Strong coupling gauge theories and effective field theories: proceedings of the 2002 international workshop

Proceedings

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International Workshop

Effective Field Theories M . Harada, Y. Kikukawa & K. Yamawaki

Proceedings

°ftne/y/}fk/J

International Workshop

Strong Coupling GaugeTheories and Effective Field Theories Editors

M. Harada Y. Kikukawa K. Yamawaki Nagoya University, Japan

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World Scientific New Jersey London • Sine Singapore • Hong Kong

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10, is obeyed explicitly in the light-front formalism.2 The light-front Fock representation is especially advantageous in the study of exclusive B decays. For example, we can write down an ex-

4

act frame-independent representation of decay matrix elements such as B —» Dip from the overlap of n' = n parton conserving wavefunctions and the overlap of n' = n — 2 from the annihilation of a quark-antiquark pair in the initial wavefunction.11 The off-diagonal n + 1 —> n — 1 contributions give a new perspective for the physics of B-decays. A semileptonic decay involves not only matrix elements where a quark changes flavor, but also a contribution where the leptonic pair is created from the annihilation of a qq' pair within the Fock states of the initial B wavefunction. The semileptonic decay thus can occur from the annihilation of a nonvalence quark-antiquark pair in the initial hadron. Intrinsic charm | bucc) states of the B meson, although small in probability, can play an important role in its weak decays because they facilitate CKM-favored weak decays. 12 The "handbag" contribution to the leading-twist off-forward parton distributions measured in deeply virtual Compton scattering has a similar light-front wavefunction representation as overlap integrals of light-front wavefunctions.13'14 In the case of hadronic amplitudes involving a hard momentum transfer Q, it is often possible to expand the quark-gluon scattering amplitude as a function of k\/Q2. The leading-twist contribution then can be computed from a hard-scattering amplitude TH where the external quarks and gluons emanating from each hadron can be taken as collinear. The convolution with the light-front wavefunction and integration Uid2kj_i over the relative transverse momentum projects out only the Lz = 0 component of the light-front wavefunctions. This leads to hadron spin selection rules such as hadron helicity conservation. 15 Furthermore, only the minimum number of quark and gluon quanta contribute at leading order in l/Q2- The nominal scaling of hard hadron scattering amplitudes at leading twist then obeys dimensional counting rules. 16,17,18 Recently these rules have been derived to all orders in the gauge coupling in conformal QCD and large Nc using gauge/string duality. 19 There is also evidence from hadronic r decays that the QCD coupling approaches an infrared fixed-point at low scales.20 This may explain the empirical success of conformal approximations to QCD. The distribution amplitudes <j)(xi,Q) which appear in factorization formulae for hard exclusive processes are the valence LF Fock wavefunctions integrated over the relative transverse momenta up to the resolution scale Q.21 These quantities specify how a hadron shares its longitudinal momentum among its valence quarks; they control virtually all exclusive processes involving a hard scale Q, including form factors, Compton scattering, semiexclusive processes,22 and photoproduction at large momentum transfer, as well as the decay of a heavy hadron into specific final states. 23,24

5

The quark and gluon probability distributions q%{x,Q) and g{x,Q) of a hadron can be computed from the absolute squares of the light-front wavefunctions, integrated over the transverse momentum. All helicity distributions are thus encoded in terms of the light-front wavefunctions. The DGLAP evolution of the structure functions can be derived from the high fc_i_ properties of the light-front wavefunctions. Thus given the light-front wavefunctions, one can compute 21 all of the leading twist helicity and transversity distributions measured in polarized deep inelastic lepton scattering. Similarly, the transversity distributions and off-diagonal helicity convolutions are defined as a density matrix of the light-front wavefunctions. However, it is not true that the leading-twist structure functions Fi(x,Q2) measured in deep inelastic lepton scattering are identical to the quark and gluon distributions. It is usually assumed, following the parton model, that the F2 structure function measured in neutral current deep inelastic lepton scattering is at leading order in l/Q2 simply F2(x,Q2) = ^2qe2xq(x,Q2), where x = Xbj — Q2/2p • q and q{x,Q) can be computed from the absolute square of the proton's light-front wavefunction. Recent work by Hoyer, Marchal, Peigne, Sannino, and myself shows that this standard identification is wrong. 25 In fact, one cannot neglect the Wilson line integral between currents in the current correlator even in light-cone gauge. In the case of light-cone gauge, the Wilson line involves the transverse gluon field A± not A+ . 26 Gluon exchange between the fast, outgoing partons and the target spectators affects the leading-twist structure functions in a profound way. The final-state interactions lead to the Bjorken-scaling diffractive component j*p —> pX of deep inelastic scattering. The diffractive scattering of the fast outgoing quarks on spectators in the target in turn causes shadowing in the DIS cross section. Thus the depletion of the nuclear structure functions is not intrinsic to the wave function of the nucleus, but is a coherent effect arising from the destructive interference of diffractive channels induced by final-state interactions. Similarly, the effective Pomeron distribution of a hadron is not derived from its light-front wavefunction and thus is not a universal property. Many properties involving parton transverse momentum are also affected by the Wilson line. 27 Measurements from the HERMES and SMC collaborations show a remarkably large single-spin asymmetry in semi-inclusive pion leptoproduction 7* (q)p —> nX when the proton is polarized normal to the photon-topion production plane. Hwang, Schmidt, and I 28 have shown that final-

