > L£V* = V£rf , N > VN , where L[R] are the spacetime independent SU(2) matrices corresponding to global transformations in SU(2)L[SU(2)R] and V — V(n(x)) is an SU(2) matrix describing a vectorial transformation of the nucleons. For the lagrangian of Eq. (3.1), use Noether's theorem to generate the SU(2) axialvector current,
where ^ is the 'square root' of U (cf. Eq. (3.1)), and thereby show that the axialvector coupling constant for beta decay is given by g\ — gA2) Nonleptonic radiative hadronic decays a) In addition to nonleptonic pion emission B —• B'TT, there exists the nonleptonic radiative mode B —• B'^. Show that the most general form for the radiative decay amplitude is + where C, D are respectively the Ml, El transition moments. TUB
348
XII Baryon properties b) Show that the rate for radiative decay of an unpolarized baryon B is given by = 2a ( m B ^ m B / ) ( m B + mB'){\C c) Demonstrate that if baryon B is polarized with polarization vector P#, the differential decay rate is dvB^Bfl
=
i r^ ^
{1 +
pB.eiaB,
)
where aBi1 = 2Re (C*D)/[C2+.D2] is the asymmetry parameter and q is the photon direction. d) Using Eq. (6.8) for the weak BB couplings, show that in the limit of SU(3) symmetry the only nonvanishing contribution to the decay amplitudes comes from the baryon pole piece of the M l amplitude C. What does the experimental result a^ — — 0.83 ± 0.12 from E + radiative decay indicate about the validity of SU(3) invariance in these decays? 3) CP violation and nonleptonic hyperon decay Although the AS = 1 hamiltonian of the Standard Model contains a CPviolating component, there is no practical way to see this in any single hyperon decay mode. Rather, one must compare the decays of hyperons with those of antihyperons [DoHP 86]. In the presence of CP violation, there are two sources of phases in the weak matrix elements, e.g. for the A decay modes,  = A, e*l eiS" + A3 e^
where the isospin (I) subscripts '1,3' stand for A / = 1/2, 3/2, the angular momentum (J) superscripts 'S, P ' stand for *Swaves or Pwaves, Aj are real amplitudes, 6j are strong final state phases and (pj are the weak CP violating phases. Observe that there are three small parameters in these amplitudes  the weak phases (pj, the strong phases 6j ~ 10°, and the ratio of A / = 3/2 to A / = 1/2 effects. To leading order in these quantities, show that one has the CPodd observables, P\~/3
——
=
.
/
<
? p \
a + a
= sin (
r
p7r ~ J W
=
.
/
<
? p \ .
AXAS sin(gf  6j) sin(yf  yf)
A 2
I V I V
"
/ r ?
r P \
_ _ =  sin (
2
^
S2
n.p + iftp
Problems
349
A hierarchy is apparent in these three signals. The (3 + (3 asymmetry requires only the weak phase, the a + a asymmetry requires both the weak and final state phases, while F — F has both phases plus a A/ = 3/2 suppression. Present experiments are not sufficiently sensitive to test for CP violation in these observables at the required accuracy.
XIII Hadron spectroscopy
Studies of hadron masses, and of both strong and electromagnetic decays of hadrons, provide insights regarding QCD dynamics over a variety of distance scales. Among various possible theoretical approaches, the potential model has most heavily been employed in this area. We shall start our discussion by considering heavyquark bound states, which begin to approximate truly nonrelativistic systems and for which the potential model is expected to provide a suitable basis for discussion. XIII1 The charmonium and bottomonium systems Quarkonium is the bound state of a heavy quark Q with itsantiparticle. Two such systems, charmonium (cc) and bottomonium (bb) have been the subject of much experimental and theoretical study. Due to weak decay of the top quark, the (as yet undiscovered) ti system will almost certainly have rather different properties from these, and thus constitutes a special case (c/. Sect. XIV2).
Table XIII1. Nomenclature for Swave and Pwave states in the cc and bb systems.
L S Charmonium 0 1
1 0
tp(nS)a
Vc(nS)
Vb(nS)
1
XcAnP) hc(nP)
XbAnP) hb(nP)
0 a
Bottomonium
?(nS)
For historical reasons, the spinone charmonium ground state is called J/X/J.
350
XIII1 The charmonium and bottomonium systems 4.0

?
35
_
(
fl(2S)^
3.0
_ Tl(IS) +
jPc=
0

%<1P)
hc(lP),
/ Jz \] (
/t,«
CD
351
—^XodP) > —
_
"
J
J/y(IS)
r

,
—
0

,••
2 *+
Fig. XIII1 The lowlying spectrum of charmonium. Since the quarkonium systems are quarkantiquark composites, we shall employ the set of quantum numbers n, L, S, J introduced in Sect. XI2. One generally refers to the individual quarkonium levels with the nomenclature of Table XIII1, although the nL identification is sometimes replaced by either the degree of excitation or the mass, e.g. I/J(2S) is called ip' or '0(3686). The n2S+lLj spectroscopic notation is also invoked on occasion. Figs. XIII1,2 give a summary of the lightest observed cc and bb states. At present, there are no firm candidates for lP\ states in either system and only one reasonably firm Dwave assignment, tjj{377Q). The greatest observed degrees of excitation come from the ijj{nS) and Y(nS) radial towers, reaching up to n = 6 for the T system. Excitation energies are relatively small on the scale of the bottomonium reduced mass //& ~ 2.5 GeV, but not that of charmonium /i c ~ 0.8 GeV. By far, the greater part of the theoretical effort on interpreting quarkonium systems has been performed in the context of nonrelativistic quantum mechanics [QuR 79].
2MB
TK(1S>
Fig. XIII2 The lowlying spectrum of bottomonium
352
XIII Hadron spectroscopy
Thus, quarkonium mass values are expressed as ™>[nLSJ] = 2 M Q + E[nLSJ]
,
(1.1)
where E\nLSJ] ls obtained by solving the Schrodinger equation for a particle of reduced mass /XQ = MQ/2 moving in the field of an assumed potential energy function. In the following, we shall consider several aspects of quarkonium systems. Lattice studies: The ultimate aim of latticegauge studies is to show that the potential picture is a consequence of QCD, and to even specify the quarkantiquark potential itself. Although this program is far from completion, results of lattice simulations are consistent with parameterizing the longrange part of the static QQ potential in pure (i.e. without light dynamic quarks) 5/7(3) gauge theory as [Has 87, Fu 87] V{r) = br
+ V0 , (1.2) r where a, 6, Vo are constants and the color dependence between quark and antiquark is that in Eq. (XI2.4). As noted in Sect. XI2, the linear 'fer' term models a colorflux tube of constant energy density. The coefficient b is commonly described in the latticegauge literature as the string tension, in reference to the string model of hadrons, and its value is estimated from a string model relation involving the typical slope OL of a hadronic Regge trajectory (cf. Table XIII3), b = ( 2 W ) " 1 ~ 0.18 GeV2 .
(1.3)
This is equivalent to a restoring force of about 16 tons! Numerical studies imply a relation between the string tension b and the confinement scale A.j^ of QCD [Fu 87], A
Ws = (0.318 ±0.058)>/6 ^ 0.13 ± 0.02 GeV .
(1.4)
If one goes beyond pure gauge theory by including the effects of a light dynamic quark q, the longrange potential between QQ becomes a shortrange interaction between Qq and Qq [Ze 88]. That is, the original interaction between the color charge carried by a heavy quarkantiquark pair becomes screened by the creation of a light quarkantiquark pair in the color field. Phenomenological potentials: The spectra of quarkonium states already hints at the radial dependence of the QQ potential, with the progression in nL levels suggesting an interaction which lies 'between' coulomb and harmonic oscillator potentials, as depicted in Fig. XIII3. In practice,
XIII1 The charmonium and bottomonium systems
ol Oscillator
is Quarkonium
353
*i Coulomb
Fig. XIII3 Energy levels of various potential functions phenomenological studies of quarkonium are carried out by adopting an assumed potential energy function in accord with this behavior, e.g.* 64?r2
• { [ ^ ( l + ^/A2))]1}
\ ~ 0.4 GeV} ,
,b~ 0.18 GeV2 \ \ \ , fc~~ 6.87 6.87 GeV 1 cr \d~0.1 J ' (1.5) where J7 {...} denotes a Fourier transform. The first two of the potentials in Eq. (1.5) are commonly called the 'Richardson' [Ri 79] and 'Cornell' [EiGKLY 80] potentials respectively. They are constructed to mimic QCD by exhibiting a linear confining potential at long distances and single gluon exchange at short distances. The Richardson potential even incorporates the asymptotic freedom property for the strong interaction coupling. The third is a power law potential [Ma 81] which, although not motivated by QCD, can be of use in analytical work or in obtaining simple scaling laws. The power law potential also serves as a reminder of how alternative forms can achieve a reasonable success in fitting bb and cc spectra, which after all, are primarily sensitive to the limited length scale 0.25 < r(fm) < 1. From the viewpoint of phenomenologyL it is ultimately more useful to appreciate the general features of the QQ static potential than to dwell on the relative virtues and shortcomings of individual models. In this regard, a study of spin dependence is instructive. .,, , i V(r) = {
, , bra/r
br — a/r
Spin dependence: In order to analyze spindependent effects in quarkonium without detailed a priori knowledge of the inter quark potential, we assume an interaction suggested by the QED interaction of Eq. (V1.16) * The second and third potentials provide fits only up to an additive constant.
354
XIII Hadron spectroscopy
but allowing for a more general vertex structure [Ja 76], 2
Ti ,
(r = 1,7^,75,7^75,^) ,
(1.6)
where Vi(q2) represents the propagator of an exchanged quantum. The nonrelativistic limit of this expression, expanded in inverse powers of the heavyquark masses [EiF 81], yields a sum of static and spindependent contributions, general = V0 + Fspin = V0 + Vso + Ken + V8a .
(1.7a)
The potential VSp[n is seen to generally contain spinorbit (VsO), tensor (Ken), and spinspin (V88) components, SQ
• r x pg 2M*
Sg • r x ] 2M"
SQ
H Vten
=
I r
• r x pg — SQ • r x M
Q MQ
r
(1.7b)
* S
Q'SQ
where r = YQ—TQ. The quantity Vsp[n is expressed in terms of Vb, its radial derivative V^, and additional contributions K(r), V({r) (i = 1 , . . . , 4). Referring to Table XIII2, we see that the QED BreitFermi potential represents a special case, with all nonzero contributions expressed in terms of the static coulomb potential. Also appearing in Table XIII2 are general forms for vector and scalar vertices. The less interesting pseudoscalar vertex (which does not lead to a static potential) and the axialvector and tensor vertices (which have only a leading spinspin interaction) are not included. Let us apply Eqs. (1.7a,b) to quarkonium by working in the QQ centerofmass frame with equal constituent quark masses MQ — MQ = M. The
Table XIII2 . Spindependent potentials
v0
Vx
v2
Vs
v4
7M 7^
—a/r VV
0 0
1
vs
—a/r Vv 0
3a/r 3 V ^ / r  VV 0
2 V 2 VV 0
Interaction r QED Vector Scalar
Vs
XIII1
The charmonium and bottomonium systems
355
spindependent potential of Eq. (1.7b) then simplifies to
~ 'flL •
s +
b [
S

(18)
For the purpose of obtaining masssplitting relations, we require the computation of various expectation values in the basis of ^[LSJ] states,
(3Po) ,
2
2(2L  1)<2L . _,
4
"13/4
,
. (1 9)
(%).
'
For example, these expectation values imply the following mass relations for the triplet P states: ™>(*P2) = m\ ras_o  ^mten , 5 lit \
JL~^ I
ill
Tllig
Q
" ''1%QJ\ 5
ra(3P0) = fn  2ra s _ o  2mten
.
The mass formulae in Eq. (1.10) can be used to test whether the longrange confining potential transforms as a fourscalar (Vs) or instead as a fourvector (VV) [BeCDK 79]. For definiteness, we consider a simple modification of the Cornell model in which the 'scalar vs vector' issue is cast in terms of a parameter £ (0 < £ < 1), Vs = (1 — £)br ,
Vy = £br
.
(1H)
r It follows from Eq. (1.7b) and Table XIII2 that the spinorbit and tensor mass contributions are then given by
¥
(1.12)
and upon defining A = &(r~ 1 )/a(r~ 3 ), we form the ratio _ m(3P2)  mjZPi) " m( 3 Pi)  m(sP0)
2 16  19£A  5A 5 8 + 5£A  A '
(
'
356
XIII Hadron spectroscopy
This can be compared with data from the Upsilon Pwave n = 1,2 states, of
0.
Mo. .70
(L14)
M2P)) .
For vector confinement (f ~ 1), Eq. (1.13) is in accord with the experimental values of Eq. (1.14) only for A ~ 0, whereas scalar confinement produces reasonable agreement for a much larger range, 0.4 < A < 1.0. Moreover, the Cornell model suggests that the latter values for A are rather more reasonable than the former, and so supports the conclusion that confinement is produced by a longrange, fourscalar interaction. Transitions in quarkonium All quarkonium states are unstable. Among the decay mechanisms are annihilation processes, hadronic transitions, and radiative transitions. Roughly speaking, the lightest quarkonium states are relatively narrow, but those lying above the heavyflavor threshold, defined as twice the mass of the lightest heavyflavored meson and depicted by dashed lines in Figs. XIIIl(a,b), are rather broader. This pattern is particularly apparent for the SS\ states  below the heavyflavor threshold, widths are typically tens of keV, whereas above, they are tens of MeV. The primary reason for this difference is that above the heavyflavor threshold, quarkonium can rapidly 'fall apart' into a pair of heavyflavored mesons, e.g. T[45] —> BB, whereas below, this mode is kinematically forbidden. In the following, we shall describe only decays which occur beneath the heavyflavor threshold, and shall limit our discussion to annihilation processes and hadronic decays. Radiative electric and magnetic dipole transitions are adequately described in quantum mechanics textbooks. Annihilation transitions: To motivate a procedure for computing annihilation rates in quarkonium, let us consider the simple case of a charged lepton of mass m moving nonrelativistically with its antiparticle in a 1So state, and undergoing a transition to a twophoton final state.* First we
666 or
66 or yy Q.
(a)
(b)
CO
Fig. XIII4 Decay of quarkonium through annihilation. The 1S'o (3S±) states have even (odd) charge conjugation, and can therefore give rise to even (odd) numbers of photons in an annihilation process.
XIII1 The charmonium and bottomonium systems
357
write down the invariant amplitude for the pair annihilation process,
(1.15) for momentum eigenstates. In the lepton rest frame, we are free to choose, the transverse gauge e\  p = t\ • p = 0, i.e. e\2 — 0. Since SS\ states can make no contribution to the twophoton mode, we can compute the squaredamplitude for a ^ o transition by summing over over initial state spins,
^[
+
£ * ! "»']
(1J6)
where u)\£ are the photon energies in the lepton rest frame. Near threshold the photons emerge back to back, and the differential cross section is found to be
Likewise, near threshold a sum on photon polarizations gives  (ej • e^)2)thr = 2 ,
(1.18)
and upon integrating over half the solid angle (due to photon indistinguishability) we obtain the cross section,
* = *P .
(1.19)
This is the transition rate per incident flux of antileptons. Since the flux is just the antilepton velocity v+ times a unit lepton density, we interpret v+a as the transition rate for a density of one lepton per volume. For a bound state with radial quantum number n and wavefunction \P n(x), the density is ^ n (0) 2 and the lowestorder expression for the electromagnetic decay rate D ^ p S o ] becomes
^
^M) Tib
.
(1.20)
The corresponding rate for 77 emission from xSo states of the bb (T) system is obtained from Eq. (1.20) by including a factor e\ — 1/81, which accounts for the fequark charge, and a color factor of three. Determination of the twogluon emission is found similarly (cf. Fig. XIII4(a)) provided the gluons are taken to be massless free particles, and is left for a problem at the end of the chapter. Including also the effects of
358
XIII Hadron spectroscopy
QCD radiative corrections, referred to a common renormalization point HR = mb, we have [KwQR 87] 2
5o] =
vM0) 2 L  3.4 , .< 1+4A
—3(^p— L
—^\ •
(L21)
Decays can also occur from the nzS\ states.* The single photon intermediate state of Fig. XIII4(b) leads to emission of a lepton pair, whereas Fig. XIII4(c) describes final states consisting of three gluons, two gluons and a photon, or three photons. For such a threeparticle final state, there are six Feynman diagrams per amplitude and threeparticle phase space to contend with. Upon including QCD radiative corrections, the results are [KwQR 87]
2
" 12  6 ^—J '
2187(2m6) 128(7r 2 9)aa 2 (m 6 )[^ r a (0) 2 [
I
i
(L22)
^
The QCD contributions in Eq. (1.22) are of interest in several respects. They contribute, on the whole, with rather sizeable coefficients and can substantially affect the annihilation rates. Also, they have come to be used as one of several standard inputs for phenomenological determinations of as. To eliminate the modeldependent factors \t n (0) 2 , one works with ratios of annihilation rates, a 5as(mb)
w
9
as(mb)\ V"
Tra2
' "'"
7T J 2 ( Mr\ \2mh)
(1.23) In reality, there are a number of theoretical and experimental concerns which make the extraction of ^(mb) a rather more subtle process than it might at first appear: (i) the contribution of * n (0) 2 in Eqs. (1.21),(1.22) * There are annihilations from higher partial waves as well. These involve derivatives of the wavefunction at the origin.
XIII1 The charmonium and bottomonium systems
359
as a strictly multiplicative factor is a consequence of the nonrelativistic approximation and may be affected by relativistic corrections, (ii) there is no assurance that O(as)2 terms are nonnegligible, particularly in the light of the large first order corrections, (iii) experiments see not gluons but rather gluon jets, and at the mass scale of the upsilon system, jets are not particularly welldefined, and (iv) the 7 spectrum observed in the jgg mode is softer than that predicted by perturbative QCD, implying the presence of important nonperturbative effects. Nevertheless, determinations of this type lead to the central value (and its uncertainties) A^ = 16Olgo° MeV as extracted from upsilon data and cited earlier in Table II—3. This example indicates how demanding a task it is to obtain a precise experimental determination of as(q2). Hadron transitions: The transitions V —• V + TT0 and V * V + 77 involving the decay of an excited 3 5i quarkonium level (V7) down to the S S\ ground state (V) are interesting because they are forbidden in the limits of flavor SU(2) and SU(3) symmetry respectively. Their rates are therefore governed by quark mass differences, and a ratio of such rates provides a determination of quark mass ratios. There is a modest theoretical subtlety in extracting the rates, as degenerate perturbation theory must be used [IoS 80]. The leadingorder effective lagrangian for these Pwave transitions must be linear in the quark mass matrix m, ^ IV (m(U = C [{pid — TTlu)—^ + (2i7ls — TTld —
TYly)—^
+ . . .]
(1.24) where c is a constant. Here, n^ and r)$ are the pure SU(S) states which appear prior to mixing 7T° = cos 9 7T3 + sin 6 % ,
77 = — sin 0 TT3 + cos 9 rjs ,
(1.25)
where tan 9 ~9 = y/3(rrid — mu)/[2(2ms — rrid — mu)] describes the quark mixing. Upon calculating the transition amplitudes and then substituting for the small mixing angle 9, we obtain u\ = ——(rrid —
2>/2
Mo 2Mp
— mu)9 H
2ms
Tnu) ,
rridmu (1.26)
360
XIII Hadron spectroscopy
where Mo = ic e^^k^kf^p. Q
Ty'^vrj
The ratio of decay rates is found to be 27 m mddm u u m
16 ms — m
p^
p^
We can extract a quark mass ratio from charmonium data involving ip{2S)  • J / ^ transitions. Prom the measured value Q = 0.037 ± 0.009 [RPP90], wefind md mu
~ = 0.0336 ± 0.004 , (1.28) ms — m which is about 40% larger than the same ratio extracted from pion and kaon masses (c/. Eq. (VII1.10)). XIII2 Light mesons and baryons In the quark model, the light baryons and mesons are Q 3 and QQ combinations of the u, d, s quarks. The resulting spectrum is very rich, containing both orbital and radial excitations of the L = 0 ground state hadrons. For mesons, the Q and Q spins couple to the total spins S = 0,1, and each (L, S) combination occurs in the nine flavor configurations of the flavorSU(3) multiplets 8, 1. Analogous statements can be made for baryon states. In the face of such complex spectra, we are mainly interested in the regularities that allow us to extract the essential physics. A tour through the data base in [RPP 90] reveals some general patterns.* Both radial and orbital excitations of the light hadrons appear 0.5 —> 0.7 GeV above the ground states. As pointed out in Sect. XI1, this indicates that the light quarks move relativistically. Other striking regularities are (i) the existence of quasidegenerate supermultiplets of particles with differing flavors and equal (or adjoining) spins, and (ii) excitations of a given flavor having increasingly large mass (M) and angular momentum (J) values which obey J = a'M2 + Jo. SU(6) classification of the light hadrons To the extent that the potential is spinindependent and we work in the limit of equal u,d,s mass, the quark hamiltonian is invariant under flavorSU(3) and spin*SC/(2) transformations. To lowest order, hadrons are thus placed in irreducible representations of *S/7(6), and quarks are assigned to the fundamental representation 6, 6 = « d T
s"[ u[
d[
si)
.
(2.1)
* Our discussion will focus on hadron masses. Strong and electromagnetic transitions are described in[LeOPR88].
XIII2 Light mesons and baryons
361
We can also write the SU(6) quark multiplet in terms of the SU(3) flavor representation and the spin multiplicity as 6 = (3,2). Although the SU(6) invariant limit forms a convenient basis for a classification of the meson and baryon states, it cannot be a full symmetry of Nature since the spin is a spacetime property of particles whereas SU(3) flavor symmetry is not. Thus it is impossible to unite the flavor and spin symmetries in a relativistically invariant manner [CoM 67]. Although we shall avoid making detailed predictions based on SU(6), it is nonetheless useful in organizing the multitude of observed hadronic levels. Meson supermultiplets: The L=0 QQ composites are contained in the SU(6) group product 6 x 6* = 35 © 1, where the representations 35, 1 have flavorspin content 35 = (8,3) © (8,1) ©(1,3),
1 = (1,1) .
(2.2)
The L=0 ground state consists of a vector octet, a pseudoscalar octet, a vector singlet, and a pseudoscalar singlet. For excited states, the meson supermultiplets constitute an SU(6) x O(3) spectrum of particles. The O(3) label refers to how the total angular momentum is obtained from J = L + S, giving rise to the pattern of rotational excitations displayed previously in Table XI3. Roughly speaking, mesons occur in mass bands having a common degree of radial and/or orbital excitation. Fig. XIII5 provides a view of the mass spectrum for the lightest mesons. The SU(6) x O(3) structure of the ground state and a sequence of orbitally excited states are observed to the extent that sufficient data is available for particle assignments to be made. Note that the Swave QQ states are all accounted for, but gaps appear in all higher partial waves. Even after many years of study, meson phenomenology below 2 GeV is far from complete! Baryon supermultiplets: The SU(6) baryon multiplet structure arises from the Q 3 group product (6 x 6) x 6 = (21©15) x 6 = 56©70©70©20, and has flavorspin content 56 = (10,4) ©(8,2), 70 = (8,4) © (10,2) © (8,2) © (1,2) , 20 = (8,2) ©(1,4).
(2.3)
A threequark system must adhere to the constraint of Fermi statistics. Each baryon state vector is thus antisymmetric under the interchange of any two quarks. A Youngtableaux analysis of the above group product reveals that the spinflavor parts of the 56, 70, and 20 multiplets are respectively symmetric, mixed, and antisymmetric under interchange of pairs of quarks. Since the color part of any Q 3 colorsinglet state vector
XIII Hadron spectroscopy
362 NONET:
[J
1= 1/2 1 0
_
Cnn, ss) well established good evidence weak evidence
L=4
L=3
L=2
L1
Fig. XIII5 Spectrum of the light mesons
is antisymmetric under interchange of any two quarks, the 56plet has a totally symmetric space wavefunction, with zero orbital angular momentum between each quarkpair. The 70 and 20 multiplets require either radial excitations and/or orbital excitations. Recall the characterization of the baryon spectrum in terms of the basis defined by an independent pair of oscillators (c/. Eq. (XI2.12)). In this context, a standard notation for a baryon supermultiplet is (R, L^), where R labels the 577(6) representation, V is the parity, N labels the number of oscillator quanta and L is the orbital angular momentum quantum number (cf. Sect. XI2). Like meson masses, baryon masses tend to cluster in bands having a common value of N. The first three bands are shown in Fig. XIII6, and effects of SU(6) breaking are displayed for the first two. The lowest lying SU(6) x O(3) supermultiplet is the positive parity (56, 0Q~), having content as in Eq. (2.3). Next comes the negative parity (70,1J~) supermultiplet. This contains more states than the 70plet shown in Eq. (2.3) because the extension from L = 0 to L = 1 requires addition of
XIII2 Light mesons and baryons
363
3/2
(56, Oj) (70,
1/2
81/2 83/2
(56, 0 0 )
Fig. XIII6 The lowlying baryon spectrum. angular momenta, (10, 2) > (10,4)  (10,2) , (8,4)  (8,6)  (8,4) (8,2) (2.4) (1, 2) > (1,4)  (1, 2) , (8,2) > (8,4)  (8, 2) The number of supermultiplets grows per unit of excitation thereafter. There are five SU(6) multiplets in the N = 2 band, (56,2^), (56,0£), (70,2j), (50, Oj), and (20, l£). Recall that the baryonic interquark potential was expressed in Eq. (XI2.10) as V = Vosc + [/, where V^sc is the potential energy of a harmonic oscillator and U = V — Vosc. If the potential energy were purely VOSc^ the supermultiplets within the N = 2 band would all be degenerate. In the potential model, assuming that the largest part of U is purely radial, this degeneracy is removed by the firstorder perturbative effect of [/, and the splittings in the N = 2 band are shown at the top of Fig. XIII6. Aside from choosing the (56, O^) supermultiplet to have the lowest mass, one finds the pattern of splitting to be as shown in Fig. XIII6, independent of the particular form of U [HeK 83]. trajectories It is natural to classify together a ground state hadron and its rotational excitations, e.g. the isospin onehalf baryons ^V(939)j=i/2 (the nucleon), JV(1680)j=5/25 and 7V(2220)j=9/2 Although no higher spin entries have
364
XIII Hadron spectroscopy
been detected in this particular set of nucleonic states (presumably due to experimental limitations), there is no theoretical reason to expect any such sequence to end. The data base in [RPP 90] contains a number of similar structures, each characteristically containing three or four members. Each such collection of states is said to belong to a given Regge trajectory. To see how this concept arises, let us consider the simplest case of two spinless particles with scattering amplitude f(E, z) {i.e. da/dVt = \f{E,z)\2), where E is the energy and z = cos# is the scattering angle. It turns out that analytic properties of the scattering amplitude in the complex angular momentum (J) plane are of interest. One may obtain a representation of f(E, z) in the complex Jplane by converting the partial wave expansion into a socalled WatsonSommerfeld transform,
f(E, z) = 2^(Y (2£ + l)a(E, l)Pt(z)
(2.5)
(bdJ (2J + l)a(E, J)Pj(z) , T 2m r where Pa is a Legendre polynomial and C is a contour enclosing the nonnegative integers. Suppose that as C is deformed away from the Re J axis to, say, a line of constant Re J, a pole in the partial wave amplitude a(E, J) is encountered. Such a singularity is referred to as a Regge pole and contributes (c/. Eq. (2.5)) to the full scattering amplitude as sin(7ra[£;])
' "• '
where OL[E] is the energydependent pole position in the complex Jplane and j3[E] is the pole residue. Table XIII3. Regge trajectories. Trajectory
N
Slope a
N
3 3 3 3 2 3 4 4 4
0.99 0.92 0.94 1.1 0.91 0.72 0.84 0.69 0.86
A A E E* 7T
P K K* a
In units of GeV  2
Jintercept 0.34 0.07 0.64 1.2 0.24 0.05 0.54 0.22 0.29
XIII2 Light mesons and baryons
365
The Reggepole contribution of Eq. (2.6) can manifest itself physically in both the direct channel as a resonance and a crossed channel as an exchanged particle. Here, we discuss just the former case by demonstrating how a given Regge pole can be related to a sequence of rotational excitations. Suppose that at some energy £"#, the real part of the pole position equals a nonnegative integer £, i.e. Re a[Eji\ = £. Then with the aid of the identity,
i r1 we can infer from Eq. (2.6) the BreitWigner resonance form (Rg.ple.) _
e
fi 1 T/2 7T (a[E]  £)(a[E] + £ + 1) " E  ER + iV/2
l
*}
provided Re OL[ER\ » Im a[En]. A physical resonance thus appears if OL\E] passes near a nonnegative integer, and if the Regge pole moves to everincreasing J values in the complex Jplane as the energy E is increased, it generates a tower of highspin states. Except in instances of socalled exchange degeneracy, parity dictates that there be two units of angular momentum between members of a given trajectory. In this manner, a single Regge pole in the angular momentum plane gives rise to the collection of physical states called a Regge trajectory. A plot of the angular momentum vs squaredmass for the states on any meson or baryon trajectory reveals the linear behavior, J~a'm2
+ J0 .
(2.9)
A compilation of slopes (a/) and intercepts (Jo) appears in Table XIII3, with each trajectory labeled by its ground state hadron. Such linearly rising trajectories have been interpreted as a consequence of QCD [JoT 76]. In this picture, hadrons undergoing highly excited rotational motion come to approach colorflux tubes, whereupon it becomes possible to relate the angular momentum of rotation to the energy contained in the color field. This line of reasoning leads to the behavior of Eq. (2.9), and accounts for the universality seen in the slope values displayed in Table XIII3.
SU(6) breaking effects Although an SU(Q) invariant hamiltonian provides a convenient basis for describing light hadrons, the physical spectrum exhibits substantial departures from the mass degeneracies which occur in this overly symmetric picture. In the following, we shall consider some simple models for explaining the many SU(6) breaking effects observed in the real world. The QCD BreitFermi model: If one ascribes the nonconfining part of the quark interaction to singlegluon exchange, the nonrelativistic limit
366
XIII Hadron spectroscopy
yields the 'QCD BreitFermi potential' [DeGG 75] ^one—gluon — ~~
where as is the strong fine structure constant, r = ry, and k denotes the color dependence of the potential (c/. Sect. XI2) with k = 1 (1/2) for mesons (baryons). In keeping with the potential model, the mass parameters {Mi} are interpreted as constituent quark masses. Although the QCD BreitFermi model incorporates SU(6) breaking by means of both quark mass splittings and spindependent interactions, it lacks a rigorous theoretical foundation. One might argue on the grounds of asymptotic freedom that Eq. (2.10) does justice to physics at very short distances (in the approximation that as is constant), but there is no reason to believe that it suffices at intermediate length scales. It also does not account for mixing between isoscalar mesons, so such states must be considered separately. Meson masses: The gluonexchange model can be used to obtain information on constituent quark mass. In the following, we shall temporarily ignore the minor effect of isospin breaking by working with M = (Mu + Md)/2. To compute meson masses, we take the expectation value of the full hamiltonian between SU(6) eigenstates, specifically the L = 0 QQ states.* Although the form of Eq. (2.10) implies the presence of spinspin, spinorbit, and tensor interactions, the spinorbit and tensor terms do not contribute here because each quark pair moves in an 5wave, and it is the spinspin (hyperfine) interaction which lifts the vector meson states relative to the pseudosclar mesons. We can parameterize the nonisoscalar L = 0 meson masses as \ Q
+ ^ Q Q ( S Q * S Q)
»
( 2H)
* An analysis of spindependence in the L = 1 states is the subject of a problem at the end of the chapter (cf. Prob. XIII3)).
XIII2 Light mesons and baryons
367
where n and ns are the number of nonstrange and strange consituents respectively, and HQQ is the hyperfine mass parameter defined in Eq. (XII3.12). One consequence of Eq. (2.11) is a relation involving the mass ratio M/Ms. Fitting the four masses TT(138), K(496), p(770), K*(892) to the parameters in Eq. (2.11) yields nip — m^
=7xgg == M&M nn
~ 0.63 .
(2.12)
s
The origin of this result lies in the inverse dependence of the hyperfine interaction upon constituent quark mass, which affects the mass splitting between S = 1 and S = 0 mesons differently for strange and nonstrange mesons. The numerical value of M/Ms in Eq. (2.12) graphically demonstrates the difference between constituent quark masses and current quark masses, the latter having a mass ratio of about 0.04 (cf. Eq. (VII1.6a)). In earlier sections of this book which stressed the role of chiral symmetry, the pion was given a special status as a quasiGoldstone particle. In the QQ model, the small pion mass is seen to be a consequence of severe cancelation between the spinindependent and spindependent contributions. However, the parameterization of Eq. (2.11) cannot explain the large 7/(960) mass. In addition to the SU(6) symmetry breaking effects of mass and spin, there is an additive contribution present in the isoscalar channel which is induced by quarkantiquark annihilation into gluons. In the basis of u, d, s quark flavor states, this annihilation process produces a 3 x 3 mass matrix of the form '2M,, + X X X \ (2.13) where for C = +1(—1) mesons, X is the twogluon (threegluon) annihilation amplitude, and for simplicity we display just the quark mass contribution (2Mi) as the nonmixing mass contribution. The annihilation process is a shortrange phenomenon, so the magnitude of X depends sharply on the orbital angular momentum L of the QQ system. For L ^ 0 waves (where the wavefunction vanishes at zero relative separation), and C = — 1 channels (where the annihilation amplitude is suppressed by the three powers of gluon coupling), we expect Ms — M » X. In this limit, diagonalization of Eq. (2.13) yields to leading order the set of basis states (uu ± dd)/y/2 and ss. Only the L = 0 pseudoscalar channel experiences opposite limit X ^> Ms—M, wherein to leading order the basis vectors are the SU(3) singlet state (uu + dd + ss)/y/3 and octet states (uu — dd)/y/2, (uu + dd — 2ss)/y/6. The overall picture that emerges is one of relatively
368
XIII Hadron spectroscopy
unmixed light pseudoscalar states, and heavily mixed vector, tensor, etc. states. Baryon masses: Applying the onegluon exchange potential to the ground state baryons of (56, OQ") yields a mass formula analogous to Eq. (2.11),
For the system of l/2 + and 3/2 + (iospinaveraged) baryons, there are eight mass values and since the above mass formula contains five parameters, one should obtain three relations. The additional perturbative assumption Hss — Wns = tins — Wnn for the hyperfine mass parameters yields the wellknown GellMannOkubo relation of Eq. (XII3.10) for the 1/2+ baryons and the equal spacing rule for 3/2 + states, — 771A = mz* — rriz*
UIQ
ra^*
.
(Z. l o )
(Expt. 153 MeV = 149 MeV = 139 MeV ) A third relation which relates the 3/2 + and 1/2+ masses and is independent of further perturbative assumptions has the form 3raA  raE  2mN = 2(raE*  raA) (Expt. : 276 MeV  305 MeV) In addition, one can obtain estimates for M/Ms, among them M
=
2(mE» Q
^
M8 2ms* + ms  3mA ' ' . ^ M mE*  m s — = ~ 0.65 , both in accord withMEq. (2.12). s Isospin breaking effects: The above description of SU(6) breaking assumes isospin conservation. In fact, hadrons exhibit small mass splittings within isospin multiplets, arising from electromagnet ism and the u — d mass difference. In the pion and kaon systems, we were able to use chiral £(7(3) symmetry to isolate each of these separately. Unfortunately, this is not possible in general, and models are required to address this issue. There are a few consequences which follow purely from symmetry considerations. Since the mass difference mu — rrtd is A/ = 1, the A7 = 2 combinations mE+ + raE  2raEo = 1.7 ± 0.1 MeV , rap+  mpo = 0.3 ± 2.2 MeV , (2.18)
XIII2 Light mesons and baryons
369
arise only from the electromagnetic interaction. In addition, both electromagnetic and quark mass contributions satisfy the ColemanGlashow relation [CoG 64], raE+  raE +mnmp + m E   mE o = 0 [Expt. 0.4 ± 0.6 MeV = 0 ] . For electromagnet ism, this is a consequence of the [/spin singlet character of the current, whereas for quark masses it follows from the A/ = 1 and SU(3)octet character of the current. We proceed further by using a simple model, based on the QED coulomb and hyperfine effects, to describe the electromagnetic interaction of quarks, Ara cou l = v4coul 2^
QiQj
MM
Sl
S<7
'
where Acouh ^hyp axe constants, {Qi} are quark electric charges, and the sums are taken over constituent quarks. In Arahyp, we shall neglect further isospin breaking in the masses and use Mu = M^ = M, and assume electromagnetic selfinteractions of a quark to be already accounted for in the mass parameter of that quark. For any values of *AcOui a n d "4hyp> this model contains the sum rule (mnmp)em
= —(ra E + + m E  2ra E o) = 0.57±0.03 MeV , (2.21) o leaving the excess due to the quark mass difference, (mn — mp)qm =
• (n\uu — dd\n)
• (p\uu — dd\p)
— mu)(dm + fm)Zo = (mn  mp)  (mn  mp)em = 1.86 ± 0.03 MeV ,
(2.22) where the second line in the above uses the parameterization of hyperon mass splittings given in Eq. (XII3.8). To the extent that this estimate of quark mass differences is meaningful, one obtains the mass ratio, md
~m" =
ms — m
(m
" "
m
^
rriE — m
m
~ 0.015 ,
(2.23)
E
to be compared to the chiralsymmetry extraction from meson masses which yielded 0.023 (cf. Eq. (VII1.10)). With further neglect of terms O(a(Ms — M)) in the hyperfine interaction, this exercise can be repeated
370
XIII Hadron spectroscopy
for vector mesons to yield 2 (mK*o  mK*+)em = —(m p +  mpo) = 0.2 ± 1 . 5 MeV , o
(mK.o  mK*+)qm  (mK*o  mK*+)  {mK,o 
= 6.5 ± 1.9 MeV ,
md
~m" =
7Tls — Vh
mK
'° ~mK'+
mK*+)em
^' '
= 0.053 ± 0.016 .
TYlK* — flip
The additional assumption that the constants *AcOui a n d *4hyp a r ^ the same in the decuplet baryons and the octet baryons, as is true in the SU(6) limit, leads to 5 (mA++  raAo)em = ^ ( ^ E + + m E  — 2mEo) = 2.8 ± 0.2 MeV , (mA++  m A o) qm = (mA++  mAo)  (mA++  mAo)em = 5.5 ± 0.4 MeV , ^' ' ?7ls — 771
2
±
Of course, the spread of values for the mass ratios raises a concern about the validity of this simple model. However, all methods of calculation agree on the smallness of the ratio (ra^ — mu)/(ms — rh). XIII3 The heavyquark limit In the quark description, a heavyflavored hadron contains at least one of the heavy quarks c, 6, t. It is possible to describe such heavy systems with dynamical models like those employed for the light hadrons [DeGG 75, IzDS 82]. However, while such models are often valuable, it is always preferable to have a valid approximation scheme which follows directly from QCD. In this regard, a study of the heavyquark limit (rriQ —+ oo) in which the theory is expanded in powers of m^ 1 is proving useful in analytic and lattice studies of meson spectroscopy and in the area of weak decays. Heavyflavored hadrons in the quark model The spectroscopy of heavyflavored hadrons should qualitatively follow that of the light hadronic spectrum, with states containing a single heavyquark Q occurring as either mesons (Qq) or baryons (Qqiq2)> The lowest energy state for a given hadronic flavor will have zero orbital angular momentum between the quarks, leading to ground state spin values S — 0,1 for mesons and S = 1/2,3/2 for baryons. The hyperfine interaction will lower the 5 = 0 meson and S = 1/2 baryon masses, and both orbital and radial hadronic excitations of the ground state will be present.
XIII3 The heavyquark limit
371
Although it is possible to contemplate extended flavor transformations which involve interchange of the light and heavy quarks, e.g. as in the S77(4) of the light and charmed hadrons, such symmetries are so badly broken by the difference in energy scales MQ » M q and MQ » AQCD as to be rendered useless. However, the SU(3) and SU(2) flavor symmetries associated with the light hadrons are still viable, but multiplet patterns become modified. The mesons Qq will exist in the SU(3) multiplet 3*, whereas in the baryonic Qqiq2 configurations the two light quarks <7i,
M nip —
 mB mp —
where M = (Mu + M^)/2. These findings depend to some extent on how the fit is done, e.g. with mesons or with baryons, and we leave further study for Prob. XIII4.
Charmed baryons
Charmed mesons
2.2 M (GeV) 2 _ = D *
2.8 s
2.6
2.4 1 ft
Fig. XIII7 Spectrum of charmed (a) mesons, (b) baryons
372
XIII Hadron spectroscopy Spectroscopy in the TUQ —>
oo limit
In a hadron which contains a single heavy quark Q along with light degrees of freedom, the heavy quark is essentially static. The best analogy is with atoms, where the nucleus can in the first approximation be treated as a static, electricallycharged source. Likewise, for heavy hadrons the heavy quark is a static source with color charge, and the light degrees of freedom provide a nonstatic hadronic environment around Q. This scenario can be formalized by partitioning the heavy quark lagrangian as [CaL 86, Ei 88, LeT 88] CQ = ${%$$
mQ) I/J = C0 + £ space  rnQ) ^ ,
£space =  2 ^
D^
where D^ip is the covariant derivative of SU(3)C. Since the spatial 7 matrices connect upper and lower components, we see that the effect of £Space
is
O(mQ1)'
Observe that the static lagrangian Co of Eq. (3.2) is invariant under spin rotations of the heavy quark Q. In the world defined by £0, with both O(AQCD/MQ) effects and O((XS(MQ)) effects (associated with hardgluon exchange) ignored, heavy hadronic energy levels and couplings are constrained by the SU{2) spin symmetry. It is helpful to visualize the situation. A heavy flavored hadron of spin S will contain a static quark Q having a constant spin vector SQ (with SQ = 1/2) and light degrees of freedom having a constant angular momentum vector J^ = S — SQ.* For a meson of this type, we assume that J(> behaves as it does in the quark model, with J# = 1/2 in the ground state and Jt — L ± 1/2 for L > 0 rotational excitations. Prom the decoupling of the heavy quark spin, it follows that there will be a twofold degeneracy between mesons having spin values S = Ji ± 1/2. The meson L = 0 ground state will have Ji = 1/2 and thus degenerate states with S = 0,1. The L = 1 first rotational excitation with Ji = 1/2 will give rise to degenerate S = 0,1 levels, whereas for Ji = 3/2 one obtains degenerate levels having S = 1,2. Moreover, the energy differences between different levels should be independent of heavy quark flavor. Analogous conditions hold for heavy flavored baryons and hadronic transitions between levels of differing L can be similarly analyzed. Let us explicitly demonstrate that the splitting between the J p = 1~ and Jp = 0~ states of a Qq meson must vanish in the limit of infinite quark mass. We note that the mathematical condition for spinAlthough the light degree(s) of freedom in the simple quark model is an antiquark q for mesons and two quarks (71*72 for baryons, the physical (i.e. actual) light degrees of freedom could entail unlimited numbers of gluons and/or quarkantiquark pairs.
XIII3 The heavyquark limit
373
independence is "n = 0 ,
(3.3)
where S® is the generator of spin rotations about the 3axis for quark Q and Ho is the hamiltonian obtained from Co Since the action of S3 on a 0~ state produces a 1~ state, i.e. \MX~) = 2S3 M 0 ), we then have = 2SsQHo\Mo) = mo\M1) ,
Ho\M1)=m1\M1)
(3.4)
implying that mx — m0 —• 0 as TTIQ —• 00. Another consequence of working in the static limit of Co is that the propagator, Soo(x,y), of the heavy quark in an external field can be determined exactly. Prom the defining equations, y)
(A) = do+ igsAoX/2) , (3.5)
one has the solution 500(0:, y) = iP(x0, yo)6{3)(x  y)
(36) where P(xo, yo) is the pathordered exponential along the time direction,
P(x0, w) = Pexp \& r dt\. Ao(x,t)l .
(3.7)
In this approximation, the heavy quark is static at point x and the only timedependence is that of a phase. This discussion can be generalized to a frame where the heavy quark is moving at a fixed velocity v, described by a velocity four vector v^ = pV/rriQ, with v^v^ = 1. One can define projection operators Tv±=l{l±i})
,
(3.8)
where T^± = Tv±, Tv±Tvzp = 0, and Fv+ + Tv = 1. The Tv± generalize the usual projection of 'upper' and 'lower' components into the moving frame. A quark moving with velocity v will have the leading description of its wavefunction contained in the 'upper' component described by a field hv [Ge 90, Wi 91], Tv ^
= eimQv'xhv(x)
,
(3.9)
where the main dependence on the quark mass has been factored out, and hv obviously satisfies Tv+hv = hv. Substituting into the Dirac lagrangian,
374
XIII Hadron spectroscopy
neglecting lower components, and using Tv+Iprv+ = v • D yields  mQ)
 mQ) t/> ~
= hviv • Dhv , (3.10)
which generates the lowest order equation of motion v • Dhv = 0. This approximation can be systematically improved by inclusion of a 'lower' component for the heavyquark field [EiH 90a,b, Lu 90, GeGW 90], V V = eimQvx£v(x)
,
(3.11)
with Tv£v = £v. The equations of motion allow us to solve for £v by following the sequence of steps, 0 = (iIp  mq) tf> = {ilp  mQ) eimQv'x
[hv + lv] xIp) [hv + £v]
x
[{2mQ
which yields lv and ip as
ilphv]
,
(3.12)
(3.13) eimQvx
=
j
Inserting these forms into Eq. (3.10) and using Tv+hv = hv and Eq. (Ill— 3.50) for^Z) yields L,v — tiv
m
Q
v
D
•

