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> Q2 and must be introduced so that the integrals involved in these corrections are convergent. Here . ' . denotes higher order corrections and 0 is the momentum carried by a gluon at quark-quark-gluon vertex which defines g s ( Q 2 ) .It is convenient to rewrite Eq. (48a) as
(7.4813)
(7.48~)
We now eliminate the unobserved " bare" coupling constant a , and ~ the cut-off X2 by making a subtraction at Q2 = p2. Thus we obtain a,-1 ( Q 2 )- a ; ' ( p 2 ) = bln-Q 2 P2
(7.48d)
233
Quantum Chromodynamics (QCD)
with b = 87rbo. Or 1
as(Q2) =
+
a ~ ~ ( pb In~Q)2 / p 2
(7.48e)
The constant b is evaluated in Appendix B and is given by 1
b = - (11 47r
-
2 gnf)
(7.48f)
where nf is the number of effective quark flavors. Another way of writing Eq. (48e) is
ai1(Q2)= bln-
Q2
A&D
where
a,-1 ( p ) - blnp' = -blnA&D.
Thus finally we have (7.48h) and we see the running of a,(Q2)with Q2. A Q ~ Dis the QCD scale factor which effectively defines the energy scale at which the running coupling constant attains its maximum value. AQCD can be determined from experiment. For gnf < 11, it is clear from Eq. (48b) or (48h) that a,(Q2)decreases as Q2 increases and approaches zero as Q2 --t 00 or T- + 0. This is known as the asympotic freedom property of QCD.This is due to the factor 11 in Eq. (48f) or (48h) and arises due to the self-interaction of gluons (see Appendix B). We now discuss the experimental determination of the coupling constant a,(Q2) at various values of Q2 from different reactions, starting from the lowest value of 0.
Color, Gauge Principle and Quantum Chromodynamics
234
The rates of quarkonium decay, in particular the ratio of [see, in particular Eq. (8.45)] provides a deterrates r,p/l?ggg at rnJ/,,,and rnr for the charmonium and mination of a, bottomonium states and give
(m)
"3(mJ/$)= 0.216 k 0.024, a , ( m ~=) 0.178 f 0.005
(7.49a)
where we have used
R,
( '4 )
r =
(
T
-+
hadrons
The value of a, obtained from the scaling violations in deep inelastic lepton-nucleon scattering [see Chap. 141 gives a,
(@
= 2.6
GeV = 0.264
0.101.
(7.4913)
The order a, corrections to the total hadronic cross-section in e-e+ annihilation in the ratio (3) modify it from Eq. (6) to
By fitting the value of R at 4 = 34 GeV shown in Fig. 3 one obtains ~ ~ ( GeV) 3 4 = 0.142 f 0.03. (7.49c) Finally from the semi-leptonic branching ratio R, for the inclusive decay T -+ u, hadrons, one obtains
+
a,(?%)= 0.35 f 0.03.
235
Quantum Chromodynamics (QCD)
(in 0.04
0.1
I'
'
scheme, in GeV) 0.5
0.2
"
--c
Average e+e- rates "
e+e- event shapes
__o__
-
Fragmentation Z width
-0-
Small x structure functions ep event rhapes c
Deep Inelastic S c e g (DIS) PoluizedDIS t decays
=
QQ L a t t i c G rdpy 0.10
0.12
0.14
a,(Mz)
Figure 5 Summary of the values of a , ( m ~ )and A(5) from various processes. The values shown indicate the process and the measured value of as extrapolated upto p = mZ. The error shown is the total error including theoretical uncertainties.
Figure 5 shows the values of cr,(m,) deduced from the various experiments. Figure 6 clearly shows the experimental evidance for the running of ~ , ( p i.e. ) decrease of the coupling constant as increases as indicated by Eq. (48). An average of the p = values in Fig. 5 gives Qs(rnz)= 0.119 f 0.002
which corresponds to
(7.50) The LEP / SLC value for a,(rnz) is 0.124 f 0.004.
Color, Gauge Principle and Quantum Chromodynamics
236
0.4
0.3
am 0.2
0.1
0.0
1
2
6
10
20
50
100
200
1 (GeV)
Figure 6 Summary of the values of a,(p) a t the values of p where they are measured. The figure shows clearly the decrease in a s ( p ) with increasing p.
Hadron Spectroscopy
Figure 7 system.
7.4
237
Diagram generating one-gluon exchange potential for q?j
Hadron Spectroscopy
7.4.1 One gluon exchange potential All known hadrons are color singlets. Just as an exchange of photon gives force of repulsion between like charges and force of attraction between unlike charges, the exchange of gluon gives force of attraction between color singlet states. The exchange of gluons can provide binding between quarks in a hadron. For qij system (meson), the color electric potential due to one gluon exchange diagram [see Fig. 71 is given by:
The factors L6' and $6: in the initial and final states arise due to normalize color singlet totally symmetric wave function for the qq system. The minus sign arises due to the coupling of a vector particle to the antiquark. Here i ,j are flavor indices and a, b, c, d are color indices. Since T T ( X A X , ) = T r ( X A X A ) = 16, we get
ea
(7.52)
Color, Gauge Principle and Quantum Chromodynamics
238
Figure 8 Diagram generating one-gluon exchange two-body potential for three quarks (baryon) system.
For three quarks system (baryon), one gluon exchange diagram (Fig. 8) gives the following two-body potential
3
The factors and arise due to the fact that three-quark color wave function is totally antisymmetric in color indices. Using Eebd - StS,d - S,dS,b, and T ~ X= A0, we get
(7.5313) Note the important fact that in both cases, we get an attractive potential. We also note that V,? = 2x39 for color singlet states. Thus we can write the two-body one-gluon exchange potential as
(7.54) Since the running coupling constant as becomes smaller as we decrease the distance, the effective potential Kj approaches the lowest order one-gluon exchange potential given in Eq. (53) as T + 0.
Hadron Spectroscopy
239
Now in momentum space, we can write the potential in QCD perturbation theory for small distances (. < 0.1 fm) as
(7.55) where V(T) is the Fourier transform of V(q2) and q2 is the momentum conjugate to T-. The running coupling cys(q2)in QCD is given by Eq. (48h). We conclude that for short distances, one can use the one gluon exchange potential, taking into account the running coupling constant as(q2). 7.4.2 Long range QCD motivated potential The second regime, i.e. for large T-, QCD perturbation theory breaks down and we have the confinement of the quarks. Thus unlike the short range part of the potential, the long range part cannot be calculated on perturbative QCD as the QCD constants become large in this region. Perturbative QCD gives no hint of intrinsically nonperturbative phenomena such as color confinement. One may look for the origin of this yet unsatisfactorily explained phenomena. There are many pictures which support the existence of a linear confining term. One of these is discussed below:
The string picture of hadrons: This picture is depicted in Figs. 9 and 10. A string carries color indices at its ends. Gauge invariance implies that each site must be a color-singlet. Thus, an allowed configuration of a quark and an antiquark on adjacent sites is the one in which the quark and antiquark are linked by a string so that the color index of quark (antiquark) and the color index of the string at that end are contracted to form a color singlet. When a quark and an antiquark are far apart, many strings have to be excited to connect the two sites [see Fig, lo]. When there is enough energy available to create a new qq pair, the system breaks up permitting the formation of two color singlets. Calculation based on this theory shows that the energy stored in this configuration is:
240
Color, Gauge Principle and Quantum Chromodynamics
Figure 9
String picture of qq.
Figure 10 String separation of a quark-antiquark pair.
L E = Toa
for L
>> a,
where L is the quark-antiquark separation and TOis the string tension. To isolate a quark for example, the antiquark in the above illustration has to be removed to infinity; it clearly takes an infinite amount of energy to do this. This is the basis of color confinement. The confining potential is of the form:
V(T) N constant x
T,
for T > 1/M, where M is a typical hadronic mass scale. Thus $ is of order of the hadron size of 1 fm = 5 GeV-' so that M M 200 MeV. The confining potential is spin and flavor independent. This picture is supported by the observation that hadrons of a given internal symmetry quantum number but different spins obey a simple spin ( J ) - mass ( M ) straight line relation i.e. we say that they lie on linear Regge trajectories, an example of which is displayed in Fig. 11.
241
Hadron Spectroscopy
spin
h 1 2 Figure 11 Regge trajectories for non-strange (I = 1) and strange ( I = 1/2) bosons.
For the families of hadrons composed entirely of light quarks, the above mentioned relation between J and M 2 for Regge trajectories is given by: (7.56a) J (M 2 ) = QO Q’M’,
+
with
a’
“N
0.8 - 0.9(GeV/c2)-’.
(7.56b)
The connection between linear energy density and the linear Regge trajectory is provided by the string model formulated by Nambu. We consider a massless (and for simplicity spinless) quark and antiquark connected by a string of length T O , which is characterized by an energy per unit length 0. The situation is sketched
242
Color, Gauge Principle and Quantum Chromodynamics
below:
For a given value of length ro,the largest achievable angular momentum J occurs when the ends of the string move with the velocity of light. In these circumstances, the speed at any point along the string at a distance r from the center will be: ( p = V/C)
The total mass of the system is then:
M=21
dra
7r
(7.57a)
JW- 2
= arg-,
while the orbital angular momentum of the string is: TOPdr a rP(r)
J=2L
27r
JW= aro-8 .
(7.57b)
Using the relation (57a), one finds that: J=-
M2
(7.58a)
27ra ’
which corresponds to a linear Regge trajectory with (7.58b) This connection yields: 0.18 GeV2 0.20 GeV2
ora =
0.9 GeVP2 0.8 G e V 2
(7.59) ’
Hadron Spectroscopy
243
This heuristic estimate of the energy density suggests that at a separation of the order of 1 fm, we may characterize the interquark interaction by the linear potential
V ( r )= ur.
(7.60)
The lattice gauge theory calculations also support the linear form for the long range part of the QCD potential, Thus phenomenological potential of the form (7.61) can be used for heavy quarks. The Cornell potential
K V ( r )= --r
r ++ c, a2
(7.62a)
where
K
= 0.48, a = 2.34(GeV)-'
and C = -0.25
(7.62b)
has been used successfully to describe mass spectrum of charmonium and bottomonium systems [see Chap. 81. Note that value of a (s-)-) in Eq. (62b) is consistent with the value of u stated above [cf. Eq. (59)]. The purely phenomenological potentials of the form and ~ ( r=)a + bro.l (7.63a) and
V ( r )= C l n r
(7.63b)
have also been used successfully for cF and bb systems. 7.4.3 Spin-spin interaction Finally, we note that a spin 1/2 charged particle of charge eQi has a magnetic momentum e,ui = $oi. In quantum mechanics, the energy splitting between S-states (zero orbital angular momentum) is given by two-particle operator (Fermi contact term) (7.64)
244
Color, Gauge Principle and Quantum Chromodynamics
Similarly in QCD, we have eight color-magnetic moments
The analogous two-particle interaction for QCD is then given by
(7.66) Again for a color singlet system
(7.67) Eq. (67) would immediately give m(3S1) > m('So) [for example mp > m,] in agreement with the experimental result. This supports the fact that gluons are spin 1 particles.
7.5
The Mass Spectrum
The one gluon exchange potential is obtained by summing over all possible quark indices in Ky in a multiquark system like qij and qqq. Thus
=
CyF.
(7.68)
i>j
The potential Vc for S-states is found to be [in non-relativistic limit keeping terms up to (p2/m2)]
The Mass Spectrum
245
(7.69) The first term on the ri ht hand side is the potential in the extreme non relativistic limit = 0); spin dependent term is due to the color magnetic moments interaction as mentioned previously. For S-states,
f
Now our Hamiltonian, including the rest masses of the quarks can be written as
where
p- 2. = - t i 2 0 2
(7.72)
Here Vc(r)is the confining potential, VG(T) is the one gluon exchange potential given in Eq. (69), i is the quark flavor index, i.e. i = u , d , s for ordinary hadrons. We will take mu = m d . In order to discuss the mass spectrum of hadrons, we have to take the expectation value of the Hamiltonian H(r) with respect to the relevant wave functions of the hadrons. The wave function is the product of three parts viz. unitary spin, spin and space parts. For s-wave, we write the space function as Qs(r). Let us first take the expectation value of H(r) with respect to Qs(r),we have
(7.73)
Color, Gauge Principle and Quantum Chromodynamics
246
where (7.74a) (7.7413) (7.74c) (7.74d)
Note that the mass operator M is still an operator in unitary spin and spin space. The parameters a , Ao, d , b and c may be different for L = O(qq) meson and L = O(qqq) baryon systems. We first apply the mass formula (73) to pseudoscalar meson system.
7.5.1 Meson mass spectrum From Eq. (73), the mass operator for S-wave mesons can be written as
(7.75) where
(7.76)
Indices 1 and 2 refer to the constituent antiquark and quark respectively. For vector gluon k , = Now
-4.
s1 * s 2 =
{ 's
spin triplet state S = 1: vector meson spin singlet state S = 0: pseudoscalar meson.
The Mass Spectrum
247
-$,
as for vector gluons, it is clear from Eqs. (75)
Thus if k, = and (76) that
m(3S1)> m(lS0)
(7.77)
i.e. vector meson mass is greater than the corresponding pseudoscalar mass in agreement with experimental observation. If gluons were scalar particles, then s1 s2 term would be absent so that m(3S1) = rn('S0) in disagreement with the experimental observation. For pseudoscalar gluons, lc, = since pseudoscalar coupling is the same for antiquarks. In this case we would have m(3S1)< m(lSo),again in disagreement with the experimental result. We conclude that the experimental results about meson spectrum support the fact that gluons are vector particles and are thus quanta of QCD. From Eq. (75), we can write down the masses of vector and pseudoscalar mesons. For example (with mu = md): e
i,
For K' and K , replace 2mu by m,
+ m,, 2/mu by (k+ k),
(A+ $)
and 2/m: by in rnp and m, respectively. mp = m, and for rn4 replace mu by m, in the expression for mp. From Eq. (78), we have the following results 1 b Y'
mumS
m, mp-m, mk-mK m&- mK mp - m7r
=
mP
(7.79)
64 1 -asdT 3mE 9 mu 1 16 - 64 = d = -a,d3mum, 9 mums mu - M 0.66 (Expt 0.64), m8 16
= -d
-
(7.80) (7.81) (7.82)
248
Color, Gauge Principle and Quantum Chromodynamics
where we have used for mu and m,, the values of constituent quark masses [mu= 336 MeV, m, = 510 MeV] obtained for the magnetic moments of baryons (see Chap. 6). We also obtain
(7.83)
z:;:;
where X = is the SU(3) symmetry breaking parameter. Hence to order A, we recover the Gell-Mann-Okiibo mass formula with ideal mixing angle between w8 and w1. For pseudoscalar mesons qns and q,, we get mqns - mlr mvs = (2mK
- m,)
+ 0(X2).
(7.84a) (7.84b)
These formulae are badly broken. Thus the above analysis breaks down for J = 0 mesons, q and 7'. The reason for this is that our Hamiltonian does not take into account quark-antiquark annihilation into gluons. The lowest order annihilation diagram is shown in Fig. 12. This diagram contributes only to 'So state, because of charge conjugation conservation. Since gluons do not carry any flavor, therefore it contributes to I = Y = 0, 'SOstates only. This diagram is relevant only for 9 and q' mesons, and is of order O(a:). For I = Y = 0 vector bosons, the diagram with threegluon exchange contributes, which is of order O ( a i )and hence can be neglected. We now take into account the diagram of Fig. 12 for pseudoscalar mesons. If ua, d d and SS can annihilate with an amplitude A, which we assume to be SU(3) invariant, then there will be an additional contribution to the mass matrix, which in the u U , dd
The Mass Spectrum
249
Figure 12 The qij annihilation diagram for 'So state through two gluons.
and
SS
basis is given by
A A A (7.85)
A A A Taking into account Eq. (84) and the fact that lqs) = IsS), Iqns) = I(u0 + d z ) ) , IT' ) = I(ua - dd)), we get in no, qns and qs basis, the mass matrix
&
&
0 0
0
0
m,+2A fiA
JZA 2m~-m,+A
).
(7.86)
From Eq. (86),we note that we have to diagonalize the mass matrix
M
+M
+
Man, =
). ( m$jA 2mK .\/ZA m, + A -
(7.87)
For this purpose, we define the physical states as (see Problem 5.15) 17) = cos 4 Iqns) - sin $1175) 177') = sin 4 Iqns) C O 4 ~ 17s)
+
*
(7.88)
250
Color, Gauge Principle and Quantum Chromodynamics
Then the mass eigenvalues are given by mqmqt = m,(2mK
+
- m,)
+ A(4mK - m,)
+
mq+mm,/= mqns m,, = 2 m ~ 3A.
(7.89)
Using the experimental values for q and q’, we can determine A. The mass scale A comes out to be x 172 MeV, a rather low value compared to mq and mqt which is both interesting and reasonable. To conclude, we have shown that mass spectrum of vector mesons can be explained successfully. With the addition of annihilation diagram, the pseudoscalar meson mass spectrum can also be understood. 7.5.2 Baryon mass spectrum In order to discuss the mass spectrum of the baryons, it is convenient to first calculate the matrix elements of the spin operator 1
= zmimjsz * sj
(7.90)
between spin states. The eigenvalues of si . s j are 1/4 and -3/4 for spin triplet, and singlet states respectively. Therefore, 1
si * sj
ITT)
=
4 ITT)
si *
ILL)
=
4 ILL)
sj
1
(7.91a)
(7.91b)
From Eq. (91), we get si sj I z T j L )
=
sz ’ sj I z L j T )
=
*
-41 lzy)+ 1 I z y ) 1 1 -- l z y ) + p j l ) . 4
(7.92)
The Mass Spectrum
25 1
The spin wave functions for baryons are given in Table 6.3 and Eq. (6.8). Using these wave functions, we get with the help of Eqs. (91) and (92) for baryons with s, =
;+
=
c
-sz
1 *
sj
i> j mimj
Juud)
4:
(--A) I(tI + IT) T
-2lTt.l.)
1
3 4m2
= --+p).
(7.93a)
Similarly we get (7.9313)
4 4
(7.93c) (7.93d)
where we have used
(7.94)
Color, Gauge Principle and Quantum Chromodynamics
252
%’
For baryons, we take s, of O,,. Now
= 3/2
and calculate the matrix elements
(7.95a) Similarly we get
R,,
p*+>
= 1
as,lp*o)
=
n,,(n-)
=
(-
2 4 mums
+
5)[c*+)
(7.9513) (7.95c)
+r), 4 m:
(7.95d)
where we have used (7.96a) (7.96b) (7.96~) Since the spin-spin interaction term from Eqs. (73) and (90) is, 3
(7.97)
we have from Eqs. (93) and (95):
in agreement with experimental observations. For gluons with color, k, = -2/3; if gluons do not carry color, then k, = 1 instead of -2/3 and we would get results in contradiction with experimental values. This supports that the vector gluons carry color.
The Mass Spectrum
253
The spin dependent term R,, splits the masses of baryons with the same quark content, but with different spin. Thus, we get from Eqs. (93), (95) and (97):
rna-m, mc-mA
d 8m2 16 d = -- (1 3 me =
2) (7.98)
where d =
y. From Eqs. (98), we get
mz* - mz me8 - me 2mc. mc - 3 m 2 (mA- mp) mc-mA 2 = - (1 mA-mp 3
+
1
(expt 1.12) (7.99a)
=
1
(expt 1.04) (7.9913)
=
0.23
(expt 0.26). (7.99~)
=
~
2)
In the above derivation, the effects of wave function distortion due to symmetry breaking by quark effective masses have been neglected. These effects will give slight deviations from unity in the relations (99a, b). We now discuss the baryon masses of same spin, using Eqs. (73), (93) and (98). We can write the baryon mass formula:
m = (ml+rn2+ms)+a
+36
1 1 + c (-m1m2 +m2m3
1
m3m1 (7.100)
254
Color, Gauge Principle and Quantum Chromodynamics
where C = ia,c, 6 = - ~ a , b . Introducing the SU(3) breaking parameter X = (m, - m,)/(m, mu)and writing mo = f(m, mu) and retaining only the first order term in A, we get
+
+
mp = A + X mA = A + X mL = A + X V I , ~=
A+X
(7.101)
where
mo+b+-+$-+-a 'd)+A0 mo mo 3 m i
(7.102)
From Eq. (101), we get the Gell-Mann-Okubo mass formula rn,+mz - mc+3m~ 2 2
(7.103)
We conclude that both the meson and baryon mass spectra can be explained quite well in QCD. In this simple picture, we have used non-relativistic quantum mechanics for u , d and s quarks. Although this approximation is not so good for these quarks (a their masses are less than 1/2 GeV) and at this energy scale QCD perturbation theory may not be a good approximation, even then the results are good.
Bibliography
7.0
255
Bibliography
A. General 1. E. Abers and B. W. Lee, Gauge Theories, Phys. Rep. 9C, 1 (1973). 2. B. W. Lee, ”Particle Physics”, in Physics and Contemporary Needs, Vol. 1 (Ed. Riazuddin), 321, Plenum Press, New York (1977). 3. K. Huang, Quarks, Leptons and Gauge Fields, World Scientific, Singapore (1982). 4. K. Moriyasu, An Elementary Primer for Gauge Theory, World Scientific, Singapore (1983). 5. C. Quigg, Gauge Theories of the Strong, Weak and Electromagnetic Interactions, Benjamin/Cummings, Reading Massachusetts, (1983). 6. T&Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, Clarendon Press, Oxford (1984). 7. M. Chaichian and N. F. Nelipa, Introduction to Gauge Field Theories, Springer-Verlag (1984). 8. T. D. Lee, Particle Physics and Introduction to Field Theory (revised edition), Harwood Academic, New York (1988). 9. J. J. R. Aichison and A. J. G. Hey, Gauge Theories in Particle Physics (2nd edition), Adam Hilger, Bristol, England (1988). 10. C. H. Llewellyn Smith, Particle Phenomenology: The Standard Model, OUTP-90-16P, The Proceedings of the 1989 Scottish Universities Summer School: Physics of the Early Universe. 11. R. E. Marshak, Conceptual Foundations of Modern Particle Physics, World Scientific (1992). 12. M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, Mass (1995).
B. For Sec. 7.2.1 1. D. Bohm and H. J. Hiley, I1 Nuovo Cimento 52A, 295 (1979).
Color, Gauge Principle and Quantum Chromodynamics
256
C . QCD 1. E. Reya, Perturbative Quantum Chromodynamics Phys. Rep. 69 C, 195 (1981). 2. A. H. Mueller, Perturbative QCD at High Energies, Phys. Rep. 73 C, 237 (1981). 3. G. Altarelli, Partons in Quantum Chromodynamics, Phys. Rep. 81 C, 1 (1982) 4. F. Wilczek, Quantum Chromodynamics: the Modern Theory of the Strong Interaction, Ann. Rev. Nucl. and Part. Sci. 32, 177 (1982). 5. D. W. Duke and R. G. Roberts, Phys. Rep. 120, 275 (1985). 6. T. Muta, Foundation of Quantum Chromodynamics, World Scientific, Singapore (1987). 7. R. D. Field, Applications of Perturbative QCD, Addison-Wesley (1989). 8. Perturbative Quantum Chromodynamics, Editor: A. H. Mueller, World Scientific, Singapore (1989). 9. M. Creutz, Quarks, Gluons and Lattices, Cambridge University Press (1983). 10. Ref. 12 in A above
D. For Sec. 7.3.2 1. G. Altarelli, Experimental Tests of Perturbative QCD, Ann. Rev. Nucl. and Part. Sci, 39, 357 (1989). 2. Ref. 1 2 in A above
E. For Figs. 3 and 5 and 6 1. Particle Data Group, The European Phys. J. C3, 1-4 (1998).
F. Hadron spectroscopy 1. A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D12, 147 (1976). 2. B. W. Lee, Ref. 2 in A above. 3. 0. W. Greenberg,
Bibliography
257
Ann. Rev. Nucl. Part. Sci. 28, 327 (1978). 4. F. E. Close, An Introduction to Quarks and Partons, Academic Press, New York, 1979. 5. C. Quigg, "Models for Hadrons", in Gauge Theories in High Energy Physics, edited by M. K. Gaillard and R. Stora (Les Houches, 1981), North-Holland, Amsterdam, 1983, p. 645. 6. J. L. Rosner, "Quark Models", in Techniques and Concepts of High Energy Physics (St. Croix, 1980), edited by T. Ferbel, Plenum, New York, 1981.
G . For Sec. 7.4.2 1. B.W. Lee, Ref. 2 in A above. 2. C. Quigg, Quantum Chromodynamics near the Confinement Limit, FERMILAB-Conf-85/126-T (1985). 3. Y. Nambu, Phys. Rev. D10, 4262 (1974). 4. S. Mandelstam, In Proc. 1979 Int. Sym. on Lepton and Photon Interaction at High Energies (ed. T. B. W. Kirk and H. D. I. Abarbanel). Fermi Lab., Batavia, Illinois (1979). 5. S. Gasiorowicz and J. L. Rosner, Am. J. Phys. 49, 954 (1981).
Chapter 8 HEAVY FLAVORS 8.1
Discovery of Charm
The J/\k was discovered in 1974 in the reaction
p+Be-e'e-+X at fi = 7.6 GeV. A narrow peak at rn(e+e-) = 3.1 GeV was found. It was also seen in e+e- collision at fi = 3.105 GeV in the following reactions e-e+ e-e+ e-e+
---+
e-e+ p-p+
hadrons.
The width of the resonance was very narrow. It was less than the energy spread of the beam, r 5 3 MeV. For this reason, the width cannot be read off directly from resonance curve. The resonant cross section for any final state f :
e-e'
--+
J/Q
-+
f
is given by the Breit-Wigner formula [cf. Eq.(4.52)]:
259
Heavy Flavors
260
where J is the spin of the resonance, m is its mass, is the spin of electron or positron and
s1
= sg = 1/2
Here k = IkJis the center of mass momentum. r is the total width, I?, and rf are the partial widths into e-e+ and f respectively. We can write Eq.(l) as
Since the resonance is very narrow, r is very small and it is a good approximation to replace the denominator in Eq.(3) by the 6-function $V(& - m ) and then integration of Eq. (3) gives
Now Cf oef = otot,CfI'p = the process
and assuming I?, = F p , we have for
We also have for the total cross section
Assuming the spin of the resonance J / @ , J = 1, we determine the widths re = rp and the total decay with r. Since r = re rlL r h , we can also determine the hadronic decay with I'h. The experimental values for these decay widths are given below:
+
+
m(J/\k) = 3096.8850.04 MeV, re = rp= 5.26 f 0.37 keV, r = 8 7 & 5 keV.
Discovery of Charm
261
The J / 9 spin-parity can be determined from a study of the interference between e-e+ -+ y + p-p+ and e-e+ -+ 9 -+ p-p+. The cross section for the QED process e-e+ y --t p-p+ is well known [Eq. (78) of Appendix A] and is given by
-
(s
>> m:,
m;)
da Q2 _ -- -(I
dR
4s
+ C O S ~e)
If the spin-parity of J / @ is that of photon viz. I-, then the angular distribution would not change by the interference between QED amplitude and the resonant amplitude. In fact, experimentally, it was found to be (1 cos20 ) near the resonance, clearly establishing the spin-parity of J / 9 to be 1-.
+
8.1.1 Isospin
Experimentally the decay J / Q -+ p p occurs with a branching ratio Fpp/r = (0.214 f O.OlO)%, which is too large to be explained by the electromagnetic effects. Now p p can have only I = 0 or I = 1. Thus the isospin of J / Q is either 0 or 1. If J / Q has I = 1, then the decay J / @ + po7ro is forbidden while for I = 0 (see problem l),we have
to be compared with the experimental value of 0.494 f0.068. Thus the isospin of J / Q is 0. Now G-parity is given by G = (-l)'C, where C is the charge conjugation parity of J / Q . Since I = 0, therefore, G = C. The allowed decay J / 9 --t poxo fixes its Cparity to be C = (-l)(+l)= -1. Hence G = -1 for J / @ . 8.1.2 SU(3) classification Due to C-invariance, the VPP coupling is F-type (see Sec. 5.8.2) which is not possible if V is an SU(3) singlet. Thus an SU(3)
Heavy Flavors
262
singlet vector meson cannot decay into two pseudoscalar mesons belonging to the same SU(3) multiplet. In particular if J / Q is an SU(3) singlet, then J / 6 -+ K K is forbidden while J / Q --t K*K or J/\k 4K * K is allowed by C-invariance. Experimentally one finds
which shows that J / 6 is an SU(3) singlet. If J / Q is an SU(3) singlet, then its invariant coupling with PV is given by
\k,TT(PV)
=
Q[ pOnO
+ p - r + + p+n- + fT*-I(+ + K * + K (8.10)
(J'Q r(
Hence we have
--t
J/Q
+ KK*)
=
1 (phase space correction)
=
1.2
(8.11)
to be compared with the experimental value 1.3910.12. To summarize, the J / 6 resonance is an SU(3) singlet. with J p c = 1--, G = -1 and I = 0. 8.2
Charm
Although J / Q itself does not carry any new quantum number, its unusually narrow width in spite of large available phase space suggests that it is a bound state of cC , where c is a quark with a flavor which is outside the three flavors u, d and s of SU(3). This new flavor is called charm. The quark c is assigned a new quantum number C = 1 and C = 0 for u,d and s quarks. Thus to take this quantum number into account, the Gell-Mann-Nishijima relation would be modified to 1 (8.12) Q = 1 3 2(Y C ) .
+
+
263
Charm
Figure 1
allowed by 021
For the charmed quark c, C = 1, 13 = 0, Y = B = 1/3. Thus the charge of charmed quark is 2/3 and its mass rn, FZ ~ V L J ,=~ 1.55 GeV. (87 @ keV compared to 100 MeV The narrow width of ,I/ for p ) can be qualitatively understood by the 021 rule, just as the suppression of 4 --f 37r compared to 4 3 K K is explained (see Sec. 5.5.9) by this rule. Thus the decay depicted in Fig. 1 is allowed but that shown in Fig. 2 is suppressed by 021 rule. But the decay J / $ D D shown in Fig. 1 is not allowed energetically since mJpp < 2 m ~ . --f
8.2.1
Charmed mesons
The charmed quark c can form bound states with Q, where q = u, d, s. The low lying bound states such as cq have been found experimentally and are listed in Table 1. The C = 1 states ( D + , Do)D,’ form an SU(3) triplet (3); (D+, Do) form an isospin doublet. Similarly C = -1 states q C : (DO, W ) D ; form an SU(3) triplet (3). The states D*and 0; are unstable and decay strongly and
Heavy Flavors
264
C
.t-> C
Figure 2
J/$J suppressed by 021 rule. Table 8.1
Charm
265
Table 8.2
JP 0001112+ or 1+ 2+ or 1+
radiatively. For example
8.2.2 The fi'h quark flavor: Bottom mesons Fifth quark was discovered, when in 1977 the upsilon meson T(J p c = 1--) was found experimentally as a narrow resonance at Fermi Lab. with mass N 9.5 GeV. This was later confirmed in e+eexperiments at DESY and CESR which determined its mass to be 9460 f 10 MeV and also its width. The updated parameters of this resonance [from the Particle Data Group Tables] are mass 9460.37 f 0.21 MeV and width 52.5 f 1.8 keV. Again the narrow width in spite of large phase space available suggests the existence of a fifth quark flavor called beauty, with a new quantum number
Heavy Flavors
266
B = -1 for the bottom (b) quark. With this assignment the formula Q = I3 1/2(Y B C) would give the charge of b quark the value -1/3(13 = 0). The mass of b quark. is expected to be around 4.9 GeV as suggested by the Y mass which is regarded as a 3S1bound state of bb. Thus one would expect particles with B = *l, such as bq or qb. The lowest lying bound states bq and qb have been found experimentally, they are given in Table 2. The B = -1 states (Bo,B-)B: form an SU(3) triplet (3) and B = +1 states ( B + ,B0)B: form another triplet (3). 8.2.3 The sixth quark flavor: The top The top quark t with Q = 2/3 and new flavor T = 1 was expected on theoretical grounds. It was first found experimentally in 1996; its mass is mt = 175 f 6 GeV. Since (t, b) form a weak doublet, it decays weakly to W+ b, i.e.
+
+ +
+
t
3
w++ b.
The predicted decay (see Eq. (15.27)) rate is
where we have neglected the b quark mass compared to mw and mt. Taking m, = 175 GeV, m w = 80 GeV, GF = 1.166 x GeVP2, we get
l?
M
1.56 GeV.
(8.14)
If QCD correction is taken into account, then
r M 1.43GeV which gives the life time of
T
(8.15)
to be
= 4.60 x
(8.16)
Thus t quark decays before it can form bound states such as tf and
tq.