6

state interactions from gluon exchange between the outgoing quark and the target spectator system lead to single-spin asymmetries in deep inelastic lepton-proton scattering at leading twist in perturbative QCD; i.e., the rescattering corrections are not power-law suppressed at large photon virtually Q2 at fixed x^j. The existence of such single-spin asymmetries requires a phase difference between two amplitudes coupling the proton target with Jp = ± ^ to the same final-state, the same amplitudes which are necessary to produce a nonzero proton anomalous magnetic moment. The single-spin asymmetry which arises from such final-state interactions does not factorize into a product of distribution function and fragmentation function, and it is not related to the transversity distribution Sq(x, Q) which correlates transversely polarized quarks with the spin of the transversely polarized target nucleon. In general all measures of quark and gluon transverse momentum require consideration of final-state interactions as incorporated in the Wilson line. These effects highlight the unexpected importance of final- and initialstate interactions in QCD observables—they lead to leading-twist singlespin asymmetries, diffraction, and nuclear shadowing, phenomena not included in the light-front wavefunctions of the target. Alternatively, as discussed by Belitsky, Ji, and Yuan, 26 one can augment the light-front wavefunctions by including the phases induced by initial and final state interactions. Such wavefunctions correspond to solving the light-front bound state equation in an external field. 2. The Light-Front Quantization of QCD In Dirac's "Front Form" 29 , the generator of light-front time translations is P~ = i-§f- Boundary conditions are set on the transverse plane labelled by x± and x~ — z — ct. Given the Lagrangian of a quantum field theory, P~ can be constructed as an operator on the Fock basis, the eigenstates of the free theory. Since each particle in the Fock basis is on its mass shell, k~ = k° — k3 = 1k^n , and its energy k° — \{k+ + k~) is positive, only particles with positive momenta k+ = k° + k3 > 0 can occur in the Fock basis. Since the total plus momentum P+ — Yin k^ is conserved, the light-cone vacuum cannot have any particle content. The Heisenberg equation on the light-front is HhC\%

= M 2 |tf> .

(3)

The operator HLC = P+P~ - P±, the "light -cone Hamiltonian", is frameindependent. This can in principle be solved by diagonalizing the matrix

7

{n\Hic\fn)

0J

i the free Fock basis:

30

J3(n|//ic|m)(m|V)=M2(n|*) .

(4)

m

The eigenvalues {M 2 } of HLC = H%c + ^LC give the squared invariant masses of the bound and continuum spectrum of the theory. The lightfront Fock space is the eigenstates of the free light-front Hamiltonian; i.e., it is a Hilbert space of non-interacting quarks and gluons, each of which satisfy k2 = m2 and AT = ^ ^ 1 > 0. The projections {(n|#)} of the eigensolution on the n-particle Fock states provide the light-front wavefunctions. Thus solving a quantum field theory is equivalent to solving a coupled many-body quantum mechanical problem: "

M2-V

m

2

, 1,2 L

?/

^n = T

/ {n\VLC\n')i>n.

(5)

where the convolution and sum is understood over the Fock number, transverse momenta, plus momenta, and helicity of the intermediate states. Light-front wavefunctions are also related to momentum-space BetheSalpeter wavefunctions by integrating over the relative momenta k~ = k° — kz since this projects out the dynamics at x+ = 0. A review of the development of light-front quantization of QCD and other quantum field theories is given in the references.30 The light-front quantization of gauge theory can be most conveniently carried out in the light-cone gauge A+ = A° + Az = 0. In this gauge the A" field becomes a dependent degree of freedom, and it can be eliminated from the Hamiltonian in favor of a set of specific instantaneous light-front time interactions. In fact in QCD{\ + 1) theory, this instantaneous interaction provides the confining linear x~ interaction between quarks. In 3 + 1 dimensions, the transverse field A1 propagates massless spin-one gluon quanta with polarization vectors 21 which satisfy both the gauge condition e^ = 0 and the Lorentz condition k • e — 0. Prem Srivastava and I 31 have presented a new systematic study of light-front-quantized gauge theory in light-cone gauge using a DysonWick S-matrix expansion based on light-front-time-ordered products. The Dirac bracket method is used to identify the independent field degrees of freedom.32 In our analysis one imposes the light-cone gauge condition as a linear constraint using a Lagrange multiplier, rather than a quadratic form. We then find that the LF-quantized free gauge theory simultaneously satisfies the covariant gauge condition d • A = 0 as an operator condition as

8

well as the LC gauge condition. The gluon propagator has the form = ^ | d » f c e - * -

f ^ §

(6)

^—^2 "M"-

(7)

where we have defined Dp,(k) = £>„„(*) = -5>6.7'8>9. The first three papers present results of lattice measurements while the next three papers deal with interpretation and implications of these results.