(3.14) which is the desired expansion in terms of the heavy quark mass. Because the last term in this expression is second order in v • D and noting that v • Dhv = 0 to lowest order, it will not contribute to matrix elements at order 1/rriQ and can be dropped. The lagrangian of Eq. (3.14) corresponds to a quark moving at fixed velocity. Antiquark solutions can be constructed with the mass dependence e+tmQv'x^ with the result

i
v
•
D

(3.15) where the field kv satisfies Tvkv = kv. It is legitimate to neglect the production of heavy QQ pairs. However, one should superpose the la
XIII4 Nonconventional hadron states
375
grangians for different velocities in a Lorentz invariant fashion,
C = J d4v 6{v^  l)%0) [/# + £$]
=f^
The nature of the approximation at this stage is more of a classical limit rather than a nonrelativistic limit. To be sure, for any given quark one can work in the quark's rest frame, in which case the quark will be nonrelativistic. However, when external currents act on the fields, as will be considered in Sect. XIV2, transitions form one frame to another occur for which Av is not small. On the other hand, the result can be said to be classical because quantum corrections have not yet been included and these can renormalize the coefficients in Ly. Also, diagrams involving the exchange of hard gluons can produce nonstatic intermediate states. Such corrections can be accounted for in perturbation theory [Wi 91].
XIII4 Nonconventional hadron states Many suggestions have been made regarding the possibility of hadronic states beyond those predicted by the simple quark model of QQ and Qs configurations. The study of such states is hampered by the fact that we still have very little idea why the quark model works. QCD at low energy is a strongly interacting field theory, and we would expect a very rich and complicated description of hadronic structure. That the result should be describable in terms of a simple QQ and Q3 picture as even a first approximation remains a mystery. Quark models have been popular because they seem to work phenomenologically, not because they are a controlled approximation to QCD. This weakness becomes all the more evident when one tries to generalize quark model ideas to new areas. Much of the theoretical work on nonconventional states has involved the concept of a constituent gluon G, analogous to a constituent quark Q, and we shall cast our discussion with respect to this degree of freedom.* It is clear that there should be a cost in energy to excite a constituent gluon. The energy should not be extremely large, else it would be difficult to understand the early onset of scaling in deep inelastic scattering. However, it cannot be less than the uncertainty principle bound on a massless particle confined to a radius R ~ 1 fm of E = p>\/3/R ~ 342 MeV (cf. Sect. XI1). Calculations have tended to use an effective gluon 'mass' in the range 0.5 < MG (GeV) < 0.6. The basic idea of confinement is that only colorsinglet states exist as physical hadrons. If we identify those states which are color singlets * However, it should be understood that such a concept has not been shown to follow rigorously from QCD, nor indeed is a configuration of definite numbers of consitituent gluons a gauge invariant entity (cf. Sect. X2).
376
XIII Hadron spectroscopy
and which contain few quark or gluon quanta, we can easily find other possible configurations besides QQ and Q3. Some of the more wellknown examples are 1) Gluonia (or glueballs)  quarkless G2 or G3 states, which we shall discuss in more detail below, 2) Hybrids  colorsinglet mixtures of constituent quarks and gluons like QQG mesons or Q3G baryons, 3) Dibaryons  sixquark configurations in which the quarks have similar spatial wavefunctions rather than two separate threequark clusters, 4) Meson molecules  loosely bound deuteronlike composites of mesons. A convenient framework for describing the quantum numbers of possible hadronic states is obtained by considering gaugeinvariant, colorsinglet operators of low dimension [JaJR 86], as was discussed in Sect. XI1. Table XIII4 lists all such operators up to dimension five which can be constructed from quark fields, the QCD covariant derivative, and the gluon field strength, denoted respectively by q, X>, and F. Also appearing in Table XIII4 is the collection of JPC quantum numbers associated with each such operator. Particular spinparity values are obtained from these operators by choosing indices in appropriate combinations.
Gluonia The existence of a gluon degree of freedom in hadrons is beyond dispute, with evidence from deep inelastic lepton scattering and jet structure in hadronhadron collisions. However, trying to predict the properties of a new class of hadrons whose primary ingredient is gluonic is nontrivial. Hypothetically, if quarks could be removed from QCD the resulting hadron spectrum would consist only of gluonia (or 'glueballs'). Gluonic configurations should be signaled by the existence of extra states beyond the expected nonets of QQ hadrons. However, mixing with QQ hadrons is generally possible (cf. Sect. X2). Although predicted by Table XIII4. Gaugeinvariant colorsinglet interpolating fields. Operator Dimension qTq qTVq FF grgF FVF
3 4 4 5 5
JPC 2++,2"± 0 + + 2 + + 0—^ 2—*" 0±+,0±,l±+,l±,2±+,2±
XIII4 Nonconventional hadron states
377
the 1/NC expansion to be suppressed, such mixing effects serve to cloud the interpretation of data visavis gluonium states. Referring to the interpolating fields mentioned above, we see that for gluons the gaugeinvariant combinations pa.Fx
™ piLv
1
£
[iv a
r
5
fiX
1
pa
av •>
c
ILV
L
fi»v a
?
Fa^Fx
L
[i\Lav
(AW \^'LJ
can be formed out of two factors of a gluon fieldstrength tensor F^v or its dual F%v. The spin, parity, and charge conjugation carried by these these l l operators are respectively JPC = 0 + + , 2~~~ ~, 0~+, 2~ + , and are thus the quantum numbers expected for the lightest glueballs,* i.e. such operators acting on the vacuum state produce states with these quantum numbers. Although there is no a priori guarantee that one obtains a single particle state (e.g. a 2 + + operator could in principle create two 0 + + glueballs in a .Dwave), the simplicity of the operators leads one to suspect that this will be the case. There is one, somewhat controversial, construct missing from the above list. Two massive spinone particles in an Swave can have JPC = 1~+ as well as JPC = 0"f+,2+"l~, and some models predict such a gluonium state. However, a 1~+ combination of two massless onshell vector particles is forbidden by a combination of gauge invariance plus rotational symmetry [Ya 50]. The lack of a 1~+ gaugeinvariant, twofield operator is an indication of this. Aside from a list of quantum numbers and some guidance as to relative mass values, theory does not provide a very clear profile of gluonium phenomenology. Latticegauge methods offer the most hope for future progress. At present, they predict in a quarkless version of QCD that the lightest glueball will be a 0 + + state of mass 1.2 ± 0.3 GeV and that the 2++ glueball is 1.5 ± 0.1 times heavier [BaK 89]. Gluonium states would be classified as flavor5C/(3) singlets. One process expected to lead to direct production of glueballs is the radiative decay of J/ift, which takes place in QCD through the annihilation process cc —> jGG (as in Fig. XIII4(c)). The two gluons emerge in a color singlet configuration, and by varying the energy of the finalstate photon, all masses between m = 0 and m ~ Mj/^ can be probed. A glueball should thus be revealed as a resonance M in the decay J/ip —• 7M. Of course, QQ states can be produced as well. In order to distinguish gluonia from such neutral quark states, a measure SM? whimsically called the 'stickiness' of meson M, has been introduced [Ch 84], T
JI^M
p.s. [77 > M]
Gluonic operators with three fieldstrength tensors produce states with JPC — 0 ± + , , 2 ± + , 1 ± + , 2 ±  , 3^"~. Because of the extra gluon field, one expects these states to be somewhat heavier.
378
XIII Hadron spectroscopy
where 'p.s.' stands for the available phase space. A glueball would be expected to couple strongly to GG but not to_77, and thus to have a high relative value of stickiness compared to a QQ composite. Let us consider two states, among others under active investigation, which are strongly produced in J/ip —• 7M and have provoked much attention as possible anomalous states. These are the /2(1720) and 7/(1440), also called 0(1720) and ^(1440) respectively. Evidence to date suggests that /2(1720) (i.e. 9) has the properties of a nonQQ system. It is a resonance seen primarily in J/ij; —> 7/2(1720) —> jKK and has been assigned the quantum numbers JPC = 2 + + . The TTTT and 7777 decay modes have also been observed, with the quoted branching ratios ™  1.00 : 0.47 : 0.10
.
(4.3)
The dominance of KK is even more striking when one notes that Dwave phase space favors the TTTT mode by a factor ~ 2.6. The QQ 2++ ground states are /2(1270) and /£(1525), with dominant components (uu + dd)/\/2 and ss as deduced from their decays. If 9 were a QQ state, it would be a radial excitation of the j ^ and fy. However, the 9 is too close in mass to the /2(1525) to be its radial excitation, and the 9 decay pattern requires its interpretation as largely ss, so that it cannot be the radial excitation of the /2(1270). Besides, there is some experimental evidence identifying the radial excitation of /2(1270) at 1800 MeV. The 9 has not been seen in other hadronic reactions where /2(1270) and /2(1525) stand out strongly, and its stickiness is remarkable, Sf : Sr : Se = 1 : 3 : > 20 .
(4.4)
Even if one accepts that 9 is not a QQ hadron, a firm identification of its primary content is still difficult. The high stickiness and lack of any known multiplet partners favor a glueball interpretation, but its decays do not seem SU(3) symmetric. Additional knowledge of the JPC = 2 + + spectrum and further experimentation will be required to clarify this issue. Indeed, a new analysis [Du 92] reports evidence that JPC = 0 + + for the 9. If this turns out to be correct, the difficulties of accommodating the 9 as a QQ state may not be as severe. The other interesting glueball candidate is the iota, ^(1440). It is the JPC = 0~+ state with the largest branching fraction in J/ip radiative decays, but appears as a rather obscure resonance in hadronic reactions. In comparison with ry(549) and 7/(960), it has stickiness ratios S^ : Srj> : SL = 1 : 4 : > 45 ,
(4.5)
which would favor a glueball interpretation. However, the situation regarding hadronic experiments which probe the JPC — 0~ + spectrum in the range 1.01.7 GeV is presently so confused that even QQ states cannot be identified with any certainty. As with all gluonium states, the
XIII4 Nonconventional hadron states
379
situation will continue to be distressingly vague without a good deal of additional experimental and phenomenological guidance. Additional nonconventional states There is a widespread belief that gluonium states must appear in the spectrum of the QCD hamiltonian. For other kinds of nonconventional configurations, it is far more difficult to reach a meaningful consensus, although experimental efforts to detect such states continue. Let us briefly review several such possibilities. (i) Hybrids: Prom Table XIII4, we see that among the QQG meson hybrids is one with the quantum numbers JPC = 1~+. This wouldbe hadron is of particular interest because comparison with Table XI3 reveals that it cannot be a QQ configuration. Model calculations suggest that the lightest such state should be isovector, with mass in the range 1.52.0 GeV, and that such states may largely decouple from L = 0 QQ meson final states. A study of Q^G baryon hybrids reveals that none of the states is exotic in the sense of lying outside the usual Q 3 spectrum [GoHK 83]. (ii) Dibaryons: The most remarkable aspect learned yet about the dibaryon states is how much sixquark configurations are restricted by FermiDirac statistics. Table XIII5 lists the possible sixquark SU(3) multiplets along with their spin values [Ja 77]. Of this collection of states, most attention has been given to the spinless 5C/(3)singlet state, called the Hdibaryon. This particle, which has strangeness S = — 2 and isospin / = 0, is predicted to be the lightest dibaryon, and to be unstable to weak decay. Although evidence for the H is limited to observation of a neutral object decaying to p + E~ [ShSKM 90], experimental searches continue. (iii) Hadronic molecules: Together with the glueball candidates discussed earlier, another possible interpretation of observed particles as Table XIII5. Spectroscopy of sixquark configurations. 5/7(6) of colorspin 5(7(3) of flavor Spin 490 896 280 175 189 35 1
1 8 10 10* 27 35 28
0 1,2 1 1,3 0,2 1 0
380
XIII Hadron spectroscopy
nonconventional hadrons occurs with the isovector ao(98O) and isoscalar /o(975) mesons. Nominally, these particles have the quantum numbers of "Hie L = 1 sector of the QQ model, and their near equality in mass suggests an internal composition similar to that of the p(770) and u;(783), i.e. orthogonal configurations of nonstrange quarkantiquark pairs. However, among properties which argue against this are their relatively strong coupling to modes which contain strange quarks, their narrowerthanexpected widths, and their 77 couplings1 The proximity of the KK threshold and the importance of the KK modes has motivated their interpretation as KK molecules [Wil 83]. Unfortunately, interpretation of scattering data near the 1 GeV region is confused, with even the very number of isocalar states in question [AuMP 87]. Problems 1) Power law potential in quarkonium Consider an interquark potential of the form V(r) — crd. a) Use the virial theorem to determine (T)/(V) for the ground state. b) Given the form E2s  Eis = f{d)Mdl^2+d\ where M is the reduced mass, determine d from the observed mass differences in the cc and bb systems, using Eq. (3.1) to supply heavyquark mass values. c) Assuming this model is used to fit the spinaveraged ground state cc and bb mass values, determine v 2 /c 2 for each system. 2) Quarkonium annihilation from the 1So state Modify Eq. (1.20) to obtain the leadingorder contributions appearing in Eq. (1.21). 3) Spindependence and light Pwave mesons a) Numerically determine the quantities m, ms_o,rnten of Eq. (1.10) using n = 1 T Pwave mass values. b) Repeat this evaluation for light Pwave mesons, but include a lP\ state in your analysis. Thus, you must generalize Eq. (1.10) to include a spinspin contribution mss. Choose mass values according to the assignments
f
(3P2) (3Pi) (3P0) .
Comparison of the results with the QCD BreitFermi interaction of Eq. (2.10) reveals the real world to have a smaller spinorbit effect than is (at least naively) anticipated from this model.
Problems
381
4) Mass relations involving heavy quarks a) Repeat the analysis of Eq. (3.1) but using the masses of the charmed/strange mesons DS,D*S instead. Infer a value for M/Mc by referring to the result obtained in Eq. (2.17). Compare with the determination of Eq. (3.1). b) Extend the procedure of Eqs. (2.202.25) to isospinviolating mass differences of cflavored and 6flavored hadrons.
XIV Weak interactions of heavy quarks
Heavy quarks provide a valuable guide to the study of weak interactions. Measurements of decay lifetimes and of semileptonic decay spectra of heavy, flavored mesons* yield information on elements of the KM mixing matrix, as does the observation of particleantiparticle mixing like that in the BdBd complex. Finally, detection of CPviolating signals in heavyquark systems has the potential to become the breakthrough that has been sought since the discovery of this phenomenon in the kaon system. XIV—1 Heavyquark lifetime and semileptonic decays Of the weak interaction properties associated with heavy quarks, the lifetimes and semileptonic decays are the most amenable to theoretical analysis, and it is with these that we begin. The spectator model Consider the weak beta decay, Q —> qeve, of an isolated heavy quark Q into a lighter quark q. By analogy with muon decay, this proceeds with decay rate (if radiative corrections are ignored) _
5 G\^F" mlQ
/0VmQ) > f(x) = 1  8x2 + 8x6  xs  24x 4 lnx ,
(1.1)
where j{x) is the phase space factor the first terms of which were already encountered in our discussion of muon decay in Sect. V2. Under what circumstances would this be a good representation for the beta decay of a heavy meson containing quark Ql For it to be accurate, the final state must develop independently of the other (socalled spectator) quark * Note that the conventions of [RPP 90] imply that the quantum numbers of the neutral mesons are K° = {ds),D° = {cu),B° = (db) and B° = (s6).
382
XIV1 Heavyquark lifetime and semileptonic decays
383
in the heavy meson. Experience with deep inelastic scattering suggests that this occurs when the recoiling quark q carries energy and momentum larger than typical hadronic scales, i.e. in the range Eq > 11.5 GeV. In D decays, the average lightquark energy is (Eq) ~ mo/S ~ 0.5 GeV, so that this approximation is suspect. It should be considerably better in B decays, but still not perfect. Let us first explore the consequences of adopting the spectator model. If we neglect KM suppressed modes, the main decay channels for b quarks are b —> cud, ccs, c£i?£ (£ = e,/i, r), while for cquarks they are restricted to c —• sdu, SJLV^, seve. Relative to the lepton modes, each hadronic decay channel picks up an additional factor of 3 upon summing over the final state colors. Two of the J3meson final states (ccs and CTUT) have significant phase space suppressions (reducing them to about 20% of the cud mode) due to the heavy masses involved. The simplest spectator model then predicts branching ratios Sv e X
=* r ^
3 + 2
= 0.2 , x
(12)
3 x (1 + 0.2)+ 2 + 0.2 where X denotes a sum over the remaining final state particles. Also, this picture predicts the absolute rates of the D and B decays to be
TD = [s^fer I*y2/(*c)l
 11 x 10"12s , 1
.„
__io
(l3a) 0.05"
vch
(1.3b)
where f(xc) ~ 0.7 and /(#&) ^ 0.5 are phase space corrections. For definiteness, we have taken mc — 1.5 MeV and nib = 4.9 GeV in the above. However, note the quintic dependence on quark mass; the B lifetime prediction would be 10% lower if ra& = 5.0 GeV were used! For D decays, the D+ and JD°lifetimes differ by a factor of 2.5, rD+ = (10.62 ±0.28) x 10~ 13 sec , 3 2 + = 2.52 ± 0 . 0 9 , rDo
(L4)
whereas the spectator model requires them to be equal. This failure is not surprising, as the D meson mass lies in the region of strong hadronic resonances and final state interactions seriously disturb the spectator picture. Thus we expect the spectator model to reveal only gross features of the D system. It is remarkable, given its simplicity, that the spectator model predicts (roughly) the correct magnitudes of the lifetime and of
384
XIV Weak interactions of heavy quarks
the inclusive branching ratios, B r D o ^ e X = (7.7±l.l)%,
BxD+^ei?eX = (19.2 ± 2.2)% . (1.5) +
We see that the decays of the D correspond more closely to the spectator predictions than do those of the D°. For B mesons, the spectator model is likely to be close enough to reality that some small extra effects can be added in order to render it more realistic. These include QCD corrections of two kinds. First, the nonleptonic hamiltonian picks up short distance corrections of the form described in Sect. VIII3. These are smaller in magnitude for b decays, because the strong coupling as is evaluated at a higher mass scale. In addition, there are also the QCD radiative corrections associated with the decay of the heavy quark, including bremsstrahlung from the quarks. Besides the QCD correction, one can add bound state corrections to account for the fact that the b quark is not sitting at rest in the B meson, but has some spread in its momentum space wavefunction. The combined effects of all these corrections produce small modifications in the predictions, which are not easy to express analytically. However, for a numerical example we cite a model [AlCCMM 82] which employs a Gaussian momentum distribution for the motion of the bound quarks and includes QCD radiative corrections, obtaining 2
0.05 1.2 x 10"12sec . BrB+epex ^ 13% , and rB =
(1.6)
Observe how close this QCD corrected version of the spectator model is to the free quark result of Eq. (1.3b). The leptonic branching ratio is close to the experimental value* [St 90] BiB>evex = (10.8 ± 0 . 5 ) % , while the measured lifetime [Dan 92] r*xpt = (1.26 ± 0.07) x 10"12 sec
(1.7) (1.8)
can be used in Eq. (1.6) to extract a value for the KM matrix element, F cb  0.049 .
(1.9)
Beyond the spectator model What are the possibilities for making an estimate which is better than the spectator approximation? For the total rate, there is absolutely no * There remains some model dependence in the experimental result due to the need to separate the signal from the B —• DX, D —*• Xev background. The number quoted uses the [AlCCMM 82] model in this process.
XIV1 Heavyquark lifetime and semileptonic decays
385
hope of reliably calculating and summing all the individual nonleptonic decays. We must be content either with the spectator model or with very crude calculations of some twobody modes. For semileptonic decays, the situation is somewhat better. The data show that the quasionehadron states, i.e. D —> Keve, K*eve and B —• Deve, D*eue^ form the largest component of the semileptonic rates [Fu et al. 91],
( L 1 0 )
These transitions can be addressed by quark model calculations, so that we have an independent handle on such decays. The hadronic current matrix elements are described by form factors such as (K(p')\s7,c\D0(p))
= / + ( p + j/) / J + /  ( p  j / ) M ,
(K*(p') \S1,c\ D°(p)) = ige^pe" {p + p'f {p  p'f , (K*(pf)
\s7li75c\ D°(p)) = / l€* +e*q
(1.11)
[f2 (p + p')fi + / 3 q,] ,
with analogous definitions for the B decays. All form factors are functions of the fourmomentum transfer q2 = (p — pf)2. The physics underlying these form factors is twofold: 1) If the final state meson does not recoil, the amplitude is determined by a simple overlap of the quark wavefunctions, as described in Sect. XII2; 2) As the final meson recoils, the wavefunction overlap becomes smaller, so that the form factors fall off with increasing recoil momentum. For D decays, the KM element is known to a high degree of accuracy from the unitarity of the KM matrix. In this case, the quark model calculations serve to check whether the experimental rate can be reproduced. Most calculations do well for D —> Keve, while D —> K*eve occurs at a rate about onehalf of theoretical expectation [St 90]. For B decays involving the b —• c transition, one may assume that the models continue to be valid and thereby extract the value of V^b Fortunately, various models seem to agree with each other and with the relative amounts of D*/D production, leading to the value [WiSB 85, IsSGW 89, K6S 88], \Vch\ =0.046 ±0.007 ,
(1.12)
where the range of model dependence has been folded into the quoted error.* A preliminary attempt [Ne 91] using the heavy quark symmetry * Implicit to this analysis is the assumption that Br [T(45) —> BB) = 100%.
386
XIV Weak interactions of heavy quarks
relations, to be discussed in Sect. XIV2, also yields an identical result. This value agrees with the estimate obtained previously from the lifetime. It can be used to imply the following constraints on the KM and Wolfenstein parameterizations: s3 + s2ei6
=0.046 ±0.007
and
A = 0.95 ±0.14 .
(1.13)
In the case of nonleptonic B, D decays, we have considerably less confidence in our ability to predict the decay amplitudes. This is especially true in D nonleptonic decay because the rescattering corrections required by unitarity can play a major role. Unitarity predicts (cf Eq. (C3.14)) for the D —> / matrix element of the transition operator, Uf%^D
,
(1.14)
where n are the physically allowed intermediate states. The scattering matrix elements are evaluated at the mass of the Z?, which happens to lie in an energy range where many strong resonances lie. The scattering elements Tn+f are therefore expected to be of order unity, implying that rescattering can mask the underlying pattern of weak matrix elements. This makes calculation of D decays particularly suspect. In view of the large B mass, the situation may be better for B decays since scattering amplitudes for decay into individual modes fall off at high energy. A model based on the vacuum saturation method [BaSW 87] may thus prove useful in predicting nonleptonic B decays. Inclusive vs exclusive models for b —> ceue As can be seen from the equality of Vcb as extracted either from the spectator model or from quark model estimates of individual semileptonic modes, the two types of analysis give very similar results for the b —> ceve transition. At first this might seem surprising, since the models are quite different. However, the following observation [ShV 88] lends plausibility to the agreement. Consider the semileptonic decay of a heavy quark into another heavy quark, Q\ —• Q2zve, such that their mass difference Am is small compared to the average of their masses ((mi +rri2)/2 ^> Am), yet large compared to the QCD scale (Am » AQCD) Because of the second condition, one might use the spectator model result,
mHlW ,
(115)
where V12 is the appropriate weak mixing matrix element. However, if the first condition is satisfied, the quark recoil will be nonrelativistic. This leads to a nonrelativistic calculation of the transitions from a pseudoscalar Q\q state to pseudoscalar and to vector Q2q states. In this limit,
XIV1 Heavyquark lifetime and semileptonic decays
387
> ty^x is proportional to the normalization operator, while the axial current ^27i75^i —* tyfti^x is proportional to the spin operator. For states normalized as one then has
(7
\fc\
°~)= 2m , = 2m e}(p') ,
where m is either mi or m2. This translates into invariant form factors = (P + P')» , (1.18) which are the correct relativistic results. Using these to calculate the semileptonic decays, one finds G2
Comparing these, one sees that the sum of the pseudoscalar and vector widths exactly saturates the spectator calculations. In this combined set of limits, it seems that both types of calculations can be valid simultaneously! Application of this insight to b —> ceue decays is somewhat marginal, as the nonrelativistic condition is not well satisfied. A velocity as large as v = 0.8c is reached in portions of the decay region, although on the average a lower value is obtained. However, it is likely that the near equality of spectator versus quark model results is a remnant of the situation described above. These ideas will be extended in Sect. XIV2. The b —> ueue transition The physics of the b —• u semileptonic transition is not as simple due to the large energy release. In the b —• uev e transition, the allowed range of the energies of the electron and the upquark are shown in the Dalitz plot of Fig. XIV1. The energy release is such that the free quark picture should be a reasonable approximation. The only questionable place in the plot occurs in the lower righthand corner, where the relative momentum of the spectator quark and of the recoiling upquark is smaller than 1 —• 1.5 GeV/c 2. Here one might expect that bound state effects could become important. Spectator models have a smooth hadronic mass
388
XIV Weak interactions of heavy quarks