Heavy Baryons
267
8.3 Heavy Baryons Since u, d, s belong to the triplet representation of SU(3), the charmed and bottom baryons with spin parity belong to either triplet representation 5 or sextet representation 6 of SU(3). Using the Pauli principle, the unitary spin and spin wave functions of spin baryons can be written as
;+
4'
(8.17) (8.18) where i, j = 1, 2, 3 (ql = u, q 2 = d , q 3 = s, Q = c or b ) and the spin wave functions X M A and X M S are given in Eq.(6.8) and Table (6.3) respectively. Note that A i j belongs to triplet representation 3 of SU(3). In particular we have an isospin singlet and isospin doublet: A12 = A!(A:),
A13
-+
-0
-0
--)
= -Ec (Zb), A 2 3 = =,(=b
(8.19)
Sij belongs to the sextet representation of SU(3). In particular, we have an isospin triplet, an isospin doublet and an isospin singlet: s11
=
s12
=
s22
=
s13
=
s23
= =
s33
;+
++ (c+ b >,
c,'(m, Ac:(c,), n'o EC(Eb 1 7 n'+
&O -c
n'-
(=b
An:(n,).
(8.20)
The spin baryons also belong to the sextet representation of SU(3). They are given by Eq.(18), with X M S replaced by xs where the spin wave functions xs are given in Table (6.3). The six baryons are labelled as spin
;+
Heavy Flavors
268
In addition to C = +1 and B = -1 baryons considered above, we also have the following baryons with C = 2 and B = -2 belonging to the triplet representation of SU(3) with spin parity (3/2)+:
Finally we have singlets with C = 3 and B = -3, namely
Experimentally only one bottom baryon has been detected so far. Its mass and life time are V L A ~ = 5641 f 50 MeV and T = (1.14 f 0.08) x 1 0 - l ' ~ . Some of the charmed baryons have been discovered experimentally, they are given in Table 3. 8.4
Quarkonium
The bound system of heavy quarks QQ, Q = c, b, is called quarkonium e.g. charmonium cC, bottomonium bb, Since quarks are fermions with spin 1/2, their bound system can be written as ( Q Q ) L , s . Now S can have two values 0 and 1 with spin wave function antisymmetric and symmetric respectively. If we regard Q and Q as identical fermions which differ only in their charges, then we can state generalized Pauli principle: The wave function is antisymmetric with the exchange of particles Q and Q. Under particle exchange, we get with space coordinates exchange, a factor (-l)",with spin coordinates exchange, a factor (-l)s+l and with charge exchange, a factor C (C is called C-parity). Hence Pauli principle gives
269
Quarkonium
Table 8.3
270
Heavy Flavors
Therefore,
c = (-1y+S.
(8.24)
Hence we have the result
-1 L + S C = { +1 L + S
odd even.
(8.25)
Also for (QQ) system, the parity
P = (-l)(-l)L
= (-1)L+1
(8.26)
Let us now use the spectroscopic notation,
L =
0,
1, 2,
3,
S, P, D, El
* * . a
*
*
I
A state is completely specified as
where n is the principal quantum number and J is the total angular momentum. Thus for L = 0, we have the following states n 'So C = + 1 , n = 1 , 2 , . . . n 'So C = -1, n = 1, 2,
,
The ground state is therefore a hyperfine doublet 1 'So(O-+) and 13S1(1--).For L = 1, we have the following states
n lPJ J = + 1 , C=-1, n 3 P ~J = 0, 1, 2 C = 1, Finally, we note that for L n TI,
= 2,
1+Of+, I++, 2++.
we have the following states
'DJ J = 2 , C = + l , 2-+ 3 D J J = 1, 2, 3 ( ? = - I , I--, 2--, 3--.
Quar konium
271
It is interesting to see that the state 3D1 has the same quantum number as 3S1.They can therefore mix, but the mixing is expected to be small. The states 3PJ and ‘PI is a hyperfine quartet (degenerate), but this degeneracy is removed due to hyperfine splitting. The low lying states listed above are shown in Figs. 3 and 4. Most of these states have been discovered experimentally and they are listed in Tables 4 and 5. The transitions and decays of charmonium states are shown in Fig. 5. Similar transitions and decays occur for bottomonium bound states. From Fig. 5, we note that both A41 and El radiative transitions are possible: J/Q
+
qc+y 7, y M1 transitions 7: y (no parity change)
+
Q+y
-+
x +y
+
qc
+
9’ + Q’
9‘
x
+ +
El transitions
+ y (parity changes).
From Eq.(6.87) and Table (6.7), we get (for example) r(QI
+
VCY)
[-I
4 a 2 1 k3R 3 3m, = 2.7 keVR =
(8.27)
where R is the overlap integral defined as
and q = (m,,/m)k, k is the momentum carried by photon, mspis the mass of the spectator quark and m is the mass of the bound state. For R = 1, r is about a factor of three larger than the experimental value.
Heavy Flavors
272
-lDz _ _ _ _ _ _ . C h a r m threshold
Figure 3
The charmonium spectrum ( cE bound state).
Table 8.4 The
11, family
(cc) bound states.
Qumkonium
273
tys 0
0-+
1--
1+Figure 4
(0 I 2 ) j (1 2 3j-
274
Heavy Flavors
Table 8.5 The 'Y family (bb) bound states.
275
Quaxkonium
r
0-+
MI El
1--
1+- (0, 1,2>*
(1,2,3)-- 2-+
Figure 5 The transitions and decays of charmonium states.
For El transitions nS1 -+ n’PJ and nPJ -+ n’S1 ( J = 0, 1, 2) the decay widths can be written (cf. Eq.(6.68)) (8.29) (8.30) where
Mntn
=
x
< Q > antn,On), = (l/h)
Jom Lo(P)
- 2j2 (QT)]%o(r) &I
(r)r3dr. (8.31)
Note that j o and j , are spherical Bessel functions and R,l are radial wave functions. In order to predict these decay widths one needs to know the radial wave functions, i.e. some potential model is needed. Finally, we note that there are 22 states below B threshold as compared with eight states below charm threshold. This is a
Heavy Flavors
276
Figure 6 consequence of the fact that interquark potential is flavor independent (as expected in QCD) so that En, - Enl is the same for cC and bb. (Note that charm threshold is at about 3.74 GeV whereas B threshold is at about 10.55 GeV.) 8.5
Leptonic Decay Width of Quarkonium
The decays of 'S1(&Q)state ( V ) into charged leptons proceeds through the virtual photon as shown in Fig. 6. The scattering cross section for the QQ -+ It is given by Eq.(A.78) u =
471-cY2 1 Pl -(&>, -3 PQ
where
(8.33)
277
Leptonic Decay Width of Quarkonium
and Q is the charge of the quark Q. Now the cross section u can be written as
(8.34) where ot is the cross section for 3S1state and usis the cross-section for IS0 state. Since the photon is coupled to a conserved vector current, therefore it contributes only to spin triplet state, Thus os= 0. Hence the decay rate in the limit pl 1 (s = 4772; >> 47723 is given by --f
F
=
(incident flux) ct
=
2pQIq 8 (0) I2
4
(8.35)
where the incident flux = p i , ( 2 , 8 ~ ) = 2(lk8(o)(2,8Q. Hence from Eqs. (32) and (34), we get (8.36)
where we have put s = 4m; M m t and PQ M 0 (in the nonrelativistic limit). Taking into account the color IV >= $ C, >, we multiply Eq.(35) by a factor of three. Hence we have
(8.37) It may be pointed out that, before comparing experimental leptonic widths with their theoretical predictions, the vacuum polarization contributions to the leptonic decay width have to be removed so that r0 = reyl
- n)2
where (1 - 11)2= 0.958, 0.932 for charmonium and bottomonium respectively and then it is r0 which is to be compared with the theoretical predictions.
Heavy Flavors
278
Figure 7
8.6
Positronium
(lS0
state) decay into two photons.
Hadronic Decay Width
The decays of quarkonium states 3S1and 'So to ordinary hadrons are suppressed by the 021 rule. The narrowness of their decay widths can be explained as follows. By C-conservation 3S1state can decay in the lowest order to three gluons and thus its hadronic decay width is proportional to a: x (probability of conversion of gluons into hadrons). Since color is confined so this probability is unity. Similarly the decay of 'So into hadrons is proportional to a:, since by C-conservation it can decay into two gluons. Here analogy with positronium is in order. Positronium in 'SOstate (para positronium) decay into two photons via the diagram (Fig.
7). In the low energy limit the cross section for the above process is given by 2
(8.38)
a=;(;).
Since at = 0, we get using Eq.(34) u s
=4a=
47r
(--)
(2
.
(8.39J
Hence the decay rate
r [ls,,(e-e+)
a2 271 = IpXPs(0)124a= 1 6 ~ ~ ~ Q ~(8.40) ( 0 ) ~ ~ 4%
279
Hadronic Decay Width
For (QQ) 'So state decaying into 27, we replace e4 --t
and
4771:
-+
4,;
M
[hQ 2 e2 ] 2 = 3Q4e4
m;. Hence we get
I' ['So(mp)+ 273
(8.41)
=
For qc 2 gluons, we replace a2 by $a: in Eq. (40) [see problem 21, so that we get the hadronic decay rate ---f
The decay rate for 3Sl(e-e+) system going to 37 is given by
For the decay of 3S1(QQ) --t 39, we replace a3 by 5a:/18 [see problem 21 and (2me)2= ( 2 m ~M) m$ ~ in Eq.(43). Hence we get
r [3S,(V)
-+
hadrons] = F [3S1-+ 3g] -
160n(r2 - 9) a3 81n m$ I QS (0) I2 . (8.44)
We now apply the above results to From Eqs.(44) and (37), we get
4, J / @ and Y decays.
From Eq.(45), we get as(md)M 0.44,
as(m*) M 0.22,
a s ( m ~M)0.18,
Heavy Flavors
280
where we have used r(q5 + non-strange mesons) x 653 keV, l?(J/Q--f hadrons) M 76.5 keV, r ( T -t hadrons) x 50 keV, l?($ + e+e-) M 1.37 keV, I'(J/e + e+e-) x 5.26 keV, r(T e+e-) M 1.32 keV. From this we see a realization of the asymptotic freedom of QCD, the coupling a,(q2)falls with the increase of q2. Finally from Eqs.(42), (44) and (37), we have [with aLIS(mqc) = --f
Qs
I)
(ma
r(qC --+ hadrons) =
271~ mi 1 I? ( J / Q + hadrons) 5(r2- 9) mic Q,(md (8.46)
x 7.6MeV
(8.47)
where we have used a,(m*) % 0.22. This value is lower than the experimental value rtotx 13.2:;:; MeV for qc.
Non-Relativistic Treatment of Quarkonium
8.7
From a theoretical point of view, heavy quark system (quarkonium) is interesting because this is a relatively simple system. To a good approximation, the quark motion in this bound state should be non-relativistic. Thus we can use the Schrodinger equation for QQ systern :
ti2 (8.48) --V2Q(r) [ V ( r )- E ] Q ( ~ =)o 21-L p is the reduced mass of QQ system i.e. p = imo. For central potential, we can use the wave function:
+
$1.
(8.49)
Q(r) = ~ ( r ) Y 1 m ( 4
The radial wave function R ( r ) satisfies the equation
[-
h2 d2
--
2p
+ dr2
R ( r )-
I. v(r) -
-
]
1(1+ l)h2 R ( r ) = 0. 2pr2 (8.50)
281
Non-Relativistic Treatment of Quarkonium
If we define a radial function (8.51) then x ( r ) satisfies the equation
+
"I
- V ( r ) )- r2
= O.
(8.52)
The wave function x ( r ) is normalized as
= I
(8.53)
with the boundary conditions x ( 0 ) = 0.
(8.54)
For S-waves: (8.55) For S-waves:
xyo) = R(0)= &QS(O)
(8.56)
We now prove two important results:
1. (8.57)
Proof. From Eq. (52) for I
=0
(8.58)
Heavy Flavors
282
Therefore, (8.59) Taking the expectation value [note X(r) is real], we get dV
00
(8.60)
Integrating left, hand side by parts, we get
=
[x’(O>l2 = [R(0)I2,
(8.61)
where we have used the boundary conditions (54) - (56). Hence Eq. (60) gives
2. Virial Theorem
:(2 )
( T ) = - r-
.
(8.62)
Proof. From Eq. (59), we have (8.63) Integrating left hand side by parts and using Eqs. (54) and ( 5 5 ) , we get 00
1.h.s. = 2 1 Xl’dr.
(8.64)
283
Non-Relativistic Treatment of Quarkonium
Therefore, (8.65) Now from Eq. (58),
(Z)
=
-$ [E- < v >].
(8.66)
But
($)
= Jdm&'dr =
1
00
I-xx'l;
-
XI2dr
= -1m~'2dr.
(8.67)
Hence from Eqs. (65) - (67), we get
E - (v)= ( rdVz ) or
( T )=
;
(8.68)
dV
( I T )
*
Let us apply Eq. (62) to one gluon exchange potential V ( r )= -$a S 1. T For this case
(f)
=
;as
(;)
= -as-, 2 1
3
a
(8.69)
where a = 3/4pas is the Bohr radius. Thus, we get _w -- -2a s . c 3
(8.70)
Heavy Flavors
284
As a, decreases with mass, for sufficiently high mass v / c = [ . i ( . i + l ) - j ( j + l ) -3- - ] .
(9.99)
4 Hence for 1 = 0 states, J = S,+SQ, and we have J = 0 or 1. The corresponding 'So and 3S, states ( D ,D*) and ( B ,B') have already been considered. We will suppress the subscript, q [i.e. light flavor index] and concentrate on DJ mesons ( for B J , replace D by B ) for which 1 = 1,j = 3/2 or 1/2. Thus
J =j
+ 1/2, j - 1/2
i.e. we have the states
j = 3/2 j = 1/2
J=2,1
D;, D1
J = 1,0
D;,Do
It is useful to write down the angular momentum part of the wave functions for the four P-states. According to the angular momentum scheme outlined above, P state can be labeled as IJ A 4 j s ~ ) We . can write these states for DJ mesons:
ID* A4
= 0) =
+ 2YlOX+ + K l X +-1 3
1 [Yl-lx:l
0
&
2,
(9.100a) ID1, A4 =
+I)
1
=
- [-Y10xy
fi 1
- [Ylox;'
fi
+ y11 (x: + ZX"]
+ Yl-1 (-x: + 2x!!)]
The P-wave Heavy Mesons: Mass Spectroscopy
1
ID,, A4 = 0)
+ 2YlOXO- + Y,,x;']
Y1-1x:l
=
31 1
(9.100b)
p ; , M = *l)
1
+ y11 (x: + Yl-1 (-x:
= - [-Y10x:l
fi 1 [Ylox;' fi
_ .
ID;,M
d3 [- YI-lx:l
- YIOX-0
= 0) = 1
x"] - x"]
-
+ Y,lx;l] (9.101a)
We first discuss the masses of P-wave mesons. The four P-states are degenerate. The degeneracy between j = 3/2 and j = 1/2 states is removed by the spin-orbit coupling term S,.L in the Hamiltonian Hq given in Eq.(l). Thus we have using Eq. (99): (9.102a) where
5m mj=3/2
=
. +3mDl D2
8
(9.102b) (9.102~) (9.102d)
x
(subscript 1 on refers to 1 = 1 state). The degeneracy between the doublet D$and D1 and the doublet 0; and DOis removed by
Heavy Quark Effective Theory (HQET)
312
the term 0Q.B"in the Hamiltonian (65). For P-wave this term induces the color magnetic moment, interaction of the type
sl2= [12 (S,
+
n) (SQ- n) - 4S, . S Q ]
(9.103)
where n is a unit vector f . Then using the angular wave functions for the states D;, D1, 07, and Do given in Eqs. (101) and (102), we have
2 (s12)D;
=
4 3'
-
(Sl2)0,- 3
(9.104a)
( S ~ Z= ) ~-4. ,
(9.104b)
Hence we can write the masses for these states. We write explicitly the mass formulae for 0; and D1 mesons.
where the parameters til and dl refer to P-state similar to 6 and d for S-state. For m D ; * and moo,, replace $ X I q by - x l g , d~( D ) by 2d, ( D ) and -621 ( D ) respectively in Eq. (107). From Eqs. (106) and (107), we obtain
=
-5 (mn; - mD,) *
(9.108)
Needless to say that for b-flavor P-states, replace D by B and m, by mb. Using the experimental values for the masses, we find
The P-wave Heavy Mesons: Mass Spectroscopy
313
m ~ -; mD, = 40 MeV = m D ; , - m ~ ~Thus , . relation (108) is well satisfied. From Eqs. (108),(70),and (109) we obtain (9.109)
m ~ -moo ; = -200 MeV.
(9.111)
Also from Eq. (106)’ we get
(AIS - A i d )
+f
(xis -
x,,> = mD;, - mD;*
=
113 MeV. (9.112)
On the other hand, for the S-states m D s - m D d = (99.2 f 0.50 MeV) implying that XIs = X l d i.e. independent of light flavor. From Eq. (112), we conclude that if we use the same dl for j = 3/2 and j = 1/2 states then m D ; < mD,. If as expected m D ; > moo, then d1 ( j = 1/2) must have opposite sign to that of 21 ( j = 3/2); in that case there is no reason to believe that they have the same magnitude also. It is hard to imagine that the interaction which removes the degeneracy between j = 3/2 multiplet and j = 1/2 multiplets is strongly dependent on their j values. However, when we take into account the relative motion of heavy quark it is not reasonable to neglect the spin orbit coupling We now take this term into account. Then from the wave functions given in Eqs. (101) and (102) we find
6.
(S-L),;
= 1,
(S
*
1 3
L ) D 1 = --
(9.113)
r)
(S
*
L)D, =
-2.
(9.114)
Thus the contribution of this term can be written respectively for D,*, D1,Q, and Do as: (9.115)
Heavy Quark Effective Theory (HQET)
314
Hence we get
mo; -mD,
=
mDi -moo =
5 2mc 8- dl [I 2m,
+
i:].
(9.116) (9.117)
If we put, z1 = 4dl, i.e. the same strength for the tensor interaction and the spin orbit interaction, we obtain
m q -mo,
32 d i ( D ) 5 2mc
= -___
(9.118) Hence (9.119) Therefore mDi - mD,= 100 MeV
(9.120)
and (9.121) There is no experimental evidence for D;, and Do . Since they are broad resonances, it is not, easy to test Eq. (120) or (121). However if the spin orbit interaction for the relative motion is not taken into account and tensor interaction is not strongly dependent on the j -values, then as we have seen moo-mD; = 200 MeV; hence Do can decay into 0; by emission of the pion and this decay is a P-wave decay and can be distinguished from the S-wave decay Do -+ DT. This is an interesting possibility which can be tested experiment ally.
Decays of P-wave Mesons
9.5
315
Decays of P-wave Mesons
We now discuss the strong decays of P-wave mesons. Parity and angular momentum conservation restricts these decays to the following modes: Dg + (Dn),=,, D; --t (D*n),=,, DI, 0; -+ ( D * X ) ~ = ~ , ~ , and Do t ( D T ) , , ~ .Note that D1,0; 3 D n is forbidden due to parity conservation. It is convenient to express the decay width in terms of the helicity amplitudes (see Eq. 4.41) (9.122) In the rest frame of the decaying particle the helicity amplitudes which contribute are FO,F:, FJ,F;, and F;. In the heavy quark limit the helicity amplitudes are related as follows: j = 3/2 multiplet: n
(9.123) j = 1/2 multiplet: p10 -
-p1f l --po0 .
(9.124)
The simplest way to see this is as follows. The emission of pion by DJ would not affect the velocity of heavy quark. Thus it is the operator S, n which is relevant for these decays. If we select the direction of quantization along z-axis, (i.e. L, is taken along z-axis) then for the helicity amplitudes, the operator S 3 , e Y10 contributes. Then using the wave functions in Eqs. (101) and (102), it is easy to derive Eqs. (124) and (125) by considering the matrix elements of the type
F:
=
f (D*( D )
Y
IS3qY101DJ, A>
(9.125)
where f is the reduced amplitude. Since hadronic decays of Da are pure D-wave, it follows that D1 ---t D*n is also D-wave. For these
Heavy Quark Effective Theory (HQET)
316
decays, therefore F i (s) lps12. Similarly since the decay of DOis pure S-wave, it follows that the decay of 0; is also pure S-wave. The above restrictions are conseqiience of relations (124) and (125) which hold in the heavy quark spin symmetry limit. Hence for the decays Da 4 D x ,D; -+ D*x, and D1 + D*T , we get N
5 IP?rl;mr
M
1.1
(9.127a)
IPTlLr
5 (phase space) 3 -
M
0.44
(9.127b)
where we have used from the experimental data, IpTIDs = 503 MeV, IpnlD*T= 387 MeV, IpslDID.T = 355 MeV. In Eq. (127), the number in parenthesis is experimental value. Thus we see that this prediction of heavy quark spin symmetry is well satisfied. Experimnetally
rD; = r (D; + D*T-+ D*# + DT- + D%O) =
(9.128)
23f5MeV.
From Eq. (128) we get 1 rD 0.31 MeV
(9.129)
rD;
which gives
-
~ D -I
7.1 f 1.5 MeV (18.9+!:: MeV) .
(9.130)
This is in complete disagreement with the experimental value. This shows that the decay D1 + D*x is not pure D-wave; there may be a component, of S-wave. The S-wave widths are usually large,
Decays of P-wave Mesons
317
a small component of S-wave may be possible due to symmetry breaking, since heavy quark spin symmetry is not exact. This may be tested for B,*and B1 decays where the symmetry breaking effects are expected to be small. The decays 0 7 -+ D*x and Do t D n are S-wave decays; thus the decay widths are expected to be large i.e. in the range of few hundreds of MeV. No experimental data are available even on the masses of 0;and DO. To sum up: from the analysis of mass spectrum one cannot conclude that heavy quark spin symmetry is well established; additional experimental data on the masses of heavy hadrons are needed, Except for one prediction on the decay widths viz.
which agrees with the experimental values, the other predictions on the decay widths of heavy hadrons have to wait fot their verification till the experimental data are available.
Heavy Quark Effective Theory (HQET)
318
9.6
Bibliography 1. H. Georgi,”Heavy Quark Effective Field Theory” in Proc. Theoretical Advanced Study Institute (1991) editors R.K. Ellis, C.T. Hill, and J.D. Lykken (World Scientific Singapore, 1992). 2. Riazuddin and Fayyazuddin ”Heavy Quark Spin Symmetry” in Salamfest, eds. A. Ali, J. Ellis and S. Randjbar-Daemi, World Scientific. 3. Mark B Wise ”Heavy Flavor Theory: overview” in AIP Conference Proceedings 302, editors P. Drell and D. Rubin (AIP Press, 1993) 4. M Neubert, Phys. Rep. 245, 259 (1994); Int. J. Mod. Phys. A l l , 4173 (1996). 5 . M Neubert,”B decays and the Heavy Quark Expansion” CERNTH/97-24, hep-ph/9702375, to appear in the second edition of Heavy Flavors, edited by A.J. Buras and M. Linder (World Scientific Singapore) 6. Fayyazuddin and Riazuddin, Phys. Rev. 48, 2224(1993); Mod. Phys. Lett. A, 12, 1791(1991). 7. Particle Data Group, The European Physical Journal C3, 1-4 (1998).
The references to t h e original literature can be found in the above reviews.
Chapter 10 NEUTRINO 10.1 Introduction Experimental puzzles in the past have led to some important discoveries in Physics. Neutrino, which has spin 1/2, was invented in 1930 by Pauli as the explanation of such a puzzle, namely the conservation of angular momentum and energy in P-decay
n+p+e-, require such a particle, so that
n -+ p
+ e- + V e .
(10.1)
Its direct observation was made much later. The electron type anti-neutrinos are thus produced by the decay of pile neutrons in a fission reactor. These can be captured in hydrogen giving the reaction: Ye
+p
+
e+
+
TL,
(10.2)
whose cross-section was measured by Reines and Cowan gezp
= (11 f 2.5)
1 0 - 4 4 ~ ~ ~
(10.3)
to be compared with the theoretical value nth =
(11
1.6)
10-44Cm2.
(10.4)
Note the extreme smallness of the cross-section. It is a reflection of the fact that neutrino has only weak interaction. 319
320
Neutrino
10.2 Mass The question of neutrino mass is one of long standing. In the context of the standard model of unified electro-weak interactions (Chap. 13), there is no understanding of the origin of masses of elementary fermions. In this category the question of neutrino mass also arises. It has an added importance for the following reasons: (a) Among the elementary fermions, only the neutrinos occur asymmetrically in one (LH, left handed) helicity state (see Sec. 11.1.1) i.e. appear to be spinning clockwise as viewed by an observer. This is still an unsolved puzzle. This fact together with lepton number conservation imply that rn, = 0 (see below). However, there is no local gauge symmetry to guarantee the masslessness of neutrino and lepton number conservation in contrast to the photon where both the masslessness of photon and charge conservation are consequences of local gauge invariance of Maxwell’s equations. One may thus expect a finite mass for neutrino. But the intriguing question is why m(v,)
These neutrinos are detected through the reactions vp+n --+ p-+p, Vu + p + p + + n and v, n --+ e- + p , fie + p 3 e+ + n and are
+
Neutrino Oscillations
337
respectively called p-like and e-like events. One would expect the ratio
However this ratio has been measured in several detectors and it, is found that [MC denotes the Monte Carlo Simulated ratio]
R r ( N ( v p ) / N ( v e ) ) 06s < 1 ( N (vp)/ N ( v e ) ) M c and that it depends on the zenith angle as well, implying neutrino oscillations. The latest atmospheric v data is summarized below:
R
0.63 f 0.03 f 0.05 (Super-Kamiokande sub-GeV) R = 0.65 f 0.05 f 0.08 (Super-Kamiokande multi-GeV) =
The zenith angle distribution of R is shown in Fig. 6. One would not expect up/down asymmetry i.e. between the number of events arising from the neutrinos coming from below the earth and going upward through its center to the detector and those arising from neutrinos coming from up, since we are in a “spherical shell of v’s”. However, for multi GeV one finds for this asymmetry: up/down (e - like) = 0.93 f 0.1 f 0.02 up/down ( p - like) = 0.54-0:0,. +O 06
(10.65)
The former is consistent with no oscillations. The latter is a 6a discrepancy. The result (65) is consistent with v p 1--f v, oscillations which would imply that the former ratio to be unity. The conversion probability P,,-,, as given in Eq. (63) fits the data quit,e well for Am2 = 2.5 x lop3eV2, sin228 = 1.0.
(10.66)
Several long - base line neutrino oscillation experiments that will allow an investigation of the atmospheric neutrino range of Am2 to other channels are at present under preparation.
338
Neutrino
F .!
......., ' ..a.m.
;.:.
'
'
'.A' '1,;' '
( sin'Zd. ~ m ) '
( I
.o. 5x
.
' ' ,012' ' ' 0 : '
10-1
'01.'
)
'
.A'
cxpected zenith angle distribution
-
,A
Super Kimiokondr Prdimino
Sub-GeV
:
Y'J
5, Lf Z6*
1
0.6
-
0.4
i +
0.1
-
O . ,
'
'OS'
I
i.
rj---T-- ....
~
( s'n'2fl. A m ' )
........ -0..
.
I
-0.4
.o.,
0
( 1 .o. 5 0.3
0 4
x lo-' 0.8
) 0"
Figure 6 Zenith-angle distribution of R with neutrino oscillations parameters corresponding t o the best fit values to the Super-Kamiokande data.
( c ) Solar neutrinos
Electron type antineutrinos are produced by the decay of pile neutrons in a fission reactor: R 3 p + e- V , , Electron type neutrinos
+
Neutrino Oscillations
339
are; on the other hand, produced from reactions in the sun, called solar neutrinos. The energy of the sun is generated in the reactions of pp and CNO cycles. Energy is generated through nuclear burning involving the transitions of four protons into 4 H e : 4p + 4 H e
+ 2e+ + 2 ~ +e Q
(10.67)
where Q = 26.7 MeV is the energy release in the above transition. Thus the generation of the energy of the sun is accompanied by the emission of ue's. The total flux of the neutrinos is connect,ed to the luminosity of the sun La by the relation: (10.68) where R is the sun-earth distance, aiis the total flux of neutrinos from the source i, and Ei is the average energy. The most important sources of solar neutrinos in the p p cycle, which dominates cooler stars, particularly the sun, are the following reactions: p p --+ 2He+ve ppe- + 2Hue 7 B e e - ---t 7Liue ' B + 'B,*e+ue
E, < 0.42 MeV : E, = 1.442 MeV : E, 0.86 MeV : E, < 15 MeV :
N
On the other hand, the CNO cycle dominates hot stars and following reactions are sources of ue's:
13N
--+
l50
+ "Ne+ve .
13Cefue
The first reaction in the pp cycle is the main source of solar neutrinos. The third reaction is a source of monochromatic neutrinos. This reaction contributes about 10% to the total flux of solar neutrinos, The fourth reaction contributes only about to the
Neutrino
340
Table 10.1 The standard solar model predictions of neutrino fluxes and observed rates.
Homestake Eth( M e V )
Mode Sensitive to Observed rate
BPSSM (Expected) Ratio
PNUI 0.814 v, +37 c1 + e- +37 Ar ‘ B v ’ s ( 90%) ~
Kamiokande [106crn-ls-] 9.3, 7.5 and 7.0
SAGE and GALLEX [SNU
‘ B v’s
all 3 sources
0.232
but also to 7B, u’s
2.54 f0.20 9.3:;:: 0.273 f 0.03
2.89 f 0.42 +O 25 2.45 f 0.06-0:09 Super-Kamiokande 6.62 f 1.06 0.42 f 0.07 0.368 f O.Ol?::!g Super-Kamiokande
70.3 f7.0 1372;
0.51 f 0.06
interactions per target atom per sec BPSSM: Bahcall-Pinsonneault, Phys. Rev. Lett. 78 (1997) 67. 1SNU
=
total flux but it is the main source of high energy solar neutrinos (up to 15 MeV). Due to different, detection thresholds, solar neutrinos from different sources can be detected in different, reactions. Thus the solar neutrinos with energy > 0.814 MeV can be detected in 37CZ and those > 0.233 MeV in 71Ga. A discrepancy exists between the standard solar model (SSM) predictions of neutrino fluxes and rates observed in terrestial experiments as shown Table 1. We see from this table that in all experiments the observed event rate is
Possible Particle Physics Solutions of Solar Neutrino Problem
341
significantly smaller than the rate predicted by the standard solar model. We may thus conclude that solar neutrinos are detected, thereby establishing the solar fusion. That the observed event rate for solar neutrinos production is smaller than the predicted rate provides a cricumstantial evidence for new physics as will be discussed in the next section.
10.4 Possible Particle Physics Solutions of Solar Neutrino Problem If the experiments are correct, it is very unlikely that non-standard solar models can fit the solar neutrino data. However, there are possible particle-physics solutions, some of which are listed below: (i) Vacuum oscillations (involving 2 or 3
Y’S)
(ii) Matter induced oscillations (involving 2 or 3 v’s) (iii) Sterile neutrino (iv) Magnetic moment transitions Magnetic moment transitions need large neutrino magnetic moment which surpass upper limits on them from astrophysics [see Sec. 51. The possibilities (i) to (iii) involve new physics (nonstandard neutrino properties) in terms of modest extension of the standard electroweak theory in which neutrinos have small masses and lepton flavor is not conserved leading to neutrino oscillations. 10.4.1 Vacuum oscillations Vacuum oscillations of v, to v, give the survival probability : (10.69) where R is the earth-sun distance (cx 1.5 x lOl3crn) while T gives the production location. The results of a fit including all the latest data is shown in Fig. 7. The best fit gives the oscillation parameters
Am2 = 6.0 x 10-11eV2, sin226 = 0.96.
(10.70)
342
Neutrino
(99%) C.L.Allowed Regions
10-10
3 -
n
5PI .
v
2a -
-
!E5lZXl ClAr
+
Kamloksnde
4 experlments
+
+
SAGE
+
GALLEX
Super-Karnlokende
(exp/SSM)
-0.396 f 0.039--
SSM: Bahcall and Pisonneault 1995
Figure 7 Region in the Am2 vs sin228 plane for the vacuum solution in the solar neutrino problem
Possible Particle Physics Solutions of Solar Neutrino Problem
343
10.4.2 Possible explanation in terms of resonant matter oscilla-
tions: Makheyev-Smirnov- Wolfenstein [MS W] efect First we write the Hamiltonian in ve, u, basis [z = p or r or s(steri1e v)] : H,(lc) where H is diagonal in
basis (cf. Sec. 3):
(
sin8
U= M
(10.71)
v1 - v2
and
while Ei
= UHU-’
cos8 -sin0
(10.73)
C O S ~
22k 1 - - Then k + m7 and E2 - El = m2-m2
H,(k)= const.