distribution instead of having resonances in the low mass region. The effect of the spectator assumption is to smooth over the resonances which occur in this region of phase space. Because Vch ^> Kib> it is difficult to experimentally observe the b —> ueve transition. There are two possible strategies. In one, only the highest energy electrons are looked for. Above Ee = 2.4 GeV, there can be no electrons from b —> ceve because the heavier cquark mass kinematically rules out these energies. Thus the endpoint region Eo = 2.4 —> 2.6 GeV is uniquely sensitive to the b —> ueve transitions. Alternatively, one can look for the exclusive noncharmed decays B —• 7rePe, peue, 7T7rei/e, etc. In employing this latter technique, comparison with exclusive quark model calculations is necessary. Unfortunately, the exclusive quark model b —» u predictions do not agree with each other as well as do the b —> c results, e.g. the range of discrepancy in the B —• 7rePe transition is a factor of 7. For the method using the electron endpoint region, the exclusive quark model calculations are not sufficient. Note from Fig. XIV1 that the endpoint region involves both large and small values of uquark recoil. These configurations occur when the u and ve are produced nearly collinearly, recoiling against a high energy electron. For small u recoil, the final state may well be largely a single meson, with the uqnavk combining with the spectator quark. However, for large recoil, one expects significant inelastic nonresonant contributions, of which B —• 7T7reue is the simplest example. Quark models cannot hope to individually calculate and then sum these contributions, any more than they could sum the individual hadronic components of a quark jet. However, the spectator model can be applied to the endpoint spectrum. By definition, this model sums over all final states. The QCD and bound state corrections have been applied to the b —• uev e case [A1CCMM 82] and in particular to the endpoint region. As noted above, this method is most reliable for the largest i/quark recoil, when it is most realistic to have its production independent of the spectator quark. Alternatively, a hybrid approach has u ev
m—.—•D
Fig. XIV1
I
Q
Kinematically allowed energies in b —> ueve
XIV1 Heavyquark lifetime and semileptonic decays
389
been developed [RaDB 90] which uses the exclusive B —> Meue (M = meson) calculations for energies below 1.5 GeV and a spectator model for larger energies. This is an attempt to use both frameworks in the region in which they are valid, and thus provides a check on the importance of low energy effects on the endpoint spectrum. The hybrid method and the pure spectator approach yield endpoint spectra which are in agreement in shape and magnitude to about 30%, indicating that the basic theoretical uncertainties in the endpoint region are not large. We have seen that there are two methods of approaching the b —> ueve measurements, and two corresponding theoretical styles of analysis: (i) exclusive decays such as B —> peue can only be calculated using exclusive quark models, while (ii) inclusive measurements of the electron endpoint region can only be addressed in inclusive models such as the spectator or hybrid models. At the time of this writing, there is progress on both approaches. A few exclusive b —> u events have been reported, although not yet enough to extract rates. In addition, there are indications of b —> u events using the endpoint method [Fu et al. 90], [Al et al. 90]. Prom these, one extracts the ratio 0.10 ±0.03 , (1.20) Tr Kb where the model dependence of the two inclusive methods has been folded into the error estimate. Thus a constraint is obtained on the KM angles, 3 + S2 e%6
= 0.45 ±0.14 ,
(1.21)
or equivalently on the Wolfenstein parameters, v/p2 + r/2 = o.45±O.14 .
(1.22)
The latter is a particularly interesting constraint on the CPviolating parameter 77. The top quark Compared to the other 'heavy' quarks, the top quark will present a rather novel decay pattern. Because nit > M\y +ra&and the KM element Vtb is near unity, the dominant decay is the semiweak transition t —• b + W+. The amplitude and transition rate for this process are
llVtb12 g(xw, xb) = (1  (xw + xfc)2)1/2(l  (xw 
390
XIV Weak interactions of heavy quarks f(xw, xb) = (1  x2h)2 + x2w(l + x2h)  2x$v , an
=
(1.23)
where xw = Mw/mt d ^6 m^/rrit. For the range of top quark masses, rrit = 100, 150, 200 GeV, the width is quite large, Tt^bW+ = 0.093, 0.87, 2.4 GeV respectively and corresponds to a lifetime of r = (70., 7.6, 2.7) x 10~25 sec. For such a large tquark mass, the emitted VF+'s will be predominantly longitudinally polarized, exceeding production of transversely polarized W + 's by a factor ~ m2/M^r. This is a reflection of the large Yukawa coupling of the t quark to the (unphysical) charged Higgs scalar which becomes the longitudinal component of the W+. Other decay modes of the t quark will be highly suppressed by weak mixing factors, e.g. for the mode t —• s + W+ the suppression amounts to  Vts/Vtb>2 — 2 x 10~3. However, decays of the top quark would be quite sensitive to the presence of physics beyond the Standard Model, such as the occurrence of sufficiently light charged Higgs bosons in models with an extended Higgs structure. An interesting consequence of the large t —• b + W + quark decay rate is that for large values of m*, there will not be sufficient time for the top quark to form bound state hadrons. In view of the large top quark mass, the ti system (toponium) is nonrelativistic and sits in an effectively coulombic potential, V = — 4a s /3r. In the ground state, one finds the quark velocity vrms = 4a s /3 and atomic radius r0 = 3/(2asmt). A characteristic orbital period is then t = 27rro/frms — 97r/(4:a2smt). Using as(r0) = 0.12, we estimate t = (32, 22, 16) x 10~25 sec for mt = 100, 150, 200 GeV. In contrast, the toponium lifetime would be one half the t lifetime given above, since either t or t could decay first. These comparisons imply that a heavy top quark has an appreciable probability of decaying before completion of even a single bound state orbit. An equivalent indication of the same effect is the observation that the toponium weak decay width (twice that of a single top quark) becomes larger than the spacing between energy levels, such as E2s ~ Eis = a2mt/3 = 0.50, 0.75, 1.0 GeV for mt_= 100, 150, 200 GeV. The production cross section, say in e +e~ —• tt, will not then occur through sharp resonances. Instead, there will exist a rather broad and weak threshold enhancement, due to the attractive nature of the coulombic potential [FaK 88, StP 90]. This permits the production and decay of top quarks to be analyzed perturbatively, with I\ serving as the infrared cutoff. A heavy top quark can then provide a new laboratory for perturbative QCD studies. Our lack of understanding of the large top mass illustrates how little we actually know about the mechanism of mass generation. If all fermion masses arise from the Yukawa interaction of a single Higgs doublet, then the Yukawa coupling constants must vary by the factor gt/ge — rut/me > 1.7 x 105. There is nothing inconsistent about such a variation, but it is so
XIV2 Weak decays in the heavyquark limit
391
striking as to beg for a logical explanation, one which is presently lacking. On a more practical level, we shall see later in this chapter and also in Chap. XVI how the top quark mass contributes as a parameter to many of the flavorchanging decays and radiative corrections in the Standard Model. Given the present range of top quark mass estimates, various predictions sensitive to rrtt are typically quoted as bands of numerical values. Upon discovery of the top quark and measurement of its mass, it will be possible to test the Standard Model with greater rigor. XIV2 Weak decays in the heavyquark limit The discussion of the previous section leaned heavily on the use of models to describe quark weak decay. However, some aspects of weak transitions can be obtained in a model independent fashion through the use of the rrtQ —> oo limit which was introduced in Sect. XIII3. While the full consequences and limitations of this method are not yet clear, it provides a variety of qualitative and quantitative insights of considerable value. The heavyquark approximation manages to justify many results which have become part of the standard lore of quark models. For example, consider the decay constant of a Qq pseudoscalar meson M, (0 \q(x)ry5Q(x)\ M(p)> = iV2FMp» e^x
.
(2.1)
In the quark model one finds that FM OC (TTIM) ' • This follows from the normalization of momentum eigenstates, <M(p')M(p)> = 2Ep6^(p  p') ,
(2.2)
such that (097075QM(0)) = , _
,_
,_
(decay const defn.) ,
(quark model rein.) , (2.3) where i/>(0) is the Qq wavefunction at the origin and Nc is the number of colors. Since as TUQ —> oo, the Qq reduced mass approaches a constant value fi —• mq, we expect that ^(0) itself approaches a constant in this limit,* and the scaling behavior FM OC (UIM)~1/2 then follows immediately from Eq. (2.3). Alternatively, the dependence of FM on TTIM can be derived using the wavepacket formalism introduced in Chap. XII. * For example, in the nonrelativistic potential model, the Swave wavefunction at the origin is related to the reduced mass by ^(0) 2 = fi{dV/dr)/2irh2.
392
XIV Weak interactions of heavy quarks
This quark model result can be validated in the heavyquark limit [Ei 88]. Consider the contribution of meson M to the correlation function
C(t) = ISx (0 i4o(*,x)4(O) 0) ,
(2.4)
where AQ = qjolsQ Inserting a complete set of intermediate states and isolating the contribution of meson M, we have
C(t) = Jd3xj
{2^2E
(0 4>(*,x) M(p))(M(p) 14,(0)1 0)+... (2.5)
where the ellipses denote other intermediate states. From the definition of FM, one finds C(t) =
MmMeimMt
+
_
^
^ ^
2rriM
Alternatively, the heavy quark develops in time in this correlation function according to the static propagator of Eq. (XIII3.6), C(t) =  ^ 6  ^ ( 0 1 ^ , 0 ) 7 0 7 5 ^ , C ) ) ( l +75)70759(0)0) ,
(2.7)
with all the dynamics being contained in the light degrees of freedom. The matrix element is independent of m ^ , and the scaling behavior FM oc (m M )" 1 / 2
(2.8)
follows immediately. This technique is applicable to lattice theoretic calculations of FM There one considers euclidean (t —> —ir) correlation functions, and identifies the M contribution by the e~mMT behavior. At present, lattice calculations attempting to obtain physical results from the rriQ —• oo limit and from the lightquark limit do not agree in regions of overlap. We therefore feel that it is premature to quote theoretical values of FD, FB Another piece of quark model lore which can be justified by this correlation function is that the mass difference UIM — TTIQ approaches a constant value in the UIQ —• oo limit. This can be inferred by comparing the exponential time dependences in Eq. (2.6) and Eq. (2.7), and noting that the difference must be be independent of the heavy quark. The heavyquark limit also makes predictions [IsW 89] for transition form factors between two heavy quarks (which for definiteness we shall call b and c). Recall the lagrangian developed in Eq. (XIII3.15), the leading term of which is
Cv = hfiiv • D hfi + h®iv  D h® .
(2.9)
This lagrangian exhibits an SU{2) flavor symmetry involving rotation of hy and hv'. It is also spin independent, and thus contains an additional SU{2) spin symmetry. The two 5f/(2)'s may be combined to form an
XIV2 Weak decays in the heavyquark limit
393
5(7(4) flavorspin invariance. Physically, the internal structure of hadrons containing a heavy quark and moving at a common velocity is seen to become independent of the quark flavor and spin. This property leads to many relations between transition amplitudes. An example of a process appropriate for the heavyquark technique is the weak semileptonic transition B —• D induced by a vector current. For a static matrix element [i.e. both B and D at rest), the weak current transforms quark flavor b —• c, but leaves the remaining contents unchanged, resulting in unit wavefunction overlap. This can be seen calculationally by noting that the time component of the spatially integrated current is the conserved charge of the SU(2) flavor group mentioned above, d3x (D(p') c(:r)7o&0r) B(p)) = 6®(p  p') y/AmDmB S
= (P  P') [/+(*m) (rnD + mB) + f(tm)
(2.1UJ
(mB  mD)} ,
where tm = {TUB — mo)2 is the value of t = (p — p')2 at the point of zero recoil, and the general decomposition of a vector current matrix element, (D(p')\c7lib\B(p)) =f+(t){p + p % + /_(t) ( p  p % 
(2H)
has been used in the second line of Eq. (2.10). We have seen results similar to Eq. (2.10) in the discussion of the ShifmanVoloshin limit in the previous section. However, there the restriction ms—mi)
(PD + pi,)^ ,
where /#(0) = /D(0) = 1. Let us consider the momentum transfers ts and tsD = (PB — P'D)2 i n terms of the velocities, using p^ = mj pB = m#v and pr> = m^v, we have
{ppDf
l(lvv')
,
= {PB  PD)2 = (ms  rno)2 + 2mBm,D (l  v • v') .
(2.13)
394
XIV Weak interactions of heavy quarks
If each transition has common velocity factors, the various momentum transfers are related by
to = ^ t o = — (to*  «m) . mzB
(2.14)
TUB
In view of the normalization convention of Eq. (2.2), one must divide the state vector of particle i by \/2rrii (assuming mi » p) before applying the b <• c symmetry. Upon doing so and requiring the resulting expressions to be identical functions of the velocities v and v' leads to the identifications
{D(p'D)\cliC\D{pD)) _ (B(p'B)\Hb\B(pB))
_ (D(p'D\c7ib\B(pB))
After simple algebra, this results in the form factor relations
(2.15)
<2l6)
Although consistent with Eq. (2.13), this manages to separate out /±. The results are expressible in terms of a single function of velocity. It is notationally simpler to express the kinematic dependence using v • v1 instead of t, i.e. fi(t) —• fi(v • v'). Thus, we have
(V • v') = fD(v • vf) = J———f V mB + m D
+(v
where, aside from the constraint £(1) = 1, the function £(vv') is unknown and must thus be determined phenomenologically. If we exploit the full SU(4) flavorspin symmetry, then all of the weak current form factors involving B,B*,D and D* can be expressed in terms of £(v • vf), e.g., (D*(p'D) c7^6 B(pB)) = (D*(p'D) C7M75& B(pB)) = JmD*mBZ(v • vr) [(1 + v v%  e* • v t/J . (2.18)
XIV3 B°B° and D°D° mixing
d(s)
w
395
b
Fig. XIV2 Box diagram contribution to B meson mixing. The symmetry language is appropriate here because, similarly to the symmetry relations detailed in the first part of this book, we have related different processes even though there remains an uncalculable ingredient to be determined from experiment. The limits of validity of these expressions are not yet clear. For example, at very large recoil {i.e. v • v' large), hard gluon exchange could introduce spin dependence into the transition. These limits will be explored more fully experimentally and theoretically in the near future. The real value of the heavyquark method is that it forms a consistent approximation scheme. The leading order predictions of the type described above appear to bear some resemblance to reality and may end up being directly applicable. However, this will only become certain as the important job of working out the finiterag corrections is accomplished. There are perturbative corrections which come from interactions of hard gluons [Wi 91] plus 1/m corrections which arise from the expansion of the fields described in Eqs. (XIII3.14), (XIII3.15). As with any controlled approximation, it is the size of such corrections which will determine the limits of validity of the results. This process of analyzing corrections is now starting [Lu 90, GeGW 90]. It is likely that the theoretical insights thus gained will strongly influence the phenomenology of B decays, which until now has been dominated by the use of models. XIV3 B°B° and D°D° mixing Just as K°K° mixing can occur due to the weak interactions, so can mixing exist in the B®B®, B®B® and D°D° systems. We shall cast our discussion of heavyquark mixing in terms of B° mesons, and return at the end to the D® case. The formalism is the same in all situations and can be taken directly from the discussion of K°K° mixing in Sect. IX1. B°B° mixing The mixing occurring in B° mesons is shortdistance dominated. This is because (i) the dominant weak coupling of the 6quark is to the tquark, and (ii) the shortdistance box diagram (Fig. XIV2) grows roughly with the squared mass of the intermediate state quarks. Thus the very heavy mass of the top quark enhances this contribution.
396
XIV Weak interactions of heavy quarks
The effective hamiltonians for B® and B® mixing are given by m2tH(xt)VBOBd
^
+ h.c. ,
) 2 m%H(xt)T,BOB> + h.c. ,
(3.1)
OBd = dy^l + lh)bdrf{\ + 75)6 , 0^^(1
+75)
where TJB — 0.9 is the QCD correction and H(x t) is given in Eq. (IX1.16). The matrix elements of OBd and OBs can be parameterized analogously to that used in kaon mixing,
16 where the pseudoscalar decay constants are normalized as
(0 Idyfc&l Bd(p)> = iV2FBdPP ,
(3.3)
with a similar definition for Bs. This corresponds to the normalization Fn ~ 92 MeV. Of course our knowledge of FB is not very precise. It is known for a very heavy meson M of mass THM that FM OC (raM)~1//2, but it is neither clear what the constant of proportionality is nor at what mass this scaling law is valid. Model dependent estimates lie in the ranges (74MeV)2 < FldBBd < (200MeV)2 , FB  P  = 1.2±O.1 . FBd
(3.4)
The KM elements which contribute can be estimated using either one of the standard parameterizations, or else the unitarity relations
=  (vchv;d + KbKa)
=  (vchv;s + vuhv^) ~ vchv;s . The prediction for the overall magnitude of mixing depends strongly on the mass of the top quark. Recall the relation of Eq. (IX1.11), derived from the diagonalization of the mass matrix, = 2\M12\ ,
(3.6)
which is valid up to exclusion of the factor T12/M12 « 1. This approximation is valid here because Fi2, coming from real intermediate states, does not receive any contributions from the top quark. The ratio of mass differences is independent of the top quark and is largely determined by
XIV3 B°B° and D°D° mixing
397
the KM angles. In the Wolfenstein parameterization, one has
(3.7)
~ (0.034 ±0.009) \lpit]\2
.
Since we have constraints on p and r\ from Eq. (1.22), we find > 12 .
(3.8)
It is clear from Eqs. (3.7) (3.8) that the Standard Model requires Bs mixing to be much larger than Bd mixing. BdBd mixing has been observed, with magnitude = 0.70 ± 0.12
(3.9)
Because of the dependence on the topquark mass and on KM angles, one does not have a firm prediction at present for x&. However, one may verify that the result is consistent with what is currently known about these quantities. For example, the parameter set mt = 160 MeV, p = 77 = — 0.35 and F^ dBsd = F% yields exactly the experimental value for Xd provided the experimental value of F#d is used. This parameter set produces xs = 18. Are there longdistance contributions which can significantly modify Am for B mesons, analogous to those in the K°K° system? There are several arguments against this possibility. The 'longdistance' effects for Bd are shown in Fig. XIV3. They involve only the charm or up quarks in the intermediate states. Hence they do not have the factor 7TT H(xt), growing roughly as m^. This implies that the longdistance effects should be suppressed roughly by the factor (rriB/mt)2 ~ 4 x 10~3. In kaon decays the corresponding mass ratio of (mK/mc)2 is not as important and is in fact overcome by the large A/ = 1/2 enhancement of nonleptonic kaon decays. There is no similarly large nonleptonic enhancement in
D (a)
TT
(b)
Fig. XIV3 Longdistance contributions to B meson mixing.
XIV Weak interactions of heavy quarks
398
the B system, since the nonleptonic branching fraction is very close to its perturbative estimate. Hence we expect the above estimate of the longdistance suppression to be roughly correct. It is also the case that, in contrast to the kaon system, longdistance intermediate states occur only at the Cabibbo suppressed level for B%. Finally we note that the c and u contributions are accounted for in perturbation theory in the box diagrams. The B mass is heavy enough that the perturbative estimate should not be wildly in error. Given these considerations, we expect the short distance approximation for the B system to be valid. D°D° mixing The analysis of D°D° mixing is considerably less clear. The corresponding box diagram is shown in Fig. XIV4(a), and some possible long distance contributions are given in Fig. XIV4(b). The GIM cancelation in the intermediate state is between two light quarks d, s (the b effect is suppressed by KM angles). However, there is no compensating large mass factor here, and longdistance and shortdistance effects come out at the same order of magnitude. No reliable predictions of Amp can be made. However, the Standard Model clearly requires that ATTID/TD « 1 for the D° system because Amp is twice Cabibbo suppressed (i.e. Amp = O(X2)) while F has no such suppression. Hence, counting KM factors and noting that the GIM cancellation is a measure of the breaking of SU(3) symmetry leads one to estimate that /Ara\
~ A2 x (SU(3) breaking) ~ 10"
(3.10)
Smaller values often emerge in specific calculations. In concluding this section, it is interesting to observe the rather different patterns of behavior which occur in the meson systems exhibiting flavor mixing. The theoretical ratios of longdistance and shortdistance
c
_
w
_ u K,(TT)
d,s,b
i K,(TT)
W
(a)
(b)
Fig. XIV4. Shortdistance (a) and longdistance (b) contributions to D meson mixing.
XIV4
The unitarity triangle
399
Table XIV1. Patterns of mesonantimeson mixing. I Amiongl
Am
I Ambox I
K°K° D°D° B°dB° B°sB°
O(l)a O(102)a smalla small0
0.477 ± 0.002 <0.09 0.70 ±0.12 large0
° Theoretical expectation
(box) contributions to Am, and also the magnitude of Ara/F, are summarized in Table XIV1. XIV4 The unitarity triangle The B meson transitions form a complex system and provide much of our information on the pattern of weak mixing. The overall B lifetime and b —>• c semileptojiic decays are governed byV^b, the suppressed b —• u modes by Vn\>, B®—B® mixing by 14d> and B^—Bg mixing by V\&. Together with the V^g element, these form all of the 'interesting' sectors of weak mixing. There is a useful pictorial representation of the constraints of unitarity on these elements. Consider the effect of the unitarity relation Of the components to this equation, Vud, Vtd and Vcd are known up to corrections of second order in A = V^g, yielding Vuh\Vch
+ V*d = 0 .
(4.2)
If we treat these elements as complex vectors, this relation is equivalent to a triangle in the complex plane. In the Wolfenstein parameterization the various elements are Vch = Vts = AX2
 ir,) ,
. A^3u_e2
Vtd = XSA(1  p  u/) . (4.3)
^
Fig. XIV5 The unitarity triangle
400
XIV Weak interactions of heavy quarks
Two possible triangles are shown in Fig. XIV5 according the assumption that p < 0 (a choice which helps make Xd large) or p > 0. Note that the unitarity triangle can be constructed knowing only the magnitude of the elements  V^bU iKibl a n d l^tdl The existence of such a closed triangle is independent of the parameterization. Other unitarity triangles, corresponding to the other unitarity constraints, also exist but are either less useful than this one or are equivalent to it [Ja 89]. The unitarity triangle has an interesting connection with CP violation. If the CPviolating parameter rj vanishes, the triangle is reduced down to a line since all the angles go to either 0° or180°. In fact, the area XQA27] of this triangle is exactly the unique rephasing invariant measure of CP violation. The angles y?i, <^25 <^3 are themselves indicators of nonconservation of CP and play a role in the B studies to be described in the next section. It is reasonable to expect that the KM elements can be measured to sufficient accuracy to demonstate that the unitary triangle is not flat. Semileptonic B decays could yield accuracies to ±15% on \VC\>\ and Vub? provided some further understanding of the model dependence is achieved. In some sense Vtd is already 'measured' from B^Bj mixing, but the dependence of mt and FB are too great for a meaningful extraction. However, B®Bg mixing has essentially the same uncertainties, and they largely cancel in the ratio in Eq. (3.7). If we were able to understand the SU(S) breaking in F^s/F^d^ a measurement of Bs mixing would complete the measurement of the unitarity triangle. XIV5 CP violation in B mesons The decays of B mesons can exhibit a rich variety of CPviolating signals, some of which are rather large. It is clear that the future will bring experimental efforts to measure these observables. We consider these crucial to the ultimate verification (or falsification) of the origin of CP violation within the Standard Model. Recall that the value of e cannot be regarded as a prediction of the Standard Model because there is an unknown parameter, the KM phase <5, which must be adjusted to fit experiment. In principle the value of e'/e could be regarded as a test, but theoretical and experimental uncertainties are presently too large for this to be practical. There are other candidate theories which could also accommodate the CPodd signals in kaons without being in contradiction with any existing data. However, the Standard Model, with its single CPodd parameter, makes clear predictions for the patterns of CP violations in B decays, the observation of which would be a major triumph for the Standard Model. There is an important division in the study of CP violations for B mesons into processes which proceed using B°B° mixing and those which
XIV5 CP violation in B mesons
401
do not. We shall discuss those involving mixing first, and in the most detail, as some of these are the most reliable predictions of the Standard Model. CPodd signals induced by mixing General formalism: The time evolution of a B° or B°meson parallels that of a neutral kaon. Denoting Am = ra# — TUL and AF = T# — T^, where H (L) refers to the heavier (lighter) of the neutral B CPeigenstates, one obtains for states that start out at t = 0 being either B° or i?0,
P= g±(t)
JM12  iT12 = }.
The strategy for observing CPviolating asymmetries is to compare the decay B°(t) —> / , where / is some given final state, to that of B°(t) —>• / , where / is the CPconjugate of / ,
\I)=CV\f) .
(5.2)
Let us define the matrix elements
Af) = (f\nw\B°) ,
A(f)= (f\nw\B°) ,
A(f) = (f\nw\B°) ,
A(f)= {f\Hw\B°) ,
and their ratios,*
The decay rates for the two processes are easily found to be [BiKUS 89] ^f oc [a + /?e~ An + 7 e~^An cos Am t
1
\A(f)f2  1 +
+ R« IPU) V
* We caution the reader not to confuse the notation for these ratios with the KM element p in the Wolfenstein parameterization of Eq. (II4.24)
402
XIV Weak interactions of heavy quarks
"=w)rmi
21
V
 R e  ^ ( / ) j I , (5.5a)
and
~Art
7 e~?Art cos Am t
2"
1+ 2"
1+
(5.5b)
Any observed difference between these two quantities would indicate the presence of CP violation. Before considering some examples, there is a simplifying approximation which it is useful to make. As seen in the previous section, one expects that Myi » Fi2, so it is a good approximation to neglect Fi2 (and hence AF) in almost all cases. The one exception is the semileptonic asymmetry to be discussed below. Even if AF/F were as large as 10%, the exponential factor exp(AFt) would differ significantly from unity only after about ten lifetimes, at which time there would be extremely few particles remaining. In this approximation q/p becomes a pure phase, q/p = el(p, so that \q/p\ = 1. Decays to CPeigenstates: The most striking processes are those where the final state / is a CPeigenstate, /) = ±  / ) , such as / = ipKs, I/JKL, D+D~, 7r+7r~. In this case one has p(f) = l/p(f). Upon considering the total decay asymmetry obtained by integrating the time development
XIV5 CP violation in B mesons
403
from t = 0 to t — oo, one finds f™dt Af =
dt
(5.6)
1+ where x = Ara/F. This could be nonzero for either (or both) of two
reasons, (i) \*p(f)\\22 ^ 1, (it) Im [*p(/)] ^ 0. The cleanest analysis occurs when p(/) = 1, i.e. \A(f)\ = \A(f)\. In this case we find that P
•x*
(5.7)
An example is B® —• ipK®, which proceeds dominantly through b —* ccs, so that p(f) is basically a pure phase
v*vcb
(5.8)
Note that upon neglecting the quantity AF, one has q P
(5.9)
If we then define (5.10) the asymmetry is seen to become Af =
2x si
(5.11)
Thus for a given Ara/F, the prediction is independent of hadronic uncertainties and depends only on the phases in the KM matrix. Certainly this is a remarkably clear result which deserves to be tested. At this stage we can categorize the decays of neutral B mesons to CPeigenstates. For this purpose it is most convenient to use the Wolfenstein form of the KM matrix. In this parameterization, the elements Vtb, Kb, Vts, Vcs are all almost purely real. The Bd and Bs decays can proceed either through the ifMfavored transition b —• ccs or the if Msuppressed transitions b —» uud, b —• ccd, b —• uus. In the former category are included B% —• rfrKs and also B® —> i/j(p, iprj, DfDj. The Bs decays pick
404
XIV Weak interactions of heavy quarks
up no phase since
However, the Bd decay does have a phase since
implying (5.13b) Im pp(^lfs)l =  " ' " * r' 9 7^ 0 . IP J (1 ~~ P) ~^~ V Note that this latter number is exactly sin2
b
= sin2(^i ~ 0.5 .
(5.14)
J
Also, since Xd is close to unity, the final asymmetry in the decay Bd —> ipKs can be large. This, together with the relative observational 'cleanliness' of the ipKs mode, has justifiably made it the favored decay channel for seeking a CPviolating signal as a probe of the Standard Model. One might at first expect that \p(f)\2 = 1 is automatic if / is a CPeigenstate. However, it is possible to obtain \p(f)\ ^ 1 if there are two different ways to reach the same final state. For example, one could have the decay_B° —> TT+TT" either directly through b —• uud or indirectly through B° —• D+D~ (using b —• ccd) with a rescattering D+D~ —> 7r+7T~. In this case, we obtain I pi&n Ic
, (TT'TT')!
_ . ~ H
_i_ V*,Vu \A J I piSl ' v cd v cb I ^lnd  c 5
___\eiS° + VcjVX\Am*\eiS' ,
,w „ ,
——
where <5j and 6D are the strong interaction phase shifts. We see that there are three conditions to have p(/) ^ 1, viz. there must be two different paths to the same final state, these paths must have different strong interaction final state phases, and the two paths must also have different weak phases (i.e., arising from the KM matrix).
XIV5 CP violation in B mesons
405
Table XIV2. Standard model pattern for CP violation in B decays. Transitions Examples
Im (q/p)p(t) a
b
sin 2y?i
> CCS
b » ccd b + ?md
Brf —•
ipKs
Bs ~^ ^(f Bd ^ DD Bs
—>•
Bd
 • 7T+7T
i ^ s —> 7T
b > i m s
^ ^ 5 A s
sin 2
Ba+ir°
The KM suppressed decays can also be analyzed in terms of the angles which appear in the unitarity triangle, and are given in Table XIV2. Note that the b —> ccd and b —> uud modes can mix with each other, producing the possibility that p(/) ^ 1. Unfortunately, reliable predictions of these effects are not presently available. The predictions in Table XIV2 are given for the case p(/) = 1. It should also be pointed out that under all circumstances, asymmetries for J5j are suppressed because xs is large and all asymmetries fall off as l/xs for large xs. The physical origin of this result lies with the rapid oscillations in the B® <> B® system. Regardless of whether one starts out at t — 0 with B® or J3j, after a few oscillation lengths one has roughly equal amounts of B® and B®. This washes out the asymmetry, as expressed mathematically in Eq. (5.6). Decay to nonCPeigenstates: There may also exist CP violation in final states which are not CPeigenstates. Consider for example the final state / = D°Ks Both B% and B® can make transitions to this state, but as shown in Fig. XIV6 different KM factors contribute. In this case, p
(a)
(b)
Fig. XIV6 CP violation in B > D°KS decay.
406
XIV Weak interactions of heavy quarks
and p become respectively
p(D°Ks) = I (5.16) where the ratio A1/A2 of the two reduced matrix elements is not likely to have unit magnitude and will most likely contain a large final state interaction phase. Note that \p(f)\ = \p(f)\> In this case, we have the integrated asymmetry
/o°°* A
D°KS
=
+D°KS
dt
(5.17)
{l\p(&>Ks)\*) This prediction is not as clear as that for the ipKs mode because unknown hadronic matrix elements contribute. However, a sizeable asymmetry is likely, and the precise prediction could become more certain as one learns more about matrix elements occurring in B decay. Semileptonic asymmetries: For a final example involving mixing, let us consider CP violation in semileptonic decays. In much of our previous analysis, we have neglected the quantity F ^ . However for semileptonic decays, the whole effect vanishes if we neglect Fi2, so we must include it. For this case, only the transitions B° —• £+i/£X, B° —• QrviX (£ = e^/jt^r) can occur. The 'wrong sign' transitions in the time developments, B°(t) —> £~viX, B°(t) —> f+vgX, are then uniquely due to mixing. The appropriate formulas can be obtained from our general result Eqs. 5.5(a), 5.5(b) by the substitutions A(e~)  > 0 A(e+)  > 0
A(e)p(e~) A(e+)p(e+)
(5.18)
A(e+) = A(e~)
The integrated rate is Cdt\
L
1 p 1 p
• ' _
2 q 2 q
2 2
(5.19)
This sort of CP violation is thus solely sensitive to mixing in the mass matrix, as was the semileptonic K^ asymmetry. Unfortunately it is small
XIV5 CP violation in B mesons
407
for reasons which have little to do with CP violation. Expanding in powers of Fi2 and defining (fr = arg (F^/M^), one has Fl2
 2 ~
(5.20)
Since Fi2/Mi2 « 1, this asymmetry is always suppressed. For B®, where Fi2 might be nonnegligible, there is a further suppression in the Standard Model because the dominant contributions to Fi2 (cc intermediate states) and Myi (ti intermediate states) share the same phase. This is most easily seen using the Wolfenstein parameterization, Eq. (II4.27), where both are made real. Thus (pr is also suppressed. CPodd signals not induced by mixing Situations where CP violation occurs without the presence of mixing can occur in B*1 decays through the interference of different decay mechanisms. The requirements are the same as we saw previously in a different context, i.e., there must be two different paths to the same final state, these paths must have different strong interaction final state phases, and the two paths must also have different weak phases. Consider for example the decays B± —> K^TT0, which can take place both through the penguin diagrams of Fig. XIV7(a) and also through the usual decay process b —• uus of Fig. XIV7(b). The amplitudes, including the possibility of final state interaction phases, have the form
(521) i=u,c,t
where D is the direct amplitude, Pi are the (real) penguin direct amplitudes, and 6U6D are strong phase shifts. Then there can be a CP
(a)
(b)
Fig. XIV7 Weak transitions of the (a) penguin, (b) direct type.
408
XIV Weak interactions of heavy quarks
violating decay asymmetry of the form
Im (VisV^sVuh)
sin (S{  6D)
\A(B~ +Kifl)\2 + \A(B+All of the ingredients for this asymmetry are easily present. However, due to our inability to calculate reliably the amplitudes and phase shifts, no real prediction can be made. That is the generic problem with this class of CP tests. It may well be that a happy accident of two paths of similar magnitude and differing phases will occur, and thus large CP violation signals will be found. However if such effects are not found, we may presently hide beneath our inability to calculate rather than taking the nonobservation as an indictment of the Standard Model. To summarize, we have discussed thus far a variety of possible tests for CPviolating signals in the system of B mesons. The partial rate differences can be quite large. At first, this seems to go against the general dictum that all CP violations in the Standard Model must be proportional to a single, numerically small product of KM angles. However B decays satisfy this stricture in the sense that the mixing and decay of B mesons are in themselves proportional to small KM angles. Overall, the product of mixing, decay and CP violation does turn out to be proportional to all of these KM angles. However, in forming the asymmetry by dividing out the rates themselves, one is canceling the small KM angles, thus leaving a rather large effect. This argument also explains why there is little CP violation in D decays in the Standard Model. The CP observables must be small due to the usual product of KM angles. However, the overall decay rate itself has no small angles, so that the signal remains small. For the complex of Bd mesons, the use of mixing in channels such as ipK$ or D+D~ seems the most promising approach. However, it is still worthwhile to check all possible decay modes. There is a great deal of interest in exploring these decays experimentally, despite a realization that it will not be easy to reconstruct and measure the twobody modes for which predictions have been made. Timeintegrated correlations: There is one related issue which is worth discussing as it involves interesting quantum mechanical effects [BiS 81]. We have been discussing the B mesons as if their content at t = 0 were known. This can be accomplished in pair production experiments by watching for the decay of the 'other' B meson which is produced at the same time. By tagging its identity with some characteristic mode, such as b —• c£~ve, one can infer the identity of the B meson. However, let us consider this method in detail for the process e +e~ —• 7* —• B°B°,
XIV6 Rare decays of B mesons
409
which offers a way to produce B°'swhen running an e + e~ collider at the energy of the T(4iSr), just above threshold. In this case the BB pair must be produced in a Pwave state. If one discusses the system in terms of the mass eigenstates BJJ and BL, Bose statistics require that it be produced in the combination BHBL, because BHBH or BLBL can never be interchanged symmetrically in a Pwave. Phrased in terms of B° and i?°, the state must be a coherent superposition of the form \BB) =  ^ [£0(p)J3°(p)>  \B°(p)B°(p))]
.
(5.23)
Thus correlations are built into the wavefunction, and both B° and B° mesons can oscillate. If one works out the rates for one B to decay into tj;Ks and the other to be tagged by decaying into b —> c£~~U£ or b — one finds using the same assumption as before that
oc sinAm(t t)Im
(±p(il>Ks)
The correlations in the wavefunction have forced it to be odd under t**i. This means that if one were to integrate over all t and t the asymmetry would vanish! This happens for most methods for tagging B mesons in Pwave configurations. However, there are several ways to avoid this problem. One would be not to integrate over time, but rather to directly observe the time dependence. This couldbe done at an asymmetric e+e~ collider which could produce a moving BB centerofmass such that the B and B decay vertices would be separately visible. Another method would be to go far above threshold, where the two B mesons are produced incoherently. Alternatively one could work near threshold with the reaction e+e~ —• £ £ 7 , so that the BB pair are no longer in a Pwave state. This interesting feature influences experimental search strategies for B meson CP violation. XIV6 Rare decays of B mesons The system of Sdecays is so rich that any single mode will be 'rare' in the sense of having a small branching ratio, and thus will be difficult to observe. Despite this, there are certain classes of decays that are of sufficient interest to warrant both theoretical and experimental study. Specifically, considerable attention has been given to modes that proceed only at one loop, as in Fig. XIV8. The original hope was that, by measuring the transition rates of such processes, one could extract the mass of the top quark or perhaps observe deviations due to new physics. Those prospects seem less favorable now due to hadronic uncertainties in the transition matrix elements. However, since prediction of rare decays
410
XIV Weak interactions of heavy quarks
involves many of the techniques which we have developed for calculating weak transitions, these decays can provide a nontrivial test of our ability to apply the Standard Model. We shall focus here on one of the most accessible transitions, b —> 57. The determination of other oneloop amplitudes would proceed analogously. The quark transition b —•
57
The process b —> 87 is described by the magnetic dipole transition
x iZ(pa)
(6.2)
with Xi = mf/Myy and x
The flavor content of F
^7
s
t,C
(a)
(b)
Fig. XIV8 Oneloop decay amplitude
XIV6 Rare decays of B mesons
411
be expressed in the simple form 3QF2
2
^
where f(x) is the phase space factor given in Eq. (1.1), and factors of m s / m 6 arising from phase space and from the amplitude of Eq. (6.1) have been dropped. For mt = 150 GeV, these formulae yield F2 = 0.35 and a relative rate of 0.8 x 10~3, a rather large value. The free quark calculation can of course be improved by QCD shortdistance corrections. These produce a surprisingly large modification to the analysis of b —• 57, and the reason is instructive. The t quark is so heavy that at all scales relevant to the weak decay, its effect may be treated as a point 657 vertex, with renormalizations as in Fig. XIV9(a). However, the c quark is light on all scales from Mw to ra& so that in its renormalization one must also include the diagrams of Fig. XIV9(b), where the dot represents the b —> ccs weak Hamiltonian. That is, there is mixing between the b —> 57 vertex and the b —> ccs transition. The renormalization group procedure is similar to that described in Chap. VIII, and has been carried out by several groups [BeBM 87], [DeLTES 87], [GrOSN 90], [GrSW 90]. The result in the leading log approximation is 1 rj? 2
m
t
•
In the last expression, we have employed the form of as for five flavors. With A = 0.2 GeV and p = 1.86, we find F2 = 0.65 [F2(m2t/M^) + 0.79] ~ 0.74 .
(6.6)
The charm contribution has become dominant, and the amplitude has increased by a factor of 2. A consequence is that the dependence on the top quark mass is reduced, resulting in an amplitude change of only ±10% for mt = (150 ± 50) GeV. Unfortunately, to observe the inclusive decays B —> Xs^ at the required branching ratio appears difficult. There are many photons present
t.c
(a)
Fig. XIV9
9
(b)
Corrections to the b —»
57 vertex
412
XIV Weak interactions of heavy quarks
from background processes and it is hard to detect all states carrying strangeness. In practice one must turn to exclusive channels. The hadron transition B —> K*j At the hadronic level, the quark transition b —> 57 would be observed in channels such as B —• Kirj, KnTT'y, etc. The simplest final state occurs when the Kn system forms a resonant Jp = 1~ state, the if* (890), which in the quark model is treated as a us or ds bound state.* The difficulty lies in relating the quarklevel process to real meson transitions. If the spectator model is valid, we do not expect a large fraction of b —> 57 events to end up in B —• K*j. Since the K* emerges at high recoil, p^*  = 2.6 GeV, the spectator quark must be turned around from sitting nearly at rest to become a part of the final hadron. This is less likely than the production of a kaon plus nonresonant pions. K* production involves either the exchange of a hard gluon to transfer momentum to the spectator or the high momentum tail of the bound state wavefunctions. There is an additional complication. Besides the b —> 57 transition within the hadron, with the largest contribution shown in Fig. XIV10(a), there can also occur new diagrams, one example of which is shown in Fig. XIV10(b). Since the low mass charm plus gluon intermediate state was important in the enhancement of F2, this diagram, where the gluon connects to the spectator quark, may also be significant. Unfortunately, this diagram is most likely nonlocal, and does not appear in the b —• 57 calculation. The effects of it and others like it need to be estimated before a final conclusion can be made about B —> K*j. The B(p) —• K*(\a) + 7(q) vertex can be written in terms of two structure factors Gy and  m2K.)  (p + k)^ e*{k)
w K*
Y
(a)
B 
G
y (b)
Fig. XIV10 Contributions to B > K*j. * The B —• Kj transition is forbidden because it is a spin zero to spin zero transition.
Problems
413
where Gy (GA) is the parityconserving (parityviolating) amplitude. If the transition is due to the b —> 57 amplitude of Eq. (6.1), these two quantities can be related using the identities
v
w
(6.8)
k)  e [k){p + k)») , to obtain GA = The calculation of the amplitudes is very similar to the B —> pev e transition of semileptonic decay at the highest recoil (lowest q2). Because of this, we should expect that specific model predictions will likely vary over a wide range, since the range of predictions of exclusive channels in semileptonic b —> u transitions is large. In fact, one finds that [Wy 89] r
^ X * 7 ~ 0.05 > 0.46 .
(6.9)
If we accept this rough magnitude, then there is hope that the transition can be seen experimentally. However, the spread in such theoretical estimates is presently too large to expect firm conclusions regarding the extraction of the b —• 57 amplitude from weak decay data. Because of the similarity in the quark model between the processes B —> if*7 and B —• pev e at large p recoil momenta, there is some hope that a comparative study of these two modes may be able to reduce the hadronic uncertainties (cf. Prob. XIV2). Problems 1) Patterns of CP violation All signals of CP violation involve the interference of two or more amplitudes. Identify the origin of the interference in partial rate asymmetries for the decays (a) Bs —> (pep, (b) Bs —• p±7rZf, (c) Bj —• K*Q(p, (d) B± > p±7T°, (e) B±  • i^Tr 0 . 2) Amplitude relations in the heavyquark limit
In the heavyquark limit, a static b quark in a B meson can be described in terms of just the two upper components of its fourcomponent Dirac field. This can simplify various matrix elements or be used to relate them. Use this feature to show that the B —• if*7 matrix element of the a^v operator, (K*(e, k)\sa^b\B(p))  d""*? [A e\pp + B e^fy + e+ • p C
414
XIV Weak interactions of heavy quarks
can be related to the vector and axialvector form factors of B —> p£v£,
= %D e " " a V 4 ^ , = E e^ + e* • p [Fp» + Gfe"] ,
through A = (EkomBD)/mB , B =mBD , C = (D + G)/mB , under the assumptions of a static 6 quark and of SU(3) symmetry. In this relation, all form factors must be evaluated at the same momentum transfer, q2 = (p — k)2.
XV The Higgs boson
A central feature of the Standard Model is the spontaneous symmetry breaking in the electroweak sector which gives mass to fermions and to the W± and Z°gauge bosons. The sole physical remnant of this process is the Higgs boson. Although its couplings to the other particles in the theory are fully specified, the Higgs mass is undetermined. As a consequence, efforts to detect this particle cover the widest possible range of mass values [GuHKD 90, Ei 91].
XV—1 Mass and couplings of the Higgs boson Although a complex doublet of Higgs fields is initially introduced in the WeinbergSalam model, there remains following spontaneous symmetry breaking precisely one physical Higgs state, a neutral scalar particle if0. That is, if we define the number of degrees of freedom for Higgs and gauge boson states respectively as NJJ and NQ, then before the symmetry breaking we have NJJ = 4, NQ = 8 whereas afterwards we find NJJ = 1, NQ = 11. To obtain these values, recall that massive vector particles have three spin components whereas massless vector particles have just two. Although the total of Higgs and gaugeboson degrees of freedom remains fixed (NJJ + NQ = 12), there is a transfer of three states from the Higgs sector to the gaugeboson sector. These Higgs states become the longitudinal spin modes of the W±, Z°particles. This transfer can be displayed analytically by first performing a contact transformation to cast the two complex Higgs states <£>°,
where U(X) = exp(i* • T/V) , 415
(1.2)
416
XV The Higgs boson
and we recall that v = 1/\/21I2GF — 246 GeV. One completes the procedure with the gauge transformation,
(13)
for all fermion weak isodoublets I^L and weak isosinglets I/JR. Within this unitary gauge, the physical content of the theory is manifest, and the quantity <£' is seen to contain a single Higgs field H°. In the following, we shall employ this gauge but with the primes in Eq. (1.3) suppressed. Upon expressing the Higgs potential of Eq. (II—3.19) in terms of the field H°, we find ^
+ \H±
.
(1.4)
The first term in V is a constant energy density which can be interpreted as a contribution to the vacuum energy, (
Although the parameter fx is unknown, we can get a feeling for the scale of the Higgs vacuum energy by supposing ji ~ 0.1 TeV. The Higgs vacuum energy contributes to the cosmological constant of general relativity an amount AHiggs = 87rGNewtonf^Higgs = 0.068 cm"2. Such a term would have a remarkable effect on the geometry of spacetime, manifesting itself over a distance scale of AHiggs~1//2 — 4 cm! The present limit on the cosmological constant is A < 3 x 10~48 cm"2 [RPP 90]. There must then be some important and nontrivial physics which forces the suppression or cancellation of the vacuum energy by roughly fifty orders of magnitude. Also in Eq. (1.4) is the H° mass term, resulting in (1.6) The Higgs mass M# is not fixed because only the quantity v, but not A, is phenomenologically determined. The present lower bound, M# > 57 GeV, comes from measurements at the e +e~ collider LEP [Dav 92]. As shall be described in Chap. XVI, it is possible in principle to constrain the Higgs mass by studying the Higgs contribution to electroweak radiative corrections. Unfortunately, power law contributions of the (unknown) top quark mass tend to overpower the logarithmic Higgs contributions. There exist a number of elegant results on couplings and bounds of very light Higgs [GuHKD 90], but these are now of only historical interest for
XV1 Mass and couplings of the Higgs boson
417
the minimal Standard Model. There is presently no universally agreed upon upper bound for M#. As we shall describe in Sect. XV3, if the Higgs mass is large then the theory becomes strongly interacting. This is not a true bound on the mass, but rather a limit on our ability to calculate perturbatively. There are more subtle attempts to bound MH by using the socalled triviality of \(p4 theory [Ca 88]. This uses the result that if one renormalizes the quartic coupling A (cf. Eq. (1.4)) of tp4 theory by using a cutoff, and lets the cutoff become infinite, then one obtains a vanishing renormalized coupling, i.e. Xr —> 0, which implies MH —> 0. In the Standard Model, this final conclusion would be removed by nonzero gauge and Yukawa couplings, but A must not become too large. While extremely interesting, the possible drawback with this approach as regards phenomenology is that it requires assumptions about the presence or absence of new physics beyond the Standard Model. The nature and form of the cutoff depend on higher energy scales, which have not been probed experimentally. It is of course possible, and indeed even expected, that there will be new physics beyond the realm of the Standard Model. Therefore it makes sense phenomenologically to consider all masses up to energies where the Higgs is so broad that the very concept of an isolated resonance fails. The Higgs potential also contains cubic and quartic selfinteractions. Of more immediate phenomenological interest is the coupling between the Higgs particle and any fermion / . Prom Eq. (1.3) and Eq. (II3.20), we deduce the interaction Mw *
(1.7)
where the latter approximate form is a consequence of using 2sin# w ~ 1. The catalog of Higgs particle interactions is completed by presenting its couplings to the W± and Z° bosons. Both trilinear and quadrilinear terms are present, CWWH
= ^W'W^ [Hi + 2vH0] ^ WfW» [2eMwH0
n
LZZH =
Si ~l~ 92 ry
g
6^
ryti \ rj2
/*0 "
(1.8) where we have again employed 2sin# w ~ 1. The preceeding equations reveal the key to phenomenological searches for the Higgs particle at accelerator energies, that its coupling strength generally depends on the mass of the particle with which it interacts.
418
XV The Higgs boson
The naturalness problem Radiative corrections to the Higgs mass raise a question of the 'naturalness' of the minimal Standard Model. For contrast, consider first oneloop electromagnetic corrections to the electron mass. If we impose a cutoff Ae on the momentum flowing through the loop, the mass shift, l +   l n — 5  + ... , (1.9) rae,o J 2 7T is obtained. The magnitude of the first order correction is quite modest. Even if we take for Ae the entire mass of the observable universe, Ae ~ 1079 GeV, we obtain only the modest mass shift me ~ 1.7rae?o. This teaches us that, with logarithmic behavior, the renormalization program of absorbing divergences into renormalized parameters is not implausible. However, radiative corrections to the Higgs mass are not as tame. We display in Fig. XV1 two of the selfenergy processes which shift the Higgs boson mass. Considering for definiteness diagram (a), which involves a Higgs loop with quartic selfcoupling, we have
This expression is quadratically divergent,* £ # ~ A^, and leads to a shift of the Higgs mass,
If AH is as large as, say, the Planck mass #pianck — 1019 GeV, then in order to obtain a renormalized mass governed by the electroweak scale (MH = O(v) < 1 TeV), the parameter M # o must be negative and have a magnitude which is finetuned up to 30 decimal places! While technically possible, this is surely unnatural. This 'unnaturalness problem' has led many physicists to search for alternatives to a fundamental Higgs field, and to suggest that new physics
(a)
(b)
Fig. XV1 Some quadratically divergent Higgs selfenergies. A quadratic divergence also occcurs for the fermionantifermion loop in Fig. XVIl(b).
XV2 Production and decay of the Higgs boson
419
must exist at the TeV scale. For example, one of the proposed models of new physics is that of supersymmetry, in which the quadratic divergence in the Higgs selfenergy is removed by cancelations which occur between the contributions of fermionic and bosonic supersymmetric partners [HaK 85]. Interestingly, many supersymmetric models appear to require the presence of two complex Higgs doublets. Of the original eight real fields in the two doublets, 3 become the longitudinal degrees of freedom of the Z° and W± gauge bosons and 5 give rise to physical Higgs quanta. The latter include a charged pair (if ± ), 2 CP — 1 neutrals (H, /i), and 1 CP = — 1 neutral (A). At present, there is no evidence for these quanta as physical particles (LEP data yield mass limits of about 40 GeV on both neutral and charged particles [Dav 92]) or as virtual particles [BaHP 90]. XV2 Production and decay of the Higgs boson The present mass limit on the Higgs boson (c/. Table 14) excludes the region MH < 57 GeV. A bound of this type is obtained by comparing experimental data with theoretical expectations regarding both production and decay of the Higgs particle. In the following, we shall consider the theoretical analysis which underlies this process. Decay Each Higgs particle which is produced in an experiment will eventually decay. Because the Higgs interaction with matter is known, it is generally straightforward to compute the decay rate into any given mode. In principle, one can therefore generate a reliable profile of Higgs decay patterns. Let us begin by considering the decay of a Higgs boson to a fermionantifermion pair. It follows from Eq. (1.7) that the transition amplitude H° —» / + / is expressible in terms of the fermion mass mj and the Higgs parameter v ~ 246 GeV as A
fi(pMq)
,
(2.1)
and leads to the decay rate,
where Xf = mf/Mff and we employ the color factor Nc = 1 for leptons and Nc = 3 for quarks. We see in Eq. (2.2) the expected quadratic dependence of the decay rate on the fermion mass. The decay width for fermion emission is always a small fraction of the Higgs mass for all but the top quark mode, 1 » THQ_^JJ/MH (f ^t).
XV The Higgs boson
420
If sufficiently massive, the Higgs particle can decay into the W± and Z° gauge bosons. For VFemission, the trilinear coupling of Eq. (1.8) gives rise to the decay rate 1 16TT
Mfj v2
(2.3)
H For Z° emission, we find a similar but smaller
where xw = expression,
124
 44
)
(2.4)
where xz = An overall profile of the anticipated Higgs decay width as a function of the Higgs mass appears in Fig. XV2, where all quarks are taken as free particles. One sees a recurrent pattern, a series of shoulders associated with the opening up of new decay modes and a concomitant rapid increase in the width. Between successive shoulders, there is a steady increase as the Higgs mass increases. At the highest mass considered in Fig. XV2 (MH — 1 TeV), the Higgs decay width has increased to a value almost comparable with its mass.
Production Of all possible Higgs production mechanisms, we shall focus on just three: (i) emission by the Z° gauge boson, (ii) gluongluon fusion, and (iii) fusion. W+W
1
1
1
1
1
1
1
1
^
0° )
'
D"5