( y ) +(
-Am2 cos 28 Am’2in28 4k
Am2 sin 28 4k
0
)
(10.74)
where the first part of Eq. (74) is irrelevant for oscillations. Now in traversing matter, neutrinos interact with electrons and nucleons of intervening material and their forward coherent ‘scattering induces an effective potential energy. Such contributions of weak interaction in matter to H , arise due to Feynman diagrams shown in Fig. 8. The first diagram contributes equally to ve, vp and v, and as such is not relevant vp tt vp or v, oscillations. This gives the effective Hamiltonian[see Chap. 131: (10.75) where f = e-, p or n for which respectively I s L = -f, f , -f and Q = -1, 1, 0. The second diagram after Fierz rearrangement gives the effective Hamiltonian: (10.76)
Neutrino
344
I
I
I
I
I
I
; zo
; zo
; w-
I
I
I
Figure 8 Feynman diagrams for neutral current (n.c.) and charged current ( c x ) weak interactions which contribute t o H , for oscillations in matter
where Qe denotes the state of the medium. These diagrams give the potential energy
(10.77)
v,s= 0 where ne denotes the number of electrons per unit volume and n, that of neutrons. Then the Hamiltonian in the matter is [k III E ]
HA4(k)
=
H,(k)+Hw -Am2cQ82e
+
f
i
~
Am2sin2e 4E
Am2 sin 20
0
4E
where
n = ne for ue t--f up or u, =
1
ne - -n, for v, 2
t--f
Y,
1
~
~ (10.78)
Possible Particle Physics Solutions of Solar Neutrino Problem
345
The diagonalization to vl, u2 basis gives:
( ) ( =
CoseM -sinOM
SineM cosOM
) ( L-’ )
(10.79)
with s i n 2 B ~ = sin28-,l M 111
cos 28M = (cos 28 - A ) 1M -
(10.80)
lV
AE
=
E2-E1=-
1 21M
(10.81)
where
E A = 2 h G ~ n m
lv =
E Am2‘
-
(10.82)
(10.84)
For constant density n, the considerations of Sec. 4 give the conversion probability
(10.85) The following are useful limits:
(10.86)
A E = 2fiGFn 7r 8 M = -, u, = v2, u, 2
= U]
Neutrino
346
(iii)
n = nqe, defined by JZG,(n,),,, = 1M
HA4
-+
E Am2sin 28 0
f
(
Am2cos 20 2E
'
Am2 sin 28
Am2sin28 4E
0
(10.87)
Using the above limits, the plot, of E versus n is shown in Fig. 9. Suppose v, is created a t n o > nressay at, the center of the sun, and then it propagates out. If there is no level crossing (shown by dotted lines in Fig.9, then v(n = 0) N v, and undetectable. This conversion of v, into v, is the cause of the depletion of observable neutrinos. Now neutrinos of any energy will not go through the resonance. The resonance condition for any given neutrino energy E is:
n,resE= cos 28
Am2
(10.88)
~&GF '
We may remark here that for ve -+ v p or v, conversion,
(10.89) where Y denotes the number of electrons per nucleon and is 1/2 for ordinary matter. Then, the resonance condition (88) can be written as
=
1
1.3 x 107g/cc- cos28 2Y
[m](7). Am2
MeV
(10.90)
Evolution of Flavor Eigenstates in Matter
347
Ve
= V,
Figure 9 Plot of neutrino energy E versus density n , showing conversion of u, t o ux in matter
For pres 2 p (center of sun) = lOOg/cc, we have
Am2 (&)I1.3 x 105 (eV)2
(10.91)
’
Thus, for example, for Am2 2 6 x 10-6eV2, we will not, have 0.4MeV and the resonance will be at least at resonance for E I E = 0.8 MeV. In this case the resonance will not affect p p neutrino for which Emax= 0.44 MeV but can eliminate 7B neutrinos.
10.5 Evolution of Flavor Eigenstates in Matter The evolution of flavor eigenstates in matter is governed by the equation: (10.92)
Neutrino
348
where H ( z ) is given in Eq.(78). Note that the z dependence arises due to the z dependence of the density n, for varying density case. Using
(10.93) with
U ( X )=
cos e(x) -sinO(x)
sin e(x) COS~(X)
(10.94)
we have
(10.95) where
-
+'' ( 01 2
0
1
) + ( -74)
(10.96)
and 1 0
(10.97)
Noting that, the first part of Eq. (96) is irrelevant, for oscillations and using Eq. (81) we have
For the constant density case, O h (x) = 0 and ZM is independent of x, so that Eq. (98) has simple solutions
(10.99)
349
Evolution of Flavor Eigenstates in Matter
where we have taken z = 0 as the initial point. Then Eq.(93) gives
where we have used the boundary condition v, (0) = 0 [cf. Eq. (79)].Then the electron neutrino survival probability averaged over the detector position L (from the solar surface) is given by
P (ve --+ ve)
+
cos' ev cos' O& sin' ev sin' 1 1 = - - cos2oV cos 2 e ~ 2 2 =
OL
+
(10.101)
where t9v = 0 is the vacuum mixing angle. In general when the density n is a function of x one has to solve Eq. (98) and as a result P(v, --f ve) is given by the Parke formula:
where Bh is the initial mixing angle and Pj Landau-Zener formula. Here
_=
exp (-;y)
is the
-1
and is called the adiabaticity. In the adiabataic limit y >> 1 and Pj--f 0 and we recover the relation (101). The survival probability P ( y e + ve) as a function of E, is displayed for large and small mixing solutions in Fig.10.
350
Neutrino
Large Mixing Solution n
2 t 0.5
s
v
CL 0
1n-'
1
10
lo2
Small Mixing Solution
1.0
f
i
I
1
I I 1111
I I 1
IIll
I
1 1
lllfr
I
f 0.5
s
-.
U
L
I
n
"lo-'
I
I
1 I1111
10"
1
EV (MeV)
Figure 10 Survival probability as a function of small-angle solution
10
lo2
E, for large angle and
Evolution of Flavor Eigenstates in Matter
35 1
( 9 9 % ) C.L. Allowed Regions
ClAr
+ Kamlokande
t GALLEX t SAGE
SSM: Bahcall and PInsonneault 1
stnR(28) Figure 11 99% C.L. allowed regions in the Am2 - sin228 plane for the MSW solution t o the solar neutrino problem
Neutrino
352
We now outline the standard analysis. First determine from Kamiokande the flux of v, from 8 B that reaches the earth. Then one can understand C1 and Ga results with 85
N
78
N
0.40 S S M 0.00 S S M
pp
N
SSM
for the small mixing solution (see Fig.10). One can thus conclude that there exist neutrino oscillations that almost, totally convert, 75, v,’s into u, that, have little effect, on p p v,’s.This leads to the solution shown in Figs.11 and 12, The best-fit, solutions are 1) small-angle MSW:Active Sterile Arn2(eV2) = 5.4 x 3.5 x sin228 = 7.9 x 1 0 - ~ 2) large-angle MSW:
Arn2(eV2) = 1.7 x sin228 = 0.69 3) Vacuum oscillations [cf. Fig. 71:
Am2(eV2) = 6.0 x lo-” sin228 = 0.96 The above different interpretations may be distinguished by new experiments: Super-Kamiokande, SNO, GNO and Borexino. Solar model independent tests of the oscillations may then be feasible. We may mention here that no evidence for “Day-Night” effect
D-N -D-N
-
-0.023 f 0.020 & 0.014
Evolution of Flavor Eigenstates in Matter
353
(99%) C.L. Allowed Regtons I
I
1
I
=
ClAr
+
I
I
I 1 1 1 1
Kamiokande
4 experiments
+
GALLEX
+
1
1
8
1
SAGE
+ Super-Kamiokande 0.400*0.024 (200 days)
SSM: Bahcall and Pinsonneault 1995 10-7
I
I
I I , ,
I
I
I
I I l l
I
1
I
I
i
I I I I
100
sin'( 28) Figure 12 99% C.L. allowed regions in the Am2 - sin228 MSW solution with ye - v, conversion for the solar neutrino problem
354
Neutrino
-3
.
9 - 4
-
N
2
v
w
E
-
0 0
-5 -
-yt -8
~
-4
~
~
~
~
~
~
~
-3.5 -3 -2.5
~
"
-2
~
~
-1.5
~
~
-1'
1
'
"
'
'
-0.5, 0 log(sln 20)
'
"
~
Figure 13 99% C.L. excluded regions found from Day-Night effect for the MSW solution
'
~
"
'
~
Neutrino Magnetic Moment
355
found in Super-Kamiokande has already excluded the heart of the large-angle mixing solution (see Fig.13) In summary it appears that (10.103)
Phenomenological analyses of neutrino oscillations find it difficult to accommodate the above heirarchy of mass ranges in a threegeneration picture unless one of the experiments is sacrificed. For example, if we ignore LSND experiment, a possible solution is sin228, sin228,,
N N
1, Am;, CY 5 x Am:, N 5 x
(10.104)
consistent with the mass pattern m(vT)>> m(v,) 2 m(v,). Some suggest the remedy by introducing a fourth sterile neutrino, which may however, be disfavored by big bang nucleosynthesis (see Chap. 18). We may conclude that neutrinos have masses, neutrinos mix and oscillate, mixing angles are small, solar neutrinos are detected, and the solar fusion is established. None of the above has been convincingly proven. Nevertheless we can say that the neutrino physics provides a circumstantial evidence for physics beyond the standard model. New experiments will test new physics and establish new mass scale(s) indicative of it.
10.6 Neutrino Magnetic Moment With the definition (10.105) where /.LB is Bohr Magneton, magnetic moment interaction is
Hmag= ,U”U B. 9
(10.106)
Neutrino
356
Figure 14 The conversion of VL into V R in the solar magnetic field.
Here B is the solar magnetic field. The neutrino spin would then process in the magnetic field, some left, handed (LH) neutrinos would become RH and sterile to t,he detector as shown in Fig. 14. The conversion probability is determined by
(10.107) Now the solar magnetic field in the convective zone of thickness = (1 - 5) x lo3 gauss, so that the conversion probability is
L x 2 x 108m is B
2 x loam 4 5 . 7 9 x lO-'eV/G)(l - 5) x 103G [3 x 108m/s][6.6x 10-16eV.s] M
~ ( 0 .6 3)1010.
(10.108)
This is O( 1) if K = (0.3- 1)x 10-l' giving p, x (0.3- 1) x 10-lopB. In the standard model,
(10.109) i.e.
357
Neutrino Magnetic Moment
So if p, M 10-lop~, this would definitely indicate physics beyond the standard model. Thus the question of dipole moment of neutrino is very important. What are the other limits on it? The best laboratory limit on m, comes from reactor experiments. In addition to the usual electroweak scattering via W* and Zo bosons exchange, the process Ve
+e
--+
Ve
+e
could proceed via magnetic scattering which is large in the forward direction and for small E,. Consistency with measured crosssection requires (10.111)
More stringent limits have, however, been quoted from astrophysics: (1) Nucleosynthesis in the Early Universe Presence of p, mediates vLe- -+ vRe- scattering. If this occurs frequently in the era before the decoupling of the neutrinos, it doubles the neutrino species and increases the expansion rate of the universe, causing overabundance of helium. To avoid this, pI/ < 8.5 x 1 0 - l ' ~ ~ .
(10.112)
(2) Stellar Cooling Magnetic scattering of neutrinos produced in thermonuclear reactions may occur, flipping the helicity [VL 3 v ~ so ] that the outer regions of the star will no longer be opaque to neutrinos and cooling will proceed much faster. Applied to helium burning star in order that
where Eexotic denotes energy loss due to process of the above types while EH, denotes energy generation rate. This gives (10.113)
Neutrino
358
(3) Limit on pu from Supernova 1987A Neutrinos produced in the initial collapse state have high energies 100 MeV. These high energy neutrinos could escape following spin-flip magnetic scattering [ V L -+ VR]. Furthermore, a proportion can process back V R -+ V L in the galactic magnetic fields and the result on earth could be a signal of high energy (100 MeV) neutrino interactions in the underground detector with a high rate [note that o E2 in V , + p + e+ + n].The observance of no signal implies N
-
(10.114) In view of the above upper limits on p,,, the neutrino spin precession mechanism does not appear to be a viable solution to the solar neutrino problem.
Bibliography
359
10.7 Bibliography 1. T.D. Lee and C. S. Wu, Weak Interactions, Ann. Rev. Nucl. Sci. 15, 381 (1965). 2. R. E. Marshak, Riazuddin and C. P. Ryan, Theory of Weak Interaction in Paticle Physics, Wiley-Interscience, New York (1969). 3. Weak Interaction as Probes of Unification (VPI-1980) AIP Conference Proceedings No. 72 [Editors G. B. Collins, L. N. Chang and J. R. Ficene], AIP, New York (1981), see in particular parts IA and IIA. 4. F. Boehm and P. Vogel, Physics of Heavy Neutrinos, Cambridge Univ. Press, Cambridge, U.K. (1987). 5. R. Eicher, Nucl. Phys. B (Proc. Supp.) 3, 389 (1988). 6. Y. Totsuka, Non Accelerator Particle Physics, In Proc. of XXIV International Conference on High Energy Physics, (Editors: R. Kotthaus and J. H. Kuhn), Springer-Verlag, Heidelberg (1989), p. 282. 7. H. Daniel, Review of Trituim Experiments, in Proc. of XXIV International Conference on High Energy Physics, (Editors: R. Kotthaus and J. H. Kuhn), Springer-Verlag, Heidelberg (1989), p. 1058. 8. J. N. Bahcall and R. K. Ulrich, Rev. Mod. Phys. 60, 217 (1988); see also S. Turck-Chiez et al., Astrophys. J. 335, 415 (1988). 9. R. Davis, Jr., A. K. Mann and L. Wolfenstein, Ann. Rev. Nucl. Parti. Sci. 39, 467 (1989). 10. T. K. Huo and J. Pantalone, Rev. Mod. Phys. 61, 937 (1989). 11. J. N. Bahcall, Neutrino Astrophysics, Cambridge University Press, Cambridge, England, 1989 12. R. Kolb and M. Turner, The Early Universe, Addison and Wesley, California, 1990 13. N. Hata, Lectures delivered at BCSPIN; N. Hata and P. Langacker, Phys. Rev. D56,6107 (1997). 14. J Bahcall, Neutrinos from the Sun, Proc. of the XXV SLAC Summer Institute on Particle Physics, Aug., 4 - 15, 1997, SLACR-528, edited by A. Breaux, J. Chan, L. De Porcel and L. Dixen;
360
Neutrino
Y . Itow, Results from Super Kamiokando, ibid. 15. S. M. Bilanky, Neutrinos, Past, present, future hep-ph/9710251 16. A Balantekin, exact solutions for matter enhanced neutrino oscillation, hep - ph/9712304 17. S. Pakvasa, Neutrinos, hep-ph/9804426, Lectures delivered at the ICTP Summer School on High Energy Physics and Cosmology, June 16 - 20, 1997. 18. Particle Data Group, C. Caso et al, The European Physical Journal C 3 , 1 (1998). 19. M. T. Osaka, Recent results from SuperKamiokande, Repartuer’s talk at 29th International Conference on High Energy Physics, 23-29July 1998, Vancouver, Canada; J. Conrad, Neutrino oscillations, ibid.
Chapter 11 WEAK INTERACTIONS 11.1
V - A Interaction
In analogy with electromagnetic interaction JpAp, Fermi proposed for P-decay the interaction P J F , viz.
+ h.c.
Hint = G [ a i ( ~ ) ~ p Q 2 ( [*~(x)Y’*~(x)] ~ ) ]
(11.1)
The above interaction is for the process 241
+ 3 + ;I; (e.9. n,+ p + e- + ve).
The interaction (1) can be generalized using five Dirac bilinear co-variants. Thus the most general non-derivative foiir-fermion interaction can be written as Hint
=
c
[%(X)riQ2(X)]
[*3(2)ri(cz - C;Y5)%(X)]
a
+h.c.
(11.2)
where ri(i = S, V, T , A , P ) are the five Dirac independent matrices: 1, y p , opV,’yp75, 75. In writing Eq. (2), we have taken into account the parity violation in @-decay. is a solution of Dirac For a massless Dirac particle, if equation, then f y , Q is also its solution. WitJhout,loss of generality, we take only negative sign. Suppose particle 4 is massless, then the bilinear
*
G3(qi% (x)-+ - a3(X)r275~4 (.). 361
Weak Interactions
362
Hence for this case Ci= Ci. Thus we can write Eq. (2) as
If we identify particle 4 with the neutrino, we have the result that only left handed neutrino takes part in weak processes. This is what is observed experimentally (see below). Thus irrespective of the fact whether neutrino is massless or not, Eq. (3) will hold if we take into account the fact, that only left handed neutrinos take part in weak processes. Suppose we impose the chiral transformation for the field a3 viz. @ 3 + -y5@3, then if Hint is to be invariant under such a transformation, we have
cs = cp = CT
= 0.
Hence Eq. (3) becomes
where we put
(11.5) Further we note that if we impose trhe chiral transformation on fields or @ 2 , we have CV
=CA
(11.6)
i.e. E = 1 or V - A theory. We conclude that if one requires invariance of the four-fermion interaction under the chirality transformation of each field separately, we have the V - A theory. We have written Eq. (2) in the order 1 2 3 4. We can go to the order 3 2 1 4 by Fierz reordering theorem: 5
Ki(3214) =
C XijKj(1234). j=l
(11.7)
V - A Interaction
363
The coefficients X i j are given by the matrix
X..23 = --
1 1 4 -2
1 0
1 2
1
-4 (11.8)
where
(11.9) It is obvious that
Ki(3214) = Ki(1432). If we denote by S, V, T , A, P the five quadrilinears appearing in the order (1 2 3 4) and S', V', TI, A', P' when they appear in the order (3 2 1 4), then from Eqs. (7) and (8) we get
V'-A' S'-T'+P'
= =
V-A S-T+P
(11.10)
i.e. these combinations are invariant under Fierz rearrangement. 11.1.1 Helacity of the neutrino To obtain a direct measurement of neutrino helicity, the following reaction was studied .
1 5 2 ~ ~ ( J p = o - ) -+
+
1 5 2 ~ m i , p = , - )V,
1 --+
(152~m(JP=0+) +7) .
The main point of this experiment is that we can select those y rays from the decay of the excited state which go opposite to the v, direction (i.e., in the direction of the recoil nucleus) by having them resonance-scatter from a target of 152Sm.Balancing the spin
Weak Interactions
364
along the upward z direction (v, is assumed to be emitted along this direction), one finds that, the helicity of the downward y-ray will be the same as that for the upward v,. By measuring the circular polarization of y-ray, the experiment, fixed the helicit,y of the y-ray as negat,ive, indicating a lefbhanded v,. Thus it, is established that, only left-handed neiit,rinos take part in weak processes. 11.2
Classification of Weak Processes
(i) Purely leptonic processes The well known example is p-meson decay
In this process four well known particles p-, e - , v,, v,, called leptons, take part. The decay process is described by V- A interaction [cf. Eqs. (4)arid (6)].
c1
(11.1la)
Lr,) and L(,), are lepton currents associated respectively with p meson and its associated neutrino v, and e- and v, LYp) = fi,,YP(1 - 7 5 h L(e)p = Veyp(1 - y5)e.
(11.11b) (11.1lc)
The 7 , and 7 5 ( 2iy0y1y2y3)appearing above are the usual Dirac matrices. We write the lepton current as
L,
=
L’(”,)+ Lye).
(11.12)
Here L f denotes the hermitian conjugate of L,. One can also picture the process (1) as being mediated by a vector boson W,, the so-called weak vector boson. This is shown in Fig. 1 below:
Classification of Weak Processes
365
Figure 1 The muon decay.mediated by a W-boson.
Figure 2
Electromagnetic interaction mediated by a photon
Thus all leptonic weak processes can be described by interaction of the form
where h.c. denotes the hermitian conjugate. Note that Eq. (13) is analogous to electromagnetic interaction of say electron which is mediated by photon and is shown in Fig. 2. The interaction responsible for the process shown in Fig. 2 is the usual electromagnetic interaction
Weak Interactions
366
where a, is the photon field and
is the electromagnetic current:
(11.15)
jFm,= ey,e.
Note the similarity between Eqs. (13), (14), (llb,c) and (15) respectively. Both the electromagnetic and weak currents are vector in character, the appearance of y5 in weak current is due to the fact that parity is not conserved in weak interaction, in fact it, is violated maximally. The coupling of electromagnetic current with the photon is characterized by electric charge (related to the fine structure constant Q by $ = Q = 1/137) while that of weak current with the weak vector boson field W, is characterized by gw (related to the Fermi coupling constant, Gp by =
$
a).
(ii) Semileptonic Processes Some examples of these processes are given below
n + T+
7
-
r
ccco K+ K-
--+
+ -+
+ -+ --f
-+
From these processes, one notes the following rules: 1. The hadronic charge changes by one unit i.e. AQ = fl. 2. In the first four processes, strangeness does not change, in the last four processes it changes by one unit.
367
Classification of Weak Processes
For hadrons, Gell-Mann-Nishijima relation
Y Q=13+-
2
implies that for AQ = f l , either A13 = f l , AY = 0 or A13 = f 1 / 2 , AY = f l , if we assume that AY = 2 processes are suppressed. The processes of first kind are called hypercharge conserving processes and those of second kind are called hypercharge changing processes. In all the processes listed above, we see that either AY = 0 or AY = f l ; no weak process with lAYl > 1 is seen with the same strength as (AYI 2 1 transitions. Thus we have the selection rule AY = 0, f1, AQ = A Y . Since there are so many hadrons in nature, therefore to deal with semi-leptonic decays of each of them would be very tedious. Thus we use the simple picture of hadrons made up of quarks. The main thing about the quarks is that they are regarded as truly elementary similar to leptons. Their weak and electromagnetic interactions would then be like those of leptons. Thus in analogy with Eqs. (15) and (ll),their electromagnetic and weak currents are respectively
(11.17) while
(11.18) where d‘ = cos B,d
+ sin OCs,s’ = - sin t9,d + cos 0,s.
(11.19)
Here 8, is the Cabibbo angle; its value is t9, = 13” or sine, = 0.22. This is introduced since it is seen experimentally that, decay rates for lAYl = 1 semi-leptonic decays are suppressed by a factor of about 1/16 compared to those for AY = 0 processes. We shall
Weak Interactions
368
Figure 3
Quark level process for neutron P-decay.
deal with s’ in Chap. 13. Then in analogy with Eq. (11) or (13) the interaction responsible for fundamental processes like d -+ s --+
u+e-+D,
(11.20)
u+e-+D,
would be
or
Lw
=
-gw JkW-’”
+ h.c.
(11.21)
In this picture neutron ,&decay, A-P-decay and KO 7r++e-+De, for example, would be pictured as shown in Figs. 3-5. Not,e t,he very important, fact, that, bot,h the 1ept)onic and hadronic weak ciirrents in ( l l b , c) and (18) are charged i.e. they carry one unit of charge and t,he hadronic weak currents (18) satisfy the selection rilles IAY I 5 1 and AQ = AY. We also note that in terms of flavor SU(3) notation we can write --f
Jj
= cos BCJ’,
+ sin 8,Ji
(11.22)
369
Classification of Weak Processes
e-
\>
A
Figure 4
Quark level process for neutron A - ,f3-decay.
-I-
Figure 5
Quark level process for
Eo -+ .rrfe-De
Weak Interactions
370
where weak hadronic current is a linear combination of vector and axial vector currents involving respectively y p and ~~y~and are given by
(11.23a) (11.23b) Note also that (11.24a)
(11.24b)
(11 . 2 4 ~ )
Here q =
(
). The heavy quarks and
s' will be considered in
Chap. 13.
(iii) Non-Leptonic Processes Here no leptons are involved. The well known non-leptonic processes are:
( 11.25a)
cc+ -+ c+
-+
-+
nr-(E:) p7r"(Co+)
nr+(CI)
(11.25b)
Classification of Weak Processes
371
(11.25~) or
KO, KO K*
--t
7r+7r-, 7r07ro
-+ T * , T O ,
etc.
(11.26)
Note that all these decays are strangeness changing ([AS]= 1). Let us concentrate on the decays (25), the so-called non-leptonic decays of hyperons. If we consider the decaying particle in its rest frame, the conservation of angular momentum J gives 1 Jj, s - = Jfin;tl= e + s, 2 where C is the relative orbital angular momentum of the pion and the baryon in final state. Since spin s = l/2, l can be 0 or 1. The pion being pseudoscalar (having odd intrinsic parity), the relative parity of final state with respect to the initial state is
Pf
(-l)'(-l) = -1 odd for C = 0 = (-I)'(--I) = $1 even for = I.
=
The s-wave (l = 0) decays are parity violating while p-wave (C = 1) decays are parity conserving. Accordingly decays (25) are governed by two amplitudes, parity violating (s-wave) and parity conserving (p-wave). We can write the Lagrangian responsible for non-leptonic decays as (h)
Lw
- Lh(P.")+ Lh(P.4 - W W - --Jp GF h J hpt h.~.,
Jz
+
(11.27)
where J," is given in (18). The [AS[= 1component of (27) behaves as --sinO,cosOc{i?yp(l GF
fi
- 7 5 ) ~ -) {21yp(l - 75)d).
(11.28)
Weak Interactions
372
Now u and d belong to isospin doublet, I = 1/2 while s is isospin singlet, I = 0. Thus from the combination of angular momentum rules (isospin behaves like angular moment,iim) first, term in curly brackets in ( 2 8 ) has I = 1/2 while the second term in curly brackets has I = 0, 1. Thus the interaction contains both AI = 1/2 and 3/2 parts. Experimentally A 1 = 1/2 part predominates over A1 = 3/2 and then (25a), (25b) and (25c) respectively get, related among themselves. We shall come to these relations later. 11.2.1 p-decay Consider the p-decay p
-
+ e-
+ vp + 0,.
F'rom Eq. (4),we can write the interaction as
(11.29) The interaction writt,en in this order is called t,he charge retention order. It is easier to deal with this order in calculations. Here we have assumed 2-component neutrinos (left- handed vcLand righthanded V e ) but have allowed V -- E Ainteraction, where for V - A , E = 1 and in that, case by Fierz rearrangement we get, Eq. ( l l a ) . From Eq. (29), we can write the T-matrix for p- decay:
T =
&/Z% x [4P2)YA(l - EY5)u(P1)][U(k2)?(1
- %)?@I)] (11.30)
where pl,p2, Icl and k2 are the four momenta of p - , e-, vp and Ve respectively and u(pl),u(pz), u(k l ) and v( kz) are Dirac spinors. From Eq. (30), we get
x (MI2d3p2d3kld3k2
(11.31)
Classification of Weak Processes
373
where
We can easily calculate ]MI2using the standard trace techniques. Neglecting the neutrino masses, we get
Since neutrinos are not observed, we integrate over d3kld3kz. Performing these integrations, and writing d3p2 = 4 ~ p e E e d E ewe , get
where (11.35)
(11.36)
In evaluating the final result (34), we have gone to the rest frame of the muon:
(11.37)
Weak Interactions
374
11.2.2 Remarks (1) It, is always possible to take CV as real and take CA = E C V (e complex).
(2) The electron spectrum does not, distinguish between +1 (V - A ) or E = -1 (V + A ) interaction.
E =
(3) Any deviation from E = f l can be determined by measuring 77 in the electron spectrum. Since 7 is the coefficient of ( m e ( : y E e , > , it plays a minor role except, at low electron energies, where measurements are difficult,. The best experimental value of 7 is
v = -0.007
f 0.013
(11.38)
which is consistent, with zero.
(4) It is instructive to write the electron energy spectrum (34) as
3(W - E,)
+ 2p
-
W
me + 3v-(W Ee
- -2 3m Ee2 )
- E,)
(11.39) where p = 3/4; p is called the Michel parameter. In fact the most general interaction without assuming two-component neutrinos gives the elect,ron spectrum of the form within the square brackets. The experimental value of p = 0.7518 f 0.0026 is in excellent agreement, with p = 3/4 as given by V - E Atheory. We conclude that the two-component neutrino hypothesis is in an excellent, agreement with the experimental results. Finally integrating Eq. (34), we obtain r=7; 1 = G2 ~ P , (11.40a) where ( 1 1.40b)
Classification of Weak Processes
375
If we include O ( a ) radiative corrections
where the fine structure constant a! = -. The Fermi constant G F determined from ( ~ O C ) , using the experimental value for -rp = 2.19703 x sec, is
GF = 1.16637 x
GeVW2.
(11.41)
Decay of polarized muon We have seen that the electron spectrum cannot determine the sign of E . In order to determine E , we consider the decay of polarized muon. Let n, be the polarization vector of muon. We note that
n,2 = npn,p= -1, n, = 0.
(1 1.42)
In the rest frame of the muon m,n,o = 0; thus no = 0 and (0, n). For this case in taking the trace, we put
R
=
Using the standard trace techniques, and performing the integrations over d3k1d3k2, the differential spectrum in the asymmetry angle for p- decay is
(1 1.44) L
where y is the angle between the electron momentum and the p-spin direction and J=-
2 Re E 1 [ E l 2'
+
(11.45)
376
Weak Interactions
It is instructive to write Eq. (44) in the form
(W - E,)
+ 26
-
W-
I)-
1 m2
3 m,,
, (11.46)
where 6 = 3/4 for two-component, neutrinos viz. for V - E A theory. For a general interaction without, assuming two-component, neutrino, the asymmetry distribution in angle y is of the form given within the square brackets. The experimental value of 6 is 0.749 f0.004 in excellent, agreement, with t,wo-component,neutrino hypothesis. The experimental value of E is given by
[PI = 1.003 f 0.008.
(11.47)
11.2.3 Semi-leptonic processes For a semi-leptonic weak process we can write the interaction Hamiltonian as [cf. Eq. (21)].
-Lb’
=
H.
-
-
GF
-JA
Jz
(x)[ E ( x ) yA( 1 - 75) V , (x)]+ h . ~(11.48) .
To first order in weak int,eraction, the T-matrix for a semi-leptonic process of the type
is given by
(11.49) where Ic‘ and Ic are four momenta of electron and antineutrino. We denote four momenta of A and B by p and p’.
Classification of Weak Processes
377
(a) Baryon Decays We consider the case when A and B are spin 1/2 baryons and ( B ) J x [A ) = ( B IV, - AX1 A ) . From Lorentz invariance, the most general structure of these matrix elements is given by [q = p’ - p ]
Since the momentum transfer q = p’ - p is very small compared to the mass of A or B for the processes we are considering, we can write
Now we shall take A and B as members of the spin 1/2 baryon octet and then
J~ =
COSO,
JX+ +sino,
J:,
J:
= COSO,
J,- +sinO, J:+, (11.52)
J i , Jit are 1 f 22 and 4 f25 components of octet of
where J f and currents J ~ (2x = 1,-
- . ,8). As shown in Chap.
5
Weak Interactions
378
where g x k = if i j k F
Since F,
=
+ dijkD.
(11.53b)
J KO((),x) d 3 2 is a generator of SU(3), it, follows that
where
gvijk = ifijlc.
(11.54b)
Thus if we neglect the momentum transfer q 2 , (q2 M 0), the matrix elements (BI,IJix( B j ) are essentially determined in terms of Cabibbo angle 0, and the two reduced matrix elements F and D. Using Eqs. (53) and (54), the matrix elements of these decays are given below: Expt,. Decay Vector Axial vector Ratio value of B+B’lv, current gv current g A gA/gV
-0.340 f0.017 4312 sin 0, F-;D 0.25 Z- 4 A fisino, f0.05 x ( F - ;D) In order to test, the octet; hypothesis, we note that if we determine F and D from the first t,wo decays, we find F - D and F - ; D for the third and fourth decays in agreement, with their experimentdl values. The parametrization given in Eqs. (53b) and (54) is in excellent, agreement with experiment. Using the first two entries of the above table, we find F = 0.444 f 0.015, D = 0.823 f 0.015.
C-
-+n,
- sin0,
sin0, ( F - 0) F - D
Classification of Weak Processes
379
As an example to show how gA/gV is determined we consider the case of neutron p- decay n + p e- Ve in detail, where from Eqs. (51) and (52) we have
+ +
[gVyfi- gAyfi’Y51u(P)
(11.55)
+
with gv = cos 8,, g A = cos 8, ( F 0 ) .In the rest frame of neutron, we write k‘ G (Ee, p,), k (E,, p,), p = 0, p‘+ pe+ p, = 0. Since q is very small as compared to neutron and proton masses, we can treat them non-relativistically. Then
Let us write the leptonic part as
LP = f i ( k / ) - y f i ( l - - y 5 ) ? l ( k ) .