10 1 0

10' 1 5 10
r
T" 1 2
1
10" 1
1


1
10°
1
1
101
1
1 10 2
1
103
Higgs Mass (GeV)
Fig. XV2 Higgs decay width as a function of Higgs mass.
XV2
Production and decay of the Higgs boson
421
Fig. XV3 Amplitude for Z° + H°+ / + / . The production process which has had the greatest impact in the search for the Higgs boson is e + + e~ —• Z°—• H° + / + / ,
(2.5)
where / (/ = i^d, e, ve,...) is any kinematically allowed fermion. The amplitude associated with the production and subsequent decay of H°is displayed in Fig. XV3. A favorable aspect of this process is the presence of the rather large ZZH vertex. For example, we use the field interactions of Eqs. (1.7), (1.9) to determine the relative strength with which the Higgs boson couples to either an / / pair or a pair of Z°bosons, yielding* 9_ZZH^4MW>>1
QffH
v3 rrif
for all known fermions save the kinematically forbidden top quark. The Z° ~* ffH° transition rate, expressed relativeto that for Z°—> / / , is then computed to be a 4TT sin2 0W cos2 0W J2r
(y — 4r 2 ) + Y L
iZ
6
(2.7) J
where y = 2EH/MZ, r = MH/MZ, 7 = Tz/Mz, and fermion mass is ignored. The magnitude of the ratio in Eq. (2.7) decreases with ME, equaling approximately 10~2 for MH — 0 and 10~4 for MJJ — 40 GeV. This mode has been searched for at SLC and LEP e+e~ colliders, and its nonobservation has led to the bound on the mass of the Higgs quoted earlier. Future attainment of greater collider luminosity should allow the Higgs search to continue for even larger mass values. * Specifically, we have taken gffH ^ errif /M\y and have also denned a dimensionless coupling by extracting a factor of Mz and including a factor of 2 to allow for the gzzHMzZ^Z^H0 two ways that the Z°fields can be contracted.
XV The Higgs boson
422
At higher energies, yfs > MZ + MH, the process of Fig. XV3 can again lead to a search for the Higgs. However, now it is the Z° coupled to e+e~ which is virtual, while the final Z° is on its mass shell. The cross section,  4 s i n 2 f l w ) 2 ] p2 [p4 sin 4 0 W cos 4 9M A
2
2
2
(s
(2.8) m
2
with p = (s — m H — m z) — Am Hm z, provides an efficient way to search for the Higgs at higher energy e +e~ colliders. For Higgs production using light hadrons, the coupling between light quarks and the Higgs is so weak that the cross section is extremely suppressed. Surprisingly, however, the gluonfusion reaction gg —> H is much more favorable. This occurs through the diagram of Fig. XV4, which includes heavy quarks in the loop. The result is independent of the quark mass for mq ^> MJJ (cf. Prob. XV1) and can be expressed as a lowenergy effective lagrangian of the form (2.9) the number of
with G" being the gluon field strength tensor and heavy quarks. The decay rate for this process,
(2.10)
72TT 3
is not large enough for gluon emission to be a useful mode of observation. For quarks lighter than MJJ , a local lagrangian does not occur, and the decay rate of Eq. (2.10) is modified to a
i,
(2.11) —
H
Fig. XV4 Higgs formation via gluon fusion
/
i
XV3
The possibility of a strongly interacting Higgs sector
423
For the current range of allowed MJJ and mt values, it is the top quark which provides the dominant contribution. Gluongluon fusion is of special interest as a mechanism for Higgs production in a collider experiment, and should be the dominant Higgs production mechanism at hadron colliders for M# < 1 TeV. For the scattering oifree gluons at centerofmass energy s, one obtains the BreitWigner resonance cross section (res)
~ " y~
• y
~—n~
n
fff  S)
' n~^yy
/ ^ i c\\
+
2
where q = s/4 and J = 0 for a Higgs resonance. This expression must be folded together with the gluon distributions in the incident hadrons in order to predict the actual yield at a given collider. Numerical studies can be traced through the literature cited in [GuHKD 90]. The W+W~ fusion process W+W~ —> H° is a treelevel reaction (in contrast to gluongluon fusion) and is describable using the interaction of Eq. (1.8). The free Wboson cross section would be of the same form as Eq. (2.12) but with THo^ww taken from Eq. (2.4) and now using q2 = (s — 4M^)/4. Again, the W+W~ distributions must be supplied depending on the specific reaction. This is generally calculated in the 'effective W approximation [GuHKD 90] in which Ws are treated as partons within a hadron, being generated by the perturbative coupling to quarks. WW fusion provides an important production mechanism at e+e~ colliders, and overtakes the gluongluon fusion at hadron colliders for MH > 1 TeV. XV—3 The possibility of a strongly interacting Higgs sector As the mass of the Higgs boson becomes very large, the analysis of the Higgs sector undergoes a qualitative change. Consider for example the Higgs quartic selfcoupling A, which can be expressed as (cf Eq. (1.6)) (31) The quartic coupling is seen to grow as the square of the Higgs mass and is scaled by the energy v. For M# » i/, we enter the domain of strong coupling. A perturbative expansion in A is no longer a sensible procedure, and new methods must be employed. What are the indications that the theory has become strongly interacting? One is the width of the Higgs itself. For large MH , the and Z°Z° decay modes combine to yield the tree level decay rate
424
XV The Higgs boson
When MH ^ 1 . 4 TeV, the Higgs width is equal to its mass, a feature which surely indicates strong interaction and which calls into question even the identification of the Higgs as a resonance. Note that MH — 1.4 TeV corresponds to A ~ 16. Another indicator of strong coupling is the socalled 'unitarity violation'. Actually, any Smatrix element must satisfy the unitarity constraint for all values of the coupling constants and masses. However, this will only occur if the theory is treated to all orders. A perturbative approximation need not itself obey unitarity at a given order. Later in this section (c/. Eq. (3.21)) we shall display a calculation for which, if s ^> M\ ^> M ^ , the tree level Tmatrix element for Swave scattering of longitudinal Z° bosons becomes
This would violate the unitarity constraint (c/. Eq. (VI4.5)),
T 2
<3'4)
« < jrzSf '
for MH — 1 TeV , s ^> M  . While this indicates a flaw with the tree level calculation rather than with the full theory, it does suggest that perturbation theory fails at these energies.
The equivalence theorem One might expect that when the Higgs sector becomes strongly interacting, there would be large corrections to lowenergy observables. This turns out not to occur. There is a general 'screening' theorem [Ve 77b] which shows that lowenergy observables are shielded from large corrections arising from a large Higgs mass. For example, the oneloop correction to the gauge boson mass ratio is only logarithmic, and is made smaller by a factor of OL
12TT 12TTTT TT
( \M\V
In practice, there is no lowenergy observable which, at present sensitivities, can detect the effect of Higgs masses up to the strong coupling regime. In order to directly observe a strongly interacting Higgs sector, one needs to use the system most intimately connected with the Higgs doublet, viz. the longitudinal gauge bosons. Recall that in the original symmetric theory, the gauge bosons have only two (transverse) degrees of freedom. The longitudinal component arises only after symmetry breaking, when the gauge bosons absorb three of the four components of the
XV3 The possibility of a strongly interacting Higgs sector 425 complex Higgs doublet* (3.6) as was displayed in the unitary gauge in Eq. (1.3). In quite a physical sense, the longitudinal components of (VF+, W~, Z°) are due to the unphysical Higgs particles ((^+,
.
(3.7)
This is the key to probing the strongly interacting Higgs sector. Let us use the simple example of H —> W+W~ to illustrate how the equivalence theorem works. The relevant coupling is given in Eq. (1.8), and yields the invariant amplitude MHo^w+w
= g2MwelfM • e ? .
(3.8)
The longitudinal (e^) and two transverse (e^) polarization vectors are ,±) =  p ( 0 , l , ± i , 0 ) . (3.9) Note the factor of l/Mw in the longitudinal mode. For large Higgs mass, this leads to a great difference in the respective longitudinal and transverse decay amplitudes,
where A, A' are helicity labels, and also in the decay rates,  = {l ~
2x
w)
F
0,
where xw = Mw/MJJ As a check, observe that these sum to the total rate previously given in Eq. (2.4). The enhancement of the longitudinal/transverse ratio, as scaled by the factor Mjj/Myy, is a general feature * Comparing this representation to that in Eqs. (1.11.3), we have in the small \ limit the correspondences \3 —> <^3 and x^ ^ i^
426
XV The Higgs boson
since the transverse modes are of standard perturbative strength, while for large M# the longitudinal couplings are strongly interacting. In order to test the equivalence theorem, we next calculate the H —•
(3.12)
We then find in the notation of Eq. (3.11), r
#°^+y? = r 0 .
(3.13)
Up to the presence of M^r/s (i.e. M^/M^j) corrections, this conforms with the equivalence theorem. The equivalence theorem describes a situation almost like that covered by Haag's theorem as described in Chap. IV. Either W^ or y^, properly normalized, can serve as a good field variable for the longitudinal modes; the renaming does not change the physics. The difference with the usual application of Haag's theorem is that more than one field is involved (c/. Eq. (1.3)), and one must show that the extra field only produces suppressed contributions. The most careful proof [ChG 85] is quite involved. However, the essence can be exposed in 't HooftFeynman gauge [LeQT 77], wherein the constraint added to fix the gauge (see Sect. XVI3) is dflW£ + iMw
(3.14)
One can define the longitudinal component in momentum space by
W£(*) = e£(*)Wj(fc),
(3.15)
with e^ given in Eq. (3.9). However, at high energy, since
M
+o
we see that
where in the second line we have used the momentum space version of the gauge constraint. This displays the nature of the connection between the longitudinal W's and the Higgs doublet. Note that the equivalence theorem holds at high energy whether or not the symmetry breaking sector is strongly interacting. It may be useful even if a light Higgs boson is found, although special care may be needed in this case [BaS 90].
XV3 The possibility of a strongly interacting Higgs sector
427
Scattering of longitudinal gauge bosons The best probe of the symmetry breaking sector would be the discovery and study of the Higgs boson. However, if this particle is so heavy that its observation is difficult or even impossible, the use of longitudinal gauge bosons can be valuable. While reactions of these particles can of course be calculated directly, the computations are quite involved and contain subtle cancelations. It is generally easier, and ultimately more instructive, to transfer the problem via the equivalence theorem to the Higgs sector. If we neglect perturbative corrections due to the gauge coupling, the interaction of longitudinal gauge bosons at high energy is governed by the Higgs potential of Eq. (II3.19). Moreover, since the Higgs potential is in exact correspondence (upon ignoring fermions) with the linear sigma model of Eq. (IV1.4),* it is seen to have the symmetry structure SU(2)L x SU(2)R (cf Eq. (IV1.5)). This invariance is larger than the gauge symmetry SU(2)L X C/(l)y, containing an extra custodial SU(2) [SiSVZ 80]. As a result of the SU(2)L X SU(2)R chiral symmetry, we can adopt the effective lagrangian framework which was originally constructed for application to low energy processes to describe the scattering of longitudinal gauge bosons. The primary modification is the replacement Fv ~ 92 MeV —• v = 246 GeV. Note, however, that the Higgs application would if anything be closer to the symmetry limit since
0.015 >
=
f ^ y = 0.0007.
(3.18)
\47TVJ
v
J
2
The O(E ) predictions of the theory are summarized by the effective lagrangian 2
(3.19) where U = exp (ir • (p/v). This result is actually useful in an intermediate range of energies  the energy must be sufficiently greater than M\y for the equivalence theorem to apply, yet below MJJ to allow an expansion in powers of M^2. The simplest manifestation of the strong coupling between longitudinal gauge bosons would occur in WLWL scattering. These results are exactly given by Eq. (VI4.2,4.7) with m^ —> 0. Written out for the individual * Note that the strict equivalence theorem limit, entailing the neglect of O(mw /E) effects, corresponds to the m^ —> 0 limit of the sigma model.
428
XV The Higgs boson
channels, they are L ur/+Tj/ —
w+w^w+w
 tf
T
wtz^wizL L
L
L
u
TzLzL+zLzL = 0 ,
' T
=^ ' _ L
s
wiw,zLzL = ^ .
(320)
V
It is instructive to examine a specific calculation. In the full theory, the leading contribution to elastic ZZ scattering is s
t
u
as a consequence of Higgs exchange in each channel. For Mjj ^> s,t,u, this reduces to + t + u+
—
2
++ . J
(3.22)
The first three terms in this expression sum to give the zero of Eq. (3.20) because kinematically s + t + u = 4 M  which is neglected with respect to s in the usage of the equivalence theorem. The next set of terms of order s1 give the O{E^) corrections to the lowest order effective lagrangian, and would be reproduced through the O(E4) effective lagrangian of the linear sigma model, previously calculated in Eq. (IV2.10). For example, the amplitude for TW±ZO^\Y±Z° ^S the direct analog of the pionpion amplitude in Sect. IV1. Note also that, even though the results of Eq. (3.20) agree with the tree level perturbative calculation, they in fact are the correct results to all orders in the strong coupling when expressed in terms of the renormalized value of v. This is guaranteed by the chiral symmetry analysis, which is fully nonperturbative. The process of WLWL scattering is thus seen to share many of the phenomenological properties of the analysis of TTTT scattering, scaled up in energy by a factor of v/Fn ~ 2700. The lowest order result will violate the simplest consequences of unitarity at energies above 1.7 TeV. This can be corrected order by order in the energy expansion of chiral perturbation theory by including effective lagrangians with more derivatives. Since we are neglecting masses here and using the group /SC/(2), the program involved is even simpler than the hadronic physics described earlier in the book. If it turns out that Nature has indeed chosen a strongly interacting symmetry breaking sector, this program should prove to be extremely useful [Ch 88]. Note that the formalism extends beyond the Standard Model. The lowest order results are constrained by the custo
Problems
429
dial SU{2) symmetry to be universal.* However, at O(E4) the pattern of the correction to the lowest order results depends on the underlying theory (see Sect. VI6) and we could hope to distinguish the Standard Model from the symmetry breaking schemes by careful study of longitudinal gauge boson scattering [DobH 89, DoR 90]. Problems 1) Higgsgluon coupling Calculate the H°—> gg effective lagrangian of Eq. (2.9) using a heavy top quark in the loop. Show that the loop integral is finite and falls with increasing quark mass for mt » ra#. However, show that the result is independent of the quark mass because the coupling is proportional to mt. This violates the naive expectation of decoupling because of the growth of the coupling constant as ra^ —> oo. 2) Equivalence theorem One can see the equivalence theorem at work in the decay t —» 6VF+, which was described earlier in Sect. XIV1. Assuming a very heavy top quark, calculate the t —> bW^, t —• bW£ and t —> btp + decay rates. Show that the equivalence theorem works for this decay and calculate the O(Mw/E) corrections. Determine the ratio of WL to production.
* The results can also be extended to theories without the custodial symmetry [ChGG 87], but the answers reduce to Eq. (3.20) in the limit p —• 1.
XVI Physics of the W and Z bosons
Because it is a renormalizable theory, the Standard Model is, in principle, capable of making predictions to any level of accuracy. The process of testing the electroweak sector is now fully under way and will continue for many years, much like the systematic exploration of Quantum Electrodynamics. We shall keep our treatment of this already vast field at a relatively simple introductory level, with the intent of helping the reader obtain an overall grasp of the main issues and techniques. XVI1 Neutral weak currents at low energy Early studies of the weak interactions were confined to processes, like nuclear beta decay and muon decay, which concern just the charged weak current. Starting from the mid1970's, the field of weak interaction phenomenology was broadened by experiments involving neutral weak currents, most notably [Am et.al. 87] 1) 2) 3) 4) 5)
deep inelastic neutrino scattering from a variety of targets, neutrinoelectron scattering, neutrinoproton elastic scattering, parity violation in atoms, and cross sections and asymmetries in ee reactions.
Important work on these low energy {M^vz » g2) experiments continues. In addition, electroweak phenomenology has recently been enriched by higher energy (#2 ~ ^wz) experiments which directly probe the massive gauge bosons, particularly the very accurate Z°studies. Before turning in the next section to a discussion of W± and Z°physics, we shall first review 'traditional' neutral weak current phenomenology. 430
XVI1 Neutral weak currents at low energy
431
Neutral current effective lagrangians Recall that the neutral weak interaction between the gauge boson Z° and a fermion / is given at treelevel by* 75JJ (/)
02
n (/)
w3
J/) ,
T
(1.1)
(/)
Examples of individual g^Q and <7a0 appear in Eq. (II—3.41). To describe neutral current interactions at low energies, it is convenient to employ an effective fourfermion lagrangian, analogous to the Fermi model of charged current interactions. At treelevel, the Z°mediated interaction in the lowenergy limit is IVI
/,/ where po is the treelevel rhoparameter, C
w,0
Z,0
(1.2)
IVI
Z,0
Comparing the second of the relations in Eq. (1.2) with Eq. (V2.1), we see that po governs the relative strengths of the neutral and charged weak current effective lagrangians. In the Standard Model, it has the treelevel value unity, PQ ^ = 1. The rhoparameter is important because it is sensitive to the possible presence of physics beyond that encompassed by the Standard Model. Alternative choices for the Higgs structure can lead to different values for po (c/. Prob. XVI1). Experiments (l)(5) listed earlier involve neutrinoelectron, neutrinoquark, and parityviolating electronquark interactions. There is an effective lagrangian for each of these, among them
Cvq =  ^ Vtf{\ + Js)ut [e^kaM1 + 75)
432
XVI Physics of the W and Z bosons
where the index a = u,d,... denotes quark flavor. Of course, contributions other than neutral weak effects must not be forgotten, e.g., parityconserving eq scattering also experiences the electromagnetic interaction. In Eq. (1.4), we have implicitly included the effect of radiative corrections and thus omit the subscript '0'. Table XVI1 gives a compilation ([RPP 90]) of the radiatively corrected coefficients. Observe the presence in Table XVI1 of quantities pi and /^. At treelevel, they reduce to unity, i.e. p^o — ^i,o — 1 The pi are overall multiplicative factors and the K{ multiply the weak mixing angle, which itself has become renormalized, s^ 0 —> s\. The presence of such quantities in the effective lagrangian can be traced back to the underlying neutral current couplings,
Although the pi and Ki are generally process dependent, they also contain certain common (or universal) contributions, including terms quadratic in the topquark mass (O(G^,m^)). It is this class of electroweak corrections which shall be emphasized in the discussion to follow. Another unknown and potentially large mass parameter, M#, enters only logarithmically (e.g. <9(ln[M^/Aff ])) at the oneloop level. Determination of the weak mixing angle 0W It is convenient to parameterize neutral weak phenomena in terms of s^, and the value of s^ is often cited as the result of a given experiment. We shall use two examples to illustrate this process. Table XVI1. Radiatively corrected coefficients. Coefficient
General Form^
nu ^1 °1
( I I I 2 n Peq ^ 2 ' 3 ecl w Peq \2 S^eq^wy
^ Small additive terms are omitted.
XVI1 Neutral weak currents at low energy
433
(i) Deepinelastic neutrino scattering from isoscalar targets: Here, one measures the ratios of neutraltocharged current cross sections, _
a
vN
(1.6)
cc
vN
Under the conditions of 'deepinelastic' kinematics ([BaP 87], [Fi 89]), theoretical calculations of Rv and Rv are carried out in terms of quark, and not hadronic, degrees of freedom. Although details such as quark distribution functions are beyond the framework of this book [Ro 91], it is plausible that by working with ratios like those in Eq. (1.6), theoretical uncertainites associated with hadron structure tend to cancel. At tree level, Rv and Rv are straightforward to compute if scattering from an isoscalar target is assumed and antiquark contributions are ignored. One then obtains the simple forms (cf. Prob. XVI2), 1 5 2 4 5 = o ~ w,0+ o(l+ r 0)s W j 0 , wu 4,o
= 2x  ~ ' ' 9 (
1+ r
(1.7)
5
°) w,o >
where r = r 1 = ^ v / a S v a r e measurable quantities with treelevel values ro = f^1 = 3. Radiative corrections produce the modifications [MaS 80]
5
^
('•»>
R i?,0 where KUN and p2vN are functions of the lepton momentum transfer and we have omitted nonfactorizable contributions. When comparing the Standard Model with experiment, it is common to find in the literature a variety of values attributed to a given quantity such as s\. This has several causes, e.g. the everchanging data base, differences in the choice of theoretical parameters like mt* etc. Such is the case for the weak mixing angle, and we cite one just determination One example [Ca 92] of fits to lowenergy and highenergy electroweak data which constrain by assuming correctness of the Standard Model gives mt = 144l26l2i GeV , where the latter error bars reflect uncertainty in the Higgs mass and a central value 300 GeV is assumed. Assuming a smaller value of MH lowers the fitted value of mt.
434
XVI Physics of the W and Z bosons
here [RPP 90],
/ °'233± °003 ± °005
(mt = 10 ° G e V ) >
inei.  j Q 2 3 ( ) ± ±0 .003 ± 0.005 (mt = 200 GeV) , where the error bars are respectively experimental and theoretical and a Higgs mass MJJ = 100 GeV is assumed. For deep inelastic neutrino scattering, the O{G^ml) effects are somewhat suppressed by a cancelation between the pv^ and KVN contributions. A relatively large theoretical uncertainty arises from the contribution of the cquark threshold to the chargedcurrent cross section. The (intrinsically nonperturbative) threshold analysis is expressed in terms of a cquark mass parameter mc. Fits to muon and dimuon production data yield a rather imprecise determination for this quantity, mc = 1.31 ^QAS lFo et al 90 ]* (ii) Atomic parity violation: The Z°mediated electronnucleus interaction contains a part which is parityviolating. Consider, for example, the effect in atomic cesium. Because of the weak neutral interaction, the single valence electron in cesium contains small admixtures of Pwave in its 65 (ground) and IS (excited) states. As a consequence, there occurs a measurable IS —> 6S electricdipole transition from which one can extract information regarding the parityviolating component in the electron wavefunction [NoMW 88]. It is convenient to cast the electronquark interaction in terms of a hamiltonian density defined in the electron spin space,
£
,
(1.9)
where 75 signals the presence of parity violation and Pnucl(^) reminds us that the electron feels the effect only where the nuclear density is nonzero. The quantity Qw is the 'weak nuclear charge' to which the electron couples, and is given to lowest order by
Qwfi = 2(NuC?fi + Ndpffi)
= Z(l4slfl)+N
,
(1.10)
where Z and N are respectively the nuclear proton and neutron number. The fact that s^ 0 ~ 0.25 suppresses the proton contribution, leaving the coupling of the atomic electron to neutrons as dominant. Thus, a study of the various isotopes of a given element should prove informative. The C\"d are coefficients whose treelevel and radiatively corrected forms are
\
4
(1.11)
XVI1 Neutral weak currents at low energy
435
The extraction of Qw from experimental studies ([NoMW 88]) of Cs133 depends crucially on a theoretical understanding of atomic structure. As such effects become better understood, the central value and estimated uncertainty are modified, e.g., _ f 69.4 ± 1.5 ± 3.8 ~\ 71.04 ± 1.58 ± 0.88
Qw
[NoMW 88] , [MaR 90] ,
where the uncertainties refer respectively to statistical and theoretical contributions. A determination of the weak mixing angle obtained in this manner is [RPP 90] _ f 0.215 ± 0.007 ± 0.017 2 wlatom. par. viol. ~ j Q 2 Q 4 ± Qm? ±
5
(mt = 100 GeV)
where again MJJ = 100 GeV is assumed.
Definitions of the weak mixing angle Thus far, we have stressed the role that the weak mixing angle plays as a parameter in neutral current phenomenology. What is its status in a more formal development of electroweak theory? We shall see in Sect. XVI3 how the weak mixing angle is ordinarily treated as a derived quantity, expressible in terms of gauge boson masses or electroweak coupling constants. Such relations, introduced at treelevel, are subject to the effects of radiative corrections, and must therefore be considered in the context of a specific renormalization program. We shall briefly describe two such approaches, the onshell renormalization scheme and the class of renormalizations wherein the weak mixing angle has the status of a running coupling constant. (i) Onshell renormalization: Onshell renormalization [RoT 73, Si 80] fixes the three bare electroweak parameters pi5o, #2,0 &nd ^0 in terms of the physical quantities Mw,Mz, and a (cf. Sect. XVI3). The onshell weak mixing angle is then defined in terms of the physical gauge boson masses,
sl = \M>wlM\ .
(1.12)
Thus, the onshell weak mixing angle can be experimentally determined not only as a parameter in various neutral weak processes, but directly from Mw and Mz For the remainder of this chapter, the notation 's^' will be reserved for onshell renormalization. A great deal of effort has gone into obtaining a precise determination of the weak mixing angle, and the process is continously being updated. The gauge boson mass values in Table 14 imply [Ab et al. 90] 4
Z
=
0.227 ±0.006 ,
(1.13a)
436
XVI Physics of the W and Z bosons
where the uncertainty is due to the W± mass. The value cited in [RPP 90], based on an analysis of all data from low energy and high energy experiments and assuming MJJ = 100 GeV, is (mt = 100 GeV) , (mt = 200 GeV) . " \ 0..2189 ± 0.0002 ± 0.0004 Use of the full data base reduces the uncertainty in s^, but introduces dependence on the top quark mass via electroweak radiative corrections. If one treats po as a free parameter in fitting the data by first extracting electroweak radiative corrections and assuming MJJ = 100 GeV, the value _ f 0..2305 ± 0.0002 ± 0.0004
_ f 1.003 ± 0.004 (mt = 100 GeV) , °~~ \ 0.993 ± 0.004 (mt = 200 GeV) . ^'^ is obtained [RPP 90]. This reveals no inconsistency with the basic structure of the Standard Model, so we shall take po = 1 hereafter. Let us return to the subject of electroweak radiative corrections. Contributions to the factors pi and /c» can be classified as either independent of the external fermions (universal) or explicitly dependent on the fermion flavor (nonuniversal), P
« = l+ A p + ( A p ) 2 , where Ap and An denote universal pieces. It should be apparent that W± and Z° propagator corrections, like those in Fig.XVI1, occur independently of the external fermions and are thus 'universal'. The universal effects are of special interest because they are the primary source of the O(G^ml) radiative corrections [Ve 77a, BiH 87], and in the following we shall approximate (cf. Sect. XVI5) Ap = (Ap)t + ... ,
AK = 4(Ap)t + • • • ,
(1.16)
where
b
T
(a)
(b)
Fig. XVI1 Topquark corrections to the (a) W±, (b) Z° propagators.
XVI1 Neutral weak currents at low energy
437
Observe in Eq. (1.16) that AK is proportional to Ap. This is a result of the Standard Model; in general, these quantities are independent. In addition to Ap and AK, there is a third electroweak correction, Ar w , which is worthy of mention. This quantity modifies the treelevel relation between the gauge boson masses Mw and M\)
M\
y ^ M K l  Ar w ) '
U WJ
'
and in the Standard Model, one finds (c/. Sect. XVI5)
Arw =  4 A p .
(1.19)
S
W
By convention, Ar w does not contain photonic corrections of low frequency (q2 < M), whose effects instead are contained in G^ and in the running fine structure constant a(Mz)> Like AK, it is proportional to Ap in the Standard Model, but becomes an independent quantity in more general contexts. (ii) The weak mixing angle as a running quantity. A popular approach of this type is modified minimal subtraction (MS). Here, one denotes the weak mixing angle as s^(q2) (or sw(q2)jjg) and adopts the definition [Ma 79, MaS 81]*
We have already seen (cf. Sect. II—1) how modified minimal subtraction can be implemented in dimensional regularization for the electric charge e(q2), and one proceeds accordingly for the coupling #2(
4 = 4(Mf) * sl{Ml) ,
(1.21)
and writes for the weak neutral coupling constants,
* Yet another running coupling, s1(q2), appears in the literature [KeL 89]. In the approximation of ignoring the 'nonuniversal' corrections adopted here, the quantities s^(q2) and s2c(q2) are equivalent. t We exclude the case f = b here. See Chap. XVI5.
438
XVI Physics of the W and Z bosons
Thus, the Z° resonance amplitude evaluated at energy Mz will yield information on the combination of factors,
V^(JS  24JW) • y/pj(J£  2s2w4Q) .
(1.23)
In the approximation of studying just universal radiative corrections, the quantity pf is independent of fermion type and still contains the O(G^m^) dependence, ( p / W ^ p = 1 + (Ap)t + ... •
(1.24)
However, no analog of the correction factor K (C/. Eq. (1.5)) appears in Eqs. (1.22), (1.23) because it has been absorbed into the definition of s^. Since the relative amount of weak isospin (Tw^) and electric charge (Qei) occurring in the weak neutral current is measureable, it must follow from renormalization scheme independence that Using Eqs. (1.15), (1.16) to solve for K;, and keeping only terms of first order in an expansion in powers of (Ap)t, we obtain M2 4 ~ sl + c w(Ap)t ~ l   r ^ , (1.26) pM\ to be contrasted with the onshell version in Eq. (1.12). The introducion of p, itself containing O(G^m2) dependence, leads to a reduced dependence on the topquark mass in s^. Precision LEP data plays an important role in the determination of s^ [Al 92], 2
_2 Sw
f 0.2335 ±0.0012 \ 0.2333 ± 0.0008
{LEP) (world ave.) .
[
'
}
A tantalizing application of the running weak mixing angle involves relating physics at the Z° scale with that of a 'grand unified' theory defined at an energy EQUT The latter scale signals the existence of a gauge group undergoing spontaneous symmetry breaking to SU(3)C x SU(2)L x C/(l)y. The condition 91=92 = 93
(E = EGVT)
(1.28)
leads to a prediction [GeQW 74] for the weak mixing angle at the scale EGXJT In the grand unified theory of SU(5) [GeG 74, La 81] and its supersymmetric extension (SUSYSU(5)), the MS weak mixing angle obeys (1.29)
XVI2 Phenomenology of the W^ and 2°gauge bosons
439
At the much lower energy scale /x = Mz, this becomes reduced by a calculable amount,* (1.30)
where a = a(Mz), M j is the mass scale of the superheavy gauge bosons, and C is a constant which depends upon the number n# of Higgs doublets, (SU(S)) (1.31)
(SUSYSU(5)) . The 5/7(5) extension of the Standard Model has n# = 1, whereas the minimal supersymmetric model takes n# = 2. The 'bare bones' SU(5) model turns out to be unacceptable. It is well known to give rise to an unacceptably short proton lifetime, and recent precision data indicates that the three coupling constants of the Standard Model disagree with a single unification point if evolved according to 577(5) [AmBF 91]. Interestingly, the SUSY extension appears to succeed in both respects. The rate at which s^ 'runs' is decreased due to contributions of supersymmetric partners ('sparticles') of the known particles, and the unification scale is raised to a level (Mx — 1016 GeV) consistent with the observed proton stability. The unification condition of Eq. (1.28) is also satisfied. Studies are underway to see whether a careful analysis of supersymmetrybreaking yields insights regarding masses of the long sought SUSY 'sparticles'. XVI—2 Phenomenology of the W ± and Z°gauge bosons Experiments at SLC and LEP have provided accurate determinations of the Z°mass and decay modes. Facilities like Tevatron and LEP2 are expected to do the same for the W± bosons. In the following, we shall consider a few of the many aspects of W± and Z°physics [A1KV 89]. The emphasis will continue to be phenomenological, with effects of strong and electroweak radiative corrections included but not explicitly calculated. * Actually, Eqs. (1.30), (1.31) represent a simplification in that (i) lowestorder estimates for the renormalization group coefficients are employed, (ii) supersymmetrybreaking effects are ignored, and (iii) the fact that rat > Mz is also ignored.
440
XVI Physics of the W and Z bosons
Decays of W± into fermions The decay of a Wboson into a lepton pair is governed by the lagrangian of Eq. (II3.31),* ^g
^ l + ^ ) i
+ h.c. .
(2.1)
It is a straightforward exercise to compute the treelevel decay width,
where x = mj/M^ and we have employed Eq. (II3.43) in obtaining the x —> 0 form. There are also decays W —» q^qd) into quark modes (the superscripts i,j = 1,2,3 are generation labels), induced by
4 f =  ^ KV* tfV(l + 75)^ + h.c. ,
(2.3)
where Vy is a K M matrix element, and the index k labels color. The lowest order decay width for quark emission is
color
T
1
where x,x are mass ratios defined as above, and we assume that all emitted quarks eventually convert to hadrons. Since the tquark is too massive to be a product of VFdecay, a sum over accessible quark flavors yields J2i,j l^j 2 = 2. QCD radiative corrections modify Eq. (2.4) by a multiplicative factor <5QCD5 (2.5) = 1 + 0.038 + 0.002 + ...~
1.04 .
The numerical value is estimated from as(M%) = 0.118 ± 0.008 [Ca 92] and rif = 5. There are also smaller electroweak corrections which we shall not discuss. * Although we shall denote treelevel decay widths, cross sections, etc. with a zero superscript in this section, for the sake of notational simplicity, we shall suppress the zero subscript for bare parameters.
XVI2 Phenomenology of the W^ and 2° gauge bosons
441
If all final state masses are ignored, the predicted total width for W± decay into fermions is r (tot)
_
r (had)
r (lept)
_ .
^M—W
nA
,
n
*
3
—
(2.6)
(^f) 
An average of current UA1,UA2,CDF data [El 92] yields the value ^ 2.13 ± 0.11 GeV, which is consistent with the prediction of Eq. (2.6). In the limit of masslessfinalstate particles, the branching ratio for decay into a lepton pair Ivi is (Br)^ ~ 1/9 (£ = e, /i, r), while inclusive decay to a mode containing a positively charged quark q (q=u, c) gives (Br)9 ~ 1/3.
,29
10 10'
1
10
10
10
10
Ecm(GeV)
Fig. XVI2 Resonances in ee collisions.
442
XVI Physics of the W and Z bosons
Decays of Z° into fermions The collection of resonances observed in ee collisions as a function of the total centerofmass energy is displayed in Fig. XVI2. It is the weak interaction which dominates physics for s ~ M  , with the strong and electromagnetic effects merely supplying modest corrections. To lowest order, the decay of a Z° boson into a fermion pair / / can be conveniently expressed as (27)
where / = u,d, ve, e, Upon defining y = m^/M^ for fermion mass m, we obtain for the lowest order transition rate to a pair / / , p(0) JL
rrC\
_
NCGUM%
e T —
6TT
KJV
y/2
+
(2.8) For decay into quarks, these lowest order partial decay widths are modified by the QCD factor of Eq. (2.4) [CeG 79, ChKT 79, DiS 79]. There exist also electroweak radiative effects, which we can take into account by employing the modified neutral weak coupling constants #v and §£' of Eq. (1.22). Upon including both strong and electroweak corrections, the treelevel relation of Eq. (2.8) is replaced in the limit of massless final state fermions by C
6TTA/2
Various Z° decay widths measured at LEP are listed in Table XVI2. Overall, the agreement with Standard Model predictions is seen to be impressive.
Asymmetry measurements For the reaction e~e + —• / ~ / + , the forwardbackward asymmetry, AFB 5 refers to the relative difference between the / " traveling forward or backward relative to the incident e~ direction, ^FB = ^ F ~ ^ B ,
(2.10)
where dcosO ——
dto
,
7_i
dcosO ——
dn
.
(2.11)
XVI2
Phenomenology of the W^ and 2° gauge bosons
443
We shall consider the restricted set of final state leptons / = fi,r, and shall take the incident electron beam to be unpolarized. If the final state fermion mass is neglected, then the treelevel differential cross section at centerofmass energy EQM = s can be written in terms of direct channel photon and Z° exchange as [KiiZ 89]
+(9ie)2 + 9ie)2)(9if)2 + Si/)2)IX2](1 + cos2 0) + Sg^ g^ g^ g^ \X\2}
where x(s) is proportional to the Z° propagator, i
^2
For energies s ~ M  , formulae involving x(s) become simplified since Re x(^f) — 0 Upon performing the integrals in Eq. (2.12) and including the effects of electroweak radiative corrections, one obtains*
34
~
, Ofi
"
IQI 1
^
(/)
""•
y/iMz
2.14)
M
where
The dependence on the effective weak mixing angle s^ will occur mainly in the onresonance (s = M) asymmetries rather than in the slopes. Note, however, that all onresonance asymmetries are suppressed by the presence of the small quantity g^ . Measurement of A^ for a final state lepton £ provides a distinct probe of the couplings gv , 3a [B6H 89]. A combination of decay width and forwardbackward asymmetry data yields the values given in Table XVI2. Observe, however, that the relative phase of g\,L is not fixed by such measurements. In addition to the leptonic forwardbackward asymmetry Apg, one can probe the analogous quark asymmetry Apg. Here, the * For final state quarks, QCD radiative corrections must also be included [DjKZ 90].
444
XVI Physics of the W and Z bosons Table XVI2. Z°data from LEP [Ca 92].a Decay width Tee 1
MM
rhad Fhad/riept p. 1
inv
rtot
gie)2 gie)2 a
Experiment
Standard Model prediction6
83.0 ±0.5 83.8 ± 0.8 83.3 ±1.0 1740 ± 9 20.92 ±0.11 496.2 ± 8.8 2487 ± 10 0.0012 ± 0.0003 0.2492 ± 0.0012
83.52  • 83.78 83.52 > 83.78 83.52 > 83.78 1731 > 1742 20.71 > 20.84 496 > 497 2484 + 2496 0.0011 > 0.0013 0.2513 > 0.2518
Decay widths are expressed in units of MeV. For definiteness, the central values mt = 150 GeV and as{Mz) = 0.118 are assumed, and the band of predicted values corresponds to the range of Higgs masses, 50 < M#(GeV) < 1000.
b
preceding formulae describe physics at the parton level, but ultimately it is the charge asymmetry of hadrons which is measured. More can be learned by working with polarized fermions. For example, with s ~ M  the lowest order differential cross section for producing tau leptons of longitudinal polarization PT is dQ
a2\X\2 4s
e)2
+ gi )(giT)2 +
W
cos9 + g^2) cos*)]
(2.16) where again we have ignored (9(F 2 /M) terms. Working at the Z° peak and integrating over all production angles, we can isolate the observable (T) PO1
^