(11.57)
The amplitude F [cf. Eq. (49) and Eq. (2.75)] is given by
(11.58) We now sum over proton spin and lepton spin and define the neutron spin Sn as
1 S n = ~ x n +a X n .
(11.59)
Using Eq. (2.110), we get for the probability distribution
(11.60)
Weak Interactions
380
where
8,
is the direction of the neutron spin and
A = A =
A’
=
B =
D = The experimental data give the following values of these correlation functions,
X = -0.102 f 0.005 A’ = -0.1162f0.0012 B = 0.990 iz 0.008 D = (-0.5 f 1.4) x
If we write
J: =
( 11.62)
( g A / g V ( ,t,hen the value of X gives Z=
1gA/gvl = 1.261 f 0.004.
(11.63)
The very fact that, B is nearly 1 implies the maximum parity violation in ,&decay. The value of A’ [assuming gv and g~ are relatively real, see below] gives IgA/gvI = 1.267 f 0.014, consistent with Eq. (63). A non-zero value of D would imply t h e reversal violation in P-decay. The experimental value of D is nearly zero and show that time reversal invariance holds. If we write g A / g V = -zei6, where for q5 = 0 or T , T invariance holds, we obtain q5 = (180.07 f 0.18)’.
(11.64)
Finally, from Eq. (60), we obtain for the decay width I‘: (11.65)
Classification of Weak Processes
381
where
and f(po) =
1''
d p p2
(JGJG) = p y . -
PO
-
(11.66)
me
Since charged particles are involved, this expression of f r = fr-' is subject, to radiative corrections, which are normally incorporated into the factor f along with the first, order Coulomb corrections. These corrections change f by about 5%. The average value from direct neutron lifetime measurements is r = 888.6 f 3.5 sec.
(11.67)
Knowing [gA/gV1 , one can determine Gv = GFgV from Eq. (65). Gv can also be determined from the superallowed O+ Of pure Fermi decays for which FT value is --f
(11.68a) where (11.68b)
F here is different from f for the neutron p- decay and it must account for the stronger Coulomb effect, and for the much more subtle radiative effect,s associated with the higher electric charge. The quantity ( F T )=~3070.6 ~ f 1.6 sec from the O+ + O+decays together with the phase space factor F from Wilkinson and the value of ( g ~ / g v given I in Eq. (63) gives T = 894 f 37 sec, to be compared with the direct neutron life-time measurement given in Eq. (67).
Weak Interactions
382
Finally the Cabibbo angle (11.69) where GL = 1.16637 (13) x GeV-2 while AD and A p are the “inner” radiative corrections to both nucleon and muon AD-decay with Ap - A p = 0.023 (2). This gives
IVud(= COS~’, = 0.9744 & 0.0010.
(11.70)
sin Bc is determined from hyperon P-decays and is given by
lVusl = sine, whereas from
Ke3 decay
=
0.2176 f 0.0026
(11.71)
its value is 0.2196 f 0.0023
(b) Pseudoscalar Meson Decays
(i) Pion Decay 7r-
-+ c-
+ v,,
.!?= e,p.
For this decay, the T-matrix is given by
T
GF
= --
Jz
Here, we have p = k’
COSB‘
(0
p,+I
7r-)
+ k . Now from Lorentz invariance
Using the standard techniques of Chap. 2, the decay rate r can be easily calculated. We obtain
r (*- + e- + ve) = G$ cos2 87r Oc
f: m: m, (1
-
$)’.
(11.74)
Classification of Weak Processes
383
It thus follows that pion decays mainly to muon, its decay to electron is suppressed by a factor rn:/rn; (phase space). In the same way, we can write down the decay rate of K !+ Q; it is given --f
by
r ( K - -+ e-
2) 2
8, + ot) = G$ sin2 8n
fimi m K (1 -
. (11.75)
born the experimental values of the decay rates for pion and kaon we can determine fir and f ~We , get fn M 131 MeV and f K / f i r x 1.22. Fkom the particle data group: f, x (130.7 f 0.1 f 0.36) MeV, fK M (159.8 f 1.4 & 0.44) MeV. Remarks Suppose pion decay occurs through a vector boson W . Then we can write the decay amplitude F :
F = -gw i f n f l
-gpx
+w
yw
(11.76)
ii(k’) yx (1 - y5) ‘u (k). P2 - mw We write the W-propagator in the following form 1 p2
- m&
[(-%A
+
y )+ 1
p 2 - rnb PpPA
P 2 mi/
PpPx
I (11.77)
The first part of Eq. (77) gives the transverse part of the propagator and second part gives its longitudinal part. If we substitute Eq. (77) into Eq. (76), we find that the first part of Eq. (77) gives zero and the entire contribution comes from the second part. We get
2
=
--gw if, me m2w
qc’)
(1 - 75)21 (k).
(11.78)
Weak Interactions
384
Here we have used the Dirac equation U ( k ’ ) ( y . k’ - me) = 0 and p = k’ k. Thus we note that the longitudinal part behaves as if the decay has taken place through a scalar particle of zero mass with effective coupling g&/mk. We also note that it gives a contribution proportional to the lepton mass which is reflected in the formula (74). This is called helicity suppression.
+
(ii) Strangeness Changing Semi-Leptonic Decays As an example of these decays we consider the decay. K-
+ no
+ t- + P!,
We first, note the rule: Ad)
T
=
GF ---sinQ,
fi
= AS =
( = e,p.
1 . The T-matrix is given by
( noI Jl i -K )
The Lorentz striictme of the hadronic matrix elements is given by
(no
IpiI K - )
=
(no
- -
1v;I K - ) 1
1
(W3J2pn2pb x [f+ ( q 2 ) ( P + P‘>x + f- (a2) ( P - ?%] ?
( 11.80) where p and p’ are four-momenta of K - and T ” , q = (p’- p) and k’ and k are four momenta of e- and ve respectively. In the rest frame of K - , we have m K = w Ee E,, pT+pe p, = 0. Using the standard techniques of Chap. 2, we get
+ +
dr 1 = -G$sin28, dEP dw 4n3
~
If+
+
( q 2 ) I 2 [A+BRe[+CI[12],
(11.81)
385
Classification of Weak Processes
where
C =
w = t
=
-41[ w - w ]m;
mK+m,-m; 2 2 2 mK
f- (Q2) / f+ ( q 2 )
(11.82)
For electron, we can neglect, its mass i.e. we put, me2 M 0. Then Eq. (81) is much simplified. In this case, we get, for the electron spectrum
Here we have put
f+ (q2) M f+ (0) = f+. For this case we obtain (11.84)
In the SU(3) limit (7ro IV'l K - ) 0: if4+i553 so that f+(O) = Consider the neutral Kaon decays:
KO KO
-+
7r-+e+ fve,
-+
7r++e-
tve,
5.
AS=AQ AS=-AQ.
For the first case the hadronic matrix elements are given by
Jlt creates negative charge and S = -1.
For AS = -AQ, no such current can be written down in this conventional theory. For more details for semi-leptonic K-decays see Ref. 2.
Weak Interactions
386
Hadronic weak decays (a) Non-Leptonic Decays of Hyperons l l .Z.4
Consider the decay
B (P)
+
B’ (P’) + 7r ( k ) ‘
The Lorentz structure of the T-matrix for this process is given by
The amplitudes A and B are functions of scalars: s = (p’+ l ~ ) t ~=, ( p - P’)~. A is called the parity violating (p.v) [or s-wave] amplitude and B is called the parity conserving (p.c) [or p-wave] amplitude. In the rest frame of baryon B p’ = -k,
lp’l = Ikl = k ,
p’ = kn,
(11.86) In this frame, the amplitudes A and B are constants. In the rest frame of B
(11.87)
where x is a constant 2-component spinor. Using Eq. (87), we may write the T-matrix (11.88a) T = x’ M
x,
Classification of Weak Processes
387
where (11.88b)
We note that the p.w. amplitude A is essentially the s-wave amplitude and the p.c. amplitude B accounts for the p-wave amplitude. The decay width is given by
d r = (24’
s4( p - p’ - IC) [-TT-( M M t ) ]d3p‘
d3k.
(11.89)
Performing the integration, we get the decay width
li Pb [ / a s /+ r =2I~~I’] 21rm a
(11.90)
We now consider the decay of polarized baryon €3. Let S be the POlarization (spin) of B. Let s be the polarization of decayed baryon B’. In the rest frame of B‘, s gives the spin of B‘. The decay probability in this case is given by
dw
=
pn17s4 ( p - p’
- IC) x -1 { T r [ ( l + u ~ ~ ) M ( l + a ~ S ) ] M ~ } d ~ p ’ d ~ k .
2 (11.91) The trace can be easily evaluated and the transition rate is proportional to
where a=
2Re a,* up b S l 2
+ l%I2 ’
P=
21m a,* up b S l 2
+ bPI2
Weak Interactions
388
r=
laJ2 - b P I 2 lasI2
Q2
+ bPI2
+ p2 + y2 = 1.
(11.93)
Because of the last constraint, we can write
p
=
(1 -
sin4 1I2
y = (1 -2) cos$
4
=
tan-'
(P/r).
(11.94)
One also defines
A
= -tan-'
(D/Q).
If we do not, observe the polarization of B', we put, s
=
0 and we
get
dW/r
=
dQ.9
-[I
47r
+
Q
S n] .
(11.95)
Hence we can write the angular distribution
I B (8) = Const [ 1 + a S cos 81 ,
(11.96)
where 8 is the angle between the hyperon spin S and the decayed baryon momentum direction n. If a = 0, the angular distribution is isotropic. Q = 0 implies either a, = 0 or ap = 0. For this case parity is conserved. The anisotropy in angular distribution implies nonconservation of parity. From the angular distribution we can determine the product, as. Since the polarization S of baryon is not generally known, it, is difficult to measure Q! by this method. Further information about Q can be obtained from the polarization of decayed baryon B'. From Eq. (92), we obtain the polarization of decayed bayron B'.
(11.97)
389
Classification of Weak Processes
In particular if the original baryon B is unpolarized viz. S = 0, we get (s) = a n. ( 11.98) This equation implies that the baryon B’ obtained from the decay of unpolarized baryon B is longitudinally polarized. Thus a measurement, of this polarization allowed a direct determination of a. The experimental values for a, ,!l and y are given in the Table 1. Now a non-zero value for ,8 implies the violation of time reversal invariance in these decays. From Table 1, it is clear that ,Ll = (1- (Y’)~’~ sin $ is consistent with zero. Thus the time reversal invariance holds in these decays. P invariance implies either a, = 0 or up = 0, so that Q! = 0, p = 0. But Table 1 shows that a is non-zero. C invariance implies a = 0, ,f3 # 0; hence from Table 1, it follows that C invariance is also violated. The consequences of T and C invariance quoted above hold if we neglect the final state interactions.
(b) A1 = 1/2 Rule for Hyperon Decays The effective weak Hamiltonian responsible for ( A S ( = 1 nonleptonic decays in the conventional theory is given in Eq. (28), namely
G F sin Bc cos Bc - -
--fi
H,,
where
Hw
+ +;.
=
[JT (J”)’
+ h.c
(11.99)
*I J i .
(11.100)
+
NOW JX+ U 7~( I 75) d has I = 1, 1 3 = +1, S YA (1 75) u has I = f, 1 3 = Thus in general Hw has a mixture of A 1 = 1/2 and A 1 = 3/2. However, the most striking effect, of these decays is the approximate validity of A1 = 1/2 rule. The decays N
N
Weak Interactions
390
Table 11.1
Y
Decay
'! '! A pr-
4
"' *0
Q
dJ
0.642 & 0.013 (-6.5 f 3.5)'
A
(derived) (derived)
0.76
(864)'
0.65 f 0.05
-
-
-
+0.017 -0.980-0,0,5
(36 f 34)'
0.16
(187 f 6)'
':nr+ ' +'
0.068 z t 0.013
(167 f 20)'
0.97
(-73';F)O
:
-0.068 f 0.008
(10 f 15)'
0.98
(249':;o)'
-0.411 f 0.022
(21 zt 12)'
0.85
(218?ii)o
f 0.014
(4 f 4)O
0.89
(188 f8)"
--f
nr
c; : c+ --t
pro
'-
nr= a . 0' .zo -+
Y
- -_ -0.456
-+A+r' .. -+A+;?-c _ Y_
v
39 1
Classification of Weak Processes
with A1 = 3/2 are suppressed. A satisfactory understanding of this rule is still lacking. We now examine the consequences of A 1 = 1/2 rule in nonleptonic hyperon decays and its approximate experimental validity. Consider first the decays
A! A:
A13=1/2 A13 = -1/2
: A-+p+n:
A+n+no
AI
=
l / 2 , 3/2, *
* *
The simplest, possibility is A I = 1/2. Assuming this to be the case, the only possible isospinor which one can form is
fi T * T A = ( p
no + h fi r-, & p 'n
- fi
no)I\. (11.101)
Then for AQ = 0, we have
AO_ = -&A:.
(11.102)
Hence we get
r (A:)
= 2 r (A;) = "At.
"A!
It is clear from Table 1,
x
Q
~
O
,
(11.103a) ( 11.103b)
; experimentally
(11.103~) Thus AI = 1/2, rule is a good approximation, AI = 3/2 amplitude is very much suppressed for A-decays. An exactly similar argument gives z- = z;, (11.104a)
--
-Jz
which implies
r (z:) /r (z:)
=
a=-/a=o =
-_
-0
2 Expt : 1.639
( 1 Ezpt : ( 0*456 0.411
M
1.11). (11.104~)
Weak Interactions
392
For C-decays, assuming A1 = 1/2, the only isospinors which we can form are U
N
(z'.7r)+ibN
(11.105a)
(CX7+7.
Writing only the part, for which total charge is zero, we have
(C-T' + C'T- + COT') + b (fip COT' - n c+T- + n C + K ) .
u fi
-
h j3 C'7-r' (11.105b)
Thus we get
CI = ~ + b
CI
= a-b
Co
=
f i b
Ci
=
-JZb.
(11.106)
From Eq. (106), we get
CZ
-
C I = JzC,+.
(11.107)
The prediction can be tested as follows: In the ( u s , u p )plane if we regard C:, C: and d C , ' as vectors, then they should form a closed triangle. To sum up, in case the AI = 112 rule holds, out of 7 decays listed in Eq. (25) only four are independent. In the language of flavor SU(3) [cf. Chap. 51, the dominance of A1 = 1/2 rule is generalized to octet dominance. This can be seen as follows: u,d , s, belong to 3 representation of SU(3). ti, d, 3, belonging tJo3 representation of SU(3). Now 3 @ 3 = 8 @ 1. Thus J," in Eq. (27) belongs to an octet representation of SU(3). Hence Hkt in Eq. (27) or (28) contains 8@8=1@8@8@10@+TO@27.
393
Classification of Weak Processes
Figure 6 W-boson exchange graph for the reaction u + s -+ d + u
.
It can be seen that only 8 and 27 are relevant for the decays (25). Thus HLt contains both 8 and 27 where 8 corresponds to AI = 1/2 only while 27 contains A1 = 3/2 as well. Thus in the language of SU(3), generalization of A1 = 1/2 rule is the octet dominance. The oc'tet dominance for the current-current interaction implies an additional relation (called Lee-Sugawara relation) between s-wave decay amplitudes of (25)
+
2A (Z) A (A!)
(c) Non-leptonic
= +fi A (C:)
.
(11.108)
Hyperon Decays in Non-Relativistic
Quark Model One can recover not only the AI = 1/2 rule but also the right order of magnitude of the scale required to reproduce the s- and p-wave fits of non-leptonic hyperon decays. Consider the weak vector boson exchange graph of Fig. 6 as the analogue of the gluon exchange quark-quark scattering graph considered in Chap. 7 which quite successfully described the quark spectroscopy. The matrix elements for the process shown in Fig. 6 are of the form
Weak Interactions
394
where q = pi-pi = pi - p j . u's are Dirac spinors in Dirac space but are column vectors involving u,d , s quarks in ordinary flavor SU(3) space. a; and ,Bj'are operators which transform a v-like state into a d-like state and a s-like state into a u-like state respectively. We take the leading non-relativistic limit, of the above matrix elements. In the leading non-relativistic approximation, only yo and yi 7 5 have nonzero limits. Thus only parit,y conserving ( p . c ) part of A4 survives in the leading non-relativistic approximation and we have in this limit
(11.110)
MP'"
= 0.
The latter corresponds to a general result that (B' I(JJ>"'"I B ) = 0 as a consequence of CP and SU(3) invariance. The Fourier transform of Eq. (110) gives the effective Hw as
x (1 - ai.aj)f i 3 (r) .
(11.111)
Now it has been shown [see Sec. 12.4.21 that, in the currentalgebra approach the question of AI = 1/2 rule or octet, dominance for non-leptonic decays of baryons hinges on the matrix elements (BS
lHFl
BT)
aTSUu,
( 11.112)
which essentially determine both s- and p-wave amplitudes. Here u is a Dirac spinor for B, or B, which denotes a baryon like A, C, E,n, or p . Therefore, we have to take the matrix elements of Eq. (111) between the baryon states B, and B,. We regard the baryon state B, or B, as made up of three quarks. We take the
Classification of Weak Processes
395
spatial wave function for such states to be the same for the octet of baryons p , 71, A, C*, C‘, So, E- and denote it by Qo. Thus writing df = 1 6 3( .)Iq 0 )= p0( o ) I ,~ (11.113)
po
where r = ri - rj (i # j ) , we have to calculate the matrix elements of the operator
i>j
’
between the spin-unitary spin wave functions of the states p , n, C+, Co,A,Eo,given in Chap. 6 . We obtain (11.114a)
agoho
a2-z-
=
GF sin 8, cos &d’ (- 6) t/Z
(11.114b)
=
-d5apn
(11.114~)
=
GFsin8,cos8,df (-2d6)
(11.114d)
Jz
= 0.
(11.114e)
= -&upn expressed in Eq. (114c) enThe relation sures the A1 = 1/2 rule (or octet dominance) and hence A (CT) = 0 (which is good experimentally) in current algebra approach [see Eq. (117) below]. Once the octet dominance for ars is established we can parametrize ars in the SU(3) limit, as
Then the relations (114) immediately give
D’- -1. F‘
( 11.116)
Weak Interactions
396
Now using the current algebra relations [see Sec. 12.4.21 for the s-wave amplitudes one has
A(C$)
A
(c:)
=
1 -
=
-f f f ( U P p
d5 fir 1
+ d5 axon) (11.117)
+
Here f T is the constant which enters in T - t pfip decay. Then using Eqs. (114) and (117), we have the relations (107) and (108). Using the value of d’ as determined by the constituent quark spectroscopy [cf. Chap. 71 ,
ax+p
-27 G F sin BC cos Bc =
(mE - u L A )
8&.rras x -105 eV
(
7h2 A
1 - m/ms
)
constituent
(11.118)
for the accepted value of as (q2 M 1GeV) M 0.5.This is almost the phenomenological octet dominance scale, which together with D’/F’ M -0.86 [not, very far from the prediction (116)], are required to fit, the s- and p-wave amplitiides of hyperon decays.
11.3 Problems (1) Show that the electron spectrum in the decay of b-quark b using V
-
--f
c
+ e- + v,,
A theory is given by (neglecting the electron mass)
Problems
397
where y = - 2Ee ,
? J m = l -m,2 -. mb mi Similarly, show that for c-quark decay c
+s
+ e+ + v,
the electron spectrum is given by
dr _ -
dx
Hint: For b -+ c + e-
-G2F 5 1 6 7 ~mc ~
2(Ym
- YIZ
+ Pe, the matrix elements are
Use Eqs. (31) and (32) with the replacements (mp,me,mum,mve) (me,m,, me,mUe), E = 1 so that --f
in the rest frame of b. Performing d3p2 integration, write d3kld3k2 = kf dkl dlc2 d R and use
to perform the angular integration to obtain
398
Weak Interactions
where from
E, =
2 mb - mz - 2 mb Ee 2 (mb - me + Eecos8)
one has
s.
The int<egrationof Eu gives the result, For the second problem, the matrix elements are
GF
-
T=----[ 4 P 2 ) Yx (1 - 75) '11(P1)1 [u(ka) Yx (1 - 75) 4w] '
Jz
Results from the first can be obtained by changing k:! t-$ m,-+ms
h,mbtmc,
and then follow the same steps as in the first part. ( 2 ) Consider the decay
K
-+
37r.
Show that decay rate can be expressed as
o<x2+y2 lVxenlP(PU
(12.15)
[ 12.16)
404
Properties of Weak Hadronic Currents and Chiral Symmetry
where [since J d32 ~ “ ( x0), is the electric charge in unit of e ] it, follows, on using Eqs. (8)-(10) that
F,P(O)= 1, F;”(O)= 0.
(12.17)
Since gxVqv gives Pauli type interaction, it also follows that
F g o ) = K p , F,”(O) = K ,
(12.18)
where K~ and K , are the anomalous magnetic moments of proton and neutron respectively. np = 1.792 and K , = -1.913 in units of nuclear magneton. Hence we get from Eq. (11.50a) and Eqs. (14)-(16) that
where F y and F . are the isovector electromagnetic nucleon form factors. Their normalization follows from Eqs. (17) and (18).
Thus in particular
(12.21) Using SU(3), we can write the matrix elements of vector current &, i = 1, . * * , 8 for an octet of baryons (assuming q2 x 0):
namely the relation (11.54)
Partially Conserved Axial Vector Current Hypothesis (PCAC)
405
12.3 Partially Conserved Axial Vector Current Hypothesis (PCAC) Fkom Eq. (11.73), we have
(0 l t P ~ z ( x )r-) I
=
-ipA (0
IA;~
r-) e-ip'x
- - 1 -f,m,e 1 (2743/2 &
2 -ip.x
. (12.23)
If the axial vector current A: is conserved, then either f, = 0 or rn; = 0. Since for a physical pion rn; # 0, f r r must be zero and pion decay is forbidden. Thus A: is not conserved. Now @A: has the same quantum numbers as those for a pion. If we now put
PA:
= fnm:r-
(12.24)
then (12.25) Here ~ ( xis )the pion field operator which creates r+ or destroys r-. Equation (24) is called the PCAC hypothesis. We note from Eq. (23), that in the limit, m: 3 0, t,he axial vector current is conserved. This implies that strong interactions have an approximate symmetry which is exact in the limit of zero pion mass. Such a symmetry is called chiral symmetry. Chiral symmetry manifests itself in the existence of massless pseudoscalar mesons called Nambu-Goldstone bosons. We shall come to this point again later. Here we discuss one of the important consequences of PCAC. We apply PCAC to neutron P-decay. Fkom Eq. (11.50b), we have
406
Properties of Weak Hadronic Currents and Chiral Symmetry
We note that pion pole contributes to the form factor f A ( q z ) only. It, does not contribute to gA(q2) nor h A ( q 2 ) . Separating out the pion pole contribution, we write
(12.27) where fA(q2) is the remaining part, of fA(q2) . &om Eqs. (26) and ( 2 7 ) , we get
Now if we assume t8hatin the limit, m: is conserved, we get, 2mNgA
(q2) - f i g r N N f r
3
0, the axial vector current
+ q2 f A ( q 2 ) = 0.
(12.29)
At q2 = 0, this gives (12.30) This is called the Goldberger-Treiman (G-T) relation. Thus G-T relation is exact in the chiral symmetry limit when pion mass is zero and the axial vector current, is conserved. This relation can be easily tested as all the quantities in Eq. (30) are experimentally known. This relation is valid within 6% agreement, with experiment. On the other hand, we note that
Partially Conserved Axial Vector Current Hypothesis (PCAC)
407
Using PCAC, viz. Eq. (24), we get
(12.32) Evaluating it at q2 = 0, m$ # 0, we again get the G-T relation. We conclude that the success of the G-T relation implies that deviations from chiral symmetry or equivalently from PCAC are indeed small. Finally, using SU(3) we can write for q2 M 0 for an octet of baryons [cf. Eq. (11.53)].
(Bk(P’) IAix I Bj ( P I )
(12.33) In particular for neutron &decay, we get
(12.34)
We define a four-vector sx = .li(p)yx75u(p).
(12.36)
p . s = o , s2=-1.
(12.37)
We note that
The vector sx thus gives the spin of the proton. To see it explicitly we go to the rest frame of the proton. In ,this frame, we get from Eq. (37), SO = 0, s2 = 1. From Eq. (36), we get
s = x +ax.
(12.38)
408
Properties of Weak Hadronic Currents and Chiral Symmetry
In quark model, we can write the axial-vector current Ai, = ~ T ~ T ~ + ( We define the quantity Aq as ( 2 r l 3 F ( P (4^/xYsQI P> = &Sx.
(12.39)
In particular for A ~ =x $ ( ~ ~ Y ~ ” I ~ &yx75d), ’L we have
1 ( 2 ~ ) ~ (Pp o( A 3 X ( p=) ~ ( A-UAd)sx rn
SO
(12.40)
that
nu-
Ad = Q A
=F+
D.
(12.41)
12.4 Current Algebra and Chiral Symmetry Isospin conservation implies that st,rong interactions are invariant, under SU(2) group generated by the charges:
I i ( t )= / I 4 , ( x , t ) d 3 2 , i
=
1,2,3.
(12.42)
In the same way we can define the axial charges
I f ( t )=
/ Aio(x,
t)d32, i = 1,2,3.
(12.43)
The generators of the isospin group SU(2) satisfy the commutation relations [Iz( t ) ,lj ( t ) ]= Z E i j k I k ( t ) . (12.44) Since I:(t)’s belong to the adjoint representation of SU(2) group, we have [Ii@),I$)] = i E & ( t ) . (12.45) We obtain a closed algebraic system by requiring that,
[I,”@), q t ) ]= i&Zj/Jk(t).
(12.46)
The last relation constitutes a major theoretical assumption. The commutation relations (44)-(46) represent the algebra of the group
Current Algebra and Chiral Symmetry
409
SU(2)xSU(2) generated by the vector and axial vector charges. This group is called the chiral SU(2) group. Let us now write the part of the QCD Lagrangian [cf. Eq. (7.32)] which involves u and d quarks:
where q =
( 1)
is an isodoublet field and we have suppressed
color indices. For mu = md this Lagrangian is invariant under the isospin transformation 4 u9, (12.48) +
where U is a special unitary matrix, exp [i:Ai] , Ai being constant. The associated vector current Ep = Q$ypq is conserved. The existence of nearly degenerate isospin multiplets of hadrons shows clearly that [mu- mdl is small compared to hadron mass scale (-1 GeV). Setting m, = md = m, we can write
where we have split q into “left-handed” and “right-handed” comp onents 1T 7 5 qL,R = 2 4. It is clear that in the limit m = 0, the Lagrangian (49) would be invariant under independent ‘chiral’ isospin transformations on q L and qR: qL
= U L q L , qR
-+
URqR
and not only Kp but also the axial vector current qypy5?q would be conserved. w e note that the mass term m ( q L q R 4- q R q L ) or in general the coupling to scalar and pseudoscalar fields
410
Properties of Weak Hadronic Currents and Chiral Symmetry
would break chiral symmetry. This also demonstrates that the forces between the quarks have to be vector in nature [mediated by spin 1 gluons, cf. the term ~T,,X Gpqin Eq. (47) or Eq. (49)]. As we shall see later mu 5 MeV, m d -10 MeV (these are called current quark masses, not to be confused with constituent quark masses of order 300 MeV [cf. Chap. 61) are small compared to the hadron scale of O(1 GeV) so that chiral symmetry is nearly exact,. Now if Aix were conserved, the axial charge 1; would commute with the Hamiltonian: N
[1,5,H]= 0.
(12.50)
Hence if we define
1:
1x1)= ieijk
I&)
1
(12.51)
use of Eq. (50) would imply that the states I&) are degenerate in mass with IX,)even though they have opposite parity. This is because 1 : has negative parity. This condition can be realized in either of the two ways:
1. The Wigner-Weyl realization of SU(2) symmetry, in which case l Y k ) would consist of “parity doublets” of IX,)e.g. if IXj) were pseudoscalar mesons, l Y k ) would be scalar mesons degenerate in mass with the pseudoscalar mesons. This is not what occurs in nature and therefore chiral symmetry is not, realized in nature in this way in contrast to the ordinary isospin symmetry which is realized in this way. 2. Spontaneously broken symmetry realization of SU(2), in which case l Y k ) would consist of IXj) plus an odd number of pions with vanishing four-momentum (called soft pions), the pion being a massless “Nambu-Goldstone” boson. In particular (12.52)
the first part being valid only for single-pion transitions, while
1i 10) = 0.
(12.53)
Current Algebra and Chiral Symmetry
41 1
+
As we shall see m: would involve (mu md)/2 as a factor and so a measure of explicit chiral symmetry breaking is provided by rn:/m: x 0.03, p being the non-strange (non Nambu-Goldstone) boson next to pion. The notion of (approximate) spontaneously broken chiral symmetry has been found useful in hadron physics and has given rise to many predictions involving soft pions which are in good agreement with the data [see bibliography]. One such prediction is the Goldberger-Treiman relation (30) : (12.54) to be compared with the experimental value 0.06 f 0.01 of the left-hand side. The above considerations can be easily generalized to SU(3). Thus the QCD Lagrangian (7.32) shows an approximate global symmetry in the limit mp 0, this Lagrangian is invariant, under the group SU(3) xSU(3) generated by the charges associated with the weak currents Jip. Thus the generators of the group are (i = 1 , . * * ,8). --$
Fi
=
J
F5
=
J ~ i o ( xt ,) d 3 2 .
V , ~ ( Xt)d3a: ,
They satisfy the commutation relations
[E,F j ]
= ZfijkFk
(12.55)
[F,,F,5]
= ZfijkFf
(12.56)
= ZfijkFk.
(12.57)
[F5,F.f]
The commutation relations (55) and (56) follow from flavor SU(3), the commutation relation (57) is a new assumption. Equivalently if we define 1 1 (12.58) F,L = - (& - F:) , :?I = - (Fi+)'?I 2 2
412
Properties of Weak Hadronic Currents and Chiral Symmetry
we get
Symmetry generated by the above group is called the chiral symmetry. If (R1,Rz) is a multiplet of group SU(3) xSU(3), then under parity (12.60) (Rl, Rz) (R21R1). -+
For example ( 8 , l ) + (1,8),(3,3*) + (3*,3). This means that if this symmetry is realized as a classification symmetry, we must have parity doublets. This is not the case in nature. No parity doublets are found. This implies that, the chiral symmetry is realized in the Nambu-Goldstone mode that is to say, there are eight, bosons which in the chiral limit have zero mass. As we have already seen, pions are the Nambii-Goldstone bosons which in the chiral SU(2) xSU(2) limit are massless. The eight, pseudoscalar mesons are identified with Nambu-Goldstone bosons of chiral group. The algebra generated by Fi and E5is called the chiral dgebra. This algebra has rather rich physical content because generators of the symmetry group can be identified with observables. The matrix elements can be measured in electxoweak interactions. This in fact provides evidence for chiral symmetry [see bibliography]. 12.4.1 Explicit breaking of chiral s y m m e t r y As already seen the chiral symmetry is spontaneously broken [cf. Eq. (52)]. Another way of expressing it, is that
(0 1441 0)
# 0 =+ K5 10) # 0.
(12.61)
To see this, we note that, in the quark model, we have the following commutation relations:
[F:, 5’31
= id,jkPk
i = 0, 1, ...,8
Current Algebra and Chiral Symmetry
413
(12.62)
where Xi si = q-q, 2
Xi
Pa = 4-754 (12.63) 2 are respectively the scalar and pseudoscalar densities. We note from Eqs. (62) that
I +
(0 [PI
2P2, F:-,]
2 2 10) = 22& (0 IS01O)+ (0 iz 0) . (12.64)
Now we expect that flavor SU(3) is realized in the usual way and is not spontaneously broken [cf. Eq. (53)]. This implies that
as
Thus, if
F4fi5 10) = 0.
(12.6513)
(so)o = (Gu + Jd + ss)o # 0
(12.66)
then we have from Eq. (64):
the condition for spontaneously broken symmetry [cf. Eq. (52)]. Let us write
(uu)o=
(q0{ =
S - S ) ~=
-.(say).
(12.68)
Hence we have the result that, (SO)*# 0 which implies that, chiral symmetry is spontaneously broken and (Sg),, = 0 implying that flavor SU(3) is not spontaneously broken.
414
Properties of Weak Hadronic Currents and Chiral Symmetry
We can write the QCD Hamiltonian density [cf. Eq. (7.32)]
as
3-1
=
= 3
+ (m,uu + mddd + m,ss) 2 3-10 + 6 ( 2 m + ms)S0+ - ( m - m,) ss + (m, - r n d ) fi
s 3
(12.69)
3-10+3-1’.
The Hamiltonian density 3-10 is chiral invariant,. Here f i +md). NOW
=
(1/2)(m,
(12.70) where
H ( t )= J’d3x71(tlx). The (charge) continuity equation d F5
2 =
dt
=
/d3x
(aAio(t’ at
+ V . & ( t , X)
/d3xPAi,
(12.71)
then converts Eq. (71) into
PAix
= -2
[F:13-1’].
(12.72)
E%om Eq. (72), we have (12.73) Using Eq. (52)) namely
(12.74)
Current Algebra and Chiral Symmetry
415
we obtain
f7r
-i
[ ( ~ j l J d X A , / 0 ) +(OJa’Ai~lrj)] = -i(OI[F’, [ F , f , % ’ ] ] I O ) .
2Jz The use of PCAC relation a X A = is symmetric in i and j ] .
(
fT/&)
[q, [c5,%‘I]
(12.75) m?.lri,then gives [m?j
10)
(12.76)
where 3-1‘ [cf. Eq. (69)] is
Substituting Eq. (77) into Eq. (76) and using Eqs. (68) and (62), one obtains
(12.78) Let A be the electromagnetic contribution due to photon exchange Since T + ) K+ form a U-spin multiplet the electromagnetic to contribution to mg* is also A while it, is zero for m:o, mgo, so that adding A in Eq. (78) for r+,K + , we get
mi*.
mi,
(12.79)
416
Properties of Weak Hadronic Currents and Chiral Symmetry
Here we have used the explicit breaking of chiral symmetry in calculating the current quark mass ratios in terms of masses of pseudoscalar mesons. When quark masses go t,o zero pseudoscalar mesons become zero mass Nambu-Goldstone bosons required by spontaneously broken chiral symmetry. 12.4.2 An application of chiral symmetry to non-leptonic decays of hyperons Consider the matrix elements [where B, and B, are members of the same baryon octet]:
(Bs(P’)
I pi?,HW] 1
B T
(PI) = (Bs(P’) ( p h v - HWF,S( Bv (PI)
(12.80) where i = 1,2,3. Using Eq. (74) and its hermitian conjugate, we can write it, as
(12.81) In other words in the limit qp = ( p - P ’ ) ~ 0 [called the soft, l ( p ) ) are pion limit], if the matrix elements ( B , (p’) r i ( q ) l H ~ B, non singular, then Eq. (81) gives ---f
+
Now Hw = Hz;;”H r [cf. Chap. 111 and it can be shown that for s-waves [HZ;;”], the amplitude on the left hand side of Eq. (82) is non-singular [see below] and we have
417
Current Algebra and Chiral Symmetry
Figure 1 Pole diagram in hyperon decay.
For pwaves [ H F ] ,one can apply the result (82) to
= -iJ”
f?r
I
I
(B”(p’) [F:, H F ] B, ( p ) )
(12.84)
where the Born terms are shown in Fig. 1. These are singular for Hz;;” in the limit qp 0 where mB = mk as they behave like 1/ I r n ~- mbI but for H F they behave like 1/ ( m+.m&I ~ and are non-singular. Now as we have seen in Chap. 11 [cf. Eq. (11.28)], the [AS1 = 1 non-leptonic Hamiltonian is ---f
Hw
GF
= -sin BCcos BC [Syp(1
fi
+ y5)21][Uyp(l+ y5)d] .
(12.85)
This being the product of two left, handed currents [FR= F, satisfy [F?,Hw] = 0
+Ff]
(12.86)
418
Properties of Weak Hadronic Currents and Chiral Symmetry
Figure 2 Triangle diagram for ro-+ 27 decay.
Furthermore F, (being the generator of SU(3) flavor group) acting onlB, > or IB, > produces a member of the same octet. To illustrate this point, consider for example, IB, >= / A > and < B,( =< pl , i = m. Then fi F1+i2
111) = 0 and (nl = (PI
F1+i2.
Thus for s-wave from Eqs. (83) and (86) (12.87) Also as shown in Chap. 11, in the exact SU(3) limit ( B , IHg"l BT)= 0. Thus the p-wave non-leptonic decays are given by the Born terms which are also determined by (B, IHE"l B,) as far as weak vertices are concerned. These were the results which we employed in Sec. 3.3~ of Chap. 11.
12.5 Axial Anomaly As seen in Chap. 7, 7ro -+ 27 is given by the triangle graph of Fig. 2. In the chiral limit (mu= md = 0), this triangle graph gives a finite value for the 7ro -+ 27 amplitude:
M(r0
-+
27) = E ~ * ( J E ~ ) E ~ * ( I C ~ ) E ~ , ,(12.88) ~~C~~C~
Axial Anomaly
419
with
(12.89) where N , is the number of colors, e. g. 3, e, = 2/3, ed = -1/3 while the Goldberger-Trieman relation for ( Q I A ~ ~ with I Q ) A3p = 51 (Wpysu. - &&) gives ( f l r / l / Z ) g l r q q= m, so that Eq. (88) gives
(12.90) It is important to remark that the result (90) is unaltered by radiative corrections to the quark triangle and Eq. (90) is independent of the masses of fermions in the loop. Equation (90) gives
(12.91) which is remarkably close to experiment with only 2% PCAC correction to the amplitude. The above result is often stated in terms of contribution to the amplitude due to an axial-vector “anomalous” divergence:
(12.92) where Fpu= Opa, - auup [upbeing electromagnetic potential] and w - &puapFap. 2 Note that Eq. (92) does not arise from equations of motion (72). That is why it is called “anomalous” divergence. Combining Eqs. (72) and (92), we have
(12.93) The first term on the right-hand side of Eq. (93) vanishes in the chiral limit but it is not so for the second term. The PCAC relation x becomes for A ~ thus
(12.94)
420
Properties of Weak Hadronic Currents and Chiral Symmetry
The “anomalous” divergence equations for
?78 and
70 are
a!
d X A k= ~ -S~FPyFPy,
(12.95)
47r
where lc = 8 or 0 and
2
+ e i + e q ] = 2z,
(12.96)
Similar considerations show that in QCD, t,he flavor SU(3) singlet current,
has “anomalous” divergence (12.97) where
GP’ = -1E P @ I
2
GmP
(12.98)
and G,, involving gluon field has been defined in Chap. 7 [cf. Eq. (7.31c)l. Thus
dXAox =
8
+
+
.CPu.
[m,.z~i7~u.mddiy5d+ m,siyss]
J:: --GPu
(12.99)
It is clear from Eq. (99) that the SU(3) singlet current is not conserved in chiral SU(3)@SU(3)limit. An application of this will be considered in Chap. 14.
QCD Sum Rules
421
12.6 QCD Sum Rules We have seen in Chap. 7 that the asymptotic freedom property of QCD makes it possible to calculate processes at short, distances or for large q2, q2 being the square of the momentum transfer. On the other hand, bound states of quarks and gluons (hadrons or hadron resonances) arise because of large distance confinement effects, i.e. strong coupling effects, which cannot be treated in perturbation theory. The idea of QCD sum rules is to calculate resonance parameters (masses, width) in terms of QCD parameters (as, quarks masses and number of other matrix elements which are introduced to parametrize the non-perturbative effects). We have also seen previously that in the absence of quark masses, the QCD Lagrangian shows a global chiral symmetry i.e. it is invariant under a global s U ~ ( 3x) s U ~ ( 3group. ) But this chiral symmetry is spontaneously broken i.e. the ground state is not invariant under this symmetry. This gives rise to [q = u , d , s] [cf. Eq. (Sl)]
(0 la41 0)
#0
leading to an octet of zero mass pseudoscalar mesons (so-called Nambu-Goldstone bosons; such bosons acquire masses when QCD Lagrangian is explicitly broken by the quark mass terms). The non-vanishing of the above quark condensate is a non-perturbative effect and gives rise to power corrections to asymptotic freedom effect, which is logarithmic. The essential point of the QCD sum rules i.e. to relate QCD and non-perturbative parameters of the above type with resonance parameters, is illustrated by the simplest of sum rules i.e. for a two-point function:
A (q2) = 1 IT
s-q2
i
The left-hand side is saturated with resonance so that 1.h.s. =
C mi 9:- q2 a
(12.101)
422
Properties of Weak Hadronic Currents and Chiral Symmetry
where (gi,mi) are resonance parameters. The right-hand side is useful only for large q 2 in which limit the perturbative QCD allows us to calculate the coefficients Ci(q2) in the operator product, expansion. In practice we want, to sahrate 1.h.s. by a few low lying resonances. Thus we should use some weighting factor to suppress large s contributions on 1.h.s. This is done by using Bore1 transform of the sum rule, which introduces a weighting factor involving a mass parameter M 2 ,which should be sufficiently large to suppress non leading terms on r.h.s. of Eq. (101) but not too large in order to suppress contribution from higher hadron states on 1.h.s. Thus the problem in practice reduces to finding a region of stability point, for M 2so that a small variation in M 2will not affect the physical parameters. In this way from QCD sum rules for two-point and three-point functions, a large number of constraints on hadron spectrum have been obtained providing not only a consistency check but also a useful phenomenological information on resonance as well as QCD parameters and on (Olqq10). For details see the bibliography.
Bibliography
12.7
423
Bibliography
1. R. E. Marshak, Riazuddin and C. P. Ryan, Theory of Weak Interaction in Particle Physics, Wiley-Interscience (1969). 2. E. Commins and P. H. Bucksbaum, Weak Interactions of Leptons and Quarks, Cambridge University Press, Cambridge, England (1983). 3. H. Georgi, Weak Interactions and Modern Particle Theory, Benjamin/Cummings, New York (1984). 4. T. D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic (revised edition 1988). For current algebra and chiral symmetry, in addition to the above, see 5. S. L. Adler and R. F. Dashan, Current Algebra and Application to Particle Physics, Benjamin, New York (1968). 6. S. B. 'Ikieman, R. Jackiw and D. J. Gross, Lectures on Current Algebra and its Applications, Princeton University Press, Princeton, New Jersey (1972). 7. V. de Alfaro, S. Fubini, G. F'urlan and C. Rossetti, Current in Hadron Physics, North Holland, Amsterdam (1973). 8. M. D. Scadron, Current Algebra, PCAC and the Quark Model, Rep. Prog. Physics, 44, 213 (1981). 9. C. H. Llewellyn Smith, Particle Phenomenology: The Standard Model, Proc. of the 1989 Scottish Universities Summer School, Physics of the Early Universe, OUTD-90-160. 10. J. F. Donoghue, Light Quark Masses and Chiral Symmetry, Ann. Rev. Nucl. Part. Sci. 39, 1 (1989). 11. 3. F. Donoghue, Chiral Symmetry as an Experimental Science, CERN-TH. 5667/90, Lectures presented at International School of Low-Energy Antiproton, Erice, Jan. 1990. For QCD Sum Rules, see 12. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 and 448 (1979). 13. L. J. Reinders, QCD Sum Rules, An Introduction and Some Applications, CERN-TH-3701 (1983): Lectures presented at the
424
Properties of Weak Hadronic Currents and Chiral Symmetry
23rd Cracrow School of Theoretical Physics, Zakopane (1983). 14. S. Narison, QCD Spectral Sum Rules, World Scientific Lecture Notes in Physics-Vol. 26, World Scientific, Singapore (1990).
Chapter 13 ELECTROWEAK UNIFICATION 13.1 Introduction The Fermi theory of ,&decay cannot be the fundamental theory of weak interactions. It leads to many difficulties; it is non renormalizable theory. In this theory the scattering cross section for the process up e- -+ v, p- is given by Eq. (2.155):
+
+
O8
=
G; S .
(13.1)
IT
The above scattering is purely S-wave. Now Eq. (3.118) [XI = A2 = f1/2] gives ua = f 12F0I2[the factor 2 in the denominator is average over initial electron spin], where Fo = 7)o . Now the = f. maximum absorption occurs when qo = 0, so that IF0I2 5 Thus the partial wave unitarity gives
'::-'
5
(13.2) so that from Eq. (1)
G;
5
-s
7r
87r
S
or GFS
2Jz
51.
(13.3)
7r
Hence Fermi theory breaks down for s > ( ~ & / G F ) = (0.9 TeV)2. Therefore, we need a cut-off AF signifying new physics beyond AF 425
Electroweak Unification
426
where from Eq. (3)
A:wu
5 0.9 TeV.
(13.4)
Here PWU signifies that this has been obtained from partial wave unitarity. On the other hand if weak interactions are mediated through vector boson W , then instead of Eq. (l),we get
(2)(s+rnb)rnk 2
0,
=
32
T
s
(13.5)
which is finite for all energies, approaching the limiting value
Thus we see from Eq. (1) that the W-boson mass mw provides the cut-off AF. As we shall see mw w 80 GeV, so that mw 0, since V (4) would have no minimum if X < 0. If in V (&), p2 > 0, then we have the ordinary scalar particles of mass p and V (4) has a local minimum at, q!J = 0. Then t8hemodel is not. interest,ing. However, if p2 < 0, then V (4) has a local minimum a t 1412 = -$ or 4 = This is shown in Fig. 1. By convention, we select, the positive sign. This is a classical approximation to the vacuum expectation value of q!J
&a.
(13.12) Although the Lagrangian (11) is invariant, tinder the local gauge transformations (9) and (lo), the non-vanishing expectation
Spontaneous Gauge Symmetry Breaking
Figure 1 Effective potential V
429
( 4 ) for p2 < 0, showing local minima.
value of 4 means that the gauge symmetry is broken i.e. the vacuum or the ground state is not invariant under the gauge transformation (9b). This can be seen as follows. From Eq. (9b)
U-' (Az) $U (Az) = e 2iA2 4,
(13.13)
Therefore,
(0 IU-'
(A2)
4U (Az)lO)
= e2iA2(0 1410) .
(13.14)
If the vacuum is gauge invariant, then
u (A21 10) = 10)
(13.15)
(0 1-610) = e2iA2(0 1410 ) .
(13.16)
and Eq. (14) gives
Thus if (0 I q5 10) # 0, then e2aA2 = 1 for any A2, a contradiction. Hence U(A2) 10) # lo), and the gauge symmetry is spontaneously broken i.e. U1 x U2 -+ U1, UIis unbroken.
Electroweak Unification
430
We now show, how spontaneous symmetry breaking leads to the massive vector boson B p . It, is convenient to define a new field 4’;
(4‘)
0,
=
(13.17)
where $1 and $2 are hermitian fields with zero expectation values. The Lagrangian (11), in terms of the fields $1 and q52 has the form
L
=
1 4
--A
”
1 4
A,, - -B’” B,,
+ -21 (4g21j2) B’B,
-
2gvBL”d,$2
F’rom the Lagrangian (18), we derive some interesting results. If the gauge group U2 is a global gauge group, then we do not require the vector boson B, and from the Lagrangian (18), we see that the scalar field 41 has acquired a mass &%?, the scalar (pseudo) field 4 2 is massless, the fermion field Q has also acquired a mass$ and the vector field A, corresponding to unbroken gauge symmetry U1 is massless. Hence we have the Goldstone-Nambu theorem. A spontaneous breakdown of global symmetry leads to a massless scalar particle. But when U2 is a local gauge symmetry, then due to the presence of the term 2 g 1 ) Bpa,q$2 a straightforward interpretation of (18) is not possible. But we can eliminate this term by a field dependent gauge transformation. Actually what happens is that 13, $2 combines with B, (which has only transverse
Spontaneous Gauge Symmetry Breaking
431
components) to form a single massive spin 1 field, 8, 4 2 now becomes longitudinal mode of spin 1 field. This can explicitly be seen as follows: Choose the gauge function A2(x) to be ?$. Then under the gauge transformations
(13.19) the Lagrangian (18) becomes (removing ^):
1
L = --A 4
p’
1 APv - 4B””B,, Q
h -
--QQp
Jz
+ -21 (aJ
+ 21 (49’11~)BpB,
- eGyWA,
- g$y,y5Q!B,
1 - - (21129 p2 2
(13.20)
It is clear from Eq. (20), that, the would be Goldstone boson field q(z) has been transformed away; it has been eaten away by the field B, to give a longitudinal component. This mechanism is called the Higgs-Kibble mechanism. The massive scalar particle p is called the Higgs particle. To summarize: (1) No massless scalar boson appears. (2) A, which is associated with unbroken gauge symmetry (electric charge conservation) has zero mass. (3) The vector boson B, has acquired a mass m B = 2gv. ( 4 ) The fermion field has acquired a mass rnf = ( 5 ) Both the masses of B, and \k arise due to the same symmetry breaking mechanism. (6) A massive scalar particle with mass&%? appears. This particle is called Higgs particle. Presence of Higgs scalar is an essential feature of spontaneously broken gauge symmetry.
3.
432
Electroweak Unification
13.3 Renormalizability We give here few remarks about the renormalizability of a gauge theory. Now the fields A, and B, cannot be determined uniquely by field equations. In order to quantize these fields, one has to fix a gauge that is to say break gauge invariance. For the photon field A,, a term added to the Lagrangian for this purpose is Photon propagator is then givcri by
--$[-’
For the field B,, the gauge fixing term is
It, is so chosen that, it cancels awkward looking mixing term B”8,42 in the Lagrangian (18). [ is a parameter which determines the gauge. The propagator for the vector boson B, is given by
The field
42 has its propagator i k2 -
r2 rng
These form of propagators are expected to give a renormalizable t,heory for any finite value of [ since they have good high k2 behavior, falling like $. This is called R-gauge. The fields B, and 4 2 separately have no physical significance. In particular the poles at, k2 = t2m i are tinphysical and are canceled out in any S-matrix 00, the element,, which is also independent of [, In the limit, [ B-meson propagator becomes --f
and 4 2 propagator vanishes. This is called unitary ( U ) gauge. The renormalizability is not obvious in this gauge.
433
Electroweak Unification
13.4 Electroweak Unification As we have discussed in Chap. 11, the leptonic charged current ~. of weak interactions has the form V.7, (1 - 7 s ) e = 2 i i , ~ ~ ’ eThe corresponding hadronic charged weak current can be writken as %yp (1 - 7 5 ) d‘ = 2fi~y,di. Here d’ means that it is not, mass eigenstate. This suggests that we consider
as left handed doublets in a weak isospin space. The weak currents are then associated with weak isospin raising and lowering operators
(13.21) where $ L is any of the above doublets, r+ = (71 +ir2) and 7- = (rl - 2 7 2 ) . Let, the charges associated with these currents be Q+ and Q- . These charges generate an s U ~ ( 2 algebra )
[Q+, Q-I
= 2Q3.
(13.22)
The current associated with the charge Q3 is given by
Jl
G L 1~ T ~ ~ , Q L .
(13.23)
) The gauge transformation corresponding to the group s U ~ ( 2 is
qL(z) -,
q L
(a
(z) = exp i- . A ( r )
)
@L
(4.
(13.24)
Then the Lagrangian
W,” I
(13.25)
Electroweak Unification
434
where
D,
3,
=z
+ 2 g21- r .
(w
W, = 8,
+ igW,,
(13.26)
--r.W,) 1
'"-2
W,,
=
W,,
a,W, - a,W, - gW, x W, 1 - T . W,, = D,W, - D,W,
(13.27a)
+ 29 [W,, WU]
(13.2713)
2
= a,wu - &WP
9
is invariant under the gauge transformations: QL
(4
w, where
uQ L ( 4
-+
-+
2 uw,ut--ua,ut
(13.28a)
9
U is given in Eq.(24). For A infinit,esimal, we get QL
(x) W,
+ 22 7
A (x)) Q L (x)
-+
(1
-+
W,-AxW,--a,A.
9
1
(13.28b)
9
The gauge group s U ~ ( 2 leads ) to a neutral current, J," which is neither observed experimentally nor is identical with the electromagnetic current. It, is possible to unify weak and electromagnetic forces into a single gauge force, if we extend the gauge group to SUL(2) x U y (1).For this group we have two gauge couplings g and g' associated with SUL(2) and Uy(1) respectively. The weak hypercharge Y is defined by the relation Q = t 3 i Y = i ~ 3iY.The gauge vector bosons W*, W obelong to the adjoint, representation of SUL(2) and vector boson B, is associated with Uy(1). Fermions belong to either fundamental representation [doublet] or trivial representation [singlet]. The structure of charged
+
+
435
Electroweak Unification
weak currents suggest the following assignments: 1st generation dR
Y
-1
-2
413
113
-213
2nd generation CR
,
3rd generation
(
: ) L 1
In order to brez the gauge symme :y spontaneousl: so that weak vector bosons acquire their mass, we need a Higgs doublet $:
$=($).
Y=l.
(13.29)
The Lagrangian invariant under the local gauge transformations QL
3
e x p ( i r . A + - Yi ~ A o 2
QR
is given by
3
exp ( i ~ f iQh R ~)
(13.30)
Electroweak Unification
436
where W,, is given in Eq. (27a) and
(13.34)
a = 1 , 2 with q R l = e R or d R , ~ R =Z u ~Under . the infinitesimal gauge transformations (30), vector fields W, transform as given in Eq. (28), but B, transforms as
B,
+ B, -
1
-8 Ao. 9’
(13.35)
cL
In order to break the gauge symmetry spontaneously, assume that,
(13.36)
J-rL2/x,
where ZI = ($)o = (0 1410). In this way, not, only SUL(2) is broken but, Uy(1) is also broken, but it leaves the group V(1) corresponding to electric charge unbroken viz. s U ~ ( 2x) & ( I ) is broken to U Q ( ~ )We . can now write Eq. (29) as
(13.37) where 4+ and hermitian fields 41 and 4 2 have zero vacuum expectation values. We can select a gauge such that g5+ and b2 disappear
437
Electroweak Unification
ap+&
from the theory. Instead and a p + 2 provide longitudinal components to W* and one of neutral vector bosons respectively. Thus out, of the four gauge vector bosons, three become massive and the remaining one remains massless. This massless vector boson is the photon corresponding to unbroken UQ(1) symmetry. All this amounts to replacing given in Eq. (37) by (41 = H )
+
+=(
K$) 0
(13.38)
With Eq. (38), the following term of the Lagrangian (31)
gives LW-H
=
+g2 ( H 2 + 27)H + u2) (2WlWp- + W 3 p w3.) 8 +-9’2 ( H 2 + 2vH + B”B, 8 -& ( H 2 + 27)H + W”B,, (13.39) 4 -1a p H a p H 2
712)
71’)
where WE = (WlpfiW2p)/&. Fkom this equation, it is clear that vector bosons W’, have acquired a mass:
rnb = 29 1 2 v 2.
(13.40a)
For the neutral vect,or bosons, the mass terms in Eq. (39) give the matrix (13.40b) Since det (Ad2)= 0, therefore one of the eigenvalues of M 2 is zero. The mass matrix (40b) can be diagonalized by defining the physical
438
Electroweak Unification
fields A,, 2,
A,
=
2,
=
cos BwB, -sinOwB,
+ sin OW W3,, + cosOwW3,.
(13.41)
Then we get 2
A,: photon
mA = 0,
(13.42)
(13.43) where 9’ tanow = -
(13.44)
9
and the parameter p-
mL m i C O S ~ew
=
1.
(13.45)
The fermion masses are given by m . - h.2 -
1)
%fi
(13.46)
and Higgs boson mass is given by 2 mH = 2Xv 2 = -2p2,
(13.47)
From Eq. (31), using Eqs. (41), (44), (40a) and (46), the Lagrangian for the fermions can be written as: LF
=
Gi 27,
(9
--
8, - m i- -H ) 8i 2mW
9 -2 7’1(1 - 75) (T+W;
+ T-w;) 92
2 f i
-e
G iy p Qi Q iA,
+ 2 cos 8w
*i
7’ (gvi - Y 5 9 A i ) ~i
(13.48)
Electroweak Unification
where
9i =
, d:
439
( 7; ) ( ::) , and
= VZj dj
and
Qi
e = 9’ cosOw = g sinow =
(V : CKM matrix),
is its charge. g v i and
gvi =
(q3- 2Q3
Tf
1 2
= -Tf,
mi
Q A ~ are
sin 8,)
is the mass of ith
given by
,gAi
= q3
1 2
773 = - 7 3
(13.49a) (13.4913)
We note that the interaction part of the Lagrangian can be written as
Lint = -gsinOw J,”, A, - -(J” 2fi
W’,
+ h.c.) - ___ J z p 2, cos 8w (13.50a)
where
1
= - J3’ - sin2OW J,”, 2
-4sin2 OW ( - E 7p e
+...
2 + -fi 3
7’ u -
(13.50d)
Electroweak Unification
440
where ellipses in Eqs. (50) indicate repitition for tlhe second and third generations. For low momentiim transfer phenomena, q2 > 4~712, 4m;, p e M 1, pj 1)
da -- -{(1+cos28) dN,f dSl
[Q;-21
4s
Qf llevf 6 sin2OW cos28w Rex (4
+ u:)(vp + a;) I x ( 4 I?] + (16(v,2sin2 Ow cos2 Ow)? +case
[- 16sin24QfOw cos2Ow R e x (s)
8vevjaeaf + (16 sin2 Ow cos2 Ow)2
(13.108a)
where S
x(4 ve
vf
(13.108b)
= s-m2,+imzrz' = 2gve = -1 4sin2ow, a, = 2g.4, = -I f f = 2gvf = 2T3, - 4Qf sin2Ow, af = 2 g A f = 2T3,. (13.108~)
+
Near and on the peak, integrated cross section is dominated by 2-exchange and we get from Eq. (100):
Electroweak Unification
456
where (13.109b) and the effect, of radiative corrections are contained in 6(s), the large effects due to initial e* bremsstrahliing are represented in 6. The other radiative corrections which lead to improved Born approximation have already been discussed in Secs. 3.2 and 4. The LEP data is fitted with an additional modification i.e. by replacing s - m i i m z r z by s - m i i&rz . Note that the expression for l?j is given in Eq. (100). Thus by measuring cr:eak for a particular final state e.g. e+e- itself, one can directly obtain r e e / r ~or reer z rffand therefore r j j . These widths have already been discussed in the beginning of this section. The forward-backward asymmetry is defined as:
+
+
'
(13.110) It is clear that this asymmetry is given by cos 0 term in Eq. (108a). Near and on the 2-peak, we get, from Eqs. (110) and (108c)
(13.112) Taking into account the radiative corrections [cf. Eqs. (101) and (103)], we get, from Eq. (112) 3 (1 - 4.S;)l
[1+ (1 - 4 s g 3(1-4(l+Ak)~%)~ [l 4 (1 Ak) s?]' '
+
+
(13.113)
Decay Widths of W and Z Bosons
457
where s: = 0.23116 and
AkBo -
-do(+) _---
3 (1 - 4s2,)2 [I
= 0.01685 (0.01683 Az 0.00096) .
+ (1 - ~ s Z ) ~ ]
do(-)- 2a2
1
S
(13.115b) Hence the polarization at s = m i is given by
- --2vf af2 ' -- -2 gvf/gAf . 1 + s;f/sif up af
+
(13.116)
We can also write Eq. (116) in terms of effective mixing angle s f : 1- 4s;
Al= 2 1
+ (1 - 4s?)2'
( 13.117)
Using the value A, = 0.1431 f 0.0046 we obtain s; = 0.23201 f 0.00057 to be compared with si = 0.23116 f 0.00022.
Electroweak Unification
458
13.6 Tests of Yang-Mills Character of Gauge Bosons The vector bosons self-couplings are given by the Lagrangian (31)
Lw
1 4
= --
p,w, - 8”W, - g (W, x W,)I2.
( 13.118a)
This gives the trilinear W+W-W3 coupling as
Lw
.9
= 2-
2
[(a,w3v - avW3,)
( W - w + ”- W+Pw-”)
+ (a,w,+- avw;) ( W 3 W ” - W3W-p ” > -
(ay;
-
a,w-P > (W 3 W + ”- W.”W+”)]. (13.118b)
Using W; = sin &A, + cos O,Z,,, the above equation gives for the W+W-y and W+W-Z vertices
+ ( h- k 2 ) ,
-
(h- kz)pga?] ,
(13.118c)
p and y are the indices of polarization vectors of W - , W+,W 3respectively. On the other hand from Eq. (39), t,he Higgs
where a ,
coupling to gauge bosons is given by
LW-H
g2
= -(H
8
+ 1 1 ) ~2W:W-’ + cos21 ow Z,Zp] ~
(13.119a)
and the Yiikawa coupling of Higgs to leptons is given by
LlrH where
Jz
hl = -rnl 2)
=
1 -hll lH,
(13.119b)
a
112
= ( 2 2 / 2 G ~ ) rnl.
(13.120)
Tests of Yang-Mills Character of Gauge Bosons
459
One process in which the trilinear couplings can be tested directly is e+ e- -+ W+ W - .
+
+
In the lowest order of 9, the diagrams shown in Fig. 6 contribute to this process. We are interested in the high energy behavior of the amplitude M . The bad behavior comes from the longit,udinal polarization of W’s. For this case p = 0. The longitudinal polarization vector E: for a W-boson of four-momentum kp is given by
(13.121) gives the It is the first term in Eq. (121) viz. %which mW worst high energy behavior. The amplitude may grow with high energy due to this term, if it is not compensated. In fact 51s E + 00 the diagrams of Fig. 6 give
(13.122) (13.123)
It is clear from Eqs. (122) and (123) that there is no possibility of cancellation between MLL( u ) and MLL( b ) even if e = g sin Ow. The third diagram, arises due to trilinear couplings - a feature of gauge theory. All the three diagrams cancel the bad high energy
460
Electroweak Unification
Figure 6 Production of W-W+ pair in e-e+ collision through v,,y and 2 boson exchange.
Tests of Yang-Mills Character of Gauge Bosons
46 1
I I I
:H I
I I
I I
Figure 7 Production of W-W+ pair in e-e+ collision through Higgs boson exchange.
behavior except for the last term in Eq. (122), which gives S-wave cross-section for s >> m&,os = This is in conflict with the unitarity constraint [Eq. (2)] os 5 This conflict thus starts at,
2s.
F.
4 4
s = -7r
2
= (1.2 TeV)
.
(13.125)
GF
However, even this term is canceled by the diagram (Fig. 7) due to Higgs exchange. This is because this diagram gives for s >> m&: (13.126) Thus there is no trouble with the high energy behavior in the standard model if m$ < (1.2 TeV)2. There is similar cancellation for the amplitude MLT, which for each individual diagram goes as constant when s t 00. o(efe- ---f W - W’) depends crucially on gauge cancellation discussed above. For example, o(Y - exchange) 1rCX2S for m& 96sin4(JwmL , this would be the only contribution
Electroweak Unification
462
Behavior of ( r s M with energy E .
Figure 8
without W - W+y and W - W+Z vertices. On the other hand, with the above cancellation (13.127)
The cross section also contains the threshold factor \il which tends to 1 as s 4 00. Thus the cross section grows near the threshold and then falls like at large values of fi >> rnw. The cross section is 10-35~m2 at its maximum which occurs at about 40 GeV above W+W- threshold. The situation is shown in Fig. 8.
5
N
Upper Bound
463
13.7 Higgs Boson Mass
The Higgs potential
v (4) = p 2 d 2 +
q52 = $4
(13.128)
goes over to
1
1
1
- -Xu4 V ( H ) = (2Av2) H 2 + AvH4 - -AH4 4 4
(13.129a)
when the symmetry is spontaneously broken; p2 = -Xu2 (A > 0) . Thus we see that the Higgs boson maSs (13.129b) is arbitrary. We now discuss theoretical bounds on the Higgs boson mass. 13.8
Upper Bound
(a) Unitarity: We have seen in Sec. 13.6 that the Higgs boson contribution to the cross section for the process e-
+ e+
---f
Wz
+ WL
is given by (13.130) Then comparing it with Eq. (2), we get (13.131) This requires a “cut-off” (signaling new physics beyond As,):
I ( 4 f i ? ~ / G F ) ’ ’= ~ (1.2 TeV)
(13.132a)
Electroweak Unification
464
To avoid this conflict, the Higgs mass m~ should be such that ULH
< Ag,Wci = (1.2 TeV) .
(13.132b)
We saw at, the beginning of this chapter that, m w
(13.167b)
Theory is thus anomaly free when
Tr
A t ) A,")
- Tr
({A:,
=0
(13.168)
for all values of a, b, c.
Examples (i) Vector or vector like gauge theory:
AL = AR # 0.
(13.169)
A,L = A:
(13.170)
Af; = U-lA,RU,
(13.171)
For such theories either or where
U is a fixed unitary matrix. The gauge current is given by JL
+
=
% L ~ , A ~ Q L% ~ y , h f Q ~ GLy,A:QL + @Ry,UAtU-lQ~
=
Gy,A,Q,
=
where 9 = @L
+u-l@~,
(13.172)
Aa =
At,
(13.173)
is a pure vector. Note that in general the redefinition of @ generates y5 terms in the fermion mass matrix. Such a theory is caIled vectorlike.