^
a P r = + i +
_^ S^M
V
^
The LEP determination, A^x
= 0.134 ± 0.035 ,
(2.17b)
reveals that ^v and ^a have the same phase, in accord with the Standard Model. Yet another experiment involves the forwardbackward polarization asymmetry which combines the forwardbackward and polarization measurements defined above and, if carried out over the angular
XVI2 Phenomenology of the W^ and 2° gauge bosons
445
range  cos# < c, gives
Finally, a polarized incident beam of electrons can provide the means to measure the leftright asymmetry, ^
^
,
(2.19)
where ox (<JR) denotes the cross section for an incident lefthanded (righthanded) polarized electron, as well as to allow an extended test of the forwardbackward asymmetry ^4FB Neglecting the (Tz/Mz)2) contribution, we have for the leftright and forwardbackward asymmetries evaluated at the Z° peak,
A L R ^f e ,
AFB c Ifi [e^ep
,
(2.20)
where Pe is the electron polarization. The number of light neutrino generations Data from Z°decay has been used to obtain an accurate determination of the number of 'light' neutrinos [Tr 89]. This can be accomplished in a variety of ways. The most straightforward consists of simply comparing the total measured Z° width rt o t with the Standard Model prediction r(SM) 1
tot >
r& M) = 3 ^ + ^
+1^
,
(2.21)
where Fiept is the width for decay into charged leptons. If we attribute any disagreement between F^f and Ftot to the presence of additional neutrinos light enough to appear in Z° decay and which couple to Z° identically to the known neutrinos, then we have
r£tM)  r[ot = (3 + A A g r ^ ) + rlept + r had = ( ) (2.22) A more incisive battery of tests depends on singling out Fi nv, which is the part of the decay width associated with 'invisible' (i.e. undetectable) particles, which is ascribed to neutrino emission, Tinv = r to t — Thad — Tiept = NUTVV . (2.23) If indeed only neutrinos contribute to Finv and each neutrino contributes alike to the Z° decay width, then the number of light neutrinos is
iv, = r i n v / r w .
(2.24)
446
XVI Physics of the W and Z bosons
For the sake of reference, let us refer to the Standard Model treelevel value for Z°decay into a neutrino pair, 12TT
v
0.1658 GeV .
(2.25)
For the conventional value Nv = 3, this implies Finv ~ 0.4974 GeV. There are different strategies for determining Finv depending on the particular mix of theory and experiment to be employed. Let us describe two. If, along with Mz and Ftot, the ratio TV = Fhad/r^ is measured in the vicinity of the Z° peak, then the assumption of lepton universality together with the theoretical input F ^ ^ allows one to write •p.
p
p(SM)/o I fDf\
(<2 Of\\
An alternative method which is somewhat less dependent upon theory is, in addition to the above 7?/, Mz and Ftot, to use the measured hadronic cross section at the Z° peak, crj^ . Beginning with the BreitWigner formula (2J + 1) 7T FinFout B W = and assigning S\ = S2 = 1/2 for the incident spin values, J = 1 for the Z° spin and k2 = Af/4 for the squared momentum, we obtain a
Deak l^71" r e eFhad had "" 772"—F2
'
/o
OQ\ y Z' M )
From lepton universality, it follows that / n?eak
r inv —— tot r \Mz\ L
(2.29)
•» i n v — J
Methods such as these have been employed at the LEP facility to obtain the important result [Ca 92] Nu = 2.99 ± 0.05 .
(2.30)
The VWW vertex A sufficiently high energy ee collider can be used to produce pairs. Since this can arise from a virtual Z° or photon intermediate state*, infomation can be obtained regarding the VWW exit vertex, with V = Z°,j. The ZWW interaction is significant because it arises from * At treelevel, there is also neutrino exchange.
XVI2 Phenomenology of the W^ and 2°gauge bosons
447
the nonabelian gauge structure of the theory. The jWW vertex provides information on the electromagnetic structure of the W boson. Consider the transition V(q, cr) —> VF~(p_, A_)W+(p+, A+) describing creation of a W± pair by a neutral spinone quantum V. From Lorentz covariance, it follows that [HaHPZ 87]
f = (p.  P+r [fag"?  hvjf]
(2.31)
w
where gyww is the coupling strength and the {fiv(q2)} are form factors. Anticipating Standard Model predictions, we take
(232)
l Discrete symmetries constrain the form factors as /4V = /5V = 0 /5V = ftsv = frv = 0 / 4V = fay = fiy = 0
(C—invariance) , (Pinvariance) , (CP—invariance) .
(2.33)
The decomposition in Eq. (2.31) allows for the existence of a magnetic moment /iw and an electric quadrupole moment Qw» Accordingly, it is convenient to express the form factors {fiv} (i = 1,2,3) in terms of quantities Av and KV, 2
/iv = 1 + ^ 2 " A v , /2V = Av , /2V = 1 + ^v + Av •
(2.34)
which are directly related to /iw and Qw> 6
(l +
A
6
+ )
Q
(K7 
A
l) •
( 235)
To lowest order in the Standard Model (SM), one has 4 M = 1,
A^M = 0 ,
(V = 7,Z°) ,
(236)
as well as fffi = 0 f o r i = 4,5,6,7. The corresponding treelevel forms for the Wboson magnetic moment and electric quadrupole moment are .
(2.37)
448
XVI Physics of the W and Z bosons
In principle, it is possible that the VFboson has static properties which violate at least some of the discrete symmetries. An electric dipole moment dw or magnetic quadrupole moment Qw would be parameterized as 6
~
6
Searches for a neutron electric dipole moment can be used to place the bound \dw\ < 10~19 ecm on the W electric dipole moment [MaQ 86]. XVI—3 The quantum electroweak lagrangian In the following three sections, we shall give a simple description of how electroweak radiative corrections are calculated. We begin in this section by quantizing the classical electroweak lagrangian to obtain certain of its Feynman rules. There is also an expansion of earlier comments made in Sect. XVI1 regarding onshell renormalization. Classical electroweak theory of three fermion generations is defined by an SU(2)L X U(l)y gaugeinvariant lagrangian,* ) ,
(3.1)
where $ is the Higgs doublet and the collection {/$} of Higgsfermion coupling constants is flavornondiagonal. With spontaneous symmetry breaking, all particles but neutrinos and the photon become massive and diagonalization of the neutral gauge boson mass matrix occurs in the basis of the photon A^ and massive gauge boson Z® fields as given at treelevel by Eq. (II3.30). In addition, diagonalization of the fermion mass matrix for the threegeneration system involves three mixing angles {0i} and one phase angle 6. The physical degrees of freedom of the gauge and Higgs sectors become manifest in unitary gauge (cf. Sect. XV1),
4 $ = 4w} ({V 0 } , Wj,Z%Ap,Ho ;{mf},Mw,Mz,MH,e)
, (3.2)
where the {9i} , 6 parameters are included in the {rrtf}. Gaugefixing and ghost fields in the electroweak sector The quantum electroweak lagrangian £^ will contain, in addition to the classical lagrangian of Eq. (3.1), both gaugefixing and ghostfield contributions,
* We have replaced the Higgs parameter /J,2 by the equivalent quantity v2.
XVI3 The quantum electroweak lagrangian
449
Observe that mixing between gauge fields and unphysical Higgs fields occurs in the covariant derivative of the Higgs doublet (c/. Eq. (113.18)),^ 2
(3.4) & + h.c.+ ... . One can arrange the gaugefixing term to cancel such mixing contributions. Expressing the complex Higgs doublet in terms of the physical field Ho, unphysical fields x+> Xs a n d the vacuum expectation value v as (3.5) we write the gaugefixing contribution in the form, 2
2
1
(3.6)
 — [d^ + It is not hard to see that cancelation of the unwanted Higgsgauge mixing terms occurs for arbitrary values of the gaugefixing parameters £+,3,0Even with this cancelation, there remain in £ew~ quadratic terms containing the unphysical Higgs fields, and such terms will contribute to the propagators of these fields. As explained in App. A6, once the gaugefixing is specified as in Eq. (3.6), the structure of the FaddeevPopov lagrangian d%' of ghost fields is then determined. For the electroweak sector, it turns out that there are four ghost fields, 4 ^ = 4SWW,CB) .
(3.7)
These are associated with the four gauge fields W^, B^ which appear in the original SU(2)i x U(l)y symmetric lagrangian. A subset of electroweak Feynman rules The full set of electroweak Feynman rules is rather lengthy and we refer the reader to the detailed discussions in [B6HS 86, AoHKKM 82] or to the summary in [Ho 90]. A few of the more useful rules, expressed in terms of bare parameters are* t Mixing also occurs, of course, between the neutral gaugefieldsB^, W^. * For notational simplicity, we suppress the zero subscript in the following discussion.
450
XVI Physics of the W and Z bosons
fermion Wboson vertex: 9
ftj*
2—•
a,\
(3.8a) fermion Zboson vertex:
(3.8b) Wboson propagator iD^ '(q):
g2  M ^ + ie [
y
^
q Z + ^
l
(3.8c)
Zboson propagator iD\J{q):
(3.8d) unphysical charged Higgs propagator iA^
ql  £+M^ + ie (3.8e) In the above, (Vy) is a matrix element for quarkmixing, fl^i) are given in Eq. (II—3.41), and £z is defined by expressing the gaugefixing in the form of Eq. (3.6) but using the physical neutral fields. As seen in Eqs. (3.8c), (3.8e), each boson propagator is explicitly gauge dependent, and in particular, the propagator of the unphysical x+ van ~ ishes in the £+ —> oc limit of the unitary gauge. This is as expected, because only physical degrees of freedom appear in unitary gauge. In fact, the absence of unphysical degrees of freedom in unitary gauge would appear to be an appealing reason for carrying out the computation of radiative corrections in this gauge. However, there is a 'hidden cost'. In unitary gauge, the W± propagator of Eq. (3.8c) becomes = iunitary
9* " ^
+
XVI3
The quantum electroweak lagrangian
451
and the high energy behavior produced by the q^q^/M^r term makes this a questionable choice for doing higher order calculations. Instead, as the price for acceptable high energy behaviour, many opt to accept the presence of unphysical fields. One popular choice of gaugefixing is the 't HooftFeynman gauge, defined by setting all the gaugefixing parameters equal to unity, ^ = 1. In this gauge, the lowest order propagators for the physical gauge bosons and unphysical Higgs and ghost fields have poles at either M ^ or Mz. This condition can be maintained in higher orders by a suitable renormalization of the gaugefixing parameters. Onshell determination of electroweak parameters Two sets of electroweak parameters appear in the classical lagrangians of Eqs. (1.1), (1.2), • , . . f {fi},91,92,\,v 2 (Eq. (3.1)), Classical parameter sets =< , , (Eq. (3.2)) . \{mf},Mw,Mz,MH,e Considered as bare (input) parameters to the quantum theory, these obey the simple treelevel relations ni
(3.10) At this stage, there are several possible (equivalent) expressions for the bare weak mixing angle, e.g., I
z,o
4,o = Jfj
( 3  llb )
•
Radiative corrections will generally modify treelevel relations, and as a result, necessitate a precise definition of the weak mixing angle. Following the analysis in [Si 80], let us compare the parameter subsets (g\5o, 52,0 ? ^0) and (eo, M^o? ^z,o) Each of these bare quantities will experience a shift, i,o = 9i
891 ,
92,0 = 92 $92 ,
^
vl = v2  6v2 0
(3.12) In onshell renormalization, the theory is specified in terms of e, M\y and Mz. Moreover, the following relations are arranged to hold order by
XVI Physics of the W and Z bosons
452
order, &)
These equations constrain the effects of radiative corrections upon the parameters. By differentiating the three relations in Eq. (3.13), one finds after a modest amount of algebra the conditions, (3.14) Also, in onshell renormalization one defines the weak mixing angle in terms of the masses M\y,Mz as in Eq. (1.12). Since this relation is to be maintained to all orders, the bare value s^0 of Eq. (3.11a) will be modified by shifts in the W and Z masses, 5
w,0 — * ""
+
(3.15)
Ml)\
\
Finally, there is a technical point worth noting. For any renormalizable field theory, it makes sense to express results in terms of the most accurately measured quantities available. Thus, it is preferable in the electroweak sector to replace Mw by G^ and work with a slightly modified parameter set, a~l = 137.0359895(61) , Physical parameter set = { G^ = 1.16637(2) x 10~5 GeV~2 Mz = 91.175(21) GeV .
(3.16)
To accomplish this, the relationship G^ — G^(a :Mw,Mz,...) can be used to replace Mw by G^. At tree level, the replacement is simply 1+1
r\ V2" 2v27TQ;o \
(3.17)
We shall discuss the effect of radiative corrections upon this relation in Sect. XVI5. XVI4 Selfenergies of the massive gauge bosons It is is evident from Eq. (3.14) that the parameter shifts <5e2, SM^r and <5M§ play an important role in the study of electroweak radiative correc
XVI~4 Selfenergies of the massive gauge bosons
453
tions. We have already determined from our analysis of QED (cf Eq. (II1.33)) that
where the photon vacuum polarization H(q2) appears in Eq. (II—1.26). In this section, we shall compute the portion of SMyy and <5M arising from the fermionic vacuum polarization contributions to the W± and Z° propagators. As a consequence, we shall be able to identify the origin of the O{G^m2) propagator contributions. The charged gauge bosons W± The radiative correction experienced by a W± gauge boson propagating at momentum q is expressed in terms of a selfenergy function, ^ 2 •
(42)
Although a vectorboson propagator iD^v{q) generally contains terms proportional to g^v and to q^qv, it will suffice to study just the g^v part. As indicated at the end of Sect. II—3, the q^qv dependence is absent if the gauge boson couples to a conserved current or will give rise to suppressed contributions if the external particles have small mass. Thus, we have for the Wpropagator in 't HooftFeynman gauge, q2
q2 
(4.3) where we have substituted for the bare W mass using Eq. (3.12). Let us now calculate the loop contribution of a fermionantifermion pair /1/2 to the selfenergy Aww(q2). We begin with hh
(4.4)
where Vf1f2 = ^c^/i/ 2 1 2 f°r the case when the fermions are quarks. Aside from the occurrence of the 1 + 75 chiral factor and the nondegeneracy in fermion masses rai,ra2, the above Feynman integral is identical to the photon vacuum polarization function of Eq. (II—1.20). It is thus
454
XVI Physics of the W and Z bosons
straightforward to evaluate this quantity in dimensional regularization, and we find for the ga@ part,
3 f, 2
2^ T1
7
,
n1
f1 , , , 2 ,
(4.5) where M 2 = mlx + m^l — x). Since the VF^ boson is an unstable particle with decay rate T\y, the function Aww(q2) must be complexvalued. Let us consider its real and imaginary parts separately. It is clear from Eq. (4.5) that Re Aww(q2) is divergent. One can construct a finite quantity Aww(q2) by defining the field renormalization, W^o = (^2^)1/^2W/i, and constraining 6M^ and 6Z^ to cancel the ultraviolet divergence in Re A ww(g2), + 6Zf{q2M2v)
.
(4.6)
It follows from Eq. (4.5) that the mass shift 6M$y is fixed by ^ = Re A W ( M ^ ) ,
(4.7)
and the /1/2 contribution to the field renormalization which ensures that Aww(Myy) = 0 is easily found to be (4.8) To obtain a relation for the imaginary part of the selfenergy, we simply recall that instability in a propagating state of mass M is described by the replacement M —> M — zF/2. This produces the following modification of a propagator denominator,
q2 M2
(4.9)
q2M2
where we ignore the O(Y2) term. Comparison with Eq. (4.5) then immediately yields Im A W (A%) = MWTW
.
(4.10)
We can use Eq. (4.7) to check this relation by setting q2 = M^. If, for simplicity, we neglect the masses of the fermionantifermion pair /1/2, then the imaginary part comes from the logarithm contained in the first
XVI4 Selfenergies of the massive gauge bosons
455
of the integrals in Eq. (4.7), Im / dx x(lx)ln Jo and we obtain
= fi ie
—>   , q^=M^ 6
:
z
=
T
1 f.fr, J1J
2
7= r*
^VF
(4.11)
5
/c%
(4.12) ^
'
DV27T
where we have substituted g\ = 4\/2M^G^. This result agrees with the result of our earlier decay width calculations of Eq. (2.1) for leptons and Eq. (2.4) for quarks. The neutral gauge bosons Z°, 7 The system of neutral gauge bosons is treated analogously to the charged case except that a 2 x 2 propagator matrix appears, and the issue of particle mixing arises. Although the neutral channel was already diagonalized at treelevel (cf Eq. (II3.30)), interactions reintroduce nondiagonal propagator contributions at higher orders. The g^ part of the neutral channel inverse propagator D r L I/(^f2), diagonal at tree level,
has the renormalized form,
(4.14) Upon taking the inverse, we obtain for the individual neutral boson renormalized propagators, q2
+ j^tf) _ 4 z ( g 2 ) / ( g 2 _M2+
_
M

+
^27(g2)] _
^ 2 )
Observe that there is indeed a particlemixing propagator, Dtf%, proportional to the the reduced selfenergy Aiz(q2). It might appear from Eq. (4.15) that Z°photon mixing gives rise to a photon mass contribution. However, one arranges as a renormalization condition that Alz(0) = 0, and the photon remains massless under electroweak radiative corrections.
456
XVI Physics of the W and Z bosons
If we consider only the vacuum polarization loop contribution due to a fermion of mass ra, we obtain for the Z° selfenergy,
m2)
(4,6,
where m is the fermion mass, Nc is a quark color factor, and
( 4  17 )
The quantities in Na@ are just those expected from the coupling of fermion / to the neutral weak current. We then obtain from dimensional regularization,
 3 / dxx(l x) In
Jo
2iim{t f 2
z
, nr~ + lnV^
2
V
\
J
1 f1 , , m2  q2x(l dx In ?+ l Jo V>
x)]] J }\ . JJ (4.18) It is also easy to demonstrate that the photonZ° selfenergy A$P is proportional to Atf^ for the case of a fermion loop contribution, + 4m gi > \jIe
7
^r
where Qf is the electric charge of the fermion.
XVI—5 Examples of electroweak radiative corrections All electroweak electroweak amplitudes will be affected by radiative corrections. In this section, we shall continue to emphasize the role of O{G^ml) contributions by computing their occurrence in the parameters Ap, Ar and in the Z°bb vertex.
XVI5 Examples of electroweak radiative corrections
457
The O{GtlvrVf) contribution to AQ The quantity Ap introduced in Sect. XVI1 can be defined as a correction to the rhoparameter of Eq. (1.3),
Dz(q2 = 0)
_J =
MZ + 6MZ
or
Observe that Ap is finite since the singular terms in Eqs. (4.5), (4.18) cancel. If we set mi = mt and m
,
[TO?,
ra?
9i
xm
t]
_ g\N g\ c 647T2 rv
Substitution of Nc = 3 and G^/y/2 = g^/SM^ yields the result shown in Eqs. (1.16), (1.17). This quadratic dependence on the heavy quark mass is in striking contrast with the behavior observed for the photon selfenergy (cf. Eq. (II1.26)). In the heavy fermion limit, the photon vacuum polarization exhibits instead the decoupling result O(m^2). A technical factor which brings about this difference is that QED is a vector theory, whereas the charged and neutral weak interactions are chiral. Indeed, one can show (cf. Prob. XVI3) that the decoupling expected of a vector interaction results when lefthanded and righthanded selfenergies are averaged. However, equally important is the fact that as mt grows while mt is kept fixed, the weak doublet is being split in mass. Thus, decoupling of the top quark in the large mt limit should not be expected because if we were to integrate out the top quark, we would no longer have a renormalizable theory  the remaining low energy theory would have an incomplete weak doublet. As noted above, if both members of the doublet are taken to be equally heavy (mt =ra&—• oo), there would be no quadratic dependence on the heavy quark mass, and the decoupling theorem (cf. Sect. IV2) would be satisfied. It is the large splitting in the weak doublet which leads to the observable violation of decoupling.
458
XVI Physics of the W and Z bosons The OtG^rrit) contribution to Ar
The relation between the Fermi constant and the vector boson masses is modified by a quantity Ar, sometimes called the 'quantum correction' [Si 80, BuJ 89]. The treelevel relation of Eq. (II3.43) becomes modified by the radiative corrections of Fig. XVI3,
GR_jlo l It is to be understood in Eq. (5.4) that 'GM' is determined from the muon lifetime with the photonic corrections described in Sect. V2 already taken into account. Thus Ar contains only the remaining electroweak effects. To trace the origin of the quantum correction, we observe first the effect of the W± selfenergy on the bare relation in Eq. (5.4), A w (0) V2
1
J' K
Ml
8 92 
]
where we have taken q2 ~ 0. Next, we replace the bare parameters ^ and M ^ o by their physical forms as in Eq. (3.12). Comparison with Eq. (5.4) directly yields
Upon using Eq. (3.14) to substitute for bg\, we can rewrite this as
4 Recalling that the W± and Z° mass shifts can be related to the selfenergy functions AWW(M^V) and AZZ(M), it should be clear that Eq. (5.5)
(c)
(d)
Fig. XVI3. (a)(b) Vertex, (c) propagator, and (d) massshift counterterm corrections to muon decay
XVI5 Examples of electroweak radiative corrections
459
expresses Ar entirely in terms of calculable quantities.* Although each of the terms in Eq. (5.7) is divergent, the overall combination is finite. A number of rearrangements and algebraic steps can be used to isolate the leading contributions, and one finds Ar = Aa + Arw + (Ar) rem ,
(5.8)
where Aa
= a(Mz)
a
OL
„ n(Mf) , and Arw = ^fAp
.
(5.9)
Sw
A/9 is defined in Eq. (4.17), and (Ar) rem contains smaller finite contributions. The largest contribution to Ar is Aa, the shift in the fine structure constant. Although we have previously expressed the variation in a(q2) in terms of fermion masses (c/. Eq. (II—1.38)), the difficulty in precisely determining quark masses would appear to undermine an accurate evaluation of Aa. However, one can use dispersion relations to relate the hadronic contribution to the vacuum polarization, nhad(
The imaginary part of Uhad(q2) is expressible in terms of cross section data evaluated at invariant energy g2,
Im UhM)
= $Btf) 6
with Rfo>) = ^  h a d r o " S ) cr{ee —• fifj)
.
(5.n)
Thus, we obtain a dispersion relation for the subtracted quantity IIhad(
 n had (o) 3TT IJimii
Jsos J
s{s q2
ie)
where so denotes the point at which data becomes unavailable. For energies above so, a perturbative representation is used to approximate R(s). The result of Eq. (5.12), when added to the lepton contributions, implies a value for oTx{M\) [BuJPV 89, Pe 90]^ a" 1 (Ml) = 128.77 ±0.12 .
(5.13)
* There are additional radiative corrections, such as the 'box' diagrams, which we shall not discuss, t There are minor differences in various evaluations cited in the literature, depending on how the perturbative estimate is performed or on the particular renormalization scheme.
XVI Physics of the W and Z bosons
460
w b
~~—
*—b
(a)
(b)
Fig. XVI4 Topquark corrections to the Z°bbvertex. Our analysis of Ar performed thus far is only to first order in electroweak corrections. What is the result if higher orders are incorporated? For the running fine structure constant, the incorporation of higher orders proceeds according to a renormalization group analysis [Ma 79] and amounts to summing a geometric series. Although the generalization of Eq. (5.8) to all orders has not been rigorously given, a plausible construction exists based on evaluation of the quantum correction to second order. The second order result is suggestive of employing two independent geometric series [CoHJ 89], 1
i+AQ  4 4
1Aa
i+
.
(5.14)
As a consequence, the relation between M ^ and the parameters G^, a, and Mz becomes 1 V2 '
r
i =^
1+
1 2\/27ra
1 + 1 
I 1/2'
2\/2
•KOi