Electroweak Unification
478
In QCD, the left handed and right handed quarks belong to the fundamental representation 3 of SUc(3).Thus it, is a vector theory and is anomaly free. (ii) AL = AR = 0. In this case, fermion representation is such that anomalies cancel separately for left handed and right handed fermions. This is the case for example for S U ( 2 ) . For the fundamental representation 2 of S U ( 2 ) , A, = ir, and since Tbr, = 2&b,
+
The representation 2 is a real representation in S U ( 2 ) . But this is not, the case for S U ( n ) , n > 2, e.g. representation 3 of SU(3) is not equivalent to 3*. Thus S U ( n ) , n, > 2 is not safe in general. However, fermions belonging to an octet representation of SU(3) are anomaly free since octet representation is real. This can be seen as follows: If A, form a representation, -A: also form a representation. The negative sign arises, since matrices A: satisfy the commutation relation [A:, A:] = -i f a b c l \ z . (13.175) Hence -A: form a representation conjugate to A,. If (as in the case for real representation),
A, where
=
-U-lA:U,
(13.176)
U is a unitary matrix, t,heri
Thus in general real representations are safe. They do not produce axial anomaly. However, a safe representation need not be real. (iii) The standard model SUc(3) x SU(2) x U(1).
Axial Anomaly
479
We need to consider S U (2) x U (1) only as SUc(3) is anomaly free. The matrices Atand A: are given by
1 1 Q = -73+-Y. 2 2
(13.178)
Now T R ({Tf,
TF} 7:)
(13.179)
=
so that from Eq. (168), we have to show that
Tr ({&T~}YL) Tr YL, 2 6 a b Tr [2Q - 731
= 2bab =
= 4bab
Tr Q = 0
(13.180)
and
(13.181)
Tr [Y;] - T r [Yi]= 0 for the cancellation of anomalies. Now
( 13.182)
T r [Yi] = 8Tr [Q3] Tr [Y,"]= Tr [SO3 + 6Q 732 - 6Q2 73 - T:] .~
8TrQ3
=
+ 6TrQ - 6Tr (Q2
73)
(13.183)
But
Tr[Q3] Tr
[Q27-3]
0;
TrQ
0:
Tw3 = 0.
(13.184)
Hence for the cancellation of anomaly, we must have
Now
Tr Q = 0.
(13.185)
2 1 T r Q = [O - 1+ 3(- - -)] = 0. 3 3
(13.186)
480
Electroweak Unification
Hence in the standard model; lept,on anomalies cancel quark anomalies. Note that, in the cancellation of anomalies, color plays a crucial role. Left-handed fermions anomalies cancel among t,hemselves and so do the right-handed fermions anomalies.
Bibliography
13.13
48 1
Bibliography
1. J. C. Taylor, Gauge theories of weak interactions, Cambridge University Press, Cambridge, U. K. (1976). 2. M. A. Beg and A. Sirlin, Gauge theories of weak interactions, Ann. Rev. Nucl. Sci., 24, 379 (1974); Gauge theories of weak interactions 11, Phys. Rep. 88 C, 1 (1982). 3. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Meld Theory (Addision-Wesley, Reading, Mass. 1995). 4. G. Altarelli, “The Standard Electroweak Theory and Beyond” CERN-TH / 98-348 hepph / 9811456. 5. J. Ellis, “Beyond Standard Model for Hill walkers” CERN-TH / 98-329,hepph 9812235 6. J. L. Rosner, “New developments in precision electroweak physics” Comment Nucl. Phys. 22, 205 (1998). 7. W. Hollik, “Standard Model Theory” CERN-TH / 98-358; KATP-18-1998 hep-ph / 9811313, Plenary talk at the XXIX Int. Conf. HEP, Vancouver Canada (1998). 8. M. E. Peskin,“ Beyond standard Model” in proceedings of 1996 European School of High Energy Physics CERN 97-03, Eds: N. Ellis and M. Neubert. 9. M. Spirce and P. M Zerwar, “Electroweak Symmetry Breaking and Higgs Physics” CERL-TH / 97-379, DESY 97-261 hep-ph. / 9803257 10. Particle Data Group, The Euro. Phys. Journal 3, 1-4 (1998).
Chapter 14 DEEP INELASTIC SCATTERING 14.1 Introduction Lepton-nucleon scattering is an excellent tool to study the structure of nucleon. Electron’$nuon) scattering clearly shows that nucleon has a structure. Consider for example the scattering e+p+e‘+X.
Let E be the energy of the incident electron e and E‘ be the energy of the scattered electron. Let q = k - k’ be the momentum transfer. Then in the lab. frame, the four momenta P, k and Ic‘ of the target (proton), initial electron and the scattered electron are given by
P
f
k‘ E
(M,O), k (E’,k’).
=
(E,k)
Neglecting the mass of the lepton, we have
k - k’= EE’cosO.
(14.la)
We define another invariant v:
Mu=P.q. 483
(14.lb)
Deep Inelastic Scattering
484
In the lab. frame
u
= 40 =
( E - E').
(14.lc)
We also define the invariant mass: 2
s = px = (4
+ P ) 2= q2 + M 2 + 2Mv.
(14.ld)
Note that 2 M u + q2 2 0; for elastic Scattering 2Mv = -q2. The elastic scattering of electrons on spinless proton can be written in terms of Mott cross section:
da dR
(14.2a) Mott
where
(14.2b) The structure of the proton manifests itself in term of the form factor F ( q 2 ) .In elastic scattering proton recoils as a whole and the scattering is coherent. The form factor F ( q 2 ) measures the charge distribution of the proton, viz.
(14.3a) If we expand F ( q 2 ) in powers of q 2 , we get
F(0)
=
/ p ( r ) d 3 r= 1
ylq2=o =
-27r
(14.3b) ( r 2 )is called the mean square charge radius.
DeepInelastic Lepton-Nucleon Scattering
485
It is convenient to write the Mott cross section in the form
($),,,,
4Ta2
=
-
2e
E’
(7) E C O S 2
This is the scattering cross section for the scattering of electrons on spinless (structureless) particles of mass m. The scattering cross section for the scattering of electrons on structureless spin 1/2 particles can be calculated using the standard trace techniques and is given by [Q2 = -q2] 47m2 E‘ -da - - -cos2 dQ2 Q4 E
14.2 Deep-Inelastic Lepton-Nucleon Scattering We now consider the inelastic scattering of electrons on nucleons (see Fig. 1). For this case the matrix elements are
The cross-section is given by [cf. Chap. 21
x
m2
S(Px + k‘ - k - P ) 2 L , , W p ” ” , EE‘
where [see the Appendix A]
vin
lkl =-
E
(14.6b)
Deep Inelastic Scattering
486
Figure 1
Inelastic charged lepton-proton scattering.
(14.6~) and
Here S denotes the spin of the target and En denotes the sum over all the quantum numbers of state X and integration over d3Px. Then the differential cross-section is given by
Assuming invariance under C, P and T and conservation of the electromagnetic current = 0, the Lorentz structure of
487
DeepInelastic Lepton-Nucleon Scattering
Wp” is
Here S2 = SpSp = -1, S - P = 0 and FI and F2 are spin averaged structure functions: MWl E F1 and VWZ_= Fz while the remaining two are spin dependent structure functions. In Fig. 2, we show the plot of Q2(= -q2) versus 2 M v where we have defined the variables:
x=-
E-E’ 2Mv ’ y = E = - E Q2
Y
(14.8a)
OIyyl. Now
so that
2 P . q - Q 2 2 0 or 0 5
IC
5 1.
(14.8b)
If hadron masses are not important, F’s could not depend on
Q2and one might expect that scale invariance holds in the asymptotic (Bjorken) limit Q 2 ,v + 00 with x fixed. In the “naive” quark model (where the virtual photon interacts with point like constituents), in the limit of quark masses ---t 0, there are no dimensions and this suggests that in the asymptotic limit the structure
488
Deep Inelastic Scattering
Mu Figure 2 Plot of momentum transfer Q 2 versus energy transfer u = E - E’ in charged lepton-proton scattering, showing various kinematic regions.
489
DeepInelastic Lepton-Nucleon Scattering
functions scale:
Mwi(v, Q 2 ) vwz(v,Q 2 )
f
J'2(v,Q 2 )
91,2(v, Q 2 )
---t
91,2(z).
Fi(v,
Q2)
4
Fi(2)
+
F2(z)
(14.9)
In QCD, however, this scaling is broken but, only by logarithms of
Q2/A&m.
F'rom Eqs. (6) and (7), the spin averaged cross-section is
given by d2a dS2 dE'
Wz(v,Q 2 )
+ 2 tan'
0 2
1
-Wl(v, Q2) . (14.10a)
It is instructive to write this cross-section in the form d2a 9I WZ(V, Q2) 2 tan2 -Wl(v, Q')] . dQ2du 2 (14. lob)
+
We now define right and left, polarized cross-sections as UR,L
= (T &
A(T,
(14.11)
where d2a/dQ2dv is given in Eq. (10). In terms of the variables 2 z, y and K = (I - $)[= 1 - ? i!& +? 1 in the scaling limit and is Q2 a measure of how close one is to the limit, Q 2 -+ 001, we have d2a
- -
2(1 - y)
dx d y
1 + 2y2(K - 1)) F2]
(14.12) 2 given by and polarized asymmetry A a = CTR- a ~ / is
dAa
47ra2
- = dx dy
ME
Q4
x
[ {(':: + COSP
2 1.-
-
-(K-
)
1) g1 - y ( K - l)g2
11
(14.13)
490
Deep Inelastic Scattering
At high energies y + 0 and F2 and g1 dominate. It, may be noted that g2 has never been measured. In Eq. (13) p is the angle between k and spin quantization direction S. If the target is longitudinally polarized p = 0. The presence of the structure functions in Eq. (10) indicates that proton is not a point particle. The structure of t,he proton can be probed in two ways - one by elastic lepton-nucleon scattering and second by deep inelastic lepton-nucleon scattering. First we discuss the elastic scattering for which v = Q2/2M. For this case the structure functions are given by
where T = Q2/2M. Thus from Eq. (lo), we have
+ 27 tan2 2 [FI(Q2)+ F2(Q2)I2}.
(14.15)
The form factors for the proton are normalized to F,P(O) = 1, F l ( 0 ) = /cCp and for the neutron Fp(0) = 0, FT(0) = ten where IC* = 1.792 and K, = -1.913 are anomalous magnetic moments of the proton and the neutron respectively. Experimental data is analyzed in terms of Sachs form factors
These form factors are normalized as follows: GpE(0) = 1, GpiM(0) = pp = 2.792, Gk(0) = 0 and G&(O) = pn. In terms of GE and G M ,
DeepInelastic Lepton-Nucleon Scattering
49 1
t,he elastic scattering cross-section is given by
(14.17) The experimental data is fitted remarkably well by a single form factor
GnE(q2) = 0,
(14.18)
where rn; = 0.71 GeV2. From Eq. (15), we get [cf. Eq. (3b)l 12 (14.19) - 0.66fm2, (T$) n = 0.
(s)Mo
as Now Eqs. (14) and (15) clearly show that $$ -+ Q2 + 00 i.e. cross section rapidly falls as Q2 become large, clearly showing that the nucleon has a “diffused” structure in the elastic region. But the behavior of the structure functions W2 and W1 is quite different in the deep inelastic region. The experimental data in this re ion indicate that the cross section stays large and is of the characteristics of a point particle. This clearly order of indicates that in this region the scattering is incoherent, and is what one would expect if a nucleon consists of non-interacting or weakly interacting point like constituents called partons (quarks). This scattering region thus gives us information about the elementary constituents of nucleon, i.e. about their charges, spin and flavor. Moreover, the structure functions vW2 and MWl show Bjorken scaling i.e. vW2 and MWl -+ F ~ ( Zand ) F l ( z ) as Q2, v -+ 00 where z = is fixed. This is clearly indicated in Fig. 3 where Fz(z) is plotted against Q2 for various values of 2. The above characteristics lead to parton model of deep inelastic scattering which we now discuss.
qs)Mott,
492
Deep Iiielastic Scattering
Figure 3 The structure function Fz measured by the CERN muon experiments, ( a ) proton ( b ) nucleon in deuterium.
493
Parton Model
14.3 Parton Model Partons are quarks (spin 1/2), antiquarks (spin 1/2) and gluons (spin 1). Gluons do not, contribute here since they carry no electric charge. Thus we shall deal with spin 1/2 partons. If the target is a free quark of flavor i , of mass m and charge ei, we have from Eq. (64
+
x S4(p Q - pn).
(14.20a)
Now G h = d 4 p n 6 b i - m2]and we obtain from Eq. (20a) pno
m
--y 27r
r
m e 3 a ( p s ) y P[11+
x
S(2p . q - Q 2 ) .
4 + m]y v u ( p s ) (14.20b)
To proceed’further, we make use of t,he following identities of Dirac matrices algebra [see Appendix A],
Deep Inelastic Scattering
494
Thus the comparison with Eq. (7) gives
1 2 91i = -b(x 2 - l)ei,
g2i
(14.2213)
= 0.
Hence from Eqs. (12) and (13) for a spin 1/2 parton i,
(14.23a)
dAai dx d y
47ra2
- -e:mEcosp
-
Q4
1
1) 6(z - I).
(14.23b)
The comparison of Eq. (23) with Eqs. (12) and (13) clearly shows that if we replace 6(1 - x) in Eq. (23) by some distribution functions F ( x ) and g ( 2 ) we get Eqs. (12) and (13). Hence it follows that in the scaling region, the nucleon is behaving as if it consists of point-like constituents and the structure function Fzi(x)or Fli(2) or g l i ( 2 ) gives us the 2-distribution of point-like constituents inside the nucleon. The point-like constituents have been assumed to be free i.e. interaction between them can be neglected in the scaling region. This is compatible with QCD, as QCD is asymptotically free. More accurately one can write Fz and Fl as F2(x, Q 2 ) ,F l ( 2 , Q 2 ) ;but the dependence on Q2 is very weak (logarithmic). The following physical picture emerges. In the deep inelastic region, the virtual photon interacts in an incoherent manner and probes roughly the instantaneous construction of proton. In the center of mass frame of electron and proton, we can write (neglecting lepton mass):
k
= ( P , 0, 0, P ) , P =
[(AdZ+ P
y
21
(
P 1+ )";"
0, 0, -P
]
Parton Model
495
2Mu - Q2 4P Let us assume that the target (proton) has point-like constituents .~ called partons of flavor, i. Neglecting any parton momentxm transverse to the target, let us assume that the longitudinal momentum of a parton is given by p = x P . The time of interaction of photon is given by 1 4P 2P 7 = - = qo ~ M -vQ2 Mv(1 - X ) ' 40 =
~
-4
The energy of a parton =
M
z P ( 1 iph), 2 +m2 so
that the lifetime of virtual parton states is 1
2P For x not going to 0 or 1, T be a state at time t . In the XI and X2 basis, we can write
d
idt
I*@))
=
m2 -
;r2
) I@(t)).
(15.33b)
The solution is
a(t)
=
a(0)exp
b(t)
=
b(0)exp
[ , (m2 -2
-
; PI
-r2
(15.34)
Suppose we start, with Xo,viz. IQ(0)) = lXo), then from Eq. (25), we get,
a(0) = b(0) =
JIPI'
+ Iq12 2P
(15.35)
General Formalism
521
Hence from Eqs. (33a), (34) and (35), we get,
IV>>
+ exp [ (--im2- -rz 2 t /x2) l ) 1 IW))
{
= 2 exp(-iml-
-2P [exp (-iml
1
(15.36a)
1
1
-rip 2 + exp (-im2 - -r2) 2 t ) 1x0) -
-rl) 1 t - exp (-im2- -r2) 1 t] 1x0). 2 2 (15.3613)
Equation (36b) clearly shows the particle mixing. Similarly if we start with X o we get, at time t:
(W)) - exp
[ (-im2 -
(15.37a)
IW)
-{1 P- [exp (-iml- -rl 2
(
2 '
- [exp (-iml- -rl t + exp -im2-
(
2
) ] Ix -r2 ) t] I x-"} . 2
t - exp -im2 - -r2 t 2
O)
(15.3713)
xo
From Eqs. (36) and (37) we can determine X o and mixing. It is clear that if we start with X o , then at time t , the probability of finding the particles Xoor is given by [using Eq. ( 3 6 ~ 1 = e-rzt 2e-"t cos Amt] 4
xo
I(xoI+(~))/~
+
+
Particle Mixing and CP-Violation
522
(15.38) We define the mixing parameter r as (15.39) where T is a sufficiently long time. In the limit, T (38)' we can easily determine:
r=
2+y2 Il+&l 2 + x 2 - y 2 1 - E 2 -
--f
m,using Eq.
(15.40)
'
where x = A m / r and y = Al?/2I'. If we start, with X o , we can use Eq. (37b). Then we find
When CP-violation effects are neglected, then (15.42) The asymmetry paramet,er a (15.43) is a measure of CP-violation. We define another parameter x which is also a measure of particle mixing. Let x be the probability of X o --+ then
xo,
General Formalism
523
Thus
(15.45) Similarly, we get
We note from the definitions 2 =
9,y =
217
0 5 22&b
=
0.
(15.67a)
To leading order in A, tthis relation can be written, using Eq. (66), as v,*, &d - Kb = 0. (15.67b) The relation (67) can be represented by a triangle in the complex plane (Fig.3). In particular we note that
+
2rl (1 - PI (15.68) q2 (1 - p)2 . ii) If the quarks, t , c, u have nearly the same mass, sum of their contributions would involve ( 1 vqd Vgtb)’ which vanishes sin
2p
=
+
q=u,c,t
due to the iinitarity condition (67). Sincc rnt >> mC,it is clear that, dominant contribution comes from exchanged t -quark. iii) In the standard model all the complex phases enter through CKM matrix (see Eq. (66)). In particular arg [A(&
-+
Bd)]
= arg = arg
I$[
[ W d
y;,2]
BOBo Mixing and CP-Violation
531
Figure 3 The CKM-unitarity triangle in the Wolfenstein parameterization.
(15.69a) where
~FKM = arg(&:&b)
= p.
(15.69b)
&om the above considerations, using Eqs. (66) and (67), one finds the main results from the box diagrams:
BjBj System: M12 0: (vtbV,*,)2
mi = [AX3 (1 - p
+ iq)I2 m:
(15.70a)
while [cf. Eqs. (65) and (67a)l r12
0: ( K b
=
v,*, + vub vtd)2mi
(I& V,+,)2V L ~= [AX3 (1 - p
+ iq)J2mi.(15.70b)
Hence 1M121 >> IF121 and both A412 and I'12 have the same phase in the leading order. Thus it follows from Eq. (69):
M12 = lM12(e2ip,r12=
lr12le2@,
(15.71a)
Particle Mixing and CP-Violation
532
and
(15.71b)
B,OB,O Systems:
Hence we have
(15.73)
We also note that
(15.74) Using Eqs. (24) and (71), we get for the Bj that A 4 1 2 and l712 have the same phase).
- Bj system (noting
(15.75) From Eqs. (28) and (75),we then obtain
(15.76) Hence [cf. Eq. (73)l
Ar Am,
= d
P
If>, 77iP = f l .
If)
(15.95)
Then it follows from Eq. (83) that z ~=f l / Z f , so that Eq. (88) gives I ~ f =l 1, 43 E 6,- 8, = 0, or r, (15.96) accordingly as Eqs. (89)
T],&~ =
f 1 [cf. Eqs. (86)]. Then it follows from
Then the asymmetry is given by
Particle Mixing and CP-Violation
538
As qAP (31)is known and Am is already measured, this asymmetry measiires 4, independent of strong phases. The time integrated CP asymmetry is (15.99a) which on using Eqs. (97) gives
Af
= -qLP
I? sin 24
Am F2 Am2 X
f sin24 = -vcp
+
(15.99b)
___ 1f X 2 '
For B0 (BO) II,Ks,T A P = l l C P ($) YlCP ( K , ) = (+l)(+l) = 1 and in the standard model from the transition 6 CCS, it is easy to see that, @f = 0 since the CKM elements involved are Vz V,, which according to Eq. (66) do not, involve any CKM phase. Thus according t,o Eq. (90) +
--f
4 = P.
(15.100)
Thus from Eq. (98) the asymmetry is given directly in terms of CKM phase /3 (independent of strong phase).
A Q K (~t ) = - sin 2P sin Amt
(15.101)
and the time integrated asymmetry is A + K s = - sin20
If p
X ~
1 +x2'
( 15.102)
2! 10' and x d N 0.7 (see below), then the asymmetry IA,,+K~I N 0.16, which is much larger t8hanE N in kaon decays (see Sec. 5).
BOBo Mixing and CP-Violation
539
Finally we consider the decays [for example X- = D-, X+ =
D+l
In the standard model Bo decay into P v X - and Bo decay into e-a X + are forbidden. Then A ( f ) = 0 = A(f) i.e. zf = 0 = xf so that from Eq. (85). 1
r ( B o ( t )+ f )
+ cos Amt]
cx
IA(f)12e-rt[I
=
r (B0(t)+ f ) .
On the other hand from Eq. (84a) [replacing f by
( B V ) -+ f)
(15.103)
f]
.q Am IA(f)lZe-rtI - 2- sin - tI2 P 2
CX
1 2
-IA(f)I2 e-rt [I - cos Amt] (15.104) where we have used
(A(f) 1
= ( A ( f ) ( Integrating . Eqs. (103) and
(104)
n
=
-
2
X '
+22 =r
[cf. Eq. (40)].
(15.105)
Hence a nonzero value of S would indicate Bo and Bo mixing as in the standard model z~ = 0 = Zf = 0. But if xf and Ef are not zero due to some exotic mechanism, then 6 # 0, even if there is no mixing.
Particle Mixing and CP-Violation
540
The particle data group give for the Bj
-
B: system the
value Xd
=
=
rd
1 rd
+
F(P+
x-)+
x-)
r(p+ ryp0.172 f 0.010
x+) (15.106a)
and xd = 0.723 f 0.032.
(15.106b)
This gives a clear proof of the B: mixing in the standard model. However, in contrast, to KO - K'case (see t,he next section) 6 does not give any information about CP-violation. The value xd provides a determination of I& in terms of mt and fs 6( G J B ) which paramet,erizes the hadronic matrix ~p bLa) ( 2 elements of the four quark operator [cf. Fig. 11 yp bLc) between Do and Bo. This can be seen as follows: First we note that the dominant contribution to MI2 comes from the top quark in Fig.1 and it has been shown that [cf. Eqs. (76), (77) and (10511
(15.107a) where 1 F(x) = -
4
+ 4 ( 19 - ~ )- _32 (
1 3 x2hx - -~ l - ~ ) ~2 ( 1 - ~ )
(15.107b)
where ~8 is a QCD correction factor. The constant 19.4. This bound has been used to restrict the allowed p - q region for some representative values of [S. This results in the range N
0.2 < 7 < 0.4 0 < p < 0.4 with the best solution around p = 0.11, 7 = 0.33.
(15.110)
Particle Mixing and CP--Violation
542
Coming back to the ratio ( x c s / x din ) Eq. (109b), we see that, most of the models give ( J B , / J B ~ )= ~ 1.2 - 2. Thus taking T B ~M T B ~VZB, , Mm ~( J B~, / J ,B ~ )M 1 and constraint (110), we can safely conclude that, x, >> 2 so that, Eq. (105) gives r, = 1 and hence from Eq. (45), x, = 0.5. Any marked deviation from x, = 0.5 would indicate some new physics beyond trhe standard model. The particle data group gives xs > 0.4975, consistent, with the above standard model value. We conclude this section with the remarks that there is clear experimental evidence for Bo - Do mixing. The experimental determination of asymmetry parameter AqKS[cf. Ey. (102)] can give us information about the angle p. 15.5
CP-Violation in K°Ko System
We now apply the general formalism developed in Sec. 15.2 to the K0Ko system. Here we denote K1 and Kz as K s and KL. First we discuss hypercharge oscillations. Suppose that, at, t = O,Ko (Y = 1) is produced by the reaction 7r-p --t KoAo.The initial state is then piire Y = 1. It is clear from Eq. (36b) [with X = K ] that, a kaon beam which has been produced in a piire Y = 1 state has changed into one containing bot,h parts with Y = 1 and Y = -1. Experimentally gocan be verified through the observat,ion of hadronic signat,ure such as I?" p + 7r+ A' since 7rf Ao can only be produced by '?I and not, by KO. The probability of finding Y = -1 component, at time t in the kaon produced at, t = 0 in a pure Y = 1 state is given by Eq. (38) [I&] > rL (rs/rL
N
which is completely negligible. Hence we get, from Eq. (116) tan
2Am
QE M --
nr
(15.126)
which is clear from its derivation is a consequence of CPT invariance and the unitarity relation (65) approximated by dominant two-pion contribution. Putting the experimental values
Am = (mL - m s ) = 0.474 rs nr = rL-rs=-o.998rs we obtain the phase of
(15.127)
E
& = 43.49 f 0.08'
(15.128)
while Eq. (123b) gives 4Ej
= S;! - So
+ -7r2
N
48 f'4
( 15.129)
CP-Violation in KOKO System
547
where the numerical value is based on an analysis of Finally writing Eqs. (120) and (,121) as
7r 7-r
scattering.
the experimental measurements give:
1q+-1 $+-
(2.285f 0.019) x = (43.5 f 0.6)’ lqool = (2.275 f 0.019) x boo = (43.5 f 1.0)O A$ = $o, - $+- = (-0.1 f 0.8)’. =
Since E‘ involves the AI = [cf. Eq. (123b)l one has
x
(15.131) (15.132) (15.133)
f rule suppression factor IA2/AoI
IE’I mw] but most, theoretical calculations give
xi
Re
():
> m, mj = m, or md; E,! = Ei = E , m, Ej = 2E, where in the nonrelativistic limit, we have piit, Ei = m, and we also put, ma= md. Using Paiili representation of Dirac matrices, it, is a straightforward but long calculation to obtain the cross section CT for the scattering or annihilation processes shown in Figs. 7 and 8. Suppressing the color factor, we get, for the singlet, and triplet, scattering cross sections respectively
+
IEI2
as = -G; 81r gT
=
1
(8m2)-,
1tI2
-G; 8n
(16.131a)
91
1
(:m:)- PI
( 16.131b)
where is the incoming velocity in the initial state. Now defining the decay width as r = p3(o)12 u, (16.132) we get for the triplet state 1 8 r ("SI -+ u d-) = G i E2 - m& 8n 3
19,(0)12,
(16.133)
+
where we have put m;, = (m, ma)' M m;. For the singlet state, we get
('So
+u
1
2) = Gg t 28 m2 8n
(0)l"
(16.134)
Note the important fact that the decay width for the singlet state (D)is proportional to the square of the light, quark masses; in the spectator quark model it is proportional to m:. This is called helicity suppression.
594
Weak Decays of Heavy Flavors
Inserting back the color factors we have finally r,&h =
c% 87r
l%slZ
I V U(8~ mz) ~ ~ I@s ( ) I 2
(CI 4- 3c2)2, (16.135)
r:;n
=
87r
IV..I2 lV,dI2
(8 m 2 )[as (0)12(3C1
+ C2)’. (16.136)
It is clear from Eqs. (135) and (136), that both the exchange and annihilation diagrams are helicity suppressed, but rezch is color suppressed and is color enhanced. It is intresting to see that, for the annihilation diagram, one can get, the same result, just, by writing the T-matrix for the D, ---t hadrons in the form
rann
where J,”t and J W , are color singlet, currents with appropriate quantum numbers. Then
I(xIJW’I
x I(0 I-r,”tI a)I2 Now from Lorentz invariance
O)l2
*
(16.138)
(16.139) while
-
1 (2,)38 (Po) [ (-P2 +P, PA
(7
(P2)] ‘
SPA + P, PA) P (P2)
(16.140)
Weak Decays of Heavy Flavors
595
Hence we get
(16.141) Now from dimensional consideration
(16.142) Thus we obtain
One gets eactly the same results if in Eq. (137) one replaces (X) by Id)(see problem 3). Comparing Eq. (143) with Eq. (136) we get
(16.144) It is interesting to note that the vacuum saturation of the T-matrix for D, -+ hadrons viz. (X !Jw’”(0) (0 IJ,”t( D,) gives the same results as the annihilation diagram. From Eqs. (117), (135) and (136) and (143)’ we get
(16.145)
+
where in Eq. (146) the factor 3 (Ct Ci) appears for the reason discussed earlier in connection with Eq. (125). Using C1 =
596
Weak Decays of Hcavy Flavors
1.24, Cz = -0.48, m, M 1.50 GeV, F(m,/m,) M 0.47, fD = 200 MeV and f ~ =, 240 MeV, we get from Eq. (144)
( 16.147) ~~ Thc annihilation diagram gives negwhile I ? ~ ~ , J Iis' negligible. ligible contxibution to D s decays. Taking into account, Eq. (147)
r ( D s + hadrons) = (1.6) rSP where
rspis giver1 in Eq.
(16.148)
(117) and [cf. Eq. (121)]
+ c;)
3 (C? M 0.82. (16.150) 1.6 [3C? 2C1 Cz 3Cg] 7110 While the prediction (149) is consistent, with the experimental limit (D:),, < 20 %, the prediction (150) is not consistent, with its experimental value M 1.12. We conclude that, the contribution of the annihilation diagram is helicity suppressed, but enhancement by a factor of 1 9 2 ~ ' due to phase space and that due to color factor more than compensate the helicity slippression. However, there are still problems to explain the ratios rDt/rDoM 2.5 and I - ~ ~ , + / T ~ O M 1.12 as was discussed in Sec. 3.4. Finally the aririihilation diagram for B decays are Cabibbo suppressed and they may be neglected. 2 To i~ N
+
+
Problems
597
16.4 Problems 1. Taking into account finite width for p meson and using Eq. (17), show that
where
Hint: 2nS (s - m;)
-+
(s - ma)
+ m: r2
Considering the process e-e+
--+
y
--f
n+n-,
show that the cross section is given by (s >> 4m:) or+*- ( s ) = n a2
3s
I F,
( .)I2 (1 -
!33'2
where F, ( s ) is the electromagnetic form factor of pion:
Using Eq. (37), show that we get, back Eq. (A). From Eq. (A), find the decay rate for T- -+ n-7r0 vr through p-resonance.
Weak Decays of Heavy Flavors
598
2. Taking into account, finite width of al meson, and Eq. (18), shdw that, (taking m, = 0)
where F p x (s) =
fa, Faipn
[(s - m&) + a ma,
r].
The a l f n couplings are defined by the decay amplitude T:
where qp, E~ are polarization vectors of a1 and p, q and k are their four momenta. In order to derive (B), first show that
599
Problems
Using the experimental numbers for I‘( T - + T - po v7) and r ( a l Falp,and using = f,”= 2F; m;. 3. Using Eqs. (137) and (138) and writing + p T ) , determine
/ d3 Px 8 (P
T-)
- Px> -+
/
d3
Pl d3 P2 6 ( p - Pl - p 2 ) ,
show that
4. Writing ( O I J pwt
I D1)
f D : &p
where cP is the polarization of D;, show tjhat,,
Comparing it with Eq. (133) when multiplied by the color factor (3Cl C2)2,show that,
+
j& Hence show that
=
12 19, (o)12mp.
Weak Decays of Heavy Flavors
600
16.5 Bibliography 1. M. L. Perl, Rep. Prog. Phys. 55,653 (1992). 2. A. S. Schwarz, T physics in “Lepton and Photon Interaction” XVI Int. symposium, Ithaca NY 1993 (eds P. Drell and D. Rubin) AIP, p. 671; M. S. Witherell, Charm decay physics, ibid p. 198; M. B. Wise, Heavy flavor theory; ibid p. 253 3. Particle Data Group, Eur. J. Phys. C, 3 (1998). 4. J . L. Rosner “B Physics - A Theoretical Overview’’ Nuclear Instruments and Methods in Physics Research A 408, 308 (1998). 5. M. Neubrat, “B Decays and the Heavy-Qiiark Expansion” CERN-TH/97-24 hep-ph/ 9702375 [ To appear in the second edition of “Heavy Flavors” edited by A. ,J. Buras and M.Lindner; World Scientific] “Heavy-Quark Effective Theory and Weak Matrix Elements” CERN-TH/98-2 hep-ph/ 980 1269 [Invited talk presented at Int. Europhysics Conf. on High Energy Physics Jerusalem], Israel 19-26 Aug. 1997. 6. M. Neubrat, and B. Stech, “Non-Leptonic Weak Decays of 13 Mesons” CERN-TH/97-99 hep-ph/ 9705292 [ To appear in the second edition of “Heavy Flavors” edited by A. J. Buras and M.Lindner; World Scientific] 7. G. Buchalla and A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
Chapter 17 GRAND UNIFICATION, SUPERSYMMETRY AND STRINGS 17.1
Grand Unification
As we have seen all fundamental forces are of gauge nature. Thus they may be deduced from some generalized gauge principle. Ingradients of gauge models are (i) Choice of gauge group (ii) Choice of fundamental representations (iii) If gauge symmetry is spontaneously broken, choice of Higgs sector which generate mass parameters. Gauge principle restricts the form of interaction. Also gauge model may be renormalizable if its fermion content is such that the model is anomaly free. At low energies we have a spontaneously broken SU(2)xU(1) gauge group for electroweak forces and an exact SUc(3) gauge group for the strong quark-gluon forces. Thus the standard model involves
GI
EE
SU(2) x
U(1) x
ScU(3)
Q2
9'
gs
92
The fermion content of GI for the first generation is
601
9'
Grand Unification, Supersymmetry and Strings
602
Thus we have 15 two-component fermion states per generation. The electroweak part of GI is spontaneously broken
GI
S U ( 2 ) L x U(1) x SUc(3) + Gz
Uem(l) x SUc(3).