(5.15)
Arr,
where p — 1/(1 — Ap), and for the sake of comparison, we display the treelevel result on the first line. The Z —> bb vertex correction The preceding analyses of Ap and Ar could very well be carried out for any other electroweak observable. In most cases, we would again find important O{G^m^) radiative corrections. Thus, for example, the Z° width for decay into lepton £ {£ — e, /i, r) has the form
Tzo^a = r SL^1 1 + (A^)* + • • 1 >
(516)
and grows quadratically with increasing mt [AkBYR 86]. The origin of this effect, the oneloop ti contribution to the Z° propagator, is identical to that discussed earlier.
XVI6 Beyond the Standard Model
461
Interestingly, however, a more complete calculation reveals a slight decrease to occur in the decay rate r z o _ ^ as rrit grows. This is because, although the decay amplitude contains a (universal) propagator contribution proportional to (Ap)t, an even larger effect, the vertex correction of Fig.XVI4, contributes with opposite sign [AkBYR 86, DjKZ 90],*
r z0 _^ = I ^ _ ^ "l + i  ((Ap)* + (Av^)t^j + .. .1 ,
(5.17)
where the Z°bb vertex correction is given by ,* (h\,
20 / A
x
130 a ,
m?
(5.18)
The dJ, ss modes also contain virtual tquark vertex corrections, but they are greatly suppressed by the tiny accompanying KM factors Vti2 (z = d, s). Recalling the characterization given in Sect. XVI1 of radiative corrections as either 'universal' or 'nonuniversaP, one may interpet the Z°bb effect as a nonuniversal term which contributes as V^P/nonuniv
(5.19) Although O(rrif) corrections are the most important, O(In(TOJ/M) logarithmic dependence has been included in Eq. (5.19) because it has a nonnegligible numerical impact. XVI6 Beyond the Standard Model In this book, we have focussed on the consequences of the Standard Model. Despite intense experimental scrutiny, this theory has displayed no experimental inconsistencies to date. However, the Standard Model does have several features which many physicists consider unsatisfactory. On the one hand, there exists no understanding of the number of families, the origin of quantum number assignments, or the large number of arbitrary parameters. On the other, there are a set of problems with the 'naturalness' of the theory. We have already detailed the difficulties of obtaining small enough values of the cosmological constant, the strong CPviolating parameter 0, and the quadratic radiative corrections to the mass of the Higgs boson. It is necessary to look beyond the Standard Model if we are to achieve a resolution of these issues. In order to make progress, an experimental signal of new physics will be needed. There are a number of ongoing programmatic attempts to * Due to cancelations, the vertex correction turns out not to affect asymmetry phenomena, such as the 6quark forwardbackward asymmetry Ap^.
462
XVI Physics of the W and Z bosons
probe the limits of the Standard Model. Listing the main areas from low energy to high energy, we have 1) 2) 3) 4)
searches for neutrino mass and oscillation, rare decays of muons and kaons, CP violation, especially in B mesons, precision measurements of both W±,Z° properties and low energy neutral current phenomena, and 5) searches for new particles at high energies, particularly the Higgs boson or other indicators of the symmetrybreaking sector.
Of course, while new discoveries might arise from areas outside this list, the above items are particularly powerful probes. Let us consider each briefly. (1) Neutrino mass and oscillation: Strictly speaking, neutrino mass and oscillation phenomena can be accommodated within the Standard Model [KaGP 90], [BoV 87], [MoP 91]. One simply adds a righthanded neutrino field having very small Yukawa couplings. However, the need to postulate neutrino masses small enough to satisfy present bounds makes this an implausible strategy. Actually, there are models of new physics which naturally predict small neutrino masses. One such scheme involves the socalled 'seesaw mechanism'  if a righthanded neutrino is added to the theory, SU(2)L X U(1)Y allows there to be a (presumably large) Majorana mass* TUM as well as a (presumably normal sized) Dirac mass In the (yj,, VR) basis, the mass matrix is )
,
(6.1)
where the crucial zero in the diagonal matrix element arises because VL sits in an SU(2)i doublet and hence is prevented by local SU(2)L invariance from forming a direct Majorana mass. This mass matrix has a light eigenvalue mv — m2D/mM<, which would imply mv ~ 10~12 eV for the choices rap — rne and TUM — 1014 GeV. (2) Rare decays: Many models of new physics at high energy violate the flavor quantum numbers which are otherwise conserved in the Standard Model. These violations are thus anticipated in rare processes, and the decays of kaons and muons are generally the most sensitive probes. The evidence of nonzero rates for forbidden processes at low energy would provide evidence for new physics at the TeV scale. We have described techniques relevant to this topic in Sect. IV9 and in Sect. VII5. * Recall (cf. Sect.V4) that a Majorana mass term arises from the Lorentz invariant, but lepton number violating, coupling of a neutral fermion to its conjugate.
XVI6 Beyond the Standard Model
463
(3) CP violation: CP nonconservation enters the Standard Model through the Higgs sector by means of complex Yukawa couplings. However, this portion of the theory is in many ways the weakest, and alternate mechanisms of CP violation are being actively explored. If the Higgs sector is changed, the observed CP violation must arise in some other fashion [Le 74, Wo 86, DoHV 87, PaT 89]. The predictions of alternative models may be checked in rare kaon decays, hyperon decays, and most significantly in B decays. In certain channels, notably Bd —> J/ipKs, the theoretical prediction is firm enough (c/. Sect. XIV5) that a demonstrated lack of CP violation could be used to falsify the minimal Standard Model. (4) Precision measurements ofW±,Z° properties: In the study of the W ,Z® gauge bosons, we have the fortunate confluence of reliable theoretical methods and precise data. The Standard Model unambiguously predicts the properties of these particles. It is possible that experiments may soon find deviations from the Standard Model predictions. Should such deviations be found, it may even prove possible to identify their origin by considering the pattern of deviation. This point is sufficiently important to warrant further comment. Physics associated with a large energy scale A should affect the gauge boson selfenergies — iU1 (q) (i = 77,72, WW,ZZ). For A2 ^> g2, one would expect rapid convergence of an expansion for — iUl (q) in powers of #2/A2, yielding the following effective low energy description, ±
iYl^(q)=g^Ai
+ q2Afi) + ...
.
(6.2)
This description involves eight free parameters, A 7 7 , . . . , Afzz. However, the conditions n 77 (0) = n 7Z (0) = 0 reduce this number to six. An additional three parameters can be absorbed into the renormalization of , which experience the shifts [BaFGH 90], 6a 6_a_ a ~
, ^ '
£G> , GM " ww '
M2Z "
M\
The three remaining parameters may be chosen to be the same quantities, A/9, AK, and Ar w , that we have discussed in previous sections regarding the Standard Model. Actually, it is convenient to first replace AK by a quantity AK/ defined by gv/g& = 1  4(1 + A«>j) ,
(6.4)
where
l4
VM2z
,
(6.5)
464
XVI Physics of the W and Z bosons
and then to introduce the set [A1BJ 92]* d = Ap , s2
e2 = c20Ap +  2  ^  j A r w  2S20AK' , c
(6.6)
0 "" S0
These quantities are related to the selfenergy parameters of Eq. (6.2) by* 61 =
j^
j^
=
~MJ ~ M ^
^2 ~ ^ww
^33
6
^33 *
3 =
A'vx A
Ml ' \^*' /
J
In the limit of large mt and M#, only ei experiences power law radiative corrections, and one has for the Standard Model
3G>
SG.Mlsj
(MH\,
4 \MZ)
G,M2W (MH\ 3
12^^
\MZ)
...,
(6.8)
G,M2W
(mt\
2
\MZ)
6TT V2
One can obtain a determination of the three parameters ei52,3 from
a minimal data set consisting of Mw/Mz,
^z^th
an(
^ ^FB ky using
Table XVI3. Determination of the {c»} [A1BJ 92]. Parameter €l €2 €3
Minimal data set (  0 ..05 ± 0. 51) x 10  2 ( " 1 ..07 ± 1.00) x 10  2 (+0..07 ± 0. 86) x 10  2
Full data set (  0 .02 ± 0 .37) (o .71 ± 0 .89) (  0 .31 ± 0,.62)
X X X
10  2 10  2 10  2
An equivalent set appearing in the literature is [PeT 90]
The subscript '3', as in ^33 or A3^ refers to the field W3 = c^Z + s w A (c/. Eq. (II3.30)).
XVI6 Beyond the Standard Model
465
Eqs. (1.18), (2.9), (2.14)) as definitions. Alternatively, by folding in additional theoretical structure one can enlarge the description to encompass the full available data set. Results from using both approaches appear in Table XVI3 [A1BJ 92]. The error bars are seen to be relatively large. Roughly speaking, the analysis of [A1BJ 92] reveals no discrepancy between the Standard Model and existing experimental data. Future work should reduce the experimental uncertainties, and discovery of the tquark would eliminate the largest theoretical uncertainty associated with Standard Model predictions. Thus, the Standard Model could be found to agree with experiment to new levels of precision, or perhaps signals arising from new physics could emerge. For example, one goal of precision studies is to check whether the Higgs mechanism is really the origin of weak symmetry breaking. In practice, effects of the Higgs boson are shielded in many electroweak processes, and are thus hard to measure. However, alternative symmetrybreaking schemes, such as Technicolor [FaS 81], may not be so well hidden. All such schemes involve Goldstone bosons which become the longitudinal components of the W± and Z° bosons, but the remaining physical degrees of freedom need not be the same. If the other particles are light (which in this context means comparable in mass to the W±,Z°), they will presumably soon be discovered directly. If they are heavier, they may still be seen indirectly through their effect on the W±, Z° properties. As we have emphasized throughout this book, the effects of heavy particles can be analyzed theoretically by using effective lagrangians. These must respect the SU(2)L X U(1)Y symmetry, but may or may not include the extra custodial SU{2)L X SU(2)R invariance of the Higgs sector with doublet Higgs fields. The symmetries can be implemented by using an SU{2) matrix U for the Goldstone fields (as described in Chap. IV) with transformation U —• LUR) and covariant derivative DpU = dfj,U + ig<2W\x' f U + igiUr^B^ .
(69)
±
As an example of modelling new physics in the W ,Z° system, let us consider the effective lagrangian [PeT 90] 5132 s "*
BllVFi
^ (
v
V]\
+
I
167T
(6.10) where B^ ', F^ are respectively the field strength tensors defined in Eqs. (II—3.11), (II—3.12), and S, T are parameters introduced earlier. The first of the above effective interactions breaks the custodial symmetry (c/. Sect. XV3), and the second exhibits an SU(2)L X SU(2)R symmetry. In fact, the second is just the aio operator of Eq. (VI2.7) v
modified by aio —* —S/16TT,
L^V
—• g2F^vT
and R^v —>• g\B^vT^.
Among
466
XVI Physics of the W and Z bosons
the effects of the lagrangian £new are the selfenergy contributions 6Alz{q2) = —m 2 , SAZZ(O) = Mhx , cw which in turn manifest themselves in the radiative corrections Ar w = AriSM) + e3 . Ap = Ap(SM) + 6i ,
(6.11)
(6.12)
Of course, different models of symmetry breaking predict different values for the Ci (or equivalently of S and T), and in some cases are close to being ruled out. For example, by scaling up the QCD value of aio, it has been estimated [PeT 90, HoT 90, MaR 90] that SU(4) technicolor with a single generation of technifermions has S = 2 (i.e. 63 = 1.6 x 10~2). Measurements of W±,Z° properties at LEP and SLC and comparison with low energy processes can hope to reveal such effects, if present in Nature. (5) High energy searches: While it may be possible to obtain indications for new physics from low energy probes, there is no substitute for directly producing the new degrees of freedom and studying their properties. This can be achieved at the high energy limit of existing accelerators, such as the Tevatron, or with future, very high energy machines such as the SSC or LHC now being planned. Within the Standard Model, the top quark and the Higgs boson must be found. New physics associated with alternate symmetry breaking schemes would likely enter at the TeV energy scale. The most pressing physics goal of the very high energy colliders is the discovery of the Higgs, or the conclusive demonstration that it is not present up to 1 TeV. We have outlined in Sect. XV3 the fascinating possibility of a strongly interacting symmetrybreaking sector, which could in principle be explored by W^Wj, scattering. Such a theory should have a potentially rich resonance spectrum, and our experience with QCD would be called upon to sort out the physics of this new strongly interacting sector. If the theory remains weakly coupled, there may also be new particles, perhaps those predicted by supersymmetry. The Standard Model is a remarkable theory, representing the culmination of modern scientific attempts to understand the laws of Nature. While appreciating the power of the Standard Model, it is nonetheless appropriate to look forward to the discovery of new physics from the next layer of reality.
Problems
467
Problems 1) The rhoparameter a) Show that for an arbitrary number of Higgs multiplets ((
b) Given two Higgs fields, respectively with quantum numbers 7W= —7 W3=l/2 and J w = l , 7W3=0, and having the nonvanishing vacuum expectation values (^1/2) and (<^i), obtain a bound for \(ipi)/(
Appendix A Functional integration
In this appendix we outline the basis of functional methods which are employed in the text. Path integral techniques appear at first sight to be rather formal and abstract. However, it is remarkable how easy it is to obtain practical information from them. Very often they add insight or new results which are difficult to obtain from canonical quantization. A.I Quantum mechanical formalism Before attempting to address the full field theoretic formalism we first review the application of such techniques within the more familiar setting of nonrelativistic quantum mechanics in one spatial dimension. Unless otherwise specified we hereafter set ft = 1. Path integral propagator Simply stated, the functional integral is an alternative way of evaluating the quantity D(xf,tf;xuti) = (xfle^f^lxi)
= (xf,tf\xuU) .
(1.1)
This matrix element, usually called the propagator, is the amplitude for a particle located at position X{ and time U to be found at position xj and subsequent time tf. The propagator can also be written as a functional integral
D(xf,tf;xi,U) = JD[x(t)]eiSW» ,
(L2)
where the integration is over all histories (i.e. paths) of the system which begin at spacetime point Xi,t% and end at #/,£/. The paths are identified by specifying the coordinate x at each intermediate time t, so that the symbol V [x(t)] represents a sum over all such trajectories. The contribution of each path to the integral is weighted by the exponential involving 468
Functional integration
469
the classical action
S [x(t)} = J*f dt (jx2(t)
 V (x(t)))
(1.3)
which, since it depends on the detailed shape of #(£), is a functional of the trajectory.* Although the validity of the path integral representation, Eq. (1.2), may not be obvious, its correctness can be verified by beginning with Eq. (1.1) and breaking the time interval tf — U into N discrete steps of size e = (tf — U)/N. Using the completeness relation / JJ—oo — oc
dxn \xn)(xn\
,
one can write Eq. (1.1) as Joo \ cly /v I O
I c*/ /v
Joo
1 /
\ «*/ /V — 1 I v>
(1.4)
I **/ J\f
/ / • • • \ **/ 1 I ^
/
I *i ii /
where xo = Xi, XN = Xf. In the limit of large iV the time slices become infinitesimal, implying I
I iHf\
{x£\e
%n€
\
\x£i)
I
I
= {xe\e ^
^
^
)
+ O(e2) .
Inserting a complete set of momentum states and introducing a convergence factor e~Kp for the resulting integral over momentum, we have ii) = Hm f
^
eHtxixti)iep>I7mKp>
y 2 7 r
(1.6)
It is important to understand the difference between the concept of a function and that of a functional. A realvalued function involves the mapping from the space of real numbers onto themselves Reals <— [/ : Reals] . On the other hand, a realvalued functional such as 5 [#(£)] is a mapping from the space of functions x(t) onto real numbers Reals <— [S : x(t)] .
470
Appendix A
Upon taking the continuum limit we obtain
( rn \ N hm ijr—r)
N^oo \27Tl€/
n/
(1.7)
n=l J~c
It is clear then that we can make connection with Eq. (1.2) by identifying each path with the sequence of locations ( x i , . . . , #ATI) at times e, 2e,..., (iV — l)e. Integration over these intermediate positions is what is meant by the symbol JT> [x(t)], viz. Nl
Each trajectory has an associated exponential factor expi5 [#(£)], where the quantity ^ * ' * '  ^ e=i \
V{xt)) /
(1.9)
becomes the classical action in the limit N —> oo. We have thus demonstrated the equivalence of the operator (Eq. (1.1)) and path integral (Eq. (1.2)) representations of the propagator.* It is important to realize that in the latter all quantities are classical  no operators are involved. The path integral propagator contains a great deal of information, and there are a variety of techniques for extracting it. For example, the spatial wavefunctions and energies are all present, as can be seen by inserting a complete set of energy eigenstates { n)} into the definition of the propagator given in Eq. (1.1), x^e^nVfU)
^
1 0 )
n=0 For completeness, we note that by combining Eqs. (1.5)(1.8), one can also write the propagator in a corresponding hamiltonian path integral representation f,tf;xi,ti)=
lim /
^dx1^
dp Nl
This form is useful when one is dealing with noncartesian variables or with constrained systems.
Functional integration
471
In addition, other quantum mechanical amplitudes can be found by use of the identity t
, tf\T (x(ti)... x(tn)) \xi, U) =
where T ' is the timeordered product.
External sources An important technique involves the addition of an external source. In the quantum mechanical case this is added like an arbitrary external 'force' j(t), (xf,tf\xuti)m
= I V[x(t)]e Jti
.
(1.12)
The amplitude is now a functional of the source j(t). Prom this quantity one can obtain all matrix elements using functional differentiation, which can be defined by means of the relation
= jdt'S(tt')j(t') => ML=S{tt)
(1.13)
and yields the result we seek, (xf,tf\T(x(t1)...x(tn))\xi,ti)
For many applications it is necessary Drily to consider matrix elements between the lowest energy states (vacuum) of the quantum system. This can be accomplished in either of two ways. First it is possible to explicitly One can prove this relation by choosing a particular ordering, say U
,
and noting that
fak^f^tflXnitn) Xn (xn, tn \xn i, t n _ i ) Xn i . . . XX (^ where we have used completeness and have defined x^ = x(tk) (k = 1,2,..., n) . The amplitudes (xfc^fcl^fc — i>*fc — 1) are simply free propagators as in Eq. (1.1), and can be evaluated by means of the timeslice methods outlined above. Thus the above expression is identical to the righthand side of Eq. (1.11). In the case of a different time ordering the same result goes through provided one always places the times such that the later time always appears to the left of an earlier counterpart. However, this is simply the definition of the time ordered product and hence the proof holds in general.
472
Appendix A
project out this amplitude using the ground state wavefunction (x,t\0)=th(x)eiEot ,
(1.15)
which implies /»OO
(0\T(x(t1)...x(tn))\0)
=
J—oo
/»OO
dxf
J—oo
dxii>*0(xf)ei
(xf, tf\T (x(h)... x(tn)) \Xi,
(1.16) However, this amplitude can be isolated in a simpler fashion. If we consider the amplitude (x/,£/aJi,ii) in the unphysical limit tf —> — zr/, U ^ +in we find for large r/ + T{,
(xf,tf\xuU)
Y, (1.17)
Generalizing, we have eiE0(tfti)
which is operationally a much simpler procedure than Eq. (1.16). The generating functional We may combine all these techniques in the socalled generating functional, defined by (x/,t/ii)^(t) .
(1.19)
This has the path integral representation W[j] =
lim
tf^ioo J
V[x(t)]e
Jt
i
.
(120)
t^—^ioo
Noting that for U = iri and tf = — ir/, E { + )
(1.21)
we find that ground state matrix elements as in Eq. (1.16) can be given in terms of the generating functional W[j],
(0T(x(tl) ... xfa)) 0)  ( 
i r
^m
)
6
" gjM WVl\^ • (122)
Functional integration
473
It often happens with path integrals that formal procedures are best defined, as above, by using the imaginary time limits t —• ±ioo. However, in practice it is common instead to express the theory in terms of Minkowski spacetime. Thus, the generating functional will involve the real time limits t —> ±00. Does the dominance of the ground state contribution, as in Eq. (1.21), continue to hold? The answer is 'yes'. At an intuitive level, one understands this as a consequence of the rapid variation of the phase eiEnt  nfaelimit t —> 00. The more rapid phase variation accompanying the increased energy En of any excited state washes out its contribution relative to that of the ground state. In a more formal sense, the realtime limit is defined by an analytic continuation from imaginary time. To properly define the continuation, one must introduce appropriate cie' factors into the Greens functions in order to deal with various singularities. Beginning with the next section, we shall often employ the Minkowski formulation and thus explicitly display the 'ie' terms in our formulae. The prescription given in Eq. (1.22) represents a powerful but formal procedure for the generation of matrix elements in the presence of an arbitrary potential V(x). Unfortunately, an explicit evaluation is no more generally accessible via this route than is an exact solution of the Schrodinger equation. In practice, aside from an occasional special case, the only path integrals which can be performed exactly are those in quadratic form. However, approximation procedures are generally available. One of the most common of these is perturbation theory. Suppose that the full potential V(x) is the sum of two parts V\(x) and V2(x), where V\(x) is such that the generating functional can be evaluated exactly while V2(x) is in some sense small. Then we can write W[j] =
hm
/ V[x(t)]e
tf+ioo J
= t / fe»
c
wy
"
"W
(123)
t4—• i o o
n\ where w(O)[l_ L«y J —
i: 11111
t y —>• — ioo
PJti
f VW(tW I
X^lay^c/IJ
o
™ *
[2
~
W
^v~w, .VA/WJ
^ I
2 4
x
LZiil
J
t^ — • i o o
is the generating functional for V\(x) alone. Obviously Eq. (1.23) defines an expansion for W[j] in powers of the perturbing potential V2(x).
474
Appendix A
A.2 The harmonic oscillator It is useful to interrupt our formal development by considering the harmonic oscillator as an example of these methods. This treatment turns out to reproduce known oscillator properties with the use of functional methods, which are very similar to corresponding field theory techniques. It is most convenient to address the problem by employing Fourier transforms, /»OO J771
f e~iEtx(E) ,
x(t) = / J00
(2.1)
27r
whereby for U = — 00 and tf = +00,
Sj [x(t)} = £ ^ dt (jx\t)
(2
= L t {? * 
(2.2) with the definition x'{E) = x(E) + j(E)/ (mE2 — mu)2 + ie). An infintesimal imaginary part ie has been introduced to make the integration precise. Upon taking the inverse Fourier transform
x'(t)= I
— e~iEtx'(E) = x(t) + — I m
J—oo ^
dt'D(tt')j(t')
J—oo
, (2.3)
where 1
D(t 1') = f" ^ e  ^  ^ ) ^ we have /rn
/»oo
—
:
,oc
/
2m J00
= JeMft\
,
(2.4)
ze m
dt
,00 J00
r1
\
dt'j(t)D(tt')j(t') .
(25)
Finally, changing variables from x(t) to x'(t) we obtain the generating functional r
/°1
/Tf)j(t')
Functional integration
475
Note that the above change of variables has left the measure invariant
J
f
We can use this result to calculate arbitrary oscillator matrix elements. Thus for £2 > ti, we have for the ground state
(0\T(x(t2)x(h))\0) ={if
W[0] (
2
(2.7)
j=0
ti)
^
which, in the limit t £1, reproduces the familiar result
<0^0)
JL
(2.8)
Although only ground state expectation values have been treated thus far, it is also possible to deal with arbitrary oscillator matrix elements with this formalism by generalizing the operator relation (2.9) where
is the usual creation operator. First, however, it is convenient to use the classical relation p = mx to rewrite the operator a^ as
In a simple application, we calculate that
xl) = lim J^ (1  I £•)
(0\x(t2)x(tl)\0)
u: at, (**•*)
mu ( = hm W— 1 t 2 ^+ V 2 V
i
d \ i
— 1 —D(t u dtij m
2

2rnuj
which agrees with the result obtained by more conventional means,
(0xl)  Jt— (o\ (a + aA \l) = 7L= . x
' ' '
V 2muo \ ' V
/ ' /
J2mu
(2.13)
476
Appendix A
More complicated matrix elements can also be found, as with
W» = IT ,iLT (> + ~. IT) I1 V2H
^ at J V
w at2
64 i\2
,.
lim
A
i
8 \ /
1 +  r—
1
i
5
T—
x [/?(«!  t2)D(0) + 2D(ti  t)D(t  t2)} =
2muj
(2.14)
which agrees with
In this manner, arbitrary oscillator matrix elements can be reduced to ground state expectation values, which in turn can be determined from the generating functional W\j\. The ground state amplitude in the presence of an arbitrary source j(t) contains all the information about the harmonic oscillator. One should note the analogy of the above methods to those of quantum field theory. The coneparticle' matrix elements involving  1) have been reduced to vacuum matrix elements by use of Eq. (2.9). This is similar to the LSZ reduction of fields. As a result, all that one needs to deal with are the vacuum Green's functions. The generating functional is ideal for this purpose, as we shall see in our development of functional techniques in field theory. A.3 Field theoretic formalism One of the advantages of the functional approach to quantum mechanics is that it can be taken over with little difficulty to quantum field theory. An important difference is that instead of trajectories x(i) which pick out a particular point in space at a given time, one must deal with fields
Functional integration
477
Path integrals with fields The formal transition from quantum mechanics to field theory can be accomplished by dividing spacetime, both time and space, into a set of tiny fourdimensional cubes of volume 6t6x6y6z. Within each cube one takes the field
(3.1)
as a constant. Derivatives are defined in terms of differences between fields in neighboring blocks, e.g., ,Vji zkM + St)  ip fa,yj, zk,ti))
•
(3.2)
The lagrangian density is easily found, £(

C
iff (x*> Vj>zk> *i)> Q\M (**> Vi* zk> U))
,
(33)
and the action is written as
S ~ ^2 Sx8y6z6t C (
(3.4)
The field theory analog of the path integral can then be constructed by summing over all possible field values in each cell /»O
II /
(3.5)
i,3,k,r
Formally, in the limit in which the cell size is taken to zero this is written as
f[d
.
(3.6)
By analogy with the quantum mechanical case (cf. Eq. (1.18)), it is clear that, since the time integration for S in Eq. (3.4) is from — oo to +oo, this amplitude is to be identified with the vacuumtovacuum amplitude of the field theory, (00) = N f[dtp(x)] e i 5 ^ ) ' ^ ( * ) l
.
(3.7)
Generally, quantum field theory is formulated in terms of vacuum expectation values of time ordered products of the fields G< n >(xi,..., xn) = (0\T (ipix,)...
(3.8)
i.e., the Green's functions of the theory. By analogy with the quantum mechanical case, one is naturally led to the path integral definition
GW(X1,...,
xn) = N J [d
478
Appendix A
where N is a normalization factor. Again we emphasize that all quantities here are cnumbers and no operators are involved. In terms of a functional representation, we then have from Eqs. (3.7), (3.9),
Generating functional with fields These Green's functions can most easily be evaluated by use of the generating functional
W[j] =N f [d
(3.12)
6cp(x) '
which lets us obtain (c/. Eq. (3.9)) (3.13)
W[0] 6j(
j=o
As an example of this formalism consider the free scalar field theory £(°)(x) = dptpd^ip  —(f 2
.
(3.14)
In general we have 00
{o)
{o)
w [j]=w [o]
jTl \
n
POO
^ 2 U— n /
dxk
n=0 ' lk=l J°°
rtxk>> (3.15)
where the generating functional W^ [j] is given by = N
[d
V2 ^
*
2
*
J
V
.
(3.16)
There exist two common ways in which to handle the issue of convergence for such functional integrals, i.e. to ensure acceptable behavior for large
(3.17)
Functional integration
479
and is now convergent due to the negative argument of the exponential. Continuation back to Minkowski space then yields the desired result. Integrating by parts, we have from Eq. (3.16) W (0) [j] = N f [dip] g
= N f \d(p'] (3.18)
where Ox — Hx + m2 — ie and (pf(x) = (p(x) + / d% AF(x  y)j J (2?r)4
fc2
— m2 + ze
( n x + m 2 )A F (*  y) = £ ( 4 ) (*  2/). Note that we have used invariance of the measure (/ [d(p] = J [dip']). Finally, we recognize a factor of W[0] in Eq. (3.18), thus leading to the expression .
(3.20)
We can now determine the Green's functions for the free field theory, e.g.. L  X2)
,
3=0
(~i)4 W[0] 6J(
3=0
(3.21) More interesting is the case of a selfinteracting field theory for which the lagrangian density becomes
C{x) = \d^d^
 ^ m V + £ i n t M = 6«\v) + £int(ip) .
(3.22)
The theory is no longer exactly soluble, but one can find a perturbative solution by use of the generating functional W[j] = N f[dip(x)} J
(3.23)
480
Appendix A
As before, the Green's functions of the theory are given by W[0]
16 FT rr ~ 1
3=0
(3.24) For most purposes one requires only the connected portions of the Green's function, i.e. those diagrams which cannot be broken into two or more disjoint pieces. This is illustrated in Fig. Al which can be found by dividing the full Green's function ... tp(xn))\0)
(3.25)
into products of connected particle sectors and dividing by the vacuumtovacuum amplitude (00) in each sector. Mathematically one eliminates the disconnected diagrams by defining W[j] = eiZ^ .
(3.26)
Then one can show that Z[j] is the generating functional for connected Green's functions, = J2~\n J dXl... J dxnj(xi). °° n=o  °°
..j{xn)G^ (3.27)
where 6n
(3.28) j=o
A.4 Quadratic forms The most important example of a soluble path integral is one that is quadratic in the fields because, at least formally, it can be solved exactly. Let us consider an action quadratic in the fields, S=
f dAx
(4.1)
(c)
Fig. Al. Contributions to the fourpoint Green's function in tp4 theory: (a)(b) connected, (c)(d) disconnected.
Functional integration
481
where O is some differential operator which may contain fields distinct from (p within it. The general result for the quadratic path integral is given by
/quad = J[Mx)]e~ifdlx
^ ) O ^ ( X ) = iV[det O}'1'2
,
(4.2)
where det O is the determinant of the operator O. In order to prove this, one can expand
n
(43)
n
where (pn(x) satisfies O(pn(x) = Xn(pn(x)
and
/ dAx
(4.4)
The sum over all field values can then be performed by summing over all values of the expansion coefficients a n , r
poo
= N IJn /
d(ln
(4.5)J
„ Joo 'OO
v
poo
= N
Tl
dan
eiKa™ = Nf (det O)~ 1/2
OO and where TV, TV' are normalization constants
det O = J J \n
(4.6)
n=l
denotes, as usual, the product of operator eigenvalues. In general, some effort is required to evaluate the determinant of an operator. One valuable relation, easily proven for finite dimensional matrices and generalizable to infinite dimensional ones is* det O = exp(tr lnO) .
(4.7)
This trace now denotes a summation over spacetime points, i.e. tr lnO= fd4x (x  lnO \ x) , which is the most commonly used form in practice. * For a discrete basis, this follows from the result exp(tr lnO) = e x p Y ^ l n A n =TTexp(lnA n ) = n
n
where An, are the eigenvalues of the operator O.
(4.8)
482
Appendix A
Background field method to one loop We can illustrate one use of this result by constructing an expansion about a background field configuration (which satisfies the classical equation of motion) and retaining the quantum fluctuations up to quadratic order. Consider a scalar field theory with interaction Cmt (
(4.9)
Writing
(4.10)
leads to the generating functional W\j] = e ^ ^ / [dS
S [
+ £ int (4.12)
Integration by parts gives
H where Ox = nx + m2£^((p(x))
.
(4.14)
The functional integration can then be performed (cf. Eq. (4.5)) and we obtain W[j] = const, (det Ox)~1/2 e{*SMx)]+if#xj(x)
(4 15)
It is convenient to normalize the determinant somewhat differently by defining O0x = nx + m2 .
(4.16)
Then, suppressing the x subscript, we write (detO)~ 1/2 = const, (detO^ l O)~ 1 ' 2
,
(4.17)
where const. = (det O 0 )" 1 / 2
,
(4.18)
and O01O = l + A F C t (
.
(4.19)
Functional integration
483
Using Eq. (4.2) we have
The generating functional for connected diagrams can now be identified immediately as
Z[j] = S [cp] + Jd4x j(x)(p(x) + l IV In (l + A F C t (
^j(x)
(4.21)
The trace 'Tr' includes the integration over spacetime variables and can be interpreted as follows, Trln
 a;)C t (^) ,
(4.22)
IV [ A F C t (^) A F C t (^)] = J d*x J d4y AF(x  y)C'U {Cp{y)) x AF(y  x ) C In this manner, oneloop diagrams containing arbitrary numbers of C'(nt ((p) factors are generated. The physics associated with this approximation can be gleaned from counting arguments. The overall power of ft attached to a particular diagram can be found by noting that associated with a propagator and a vertex are the powers % and %~l respectively. There is also an overall factor of ft for each diagram. Then with the relation No. internal lines — No. internal vertices = No. loops — 1 , we see that this approximation corresponds to an expansion to one loop. The classical phase generates the tree diagram (O(h0)) contribution and the determinant yields the oneloop [O{ft1)) correction to a given amplitude. A.5 Fermion field theory Thus far, our development has been performed within the simple context of scalar fields. It is important also to consider the case of fermion fields where the requirements of antisymmetry impose interesting modifications on functional integration techniques. The key to the treatment of
484
Appendix A
anticommuting fields is the use of Grassmann variables. Thus, while ordinary cnumber quantities (hereafter denoted by roman letters a, 6,...) commute with one another, [a,a] = [a,6] = [a,c] = . . . = 0 ,
(5.1)
the Grassmann numbers (hereafter denoted by Greek letters a, /?,...) anticommute, {a,a} = {a,(3} = {a,7} = ... = 0 .
(5.2)
It follows that the square of a Grassmann quantity must vanish, a
2=/?2=72
=
_
=
Q
>
( 5 3 )
and that any function must have the general expansion f(a) = /o + ha ,
g(a, P)=go + g\a + g2p + g^ap .
Differentiation is defined correspondingly via d*_dp_ _ d0_da_
(5.4)
_
so that in the notation of Eq. (5.4) we have
£(«) = / i ,
^(«,/?)=»«.«•
(56)
Second derivatives then have the property d2
(57)
°
We must also define the concept of Grassmann integration. If we demand that integration have the property of translation invariance
Jdaf(a) = Jdaf(a + (3) ,
(5.8)
[dafi(3 = 0 or
(5.9)
it follows that f da = 0 .
The normalization in the diagonal integral can be chosen for convenience,
fdaa = l,
fdaf{a) = f1 .
(5.10)
Let us extend this formalism to a matrix notation by considering the discrete sets a = {a\,..., an} and a = {a\,..., an} of Grassmann variables. A class of integrals which commonly arises in a functional framework is W[M] = fdan...
dax dan... daxe^Ma
.
(5.11)
Functional
integration
485
As an example, t h e simple 2 x 2 case is calculated to be
W[M] =
J
da2 da\
1 + iaiMijOLj
(5.12) M22  M\2M2i)]
.
Only the final term survives the integration, and we obtain W[M] =detM .
(5.13)
This result generalizes to the n x n system [Le 82] yielding essentially the inverse of the result found for Bose fields, W^[M]Fermi = / dan ... doi\ dotn ... da\etaMa = det M , a*n.. ,da\dan..
.da\e
a
* M a oc (det M)
l
.
We can now extend this formalism to the case of fermion fields i^{x) and ^{x). Since such quantities always enter the lagrangian quadratically, the functional integral can be performed exactly to yield W[O] = j [dip] [di>] jJ*A*^x)Oi>{x)
= NdetO
.
(515)
The remaining development proceeds parallel to that given for scalar fields. Given the free field lagrangian density £o($,ii>)='ii>(x)(ipm)tl>(x) ,
(5.16)
the generating functional for the noninteracting spin onehalf field becomes
Wfafj] = / where Ox = iif)x — m+ie and f){x), rj(x) are Grassmann fields. Introducing the change of variables ij)'(x) = ip{x)  / d4y SF{x,y)r)(y) , $'(x) = rp(x)  / d4y fj(y)SF(y,x)
,
486
Appendix A
we find that an alternative form for the generating functional is
W[TJ,f}} = ( [<#'] [dtf] e> 01 e (5.19) Thus the generating functional for connected diagrams is
Zhf}} = fd4xJd4yfj(x)SF(x,y)V(y)
,
(5.20)
and the only nonvanishing connected Green's function is Xo)
(_