Also the experimental data show that
which implies that SUL (2) x U( 1) breaking predominantly occurs only through an SUL(2) Higgs doublet, or doublets. Despite the fact that the above picture is capable of providing a current phenomenological description of all the observed “low energy physics” , many questions given below remain: (i) 3 independent coupling constantx (ii) no charge quantization because of U ( 1) factor (iii) no relation between lepton and quark masses (iv) why are 3 generations identical in representation content but, vastly different in mass ? (v) why is the intergeneration mixing small ? (vi) no principle limiting the number of S U ( 2 ) generations - e , p , 7,. * . .
Grand Unification
603
Could the situation be improved ? Grand unification of electroweak and strong quark-gluon forces answer some of these questions but say nothing about, the generation problem. The basic hypothesis is that there exists a simple group G G 3 G1
SU,5(2) x U(1) x SUc(3)
which is characterized by a single coupling constant and that all interactions are generated by G. Quarks and leptons are in general members of the same multiplets of the group G. Then at some energy scale, G suffers a breakdown to G1:
The rank of G 2 4 since the rank of G1 is 4 and some possibilities for G are (i) G = SU(n)
S U ( n - 3) x U(1) x SUc(3)
e.g. S U ( 5 )
(ii) G
5
SO(n) SO(n - 6) x U(1) x SUc(3)
e.g.
SO(10) 3 SO(4) x U(1) x SUJ3) or
suL(2)x suR(2) There may be intermediate steps before reaching the righthand side. Another possibility is SO(10) 2 S U ( 5 ) x U(1). (iii) Exceptional groups E6
3 SU(3) x S U ( 3 ) x SUC(3)
E7 2 S U ( 6 ) x SUc(3)
Grand Unification, Supersymmetry and Strings
604
(iv) Any semi-simplc groiip G=G’xG”x ...
with an additional reflection symmetry will also do.
17.1.1 Q2 evolution of gauge coupling constants a,nd the grand unajication mass scale At presentJy available energies gs, g2 and g’ are very different. How then can we have G wit,h a single coupling constant, ? This is possible since due to qiiaritum radiative correct,ions g’s are Q2dependent,. Thus if we have a grand unification theory (GUT), there must, be a point, Q2 where g s , g2 and g‘ coincide. To see how this comes aboiit, let, 11s consider the Q2 evolution equation for the effective coupling constant in a general gauge theory [see Appendix B for more details]. da,‘ = h+ . . . , (17.1) d 111Q2 for Q2 > masses of fermions and gaiigc bosons biit, Q2 < M$ and (17.2) For SUc(3),C2(G) = 3, TjljAB
=
Tr
(”;’”2”) --
[number of SUc(3) triplets] (17.3)
where n,f is t,he niimbcr of quark flavors, known to be six. For SUr, (2) in the electroweak group,
[t
x - riiimber of left-handed doiiblots] , (17.4a)
605
Grand Unification
where comes from the fact that we have only left-handed couplings. Thus 1 1 TfS - -6 - ( 2 n f ) . (17.4b) Ts - 2 Ts2 Here 2nf = 12 appears since each generation has one lepton doublet and 3 quark doublets (one for each color). For U(1) group of electroweak
C2 = 0,Tf = 2 f
(fy)2
(17.5a)
I
because each fermion has either left-handed or right-handed coupling. Thus
15 - --nf. 23
d In Q2 da;' - d In Q 2 da;' - d In Q 2
(17.5b)
(17.6) -
-
(-
(17.7)
(-pf)
(17.8)
1 22 - 3 2 nf) > 0 b2, b2 = 4n 3 1 2 bl, bl = < 0. 4n
These renormalization group (RG) equations have solution -l ( Q 2 ) Qi
Hence as Q2 increases 1.
Q,
( Q 2 )decreases
= a;'
( m i ) + biln-.Q 2 m2z
(17.9)
Grand Unification, Supersymmetry and Strings
606
2. a2 ( Q 2 )also decreases but, less rapidly than as (Q2)
3. a1 ( Q 2 )increases. Thus, since a1 < a 2 < a, at available energies, at some Q2 = rn$ ,a,, a2 and a1 should coincide
cia, ( M i ) = Cia2 ( M i ) = C:al ( M i ) = aG',
(17.10)
where Cs, C2 and C1 are group theory numbers (so that, the generators of the group are properly normalized) and are of order 1. For example for SU(5), Ci = C: = Cf = 1. Mx is called the grand unification mass scale at, which one has only one free coilpling constant a ~Since . the gauge coupling constants are supposed to merge into one in GUT, the value of sin2O W , which measures the relative strengths of a1 and a2 at Q2= m i , namely
enters into the determination of a~ and Mx.Whether the three coupling constants meet at a single point Q2 = M i depends on the gauge group G. It may be noted that to include the contribution of Higgs doublet one adds - f n H in the expression given in Eqs. (7) and (8), where n H is the number of Higgs doublets. 17.1.2 General consequences of GUTs The general consequences which one would expect from GUTs are 1. G being simple, the charge operator will be a generator of the group and traceless. So if it acts on any representation of G containing quarks and leptons, it would give some relation between quark and lepton charges (sum of charges in each multiplet = 0) i.e. we would have charge quantization.
2. The fact that quarks and leptons share the same representation(s) of G, there would be relationship between quark and lepton masses.
607
Grand Unification
rn
MX
W
Figure 1 Behavior of as( Q 2 ) ,
a2
(Q') and
a1 ( Q 2 )
Q
2
versus Q 2 .
3. Since quarks and leptons share the same representation(s) of G and since gauge theories contain vector bosons linking all particles in a multiplet, there would in general be some interation changing quarks into leptons, thereby violating baryon charge ( B ) and lepton charge ( L ) conservation. At present energy scale E 1031 t,o 5 x years (mode dependent,). (17.12)
Using a3 ( m z ) and a ( m ~in) Ms renormalization scheme adopted for the definition of the coupling constants: 03 0-l
(mz) =
0,( m z ) = 0.1214 & 0.0031
( m z ) = 127.88 f 0.09
(17.13)
Grand Unification, Supersymmetry and Strings
608
as inputs, one can predict, bot,h M x and sin2Ow ( m z ) [cf. Eqs.(lO) and (11) and RG equations] in GUT models such as SU(5) with no extra scales bet,ween t,hc electxoweak scale and the GUT scale Mx.Typical predictions are sin28w = 0.215 f 0.003 and Mx M (2
f'
)
x 10''
GeV. This valiie of A4x in tiirn gives ~ ( -+ p
4 x 10"*0.7f1.2 years which contradict,s the experimental limit, ~ ( +p e+n+) 2 5 x years. Likewise the above predicted valiie of sin2 Ow (mz) differ from the present,ly deterrnined value of sin2Ow ( m z ). e+#)
M
sin2Ow
(771%)
= 0.23124 f 0.00017
(17.14)
by six standard derivations. The same mismatch between theory (single - breaking GUT models) and low energy measurements given in Eqs. (13) and (14) is observed if one uses the three effective coupling constants from their measured values to the GUT scale and above. This is shown in Fig. 2, which shows that the three couplings evolved to the GUT scale do not, meet, at, a point,. This observation and t,he others discussed in Sec. 1.2 perhaps point to the presence of new physics between the electroweak scale and the GUT scale. One such candidate is supersymmetry (SUSY), with a SUSY breaking scale somewhere between the electroweak scale and O(1) TeV. One consequence of supersymmetry (see next, section) is that bosonic particles are naturally paired with fermionic ones. Each minimal pairing is called a supermultiplet. For example: a left-handed fermion, its right-handed antiparticle, a complex boson and its conjugate form a chiral supermultiplet. On the other hand a massless vector field and a left-handed fermion form a vector super-multiplet - two transversally polarized vector boson states, plus the left handed fermion and its antiparticle. Thus for n/ = 1 supersymmetry one has the following helicity states chiral: (1/2,0) gauge: (1,1/2) , gravit,on: (2,3/2)
Grand Unification
609
Thus in the minimal siipersymmetxic extension of the st,andard model, we have t,he following particles Particle quark: q lepton: 1 photon: y weak vector boson: W weak vector boson: 2 Higgs: H gluon: G
Spin 1/2 1/2 1 1 1 0 1
Spartner squark: Q slepton: i photino: wino: W zino: higgsino: fi gluino: G
Z
Due to the presence of supersymmetric particles the RG coefficients b’s given in Eqs. (6)-(8)are modified. This modification leads to a solution such that, the couplings do meet, at a point [see Fig. 31. The unification scale in such extensions is higher than the value of M x discussed above in the context, of S U ( 5 ) model. This would imply a longer time for the proton, evading the present experimental bound. Supersymmetry is needed from another point of view, which is discussed in the next section. Before we end this section, we may mention that another popular GUT model, SO(10) [rotation group in ten dimensions in internal space with spinor representations], when broken in a single descent to SUL(2) x U(1) x SUc(3) is also in conflict, with the limit (12). However, in cont,rast, to S U ( 5 ) , SO(10) admits various symmetry breaking patterns, some containing new intermediate mass scales. One such chain of symmetry breaking is
where SUc(4) is the Pati-Salam group. Here it, is possible to avoid the conflict with the limit (12). However, there is no prediction for
610
Grand Unification, Supersymmetry and Strings
Figure 2 Running of the three gauge couplings in minimal SU(5) GUT showing disagreement with a single unification point. [ref. 51
sin2Ow;in fact its value is used to fix the intermediate mass scale mR which is of the order of 1013 GeV and being so large has no observable consequences. To conclude GUTS have several attractive features mentioned above, but their predictive power is limited. However, tha idea that quarks and leptons can be treated on an equal and that both lepton and baryon number violations are such unified theories, is now an integral part, of GUT their extension.
Grand Unification
611
60
40
a-' 20
0
Figure 3 Running of the three gauge couplings in minimal supersymmetric extension of the standard model. [ref. 51
612
Grand Unification, Supersymmetry and Strings
17.2 S u p e r s y m m e t r y a n d S t r i n g s 17.2.1 Introduction One of the main puzzles in quantum t,heory is how to reconcile General Relativity with quantum mechanics. The usual method of taking the classical Lagrangian and quantizing it fails because of insurmountable difficulties in making sense of the renormalization program, which has been so successful in other quantum field theories. In most situations the domains in which quantum field theories are interesting and the domains in which General Relativity is relevant have no overlap. General Relativity is used when dealing with massive bodies of interest a t large distance scales in astrophysics and cosmology and quantum mechanics is used at, short distance scales. However, there are situations where both theories become relevant. For instance, close to a black hole quantum effects become relevant as evidenced by Hawking radiation. When one begins to probe distances of the order of the Planck scale one expects that quantum gravitational effects will become important. The impasse in the field theoret,ic approach to gravity can be circumvented by using string theory. String theory is a novel program which replaces the plethora of particles that exist by a single string! In this approach the vibrational modes of the string correspond to different particles. Whereas in field theory it seems virtually impossible to include dynamical gravity, in string theory quite the opposite situation prevails: one cannot have string theory without gravity! This is because in the spectrum of string theory there is always a massless spin 2 field, which is naturally identified as the graviton. Another feature of string theory is that it requires supersymmetry. Even though there is no conclusive evidence at, the present, time that supersymmetry is a symmetry of the world, supersymmetry is a favored way of resolving some problems in phenomenology beyond the Standard Model. Issues such as the fine-tuning problem due to a fundamental Higgs are naturally avoided in supersymmet-
613
Supersymmetry and Strings
ric theories since the normally large radiative corrections due to a fundamental scalar Higgs are suppressed due to the presence of its fermionic partner, the Higgsino. Similarly the hierarchy problem can also be resolved in this framework. Supersymmetry is thus seen by many as a positive feature of string theory since the theory requires it and one doesn’t have to introduce it by hand. 17.2.2 Supersymmetry Spac&ime supersymmetry is a symmetry which generalizes ordinary Poincar6 symmetry by augmenting the usual generators with fermionic generators. They satisfy certain commutation relations with the bosonic generators and anti-commutation relations with the remaining fermionic ones:
[Qai,P,] = 0, [Qail
Jpl
{Qai, Q p j }
1
(17.15)
=
i(gpY)!Qpi,
=
- b i j ( ~ p C ) a ~ p p Capzij
+
+ (~sC)aj3z~j.
P, are generators of translations and Jpv are Lorent,z generators. Together they generate the Poincar6 group. The fermionic generators Q are in the Majorana representation and C is the charge conjugation matrix so that: Qai =
CapQP.
(17.16)
The index i runs over the number of supersymmetries i = 1, ..,,N. In the simplest, case N = 1, the other cases are known as extended supersymmetries. The 2 and 2’are so-called cent,ral charges, they are anti-symmetric in the indices i,j and commute with everything. They only exist when one has extended supersymmetry. One of the consequences of supersymmetry is that bosonic particles are naturally paired with fermionic ones so that the number of on-shell degrees of freedom of fermions and bosons are the same. Each minimal pairing consistent with a certain amount, of supersymmetry is called a “multiplet”. For instance, in four dimensions the smallest amount, of supersymmetry has four real fermionic
614
Grand Unification, Supersymmetry and Strings
generators and is referred to as N = 1 supersymmetry. In this case one can have an N = 1 “vector multiplet” which consists of a spin 1 gauge boson along with its supersymmetric partner, a Majorana fermion. The fermions and bosons both have two on-shell degrees of freedom. In addition to the vector multiplet, one can have a “chiral miiltiplet,” consisting of a complex scalar and its partner, a Weyl fermion. Again the degrees of freedom are the same, i.e. two. One can have up to sixteen real supersymmetries (usually refered to as M = 4 siipersymmetry) without introducing anything above spin 1 in four dimensions. Beyond that, one has to include higher spin degrees of freedom. Another useful limit?to remember is that if one restricts the highest, spin of the fields to 2, corresponding to the graviton, the maximum amount of supersymmet,ry is generated by 32 real fermionic generators (often referred to as hr = 8 supergravity). The highest space-time dimension in which a supersymmetric theory can be written down with fields with highest spin equal to 2, is 11 dimensions. This is why eleven dimensional siipergravity plays a distinguished role in supersymmetric physics. When supersymmetry is an exact, symmetry, the bosonic and fermionic partners in a multiplet have the same mass. Clearly, this is not seen in nature. For instance, there is no experimentally observed scalar with the same mass as the electron which would qualify as the electron’s supersymmet,ric partner. Phenomenological models then have to break supersyrnmetry. The mechanism of supersymmetry breaking is not well understood, however, once one assumes that siiperymmetry is broken at some high energy scale, one can incorporate in low energy models the breaking by simply introducing terms which break it. The number of such terms can be restricted to soft-breaking terms which are relevant in the infrared. These terms push the masses of the (as yet,) unobserved supersymmetric partners of the known fields up, to account for their unobserved status while carefully avoiding contradictions with well measured data. Supersymmetry is a vast area of research which deserves
615
Supersymmetry and Strings
and has received book-length accounts* In the next, subsection we will content ourselves with a simple example to illustrate the ideas touched on in our exposition.
Supersymmetric Yang-Mills: A n Example
To illustrate the basic ideas of supersymmetry we analyze a toy model: N = 1 supersymmetric Yang-Mills theory. As mentioned earlier, in a minimally supersymmetric model containing a vector field we need to introduce fermions with as many on-shell degrees of freedom as the vector field. A vector meson in d dimensions has d - 2 physical degrees of freedom, whereas a fermion field with n components has n/2 on-shell degrees of freedom. In four dimensions we need to find a fermion field with 2 on-shell degrees of freedom to match the vector field's physical polarizations. Both Weyl and Majorana fermion have 2 real on-shell degrees of freedom. Consider the following Lagrangian: (17.17) where a sum over repeated indices is implied. a is a group theory index and runs over the generators of the gauge group since all fields transform in the adjoint representation of the gauge group:
The fermionic field @ is taken to be a Majorana field: =
(17.19)
This Lagrangian is invariant under the Poincark group and local gauge transformation, in addition it enjoys a fermionic sym*See for instance, J. Wess and J. Bagger, "Supersymmetry and Supergravity" Princeton University Press (1992).
Grand Unification, Supersymmetry and Strings
616
metry:
(17.20) E is an “infinitesimal” spinor which anti-commutes with fermionic = zgPv as defields and commutes with bosonic fields. And ”/II. fined in Appendix A. This fermionic symmetry combined with the Poincard symmetry is known as N = 1 supersymmetry. We can derive equal-time (anti-)commutation relations for the fields $ and A,. There is a subtlety which needs to be mentioned here. Since the field A0 does not have a conjugate momentum one cannot quantize it in the usual way, more sophisticated methods are called for. In the following we pick the gauge A,-, = 0 and agree to impose the equation of motion of the A0 field (Gauss’ law) by hand on all physical states. In this gauge we can write down the following equal-time commutation relations:
{?434,+*@(Y)}
=
[FO4(”),A 3 4
=
Using these commutation relations and using the Majorana condition, we can write down the generators of supersymmetry in terms of the fields: 1 Qrw= -- d 3 z F ~ v ( y p U y 0 ) ~ $ ~ . (17.22)
4
One can easily verify that these generators generate the above siipersymmetry transformations in the gauge A0 = 0: [EQ, $“] =
E
1
{ Q ,$”} = --F;u~’u~ 4 (17.23)
N
1 super Yang-Mills (SYM) has some properties in common with ordinary QCD. For instance, the one-loop beta function of =
String Theory and Duality
617
this theory is given by:
(17.24) The beta function is negative implying that the theory is asymptotically free just like ordinary QCD. Also like QCD, it, is believed that SYM is confining and develops a mass gap. In addition, SYM has instantons which contribute to correlation functions.
17.3
String Theory and Duality
There are five known string theories, which are called the Type I, Type IIA, Type IIB, Heterotic S0(32), and Heterotic Ea x ES string theories. They are at first sight, very different. For instance, the Type I and the two Heterotic theories have half the sixpersymmetries of the Type I1 theories. Similarly, the Type I and Heterotic theories have non-abelian gauge groups while the others don’t,. One key feature that they do have in common is that they are all formulated in 10 dimensions. In 1995, the groundbreaking work of Hull, Townsend and Witten unified these theories. They argued that, while naively the theories had distinct properties, in many cases they were nonperturbatively the same. Many of these properties can be understood by thinking of these t,heories as limits of a single theory: “M-t heory” . The key concept unifying the string theories is called “duality”. The basic idea is simple. Consider a physical system which has two distinct descriptions A and B, say. A is then said to be dual to B, and vice versa. If the two descript,ions are different,, as they must for duality to be non-trivial, there must be mechanisms by which their apparenta disparity can be overcome. Also, their region of validity must be such that one doesn’t find any obvious contradiction. There are many different dualities. We list a few to illustrate the concept. Strong-weak coupling duality. This is a very powerful type of duality which relates a theory A, say, at strong coupling to an-
618
Grand Unification, Supersymmetry and Strings
other theory B at weak coupling. An example of this duality is provided by the Type I and SO(32) Heterotic theories in 10 dimensions. Their couplings are inversely related. Thus when one of them is strongly coupled the other is weakly coupled. Another example is that of the Type IIB theory which is self-dual under strong-weak duality. This means that, the weakly coupled theory is the same as the strongly coupled theory with some fields interchanged. T-duality. In its most general form T-duality relates string theories on different manifolds to each other. An example is of Type IIA on Rg x S' (where S' is a circle) which is dual to Type IIB on R9 x S1. The radii of the two circles are related by RA = ~ ' / R (a' B is the string tension which is the same as the 10 dimensional Planck length squared). Here we find that, two distinct, string theories on different, manifolds (different, because of their radii) are dual. Similarly, we have that Heterotic string theory on R6 x T4 (T4 is the four dimensional torus) is dual to Type IIA on R6x K3 (K3 is a Ricci flat manifold of complex dimension 2). Perhaps the most amazing dualit,ies involve M-theory. Very little is known about, M-theory and yet it, is a powerful tool in string theory. The defining featlure of M-theory is that, at, low energies it is accurately described by 11 dimensional supergravity. One duality states that M-theory on a circle of radius R is the same as type IIA string theory in 10 dimensions with coupling constant gs = ( R / l p ) 3 / 2 (where Z p is the 11 dimensional Planck length). A surprising consequence of this identification is that strongly coupled type IIA string theory develops a new dimension (since in that limit, R becomes large)! Another, similar, duality states that, M-theory on a line segment, is equivalent, to E8 x Es Heterotk string theory. One of the appeals of duality is that it, allows one to formulate the notion of non-perturbative string theory by changing the description. A key method used in establishing duality is to work with the various supergravities which capture the low-energy dynamics of string theories. The field content of supergravity consists of the massless modes of the string theory in question. For instance, the Type IIA supergravity describes the low-energy dy-
619
String Theory and Duality
namics of Type IIA string theory. It, has a number of massless fields of which the bosonic fields are as follows: q5 g,,
B,, A, Apvp
scalar dilaton graviton anti-symmetric 2-tensor abelian gauge field anti-symmetric 3-tensor
(17.25)
(17.26) The anti-symmetric fields all couple to extended objects known as p-branes. Just as a gauge field couples to a point particle) an antisymmetric (p+ 1)-tensor couples to a p-brane. An important example is B,, which couples to the Type IIA fundamental string. We can compare the above field content to that of 11 dimensional supergravity. The massless bosonic content of 11dimensional supergravity is: G,, graviton CpWp anti-symmetric 3-tensor (17.27)
At first sight it, seems to bare little resemblance to the type IIA field content. Recall) however, that M-theory on R9 x S1is supposed to be equivalent, to Type IIA string theory. When we compactify on S' and take the radius to be small we can ignore the dependence of the fields on the compact coordinate, as is usual when one performs dimensional reduction. From the ten dimensional point of view we can make the following identifications:
(17.28) (17.29)
620
Grand Unification, Supersymmetry and Strings
Thus we see that all the fields are accounted for. The dilaton serves as a coupling constant in type IIA supergravity. The usual string-frame dilaton is related. We see immediately that when the dilaton is large the radius of the circle becomes large and type IIA supergravity becomes a poor approximation for 11 dimensional supergravity. We understand this to mean that, Type IIA is a perturbative theory which is non-perturbatively equivalent, to Mtheory on S'. The spect,rum of p-branes is different, in the two theories, but they too are related as above. We illustrate this identification with a few examples. Type IIA string theory has 0-branes which couple to the gauge field A,, in M-theory they correspond to momentum modes along S'. Since momentum is qiiantized in the S1 direction in integer units of 27r/R, where R is the radius of the compact direction, the number of units is naturally identified with the number of 0-branes. A striking difference is that M-theory contains no strings. It does, however, have a 2-brane (membrane) which when wrapped on the S' appears as a string in 10 dimensions as long as one is justified in ignoring scales smaller than the radius of the compact direction. All string dualities have to satisfy consistency checks of the above kind. Fortunately there are many tests one can perform. Here the importance of a distinguished set, of states known as BPS states are particularly useful. BPS states preserve some fraction of the total space-time supersymmetxy, by virtue of which they are the lowest, mass states in their class and are guaranteed to be stable. Many of their properties can be established exactly, even when the theory is strongly coupled.
17.4
Some Important Results
Many new insights have been gained using duality. Although these areas do not, directly touch on finding phenomenologically viable models, some do demonstrate the ability to study phenomena which generically exist in realistic models. We briefly discuss some of
Conclusions
62 1
these below. In the last few years, using duality, considerable progress has been made in our understanding of gauge theories, particularly supersymmetric gauge theories. Significant, results include the demonstration of confinement and c h i d symmetry breaking in four dimensional gauge theories. String theories have been used to study black holes. One of the most exciting new results concerns the problem of black hole entropy. The Beckenstein-Hawking entropy is a thermodynamic quantity which satisfies a generalized version of the second law of thermodynamics. It has recently been given a statistical mechanical basis by relating it to microscopic states of a black hole. Recently, progress has been made in finding a connectmion between gravity and field theory. One manifestation of this has been a proposal that a quantum mechanics model known as Matrix theory captures the dynamics of M-theory. Many checks have been performed to test the ability of Matrix theory to reproduce supergravity calculations with success. Another approach known as the Maldacena conjecture has led to a radically new connection between conformal field theories and supergravity in Ads backgrounds.
17.5 Conclusions We have given just a flavor of the vast and rapidly growing area of supersymmetry and string theory dualities. The interested reader should consult review articles and books for a thorough introduction to the subject. A good place to start is the recent, book by Polchinski (J. Polchinski, “String Theory” Vols. 1 and 2, Cambridge University Press (1998)).
Grand Unification, Supersymmetry and Strings
622
17.6 Bibliography 1. M.K. Gaillard and L. Maiani “New quarks and leptons” Quarks and leptons cargees 1979, p. 443 (Ed. M. Levy et al.) Plenum Press, New York. 2. P. Langacker, “Grand Unified Theories and Proton Decay”Phys. Rep. 72C, 185 (1981). 3. A. Zee, The unity of forces in the universe, Vol. 1 World Scientific (1982). 4. R.’E. Marshak, Conceptual Foundations of Modern Particle Physics, ’World Scientific (1992). 5 . M.E. Peskin, “Beyond standard model” in proceeding of 1996 European School of High Energy Physics CERN 97-03, Eds. N. Ellis and M. Neubert. 6. J. Ellis, “Beyond Standard Model for Hillwalker” CERN-TH/98329, hepph/9812235. 7. Particle Data Group, The European Physical Journal C3, 1 (1999). 8. J. Wess and J. Bagger, “Supersymmetry and supergravity” Princeton University Press (1992). 9. J. Polchinski, “String Theory” Vols. 1 and 2, Cambridge University Press (1998). 10. S.P. Martin, A supersymmetry primer, Perspective in supersymmetry Ed. G.L. Kane, World Scientific; hep-ph/9709356.
Chapter 18
COSMOLOGY AND PARTICLE PHYSICS 18.1
Cosmological Principle ‘and Expansion of the Universe
On a sufficiently large scale, universe is homogeneous and isotropic. This is called the cosmological principle. A coordinate system in which matter is at rest at any moment is called a co-moving coordinate system. An observer in this coordinate system is called a co-moving observer. Any co-moving observer will see around himself a uniform and isotropic universe. Cosmological principle implies the existence of a universal cosmic time, since all observers see the same sequence of events with which to synchronize their clocks. In particular they all start their clocks with big bang. A homogeneous and isotropic universe is described by the Fkiedmann-Robertson-Walker (F-R-W) metric d s2 = c2dt2- R2 ( t )
1 - kr2
1
+ r2 (de2 + sin28 c@~) .
(18.1)
T , 8,$ are co-moving coordinates and the scale factor R(t) is a scale factor for distances in co-moving coordinates and describes the expansion. k is related to the 3-space curvature. With suitable choice of units for T , Ic has the values +1,0, or -1 corresponding to the closed, flat or open universe respectively. For k = 1, the spatial universe can be regarded as the surface of a sphere of radius R ( t ) in four dimensional Euclidean space. This can be seen as follows.
623
Cosmology and Particle Physics
624
Consider a sphere in four dimensional Euclidean space X;
+ X : + xi + xi = R2.
(18.2a)
The line element is
d12 = dx; + dxi
+ dxi + dzi.
(18.2b)
From Eq. (2a), we get x l d x l + xzdxz
+ x3dx3 + ~ 4 d 2 4= 0.
(18.2~)
The fourth element, dx: can be eliminated in Eq. (2b), wing Eqs. (2a) and (2c),and we obtain dl =
dr2 I-'
+ T" ( d o 2 + sin28 d4') ,
(18.3)
R2
where we have used the spherical polar coordinates X I = r cos qh sin 8, = T sin 4 sin 8, x3 = r cos 8. Putting r' = and then removing the prime, we get z2
It is instructive to use the spherical polar coordinates in four dimensional Euclidean space x1 = RsinXsinOcos4 2 2 = RsinxsinOsingl 2 3 = RsinXcosO ~4 = RCOSX.
(18.5)
Then we get d12 = R2 [dX2
dV
=
+ sin2x
(do2
+ sin20 d4')]
R3 sin2x sin 8 d x d0 d$.
(18.6a) (18.6b)
Cosmological Principle and Expansion of the Universe
625
Now using r = Rsinx, we get back Eq. (3). The radius of the sphere and its volume are given by (18.7)
V = J:/,
JI
R3sin2xsin0 dX d0 d 4
= 27r2R3.
The cosmological principle implies [cf. Eq. (l)]
e = R ( t ) T.
(18.8)
Thus the velocity of expansion is given by
(18.9) where (18.10) is called the Hubble parameter. Let, 11s denote by to the present, time and te the time at which the light was emitted from a distant, galaxy. Correspondingly we denote the detected wavelength by X and emitted (laboratory) wavelength by A, of some electromagnetic spectral line. We define the redshift
(18.11) The redshift is experimentally observed and it clearly shows that, the universe is expanding.
Cosmology and Particle Physics
626
The highest, redshift*so far discovered z = 4.89 so that, the Lyman-alpha line appears in the red part of the spectrum arround 7200 A. This implies that = (1 z ) = 5.89. In the matter dominated universe R t 2 / 3 (see below). This gives [with to M 1.5 x 10" yrs, the present age of the universe) t, 21 14 (1.5 x 10" yrs) 2 lo9 yrs. The existence of these high z-objects implies that, by the time the universe was about, lo9 yrs old, some galaxies (or at least, their inner region) had already been formed. For small time intervals since emission compared to H i ' , Eq. (11) takes the form
%
+
N
(18.12) where 1 is the distance to the source.
18.2 The Standard Model of Cosmology The model is described by two differential equations
A c2R2 8nG R2 + kc2 - -= ---p R2 3 3
+
d (pR3c2) p d ( R 3 )= 0.
(18.13) (18.14)
Here the second term in Eq. (13) is due to the curvat,ure, the third term contains cosmological constant A. Cosmological constant A is very small (1111 < 3 x rn-2) and this term is usually neglected except in the inflationary phase of expansion. G is the Newtonian gravitational constant,. In the units ti = c = 1, 2 - MP (the vePlanck mass) M 1.2 x 10'' GeV. Equation (14) expresses the energy conservation. Here p is the density of the universe and p is the isotropic pressure. Note that Eq. (13) can also be put in the form (18.15a)
The Standard Model of Cosmology
627
Differentiating Eq. (13) and using Eq. (14), one gets
_R --_1 A c R
3
--4nG ( p c 2 + 3 p ) 3 c2
(18.15b)
In addition we need the equation of state. We take this to be that for an ideal gas p = nkBT, (18.16) where n is the particle density and Icg is the Boltzmann constant. Icg = 0.86 x lo-'' MeV/K (K: Kelvin). If we take Icg = 1, then the temperature is measured in MeV. In particular 0.86 MeV= 10l°K. From Eq. (13) (A = 0), we have (18.17) where
3 H2(t) 8n G
(18.18)
is called the critical density. It is convenient, to define the density parameter of the universe P a=-.
(18.19)
Pc
Then from Eq. (17), we get
kc2 = R2 ( t ) H 2 ( t ) (R - 1) = Ri ( t ) H i ( t ) ( 0 0 - 1 ) .
(18.20)
Here the subscript, 0 denotes the present time. It, is clear from Eq. (20) that for 0 > 1, the universe is closed, for R 5 1, the universe is open. We note that for nonrelativistic gas (NR) p = mn 3 p
, the universe is closed and for qo 5 , the universe is open. We now discuss the three cases Ic = 0, Ic = 1 arid k = -1. We will now put, c = 1. First we discuss the flat universe (k = 0). From Eqs. (13) and (23), we get
R J R = (2GM)”’.
(18.27)
629
The Standard Model of Cosmology
Integration of Eq. (27) with R(t M 0) M 0 gives 9GM R3 ( t )= -t 2 = 2
3M 4l.r P ( t )'
(18.28)
where the last term in Eq. (28) follows from Eq. (23). Hence we have p-' ( t ) = 67rG
t2 (18.29)
For the closed universe k = 1, we have from Eq. (13) [A = 01 8nG pR2-1, 3
(18.30)
where we have put ( d t = R d q ) : (18.31) Integrating Eq. (30) with the help of Eq. (23)) we get R = MG(1 - C O S ~ ) t = M G ( q -sinq). Similarly for the open universe k
R
=
t
=
=
(18.32)
-1, we get,
MG(c0shq - 1) MG(sinhr1- q ) .
(18.33)
All the three cases are shown in Fig. 1. Finally we note that the age of the universe is essentially determined by the matter dominated universe. The radiation era lasts only for a few minutes. Now using [cf. Eq. (29)] (18.34)
630
Cosmology and Particle Physics
t
Figure 1 Plot of scale factor R ( t ) versus time t for closed open ( k < 0) and flat ( k = 0) universe model.
(k > 0),
we have from Eq. (13) &2
-
87~G Ri po - = - k . 3 R
(18.35)
By using Eq. (20) and Eqs. (18) and (19) for the present time, we obtain
[
& = R o H o 1-no+no-
RO1 R
lj2
.
(18.36)
The integration of Eq. ( 3 6 ) gives the age of the universe tu =
f (Go) Hi1,
(18.37a)
(18.37b) with x = R/Ro and R(t N 0) z 0. The function = 1 and < 1 is respectively given by
f (no)for Ro > 1,
The Standard Model of Cosmology 7.r
for RO 2 2 for Ro = 1 - -3, -
N
N
631
>> 1
(18.38a) (18.3813)
1 +RoInRo, for
0 0
> m ) )thermodynamic equilibrium is
maintained through the processes of decays, inverse decays and scatterings. As the universe cools and expands, the reaction rates will fail to keep up with the expansion rate and there will come a time when equilibrium will no longer be maintained. At, various stages then, depending on masses and interaction strengths, different, particles will decouple with a “freeze oiiV surviving abundance. We now determine conditions under which the statistical equilibrium is established. From dimensional analysis, the reaction rate for a typical process can be written as follows. For the decay of a X-particle, the decay rate is given by
2
is the measure of where rnx is the mass of the X-particle, ax = coupling strength of X-particle to the decay products, and gd are number of spin states for the decay channels. Note that
(18.90) The reaction rate for the scattering processes is given by
I‘= (cv) [number of target, particles per unit, volume]. (18.91) For a weak scattering process
.(
4 = gtv
(18.92)
Freeze Out
643
Since the number of target, particles per unit volume n, we can write the reaction rate for a weak process
N
(~cBT)~,
(18.93)
For lc~T> T,.
Inflation
< T,,T
= T,
I\
Figure 3 Scale factor R(t) versus t, showing the inflationary phase.
Cosmology and Particle Physics
662
18.10
Bibliography
1. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (4th Edition) and Statistical Mechanics (3rd Edition), Part I, Pergarnon Press 1985. 2. P. J. E. Peebles, Physical Cosmology, Princeton University Press, Princeton, N. J. (1971). 3. D. W. Sciama, Modern Cosmology, Cambridge University Press, Cambridge (1972). 4. S. Weinberg, Gravitation and Cosmology (Wiley: NY, 1972). 5. G . Steigrnan, Ann. Rev. Nucl. Part. Sci. 29, 313 (1979). 6. F. Wilczek, Erice Lecture on Cosmology, Proc. 1981 Int. Sch. of Subnucl. Physics, “Ettore Majorana”. 7. A. Zee, Unity of Forces in the Universe Vol. I1 (World Scientific, Singapore 1982). A collection of original papers relevant to this chapter can be found in this book. 8. M. S. Turner, Cosmology and Particle Physics, Lectures at the NATO Advanced Study Inst. Edited by T . Ferbal, Plenum Press (1985). 9. D. Denegri, The Number of Neutrino Species CERN-EP/89-72. Rev. Mod. Phys. 10. A. D. Linde, Particle Physics and Cosmology, in Proc. XXIV Int. Conf. on High Energy Physics (Editors R. Kotthaus and J. H. Khn) , Spinger-Verlag, Heidelberg, Germany (1989). 11. A.H. Guth, The Inflationary Universe, Addison-Wesley, Mass. (1997). 12. R. Kolb and M. Turner, The Early Universe, Addison and Wesley, California, 1990. 13. Particle Data Group; Eur. J. Phy. C 3, 1-794 (1998).
Appendix A QUANTUM FIELD THEORY [A SUMMARY] A.l
Spin 0 Field
Spin zero particle of mass rn is described by a field +(z) which in the absence of interactions, satisfies the Klein-Gordon equation.
In quantum mechanics, 4 ( x ) is regarded as a c-number. In quantum field theory, +(z) is a field operator which can create and annihilate the field quantum. The Fourier decomposition of
4(x) is
(A.2b) where $t(x) is hermitian conjugate of 4(z) and k . x = Icoz~k.x, ko = d m > 0. In Eq. (2), u ( k ) and b ( k ) are interpreted as follows:
663
Quantum Field Theory [A Summary]
664
at (k) : creation operator for the particle (spin 0 and mass m ) annihilation operator for the particle a (k) : (spin 0 and mass m ) bt (k) : creation operator for the antiparticle (spin 0 and mass m ) b (k) : annihilation operator for the antiparticle (spin 0 and mass m).
a (k) and b (k) satisfy the following commutation relations
S3 (k- k’)
(A.3a)
[ b ( k ) ) bt (k’)] = S3 (k- k’)
(A.3b)
[ ~ ( k ) at , (k’)]
I)’+
[a@),
=
=
[ a @ ) ,b+(k’)]= 0.
(A.3c)
If 10)denotes the vacuum state then one particle state of 4-momentum k is given by
lk) = a+(k)10) .
(A.4)
Define
N+ (k) N- (k)
= =
at@) u(k) bt(k) b ( k ) .
(A.5a) (A.5b)
It follows from the commutation relations (3) that N+ (k) and N- (k) have the eigenvalues 0, 1, 2, and are known as number operators for the particles and antiparticles. Then
-
n
=
c N+ (k) c N- (k)
= Total
number of particles
(A.6a)
k
A =
= Total number of antiparticles. (A.6b)
k
Spin 1/2 Particle
665
It may also be noted that for free fields
(A.7a)
@o)
=
i
>o
G ( x )(-27, a,
-m
7’”are Dirac matrices. We choose y” :
= 0.
(A.1Ob)
Quantum Field Theory [A Summary]
666
(A.ll) -f ’s satisfy the anticommutation relation
(A.12)
Matrices 1
I
I
y5 is also hermitian
Y5 t -7
Components 1 4
5
I
(A.13)
y5 anticommutes with yP viz.
Y5 YP -- -YP Y5*
(A.14)
In Pauli representation, yP ’s can be written as
(A.15a)
Yo =
r’
=
(; iil)
(-9;
The Fourier decomposition of
(A.15b)
=yo
Z)’
(x)is
75’7
5
(A.15~)
667
Spin 1/2 Particle
(A.16b) where E = t,
and u and
21
yo,
jj = t.
yo
(A.17)
satisfy the equations
( 7 . p - m, uT ( p ) (Yap+ m, vT ( p ) G ( p )(74- m) %(p)(y.p+m)
=
(A. 18a)
=
(A.18b) (A.18~) (A. 18d)
0 = 0 =
u ( p ) and b(p) are interpreted as follows:
creation operator of the particle wit,h momentum p and spin component T annihilation operator of the particle with momentum p and spin component r creation operator of the antiparticle with momentum p and spin component r annihilation operator of the particle with momentum p and spin component r. The operators a and b satisfy the anticommutation relation
Quantum Field Theory [A Summary]
668
and all other anticommutation relations give zero. Define number operators:
N,(+)( p ) = N,(-) ( p ) =
bL
(p) (p)
(PI bT
( p )'
(A.20a) (A.20b)
Then from the anticommutation relations (19), we have [A$*) ( p ) I 2 = N,(*) ( p ) .
(A.21)
Thus, we see that A$*) ( p ) have eigenvaliie 0 or 1. This means that each state is either empty or has a single particle of definite spin and momentum. Thus the anticommutation relations lead to description of a system of particles which obey the Pauli exclusion principle or in other words obey the Fermi-Dirac statistics. The spinors u and u satisfy the following orthogonality relations:
They also satisfy the completeness relations (A.23) where a and
p are spinor indices; a , p = 1 , 2 , 3 , 4 .
(A.24a)
(A.24b)
Spin 1/2 Particle
669
A+ ( p ) and A- ( p ) are the projection operators for particles and antiparticles respectively. One also writes y, p , = y p = $. Using the Pauli representation of y-matrices, we can write
u,( p ) = R
(A.25a)
where
(A.2 5b) (A.25~) =
ar( p ) is given by
( ;)
and
d2)=
-
r)t
u, ( p ) = ~ ( ‘ 1 Rt ~ yo = W (
where
R=
1
JGjZxj((Po + m) 1,
-0
( ).
(A.25d)
-
R,
(A.26a)
‘P).
(A.26b)
w, ( p ) is given by
( p ) = -iY2 u:: (P) Finally, we note that for free fields [pz= ypa,] 21,
p a (4
1
-
%
(igz+ m ) ,
= i = -2
s, p
(34
+
A (Z- d)
(x - d),
where
S
(X - d) =
(A.27)
(-igz - m) A (Z- d).
(A.28a)
Quantum Field Theory [A Summary]
670 -
[@ (x), @ (z’)]
A.3
.
= yo b3 (x - x’) +so=*;
(A.28~)
Trace of y-Matrices
We note that, y-matrices are traceless
T r y’l = 0, T r y5 = 0.
p = 0,1,2,3
(A.29)
Now
Tr (y’l 7”) = Tr (7”7’1).
(A.30)
Therefore, from Eq. (12), we have
T r (y’l 7”) = g’l” Tr
(i)= 4 gk”
(A.31)
and
(A.33) Now
T r (7’1y” y P ) = T r ( y P y’l 7”) y p y”y p = iPUPX y~ y5 + g”P y’l - gpp y”+ gp” yp.
(A.34) (A.35)
Therefore,
T r (y’l y” 7 P )
= 0 = T r (yP
y’l 7”).
(A.36)
From this, we generalize that trace of the product of odd numbers of y-matrices is zero. Further, we have
~’l~”y~ya+yay~yuy~=2g~ay’ly”-2g”ay~y~+2g’10 (A.37) Therefore,
Trace of y-Matrices
671
Noting that we can write 5
Y
Li 4!
= - &,pup 7,
YP y’ yp,
(A.39)
we have
Tr
(75
(75
7.) = 0
YP 7.)
=o
T r (75 y yv yp)
=0
(A.40) (A.41) (A.42)
and
%) = 42 Epvpu, (A.43) with the definition ~ 0 1 2 3= 1 and E~~~~ = E 2). .k while E~~~~ = -1. In calculations, we usually come across the matrix elements of the form
Tr
Therefore,
( 7 5 Yp Yv Yp
672
Quantum Field Theory [A Summary]
1
-
4
m1m2
X
n
[(h+ m2) 7 p (1 + a75) (h + ml)
(1 + a75)I (A.48)
Here Fz = y - h
F1
=y*h.
(A.49)
Using the formulae for the traces of y- matrices given previously, we get
Similarly for
we get
Spin 1 Field
673
we get the same value as given in Eq. (52).
A.4
Spin 1 Field
Electromagnetic field (photon) with mass m = 0. In the absence of interactions, the electromagnetic field A, (z) satisfies the field equation
@ A , (z)= 0.
(A.54)
There is an additional condition
8, A, (x)= 0.
(A.55)
The Fourier decomposition of A, (x):
(A.56) where E& (x), are four vectors called polarization vectors. ax (k)and u i ( I c ) are interpreted respectively as the annihilation and creation operator of the photon with momentum k and polarization E.; (k). They satisfy the following commutation relations
,
(k’)]
= SA A’
(k),
(k’)]
= [a: (k),
[UX (k) uit [UA
The polarization vector
S3 (k- k’),
(A.57)
~ 1(k’)] , = 0.
(A.58)
(k) satisfies the following relations
E;
EX (Ic)
*
EX’
(k)
=
6,
(A.59) (A.60)
k . 2 = 0. For transverse photon polarization, the four-vector chosen as &; = (0, EX 7
(W
(W)
&&
(k) can be (A.61)
Quantum Field Theory [A Summary]
674
so that we have
k . &x = k *x =~O
(A.62)
and
where q
=
(1,0,0,0).
A.5 Massive Spin 1 Particle A spin 1 particle of mass m is described by a vector field 4,
(z) ,
which in the absence of interactions satisfies the equation (m2
+
0 2 )
4, (x)= 0
(A.63a)
with the subsidiary condition
4, (x)= 0. The Fourier decomposition of dP (z) is given by ap
c 3
x
E$
( k ) [ax ( k ) e-zk.'
(A.63b)
+ b i ( k ) eik.']
X=l
(A.64a)
(A.64b)
Feynman Rules for S-Matrix in Momentum Space
675
ax (Ic) and bx (k) satisfy the following commutation relations:
(k’)]
=
6~
J3 (k- k’),
(A.65a)
[ b (k) ~ ,bi, (k’)]
=
6~ 1’ 63 (k- k’).
(A.65b)
[UX
(k),
u i (k) (UX(k))are creation (annihilation) operators for the particle with polarization X and momentum k. bl (k) (bx (k)) are creation (annihilation) operators for the antiparticle with polarization X and momentum k. The polarization vector E; satisfies the relation EX
*
k*&’=O
EX’ = 6x A’,
(A.66a) (A.66b)
A.6
Feynman Rules for S-Matrix in Momentum Space
For each internal photon line:
-0
For each internal fermion line:
I
For each internal pion line: For each external fermion line entering the graph, depending upon whether the line is in the initial or final state
For each external fermion line leaving the graph, depending upon whether the line is in the final or initial state
(27r) k +la
&a
*------
a
-@
+ * ,
1 k2-m?+ie
-&mT (p) or &P
+*
@T(P)
f i E T (p)
or
&
E ’ T
(PI
676
Quantum Field Theory [A Summary]
For each external photon line: For each external spin 0 meson line: For photon-fermion vertex: 0
For pion-fermion vertex:
I I I
I For photon-meson vertex:
A factor ( 2 ~64) ( ~p - p‘ f k) at each vertex
A factor (-1) for each closed fermion loop For a massive vector boson of mass mw This gives the propagator of the vector boson in unitary gauge. Further one has J d4 1 for each loop integral where the four momentum 1 is not fixed by energy-mometum conservation. Multiply by 6p = l (-1) and -1 (1) respectively for the direct and exchange term of fermion (antifermion)-fermion (antifermion) scattering. Feynman rules for a hermitian self-interacting spin 0 boson with the Lagrangian
are as follows
An Application of Feynman Rules
677
For each external line For each internal line For vertex
A factor (2?r)464(kl+ k2 - k3 - k4) at each vertex For each loop integral J d4Z statistical factors *----
# - - - -
-2!1
, 1 ,
- . - - -'I
+!
\
j etc.
- - - - - I
-----
'd
-4'
A.7
, 6-
An Application of Feynman Rules
As a simple application of Feynman rules, we consider the process e-+e+
+ p-+p+
Pl+P2
= P:+P;
+
Therefore, using the relation S = 1 i ( 2 ~64) (Pi ~ - P f )T :
V
(A.68)
Quantum Field Theory [A Summary]
678
Figure 1 One photon exchange Feynman diagram for the process e- e+ 4 p- p+.
and we put
Therefore,
Using Eqs. (51)-(53) ( u = 0), we get
+
P’, * P2 P; * Pl +Pa * P2 Pi * Pl m: Pi * P; 2 +m, p2 pl 2rn:rnE
=-
+
+
(A.71) Now s = (p1 + p 2 ) 2 = (pi +p;)2 = E
L
= k2
(A.72)
and in the center of mass system p1=
-P2 = P;
Pi = -P2I = PI
(A.73)
An Application of Feynman Rules
679
with IPI =
JG' 2
7
lP'l
=
Jq .
(A.74)
Therefore,
P', P2 Pi PI + P'z P2 P: ' PI *
*
*
-
and
IF1
2
=
e4
S2
+4s (ma
+ m;) (1 - cos28 ) + 16 m:mz
cos2e]
.
(A.76)
Hence from Eq. (2.39), we get
where
Quantum Field Theory [A Summary]
680
(A.78)
(A.79) In the relativistic limit, ,Be z 1 , ,LIP M 1 (s >> rn;,mi) ,we have da dR
- =
a =
A.8
- (l+COS20)
a2 4s
(A.80a)
47ra2 1 3 s'
(A.80b)
--
Charge Conjugation
Dirac equation in the presence of electromagnetic field is given by
(a, + i e A p )- rn] 9 (x)= 0. For the adjoint, field T, Eq. (1) can be written: [-i ( 7 ~(3,) -~ i e A,) - rn] qT (x)= 0. [i?"
(A.81)
(A.82)
Under the charge conjugation
9 (x) A , (x)
-+ +
9,(x)= u,@ (X) Uc-l A; (x)= U,A, (z)UC-'.
(A.83) (A.84)
681
Charge Conjugation
If the Dirac equation is invariant under charge conjugation then:
Now we can write Eq. (83) as
C [-i ( y p ) T (a, - i e A,)
- m]
C-'CqT (x)= 0,
(A.86)
where C is a unitary matrix, called the charge conjugation matrix. Equation (87) is identical to Eq. (86), provided that
y, A;
=
-c ( y q T c-l
=
-A
CL
(A.87) (A.88)
9 I C ( x ) = CTT(x).
(A.89)
c9-T
(A.90)
Also one can write QC(Z) =
9[IC(z)=
(x)= -yocq* -VT(x)C-1.
(A.91)
In Pauli-representation fo y-matrices for p = 0 , 2 for p = 1 , 3 .
(A.92)
Therefore we have from Eq. (88): (A.93) (A.94) Hence in this representation Q" (x)= -2y2Q'.
(A.95)
Quantum Field Theory [A Summary]
682
On the other hand, in the Weyl representation of y-matrices
(A.96) In this representation, one can write
(A.97) where
are two component left-handed and right-handed spinors. In this representation the relations (93) - (96) are again satisfied. Hence from Eq. (96), we get,
(A.99) Sometime it is convenient to write a right-handed field in terms of a left-handed antiparticle field (cf. Eq. (100)): 7 = ig2Jc'
(A. 100)
so that Eq. (98) becomes Q =
(
6 ig2J"*
)
(A.lO1)
Charge Conjugation
683
The Majorana spinor !PM is defined as
Qft
-T
!PM=C!PM
=
= cap%$.
*'Ma
Hence in the Weyl representation
(A.102) We also note that in the Weyl representation:
Y =( y 0
0
)
(A.103)
where f7p
=
(1'2) = (1,Z)
ap
=
(1, -2)
=
(1,- 2 ) .
(A.104)
Now the Dirac Lagrangian
L=
(iypa, - mD)9 + % (QTC-'!P - $ C a T ) 2
(A.105)
(where the second term in Eq. (106) is the Majorana mass term and violates lepton number conservation) can be written in terms of two component chiral fields using Eqs. (99), (loo), (104) and (106):
L =
[ct(Tpap(+ c c ' 8 p a p c c ] - m D [c*T '&g . 2 cc'
$.
tCT ( - i g 2 ) [] (A.106)
where we have used 02
02
z
($y
Quantum Field Theory [A Summary]
684
'
(fermion fields anticommute)
-
-3,("'-P
-
E ~ ' P ~ ~(partial E ~ integration).
c
Equation (107) can be put, in the compact form
where i, j =1, 2 and mij is the symmetric mass matrix
Bibliography
A.9
685
Bibliography
1. J. J. Sakurai, Advanced Quantum Mechanics, Addison- Wesley, Reading, Massachusetts (1967). 2. J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York (1964). 3. C. Itzykson and J. B. Zuber, Quantum Field Theory, McGrawHill, New York (1980). 4. M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, Massachusetts (1995).
Appendix B
RENORMALIZATION GROUP AND RUNNING COUPLING CONSTANT B.l
Feynman Rules for Quantum Chromodynamics
For canonical covariant quantization, the QCD Lagrangian given in Eq. (7.32) is written as [repeated indices imply summation]
1
Tr (TA,T B )= 2
fACD
fBCD
~ A B for ,
= c 2 (G) 6 A B
=
(TA);
the fundamental representation.
(TA); = C F
N
~ A B ,
for S U ( N ) gauge group (B.2)
N2 - 1
6; = 2N ~
-&
b:,
for S U ( N ) .
In the Lagrangian (I), (8, G A , ) ~ is the gauge fixing term, ( being the fixing parameter. The supplementary nonphysical fields, called ghosts are needed for covariant quantization in 687
Renormalization Group and Running Coupling Constant
688
order to cancel the probabilities of observing scalar (or time-like) and longitudinal gluons. Quantizing in a renormalizable gauge leads to the following Feynman rules:
gluon propagator
a
=k
b
s;IcrJ;kLLLC,~
B, p
26; A,quark propagator
Renormalization Group
(i) (ii) (iii)
689
The other factors are J 4 for each loop integral (2r) (-1) for closed fermion (ghost) loop Statistical factors like
B .2 Renormalization Group, Effective Coupling Constant and Asymptotic Freedom We now show that the self-coupling of gluons envisaged in the first term of the Lagrangian (1) has the consequences that QCD has a remarkabIe property of being asymptotically free i.e. the quark - quark force becomes weak at large momentum transfer or short distances, such as probed in deep-inelastic collision [cf. Chap. 141. In other words, the coupling constant a, depends on the momentum transfer in such a way that a, ( Q 2 )--+ 0 as Q2 --f 00. Consider the radiative corrections to quark - quark - gluon (qqG) vertex, where at one loop level these corrections are shown
Renormalization Group and Running Coupling Constant
690
in fig. 1, [Q2= - q 2 ] .
+
+
One loop corrections to quark - quark - gluon (qqG) vertex. The one loop corrections to qqG vertex shown above are infinite. One must define a high Z2- cut off (I being loop momentum) X2, so
691
Renormalization Group
that the loop integrals converge. We have then ~ A = F -2
TA?'p
r s (Q2,
(B.3a)
A, gs)
where
where - .denotes the corrections from higher order loops. Note here that the cut-off dependent logarithmic contributions from diagrams A [involving quark self-energy diagram] and the first of diagrams B [involving the quark gluon vertex function] cancel due to gauge invariance as is also the case in quantum electrodynamics. Since the theory is renormalizable, we must be able to write it as
where p is called the renormalization scale and 2, is a multiplicative renormalization constant. One may define the renormalization scale through the relation
Then neglecting
a0
in Eq. (3b)
and [cf. Eq. (3b)l
rs (Q2/X2, Thus
gs) = g J Y 2 ( X 2 / Q 2 , 9s)
.
(B.6b)
692
Renormalization Group and Running Coupling Constant
This relation expresses the basic-renormalization group property. It is more conveniently expressed through an equivalent differential equation which follows from the p-independence of r3 so that, d r S = 0, (B.8a) dP
or, using Eq. (4), (B.8b) This can be rewritten a s [I?:
( p ) = gs ( p ) ]
(B.9a)
so that, (B.9b) where Z;l2 is given in Eq. (6a). Equations (9) are known as the renormalization group equations for the effective coupling constant, gs ( p ) . Writing
Eq. (9b) gives
=
-29; [bo
+ bl g," + . . .] ,
(B.lob)
where we have used Eq. (6a). To integrate Eq. (9a), it is convenient to write it, on using Eq. (lob), as [putting p2 = Q 2 ]
693
Renormalization Group
or -1
d lnQ2 = d (l/g:)
(l/g?)-'
+ a]
(1/gi)
-1
+
a
*
-
]
.
(B.11)
If we keep only the lowest order term, we have (B.12a) where a, =
g, 6 = 87r bo. Integration of Eq. (12a) gives a;' ( Q ~ )= a;1 ( p 2 ) + bln -. Q2
(B.12b)
P2
Note that what renormalization group does is to relate the coupling constant at two different scales. We may also write Eq. (12b) as Q2
( Q ~=) bln-
(B.12c)
AbCD
where [a;' ( p 2 )= a;']
or (B.12d)
A Q ~ Dis one parameter which determines the size of a, ( Q 2 ) .It must be determined from experiment. Thus finally we have from Eq. (12)
-
1
+0
bln $AQCD
(a: ( Q 2 ) ) .
(B.13)
694
Renormalization Group and Running Coupling Constant
Table B . l Renormalization Constants
Note that we have been able to sum the leading logs here [compare (13) with a p (1 - a,, b In . in the ordinary perturbation theory]. Thus Eq. (13) goes beyond the ordinary perturbation theory. The perturbation now is with respect to 0,(Q2). We now determine b. For this purpose, we need 2, [cf. Eqs. (8) and (9)]. But we note from Fig. 1 that Z,is given by
5+
a)
where the renormalization constants Z ~ F2,1 and ~ 2 3 arise respectively from diagrams A [self-energy part of the fermions (quarks) propagator], B [vertex part for the fermion] and C [the vacuum polarization or the self-energy part of the gluon propagator]. The values of these constants are summarized in Table 1, which also includes Z1 which corresponds to the triple gluon vertex [i.e. the first of diagrams (B) with the quark lines replaced by the gluon
w]. 3
\ lines while the second is replaced by
Cz and
In this table
are defined in Eq. (1) and n.f denote the number of fermion flavors. F’rom Eq. (14) and Table 1, we have to order 9,” CF
Renormalization Group
695
Thus from Eq. (lob) [note that the gauge fixing parameter [ is canceled out] (B.16a) P ( % ) = - a d bo + o (d)]7 so that
(B.16b) (B.16~) Hence in summary, we have from Eq. (13) Q,
1
(Q2) = a-1 P
+ J- (Y c ~- i nr) In $ + 0
(0; ( Q 2 ) )
47r
(B.17a)
(B.17b) It is a very useful equation and the single parameter AQCD becomes the QCD scale which effectively defines the energy scale at which the running coupling constant, attains its maximum. A Q c D can be determined from experiments and turns out, to be [see Chap. 71 A Q ~ D=
140 & 60 MeV.
Note that for SU, (3) [C2 = 31 , (11 -
5
(B.18) nf)is positive for
nj < 11 (which is certainly true for known six quark flavors n,f = 6 ) and then a, ( Q 2 )decreases as Q2 increases. This is made possible because of coefficient, 11 which comes from the self coupling of gluons, non Abelian nature of QCD. The logarithmic deviation from asysmptotic freedom is a characteristic of QCD and the tests of the theory have to be sought to detect, logarithmic scaling violations.
696
B.3
Renormalization Group and Running Coupling Constant
Running Coupling Constant in Quantum Electrodynamics (QED)
For QED, only fermion loops (i.e. the first, of diagrams C in Fig. 1 with gluon replaced by photon and g: by e2) contribute to electric charge renormalization so that, in Table 1 only Z3 without C2 is relevant,. Note, however, that, the contributing charged fermions are e , u, d, p , c, s , T , b and t so that, e2 ( n , f / 2 in ) the expression for Z3 is replaced by
e 2 x Q ; = e 222 f where (n,,/2) are the number of generations [3 in our case] and the factor 3 outside the parenthesis is due to the color. Thus (B.19a) giving (B.19b) The equation analogues to (12) is t,hen (B.20a) giving
(B.20b) and increases with Q2 in cont,rast to a 3 ( Q 2 which ) decreases with Q2.
Let us apply Eq.(20b) for pz = m:, where a,(m;) is determined from Thompson scatkering, for example, [a = ae(rnz)= No matter how small a one has, one can always increase Q2 to a
&]
Running Coupling Constant for SU(2) Gauge Group
697
point where a,(Q2)which was given in Eq.(20b) becomes infinite [Landau ghost]. This, however, occurs [for six flavors] at
Q2 = m2exp (fa-')
GeV2
N"
(B.21)
which is even larger than Ad: M GeV2 by several orders of magnitude . Finally, we wish to remark that the formula (20) holds for m: 5 Q2 < mL. For Q2 2 we have to consider the contribution of charged W* bosons to Pem. In this case
mL,
(B.22)
47r and ae(Q2)still increases with Q2 for n f
=6
(or > 6 ) .
B.4 Running Coupling Constant for SU(2) Gauge Group For SU(2) group from Eq. (2) C2 = 2 and therefore from Eq. (16c) bSW(2) =
(-
1 22 2 - jnf 4lr 3
)
(B.23)
and correspondingly Eq. (17a) becomes
g,
g2 being the coupling constant associated with where a2 = qqW* vertex, W*, W3 being the gauge bosons associated with SU(2) gauge group. Note that for six quark flavors ( n f = 6 ) , > 4)and a2 (Q') is falling with Q2, although at a rate less than a, ( Q 2 )for the SUC (3) group.
(y
Renormalization Group and Running Coupling Constant
698
Renormalization Group Equation and High Q2 Behavior of Green's Function
B.5
Consider now in general a renormalized Green's function (propagator or vertex function or a related quantity) in QCD denoted by rR
(Pirasip>E)=
Z-' ( A 2 / p 2 ' a s ~ E )
( P i , a s O , ~ O )7
(B.25)
where 2 is a multiplicative renormalization factor and I' on the right-hand side knows nothing about p so that $ = 0. This implies that r R satisfies the renormalization group equation 1 drR
-- rR d p
_--1 d Z dP
or
where [cf. Eq. (9) and (lo)]
P
(as)
1 do, 1 = -gs 2 dp 4n
= -P-
P(gs).
(B.26b)
To simplify matters, let us work in the Landau gauge 5 = 0, then
The above equation also determines the high Q2 behavior of r B . To see this, we first note that there is another constraint on I' which
High Q2Behavior of Green’s Function
699
comes from dimensional analysis. Assume we scale all momenta in r~ (pi,0 8 , P ) 7 pi 4 A pi
D is the dimension of the Green’s function (e.g. for inverse gluon propagator r p 2 and we have D = 2). F is a dimensionless function of dimensionless variables. From Euler’s theorem for homogeneous function N
(B.29)
Put t = 1nX and combine the naive scaling equation (29) with the renormalization group equation (27) [which gives the dynamical constraint] to eliminate p& and obtain
Its general solution can be obtained by the method of characteristics. First one solves [cf. Eq. (26b) with t = Inp] (B.31) with the condition 5, (O,a,) = a,. The general solution of Eq. (30) can then be expressed in terms of that of the above differential equation. In this way one obtains t
r R
(A pa,
p ) = A D r R ( pi,8,
( t )>/I) exP [ - 2 s 0 dt’y (8, (t’))]
(B.32) What we learn from this general solution is that the behavior of Green’s functions when all momenta are scaled up is governed by 6, ( t ). Now as already seen in Sec. 2
,8 [(E,( t ) ) 2 ]= - (5, ( t ) ) 2b +
* *
(B.33a)
700
Renormalization Group and Running Coupling Constant
and similarly we can expand
Thus to solve Eq. (31) in the lowest order, we make use of Eq. (33a) and rewrite it as
dt = -
db, 2bb: [ 1 + .. .]
giving b, ( t ) =
-
1
+
as1 2bt
+ 0 (b.”).
(B.34)
Remember that t In X and this is the same functional dependence for b, as before for a, (Q2)in Sec.2. Thus noting that for large t, 6, ( t ) the use of Eqs. (33) and (34) in the first order enable us to write Eq. (32) for large t or X as
- &,
where y or yo is called the anomalous dimension of F R , which can be determined from Eq. (26b). If it, were zero, we would have obtained canonical scaling behavior A D as in the traditional parton [t lnX], we can model [cf. Chap. 141. Noting that 6, ( t ) say from Eq. (35) that the large Q2 behavior of rR (Q2) is N
&
-
High
Q2
Behavior of Green's Function
701
where the second factor can be written as
(B.37)
&
UC2 - in,] [cf. Eq. (16c)l. Thus it, is clear that the with b = [3 renormalizatlon group Tuation has enabled us to sum up terms of the form [a, (Q2)In Q2] whereas in ordinary perturbation theory we would have to deal with a power series in a, (Q2)ln Q 2 . Analogous logarithmic violation of the scaling will hold in the deep inelastic structure functions and similar physical quantities. Let us now consider some simple applications:
B. 5.1 Gluon propagator From Table 1,
13 [(-[) C, 8?r 3
2a,
2, = 1 + -
^/v =
2[(Y
where
d (-k2)
I,.
+0
c 2-
3 O f ].
4
- t) CZ - 5
x o =8n [ 1
+ O (a:)
-
(Y
-
-E)
4
[a, ( - k 2 ) ] "lVo/b .
(B.38)
(a:) (B.39)
(B.40b)
Renormalization Group and Running Coupling Constant
702
P
P
Figure 1 Fermion self-energy at one-loop level.
B.5.2 Fermion propagator
s,'
( P ) = (r' - m, -
where
c
( p ) = m,
c,
(P2)
c c,(P2)
(B.41a)
(P))1
+ (d - m,)
.
(B.4l b )
Let us define an effective or running mass through the following (B.40~) sc
( P 2 ) = (1 +
c,
(p2))-l
(B.40d)
The fermion self-energy diagram is given in Fig. 2 below. This determines C ( p ) at one-loop level. The renormalization mass mR is defined by m R =
Zm
mB,
(B.42)
where 2, is the multiplicative mass renormalization constant and is given by
High Q2Behavior of Green’s Function
703
(B.43) while from Table 1: Z3F
-’
=
(l-X2)
=
1 - 2-QS ( C F