^  m + le
(2TT) 4
which is the usual Feynman propagator. A.6 Gauge theories For our final topic, we examine gauge theories within a functional framework. We shall employ QED as the archetypical example, for which the action is 1
f
A
1 f A
— —  / d x F F^v —  / d x (A n A^ — / J
J
(6.1)
where the second line follows from the first by an integration by parts and
Of = gVnxd£dZ •
(62)
In the presence of a source j ^ , the generating functional is then
J\ = N f[dA^]eiS[A» ]+ifd4x
j A
» " .
(6.3)
Due to the bilinear form of Eq. (6.1), it would appear that one could perform the functional integration as usual, resulting in
where the inverse operator DF^(x,y)
is defined as y)
.
(6.5)
Functional integration
487
However, this is illusory since the inverse does not exist. That is, acting on Eq. (6.5) from the left with the derivative d* yields 0xDFfll/(x,y) = d^4\xy)
,
(6.6)
implying that Dp^v must be infinite. An alternative way to demonstrate that OxV is a singular operator is to observe that
O£/d£a = 0 .
(6.7) V
Thus any fourgradient d*a is an eigenfunction of Ox having eigenvalue zero, and an operator having zero eigenvalues does not possess an inverse.
Gauge fixing The occurrence of such a divergence in the generating functional of a gauge theory can be traced to gauge invariance. For QED, any gauge transformation of vector potentials (cf. Eq. (II—1.3)), A^x) > A'^x) = A^x) + d^x)
,
(6.8)
leaves the action invariant, '
x)] .
(6.9)
If we partition the full field integration [dA^\ into a component which includes only those configurations which are not related by a gauge transformation and a component [da] which denotes all possible gauge transformations, then we have
[[dAJ e i 5 ^> = f[dAp] eiS^
x f[da] .
(6.10)
But J[da] is clearly infinite and this is the origin of the problem. The solution, first given by Faddeev and Popov [FaP 67], involves finding a procedure which somehow isolates the integration over the distinctly different vector potentials A^x). In order to understand this technique, we shall first examine a finitedimensional analog [Ra 89]. Consider the functional
Z[A] = f n r dx\ e~^^XkAklXl , L / J
(6.11)
where A is an N x N matrix. Suppose that A is brought into diagonal form A D by linear transformation R, .
(6.12)
Appendix A
488
Letting y = Hx denote the coordinates in the diagonal basis, we have N rN
a2 AD
e LA I
.i=lJ
(6.13)
Suppose that the last n of the N eigenvalues belonging to A vanish. The exponential factor in Eq. (6.13) is then independent of the coordinates yNn+u •  • 5 UN and the corresponding integrations / dy^n+i • • • / dyjy diverge. This is reflected in the vanishing of detA, and causes the quantities in Eq. (6.13) to diverge. The infinity is removed if the integration is restricted to only variables associated with nonzero eigenvalues, in which case we obtain the finite result (6.14) It is possible to express ^ [ A ] as an integral over the full range of indices 1 < i < N by defining variables Z%
\ arbitrary (AT  n + 1 < i < N) , and writing for the generating functional rN Zf[A] =
(6.15)
(6.16)
.2=1
Upon tranforming back to an arbitrary set of coordinates {#;}, we obtain the useful expression Zf[A] =
'N
r
1 1 / dxi
det
dz
N
f[
6(ZJ(X)) e
^K
. (6.17)
j=Nn+l
Let us now return to the subject of gauge fields, broadening the scope of our discussion to include even nonabelian gauge theories. By analogy, corresponding to the variables 2#_ n +i,..., ZN will be the gauge degrees of freedom and the prescription of Faddeev and Popov becomes for generic gauge fields A^x), 6{Gb{A%)) det \SGb/6aa\
(6.18)
b=l
where the {a a } are gauge transformation parameters (cf. Sect. 14) and the {G^(A^)} are functions which vanish for some value of A^(x). Since
Functional integration
489
the {Gb} serve to define the gauge, such contributions to the generating functional are referred to as gaugefixing terms. The variation 6Gb/6aa signifies the response of the gaugefixing function Gb to a gauge transformation parameter aa. For any gauge theory, there are a variety of choices possible for the gaugefixing function G. In QED, one defines the axial gauge by G(A^ = n ^ , (6.19) where n^ is an arbitrary spacelike fourvector. Due to the presence of the fourvector nM, one must forego manifestly covariant Feynman rules in this approach. Thus, one often employs a covariant gaugefixing condition such as G(AtA) = dfiAtMF , (6.20) where F is an arbitrary constant. Under the gauge transformation of Eq. (6.8), we find G(AJ > G(AJ + Ha ,
(6.21)
6G/6a = U .
(6.22)
so that Referring back to the general formula of Eq. (6.18), we see in this case that det \6G/6a\ is independent of the gauge field and thus may be dropped from the functional integral. The QED generating functional then becomes
(6.23) Note that, as promised, this result is finite and leads to a photon propagator in Feynman gauge XV
''
W[0}6ju(x)6jx(yyj»= 0
%
] (2TT) (2TT)4
(6.24) The result is independent of the choice of F. Consequently, even if the constant F is evaluated to the status of a field F(x), one can functionally integrate over F(x) with an arbitrary weighting factor since this will only affect the overall normalization of the generating functional. A common choice is
J[[dF] 6(d»A, 
F(x))e~%fd4x
F
^
= «T* !"*
{d A )2
"»
,
(6.25)
490
Appendix A
where £ is a realvalued parameter. In this case, the generating functional becomes ^
.
[O.ZO)
The integrand of the above spacetime integral can be regarded as the effective lagrangian of the theory, and the gaugefixing term appears as one of its contributions. At this point, the functional integration can be carried out with impunity to obtain v)
?
( 6>27 )
where D^f is defined as
{Ux3r  (1  C 1 ) ^ ) DFvX{x y) = 8^\x
 y) .
(6.28)
We find in this way the form of the photon propagator in an arbitrary gauge, as appearing in Eq. (II—1.17).
Ghost fields In the path integral formalism, if the generating functional can be written in purely exponential form, then one can read off the lagrangian of the theory from the exponent. However, the general formula in Eq. (6.18) for a gaugefixed generating functional contains a seemingly nonexponential factor, the determinant factor det \6Gb/8aa\. A fruitful procedure, due to Faddeev and Popov, for expressing the determinant as an exponential factor is motivated by the identity (cf. Eq. (5.15)), det M = N f[dc][dc] eimc
,
(6.29)
where c, c are Grassmann fields. This identity suggests that we replace the determinant factor with an appropriate functional integration over Grassmann variables. For QED, the generating functional can then be written in the concise form
= NJ[dA,}[dc][dc} e* (6.30) As pointed out earlier, for this case the integration over c, c yields only an unimportant constant and may be discarded. However, for nonabelian gauge theory Eq. (6.30) generalizes to
W\fi\ = [ ^ a,b,d
(6.31)
Functional integration
491
where repeated indices are summed over. The quantities (,3 2 ) will generally depend upon the fields A^ themselves. Thus, the fields {ca} , {ca} will appear as degrees of freedom in the defining lagrangian of the theory. However, although coupled to the gauge fields A^ through cMc, they do not interact with any source terms and therefore can only appear in closed loops inside more complex diagrams.* Since these Grassmann quantities are unphysical, they are often called FaddeevPopov ghost fields. They are scalar, anticommuting variables which transform as members of the regular representation of the gauge group, e.g. for the gauge group SU(ri), there are n 2 — 1 of the {ca} and {ca} fields. To complete the discussion, let us determine the ghostfield contribution to the QCD lagrangian. We choose F^ = d^A^ and note the form of a gauge transformation (cf. Eqs. (I4.12),(I.4.17) with aa infinitesimal),
A*  4 f = A% + d^ab  fbaeA%ae
.
(6.33)
Then we find from a direct evaluation of dFi,/dac followed by the rescaling g^cc  • cc, gsfbaeAfiCe .
(6.34)
Upon performing an integration by parts in the first term and relabeling the indices in the second, we obtain the ghost contribution to the QCD lagrangian of Eq. (II2.25).
Problems 1) The van Vleck Determinant The semiclassical approximation to the propagator (valid as h —• 0) can be derived by expanding about the classical path. Writing
x(t) = xcl(t) + 6x(t) , we have
where
6x(t)6x(tf)
\dt2
dx2cl(t) J
Such loops must include a multiplicative factor of —1 to account for the anticommuting nature of these variables.
492
Appendix A
and we have dropped the term linear in 8x(t) by Hamilton's condition. Performing the path integration we have then S S
f trx t)  N (det
1
2
where N is a normalization constant and the quantity inside the square root is called the van Vleck determinant a) Show that this can be written in the form 6x(t)6x(tf)\ [2ni dxfdx{ J Hint: The following argument is hardly rigorous but leads to the correct answer. Write D(xf,tr,Xi,U)
= A(xf,Xi;tf
ti)eiS«( x''Xi*'u)
and use completeness to show that at equal times S(xf — xi) = D(xf, ti\ Xi, ti) — / dx A(xf, x; T)A*(xu x; T)e^s^xf^T^s^Xi^T^
,
where T is an arbitrary positive time. Now define p(xi,x;T) = dSc\{xi,x]T)/dxi so that , x\ T)  Sc\(xu x\ T) ~ (xf  Xi)p(xu x; T) . Finally, change variables from x to p and compare with the free particle result to obtain
b) Show that rxf . Sc\(xf, x{\ T) = ET + / dx^/2m(E  V{x)) J Xi
and verify that [27rti c i(t / )i d (ti) /*/ dx i^ 3 (x) J Hint: Recall that t is an independent variable, so that dt dt
Functional integration
493
We thus have the result for the semiclassical propagator i 1/2 \
m
f
iQ,
s
l dxxcl (x)\ which is identical to that found from WKB methods. 2) Propagator for the Charged Scalar Field The lagrangian density for a charged scalar field
\X , JL i —
can be written as DF(x'\x) = iix'WDn
+ m2  ie)~l\x) .
Suggestion: This is a quadratic form. Use the generating functional to integrate it. b) By expanding Dp{xf\x) as a power series in A^{x)^ show that an alternative representation for the propagator is oo
3) Functional Methods and cp4 Theory Consider a scalar field theory with the selfinteraction
a) Show that the generating functional can be written as
where the free field Feynman propagator iA^(x, y) is as in Eq. (C2.12). b) Evaluate the twopoint function to O(A2). Associate a Feynman diagram with each term of this expansion and separate the connected and disconnected diagrams. c) Calculate the connected generating functional via ] = Z0[j]  i In
494
Appendix A where z
o[j] = jd4xl
d4y j(x)iAF(x,y)j(y)
.
d) Compare the connected diagrams found in parts (b) and (c).
Appendix B Some field theoretic methods
B.I The heat kernel When using path integral techniques one must often evaluate quantities of the form H{x,r) = {x\erV\x) ,
(1.1)
where V is a differential operator and r is a parameter. In this section, we shall describe the heat kernel method by which H(X,T) is expressed as a power series in r. For example, if in d dimensions the differential operator T> is of the form V = n + m2 + V ,
(1.2)
where V is some interaction, then the heat kernel expansion for H(x, r) is ^
H{X,T)
= ——^
eTm
—^
2
[ao(x) + ai(x)r + a2(x)r2 + ...] .
(1.3)
where ai{x) are coefficients which will be determined below. Let us begin by citing the two most common occurrences of H(x,r). One is in the evaluation of the functional determinant detP = etTlnV = Jd4x ^W^W
,
(1.4)
where ' Tr' is a trace over internal variables like isospin, Dirac matrices, etc., and ' t r ' is a trace over these plus spacetime. The (generally singular) matrix element (xlnDx) appearing in Eq. (1.4) can be expressed in a variety of ways. For example, in dimensional regularization one can use the identity a
Jo
x 495
Appendix B
496 to write
(x\ \nV\x) =  f°° — (x\e~rV\ x) + C , ./o r
(1.6)
where C is a divergent constant having no physical consequences. Substituting Eq. (1.3) into the above yields (x\lnV\x)C = 
(1.7)
The divergences in the series representation arise from the Ffunction and are restricted in four dimensions to the terms ao(x), a\ (x), a
1
X
(1.8) 5=0
r(«) Jo The penultimate equality in Eq. (1.8) is obtained from repeated formal differentiation of Eq. (1.6) with respect to V. Upon expanding the H(x, r) term in £p(#>s), one arrives at the desired power series expansion of (x\ lnT>\x). This usage is applied in the next section. The other main use of the heat kernel is in the regularization of anomalies. Often one is faced with making sense of Tr (x \O(x)\ #), where O is a local operator. Although such quantities are generally singular, they can be defined in a gaugeinvariant manner by damping out the contributions of large eigenvalues, Tr (x \O(x)\ x) = lim Tr (x \O{x)e~eV\ x)
(1.9)
where V is a gaugeinvariant differential operator. Again it is only the loworder coefficients, generally those up to a2(x), which contribute in the €—• 0 limit. We employ this technique in Sects. 1113,4. As an example of heat kernel techniques, let us consider the following operator defined in d dimensions: V = dyp + m2 + a(x)
(d»
+
M))
(110)
where T^(x) and a(x) are functions and/or matrices defined in some internal symmetry space. In particular, neither FM nor a contains derivative operators. Employing a complete set of momentum eigenstates {\p)}
Some field theoretic methods
497
allows us to express the heat kernel as H{X T) =
'
J Wjd e^e^e^
,
(1.11)
where in d dimensions, use is made of the relations
x
'~x)=6{d){x
~x>)'
(L12)
Prom the identities d^^e^ip.
+ d,),
we can then write H(X,T)=
 /
f,
\p2m2]f,T[dd+a+2ipd]
The first exponential factor is simply the free field result, while all the interesting physics is in the second exponential. The latter can be Taylor expanded in powers of r, keeping those terms which contribute up to order r 2 after integration over momentum. Note that each power of p2 contributes a factor of 1/r. Thus we obtain the expansion H(X,T)
= r
^[(dd + a){d d +
cr)4pdpd] (1 15)
4T 3
— \pdpd(dd
+ a) +p
• d(dd +a)p  d
v
'
'
16r4 where terms odd in p have been dropped and we have displayed only those (9(T 3 ) and O ( T 4 ) terms which contribute to H at order r 2 after p is integrated over. To perform the integral, it is convenient to continue to euclidean momentum PE — {pi,P2>P35P4 = — ipo} Then with
498
Appendix B
the replacement p^ o^ —> —
\p^p^\ = —p\, we obtain
1 <*>
27r d/2
j
e m^
d
2Td/2
T(d/2) (27r)
JR
/ C / (*^Tr\d
1
ft
T) JP
/
O
)
Vr
T%
1
e" m 2 T
( 47r )d/2
r d/2
~~~
d
d 2
(4?r) /
TW2)
'
rf 2+1
r /
T(d/2 + l) (1.16) T(d/2)
2 4 w+i e r '
ON
v ^
Employing these relations to evaluate Eq. (1.14) gives (to second order inr), H(X,T) =
x
[l
or in the notation of Eq. (1.3), OQ(X) = 1 ,
O2(x) = ^
2
a\{x) = a , + ^ [ d ^ p ^ + ^ [dM, [^,or]] .
(1 18)
'
Fermions are treated in a similar manner. For example, the identity
i
(1.19)
allows the same technique to be used for the operator Iplp. In particular let us consider the case where
Some field theoretic methods
499
With some work, one can cast this into the form of Eq. (1.9) with the identifications
75
The values of ai{x) appearing in Eq. (1.18) can also be used in this case. The heat kernel coefficients have been worked out for more general situations [Gi 75]. B.2 Chiral renormalization and background fields In this section, we illustrate the method described above while also proving an important result for the theory of chiral symmetry. The goal is to demonstrate that all the divergences encountered at one loop can be absorbed into a renormalization of the coefficients of the O(E4) chiral lagrangian and to identify the renormalization constants. The technique used here, the backgroundfieldmethod, is of considerable interest in its own right [Sc 51, De 67, Ab 82, Bal 89] and is applicable to areas such as general relativity [BiD 82]. The basic idea of the background field method is to calculate quantum corrections about some nonvanishing field configuration Tp^
(2.1)
rather than about the zero field,* and to then compute the path integral over the fluctuation 6ip(x). The result is an effective action for (p. This effective action can be expanded in powers of Tp and applied to matrix elements at tree level, resulting in a description of scattering processes at oneloop order. In the case of the chiral lagrangian, one expands the full chiral matrix U = U + 8U ,
(2.2)
where U satisfies the classical equation of motion. Upon integration over 6U, one obtains the oneloop effective action for U. This contains a great deal of information. In particular, U can be expanded in the usual way in terms of a set of external meson fields U = exp(iAa<pa/F) * See the discussion in Appendix A4.
(a = l,...,8) .
(2.3)
Appendix B
500
Contained in Ses(U) is the effective oneloop action for arbitrary numbers of meson fields. Upon identification of renormalization constants, all processes become renormalized at the same time. Our starting point is, in the notation of Sect. IV6, the O(E2) lagrangian £ 2 = ^L Tr
+ ^L Tr
+
(2.4)
The procedure to follow is rather technical, so let us first quote the end result of the calculation. Upon performing the oneloop quantum corrections, the effective action will have the form Here the lagrangians in S^611, S™ n are the ones quoted in Sect. VI2, but now with renormalized coefficients. In particular 5Jen is the sum ST4en = Sj 8 " 5 + Sfv where, in chiral SU(3) and employing dimensional regularization, S$w is given by
Tr
16
Tr (x T C/ + + Ux1) I lL[Tr(XUi \ Tr 4
(25)
+ Ux<)]2 + ^Tr
(L^D^UD^ V
+
+ RurLrrfiyu)  7Tr /
4
with
The terms in S ^ are all of the same form as the terms in the bare lagrangian at order E4. Therefore, all the divergences can be absorbed into renormalized values of these constants. The finite remainder, 5fmte, cannot be simply expressed as a local lagrangian, but can be worked out for any given transition. When S$1V is added to the O(E*) treelevel lagrangian of Eq. (VI2.7), the result has the same form but with coefficients ,
(2.7)
Some field theoretic methods
501
where the {ji} are numbers which are given in Table Bl for both the case of chiral SU(2) and SU(3). Thus the divergences can all be absorbed into the redefined parameters and these in turn can be determined from experiment. Let us now turn to the task of obtaining this result. In applying the background field method, there are a variety of ways to parameterize 8U, and several different ones are used in the literature. The prime consideration is to maintain the unitarity property UW = 1 = (W + 6tf) (U + SU) along with WO = 1. We shall take U = UeiA ,
(2.8)
with A = AaAa representing the quantum fluctuations. This choice is made to simplify the algebra in the heat kernel renormalization approach, which we shall describe shortly. Another possible choice is U = £eir>Z
(2.9)
with 77 = \arf and ££ = {/. These two forms are related by 77 = Since in the path integral, we integrate over all values of A (or 77) at each point of spacetime, these two choices are equivalent. The expansion of the lagrangian in terms of U and A is straightforward, and we find
Tr
(D^UD^UA
= Tr (D^UD^UA
2iTr
+ Tr ID^AD^A + U^DJJ (AD^A  D^A A)1 , iTr ( A (2.10) where D^Asd^A + i ^ . A ]
.
(2.11)
Table Bl. Renormalization coefficients i SU(2)
7<
SU(3)
7<
1
2
3
4
5
6
7
8
9
10
1 12 3 32
1 6 3 16
0
l 8 1 8
l 4 3 8
3 32 11 144
0
0
0
5 48
I 6 1 4
1 6 1 4
0
502
Appendix B
Since U satisfies the equation of motion, there is no term linear in A. One may integrate various terms in the action by parts to obtain ^
(2.12)
where
rjf =  i Tr ([Aa, A6] (p%U + iU%U + ir aab = 1 Tr ({ Aa, A6} (xf£/ + U]x) + [Aa,tff£>„£/] [A6, (2.13) The action is now a simple quadratic form, and the path integral may be performed. The only potential complication is the question of interpreting the integration variables. This is referred to as the 'question of the path integral measure'. The integration over all the unitary matrices U can be accomplished by an integration over the parameters in the exponential
= N j[dAa],
(2.14)
where AT is a constant which plays no dynamical role. With this identification one obtains eiZiooP =
ndAa]eifd*x^A«(d^+<7)" bAb
= (det [d^
+ a])"
172
( 1 1 = exp I   tr In {d^df + a)\ .
(215)
Here ' t r ' indicates a trace over the spacetime indices as well as over the SU{N) indices a, b. The identification of divergences is most conveniently done by using the heat kernel expansion derived earlier in this appendix, where it is shown that all the ultraviolet divergences are contained in the first few expansion coefficients. The relevant terms are Z\ooP = ^ tr In [d^
where
t
+ <J)
Some field theoretic methods
503
For Nf flavors, the operator part of the first term in Eq. (2.16) is
+ ^ ^ IV (x]U + &X) .
Tr a = ^ TV (pfiirffl)
(2.18)
The above two traces are just those which appear in C2 , so that they can only modify Fn and ra2. The remaining terms can be worked out with a bit more algebra. Using the identity J,U]dvu\
,
(2.19)
we find for the field strength, Tab =  T r {\\a
Xb1 (\WD
U U^Djytf] +iU^L
UU
+ iR j \ \ (2.20)
This produces, for Nf flavors in chiral SU(Nf),
Tr (F^F^) = ^Tr (lu^DM.U^D^u] \u^D^U,U]Duu\] 8
+ iNfTr
VL
i
J L
J/
/
(RfU/& U^ff U + .
 Nf Tr {L^UR^U^  ^j Tr {L^W + o 1r / +\i2 1 / A Tr a1 =  Tr (DJJD^W 1 +  Tr (D U UD U W ) Tr p 8 L V / J 4 \ / +  ^ Tr (D^UD^WDJ.UD^W)
+
^
[Tr fx^T +
/ +  Tr (D^UD^U]\ Tr (\U] + f7xf)
(2.21) + !~ Tr which is not of the same form as the basic O(E4) The only operator lagrangian occurs in the first term of Tr F 2 . However, by use of Eq. (VI2.3) for SU(3), it can be written as a linear combination of our standard forms. For Nf — 3, these add up to the result previously quoted in Eq. (2.5). Here the divergence is in the parameter A. For convenience in applications, we have added some finite terms to the definitions of A. The results for Nf — 2 are also quoted in Table Bl, although some of the operators are redundant for that case.
504
Appendix B
The reader who has understood the above development as well as the standard perturbative methods presented in the main text will be prepared for the use of the background field method in the full calculation of transition amplitudes. This procedure consists of writing Vo = • + m 2
(2.22)
where m2 is the meson masssquared matrix. The oneloop action is then expanded in powers of the interaction V Zloop = \ tr i
\
1
=  tr pnPo + V^V  V^VV^V
1 .. .j . +
The first term is an uninteresting constant which may be dropped, and the remainder has the coordinate space form [AF(x  x)V{x)}
J
cTWyTr [AF(x  y)V(y)AF(y  x)V(x)\ + ... .
(2.24)
When the matrix elements of this action are taken, the result contains not only the divergent terms calculated above, but also the finite components of the oneloop amplitudes. The resulting expressions are presented fully in [GaL 84,GaL 85a]. This method allows one to calculate the oneloop corrections to many processes at the same time, and in practice is a much simpler procedure for some of the more difficult calculations.
Appendix C Useful formulae
C.I Numerics Conversion factors (h = c = &B = 1): 1 GeV" 1 = 6.582122 x 10~25 s
1 GeV = 1.16 x 10 13 K = 1.78 x 10~24 g
= 0.197327 fm Physical Constants: GM = 1.16637(2) x 10~5 GeV~ 2
G^ 1 / 2 = Mpi = 1.2 x 10 19 GeV
a " 1 = 137.0359895(61)
sin2 9V = 0.226(5)
mw = 80.6(4) GeV/c 2
mz = 91.175(21) GeV/c 2
me = 0.51099906(15) MeV/c 2
mp = 938.2723(3) MeV/c 2
F w = 92.4(2)MeV
FK = 112.7(2.1) MeV
\r)+\ = 2.268(23) x 10"
3
fX» = 2.253(24) x 10~ 3
CKM Matrix Elements: = 0.220(2)
\Vud\= 0.9744(10)
IFUSI
Vcd = 0.204(17)
\Va\ = 1.02(18)
\Vch\ = 0.046(7)
\Vuh\ = 0.005(2)
C.2 Notations and identities Metric tensor: 1 0 0 1 0 0 0 0
0 0 1 0 505
0 0 0
(2.1)
506
Appendix C
Totally antisymmetric fourtensor:
{
eiU,«f> ev'«'P' =
+1 {/i, v, a, /?} even permutation of {0,1,2,3} — 1 odd permutation 0 otherwise g^'gw'gW + g^'g^'g^' + g^'gaa'g^ (2.2)
Totally antisymmetric threetensor:
{
+1 —1 0
eOijk = eOijk
{z,j, k} even permutation of {1,2,3} odd permutation otherwise = eijk = eijk
(2.3)
Pauli matrices: ^k
k
ie^lal
( j , k 9 l = 1, 2,3) a
, 6, C, d = 1, 2)
Dirac matrices: • n i
2 s
75 = *7 7 7 7 (2.5)
7°rj7°=  r ,
(r* = 75)
Trace relations: Tr (7") = 0 Tr (75) = 0 Tr ( 7 "7 I/ ) = 4
=0
(2.6)
Useful formulae
507
Plane wave solutions: The Dirac spinor tx(p, s) is a positiveenergy eigenstate of the momentum p and energy E = >/p2 + ra2. Antifermions are described in terms of the Dirac spinor v(p, s). The adjoint solutions are denoted by u = 1^7° and v = v^j°. Note that our normalization of Dirac spinors behaves smoothly in the massless limit.
(p  m)u(p, s) = 0 u(p,s)(fim) = 0 U> + m)v(p, s) = 0 v(p, s)(p + m) = 0 ii(p, r)w(p, s) = 2m^rs v(p, r)v(p, 5) = 2mSrs ^ f (p,r)^(p,5) = 2E6rs
(2.7)
S
Gordon decomposition for a fermion of mass m: S(pVb'Mp,
 S ( p » (&±&. + ^
r t
' ) »(P,») (2.8)
Dirac representation: .
, , u(p, s) = y/E + m I
X a
p
0
\
)
v(p, 5) =
V Xs
(2.10)
Fierz relations: The anticommutativity of fermion fields and the algebra of Dirac matrices imply the (particularly useful) Fierz relations, + 75)^2 (211) 75)^2
.
508
Appendix C
Propagators: The propagators associated with fields
(x) = (O\T(M i D F X u ( x ) = (0\T
(
= f dpr)
w
\
)
A
2
4
J (27T)
e
2
2
+ ie)
2
p  M + ie (2.12)
where £ is a gaugedependent parameter. C.3 Decay lifetimes and cross sections Parameters of choice for quantum fields: The literature reveals a variety of conventions employed in quantum field theory. We can characterize all of these with certain parameters of choice, Ji, Ki, Li (i = J5,F distinguishes bosons from fermions), occurring in the normalization of charged spin zero and spin onehalf fields, H3k
^f (a(k)eihx + a\k)eikx)
/
J
f ,3
5
s
(*) = E J 7 ^ W P > M P > > ~
ipx
dt
s eip
+ (P>*MP' ) ") '
(3.1)
s
in momentum space algebraic relations, e.g.,
,r),&V, s )} =KF6rsSz(pp')
(3 2)
,
'
and in the normalization of singleparticle states
tf .
(3.3)
It is convenient to introduce an additional parameter NF to characterize the choice of fermion spinor normalization, u\p,r)u(p,8)
= NF2Ep6r8 .
(3.4)
For uniformity of notation, we also define NB = 1. The constants Ji,Ki,N{ are constrained by the canonical commutation or anticommu
Useful formulae
509
tation relations to obey
Using the above, one can express the singleparticle expectation value of the quantum mechanical probability density as (i = B,F)
(3.6)
.
The conventions employed in this book, together with the implied normalization for boson or fermion singleparticle states, are LB
= LF
= NB
= NF
= 1 ,
J B = J F = KB
= KF
= 2£(2TT) 3 ,
 p) ,
(3.7)
where r, s are spin labels. This choice, although somewhat unconventional for fermions,* has the advantages that bosons and fermions are treated symmetrically throughout the formalism, the zero mass limit presents no difficulty, and matrix elements are free of cumbersome kinematic factors. Lifetimes: Prom the decay law N(t) = iV(0)e~*/r, the mean life r is seen to be the transition rate per decaying particle, F = r" 1 = —N/N. For decay of a particle of energy E\ into a total of n — 1 bosons and/or fermions, the Smatrix amplitude can be written in terms of a reduced (or invariant) amplitude M& as
(f\S  l\i) = i(27r)4^4)(pi P2...Pn)
II n
/
„.
\l/2
(3.8) where the index k labels the individual particles as to whether they are bosons or fermions. The inverse lifetime is computed from the squared Smatrix amplitude per spacetime volume VT and incident particle density pi, integrated over final state phase space. The choice of phase space is already fixed by our analysis. Thus, defining a parameter of choice A(p) for the (momentum) phase space per particle, Phase space per particle * Another book sharing this convention is [ChL 84].
 /
AQT)
510
Appendix C
the application of completeness to Eq. (3.7) yields (p'p) = / ^ ( p '  k ) ( k  p )
=* A = KL2 = (2nfp .
(3.10)
The inverse lifetime (or decay width) is then given by
T _ 1=r =
H I (fr d\k \ \S\\\ VT
fil2 (3.11) where Z = fj • rtjl is a statistical factor accounting for the presence of rij identical particles of type j in the final state, and the sum 'int' is over internal degrees of freedom such as spin and color. Cross sections: For the reaction 1 + 2 —> 3 + . . . n, the cross section a is the transition rate per incident flux. The incident flux /i n c can be represented as fine = PiPalvi  v 2  = £^r[(pi • n?  m\ml}^
,
(3.12)
and the cross section becomes 1 1
int
(3.13) Watson's theorem: The scattering operator S is unitary, S^S = 1. Thus the transition operator T, defined by S = 1  iT, obeys i(T  7^) = T^T. With the aid of the relation (f\T^\i) = (iT/)*, we obtain the unitarity constraint for matrix elements, ,
(3.14)
where Tfi = (f\T\i). This constraint implies the existence of phase relations between the various intermediate state amplitudes. For example, consider a weak transition followed by a strong finalstate interaction for which there is a unique intermediate state identical to the final state, A —> BC — • BC , weak
strong
(3.15)
Useful formulae
511
i.e. i = A,n = / = BC. In this circumstance, timereversal invariance of the hamiltonian implies Tfi = Tjf, so the lefthand side of the unitarity relation reduces to —2Im7if and both sides of Eq. (3.14) are realvalued. Denoting the weak and strong matrix elements as Twe**w and Tse**s, it then follows that <5w = <$sC.4 Field dimension We consider a limit in which the theory is invariant under the set of scale transformations x1* —• \x^ (A > 0) of the spacetime coordinates. Associate with each such coordinate t