2. Definitions of the monopole trajectories and of vortices 2.1. Topological defects in SU(2),

U{1) and Z-z cases

The trajectories and surfaces are defined on the lattice as topological defects in projected field configurations. The definitions are not an offspring of a clear theoretical concept but rather an outcome of a long empirical search for 'effective' degrees of freedom responsible for the confinement. Our main point will be that these vacuum fluctuations have also a highly non-trivial structure in the ultraviolet. However, we need to explain first the definitions. Topological defects in gauge theories are well known of course and here we will only remind a few points. The most famous example seems to be instantons. The corresponding topological charge is defined as

QtoP = ~^JG%Gapae^»°d\

,

(1)

where G£„ is the non-Abelian field strength tensor, a is the color index, a = 1,2,3. For a field configuration with a non vanishing charge there exists a non-trivial bound on the action: Sci > \QtoP\-\

•

(2)

Instantons saturate the bound. In case of SU{2) Yang-Mills theory there are no other definitions of topological charges which would be associated with a bound like (2). In this sense instantons are the only 'natural', or 'inherent' to SU(2) topological excitations.

21

If we would restrict ourselves to a U(l) subgroup of the SU(2), instantons would not appear but instead we could discuss magnetic monopoles. The topological charge now is given in terms of the magnetic flux: QM

= s - / H • ds , Sir J

(3)

where H is the magnetic field and J ds is the integral over surface of a sphere. Note that the magnetic field in (3) does not include the field of the Dirac string. Given the magnetic charge, minimum of the energy is achieved for a spherically symmetric magnetic field. The corresponding magnetic mass diverges in the ultraviolet: oo

co

d

Mmm = ±[nWr~\[ ^-~C-^,

(4)

where a is an ultraviolet cut off, the overall constant depends in fact on the details of the cut off and we kept explicit the factor 1/e2 which is due to the Dirac quantization condition. It is convenient to translate the bound on the mass (4) into a bound on the action Smon since it is the action which controls the probability to find a fluctuation. The translation is easy once we realize that monopoles are represented by closed lines, or trajectories of a length L. Indeed, the ultraviolet divergence in the mass, see (4), implies that the monopole can be visualized as point like while conservation of the magnetic charge means that the trajectories are closed. Thus, the monopole action in case of U(\) gauge theory is bounded as: const L J

mon

_

-^2~-

>

(5)

where by a we will understand hereafter the lattice spacing. It is worth emphasizing that the bound (5) is not valid if we embed the U(l) into SU(2). In this sense monopoles are not natural topological excitations for SU(2). Indeed only in the £/(l) case both the magnetic topological charge (3) and the energy (4) are expressed in terms of the same magnetic field. And this is the reason why the bound (4) exists. In the SU{2) case, we could still define the topological charge in terms of, say, Abelian part of the full non-Abelian field. The Bianchi identities would still be there and the charge would conserve. However, the action is expressed in terms of the total non-Abelian field strength tensor and for a fixed Abelian part one may have vanishing non-Abelian field, for details see 10 . In other

22

words, topological definition of the magnetic charge in the SU(2) case can be realized on gauge copies of the trivial field A^ = 0. Finally, we can consider the Zi subgroup of the original SU(2). For the Z? gauge theory the natural topological excitations are closed surfaces (for review and further references see, e.g., 2 ) . Indeed, in this theory the links can be ±1 where / is the unit matrix. Respectively the plaquettes take on values ± 1 . Unification of all the negative plaquettes is a closed surface and the action is Syort

= COUSt-z

,

(6)

a* where A is the area of the surface. Again, the infinitely thin vortices are natural excitations only in case of Zi gauge theory. 2.2. Projected

field

configurations

Since monopoles are intrinsically a U(l) object their definition within SU{2) theory is not unique. We will discuss phenomenology of the monopoles defined through the so called maximal Abelian projection, for review see, e.g., 1. First, one fixes the gauge in such a way that charged-gluon fields, A};2 are minimized over the whole lattice:

mm]T [(4) 2 + (4)2] ,

(7)

links

where 1,2 are color indices. Then, one projects the original field configuration into the closest Abelian set by putting A*'2 = 0. After this projection one gets pure Abelian configuration of fields A3^. Finally, the monopoles are denned in terms of violations of the Bianchi identities:

JTU = dvdpe^paAl .

(8)

Clearly, singular fields A^ are involved at this point. However, all expressions are in fact well defined on the lattice. At the next step, one can project the A^ fields into the closest Zi configuration and define surfaces constructed on negative plaquettes in these projected configurations. Thus, both the monopole trajectories and the vortices are defined as infinitely thin. This does not automatically mean of course that the corresponding 51/(2) fields are singular. It could well be so that the singularity of the projected associated with the magnetic charge is an artifact of the projection. This view seems to dominate the literature. We shall postpone further discussion of this point until the data are presented.

23

3. Notion of fine tuning 3.1. Action-entropy

balance

Let us concentrate for a moment on the case of 17(1) gauge theory. The 'natural' topological excitations are then monopoles. However, the action (5) is ultraviolet divergent and, at first sight, the monopole contribution is enormously suppressed in the limit a —> 0. This is actually not true. The point is that the entropy is also divergent in the limit a —» 0 (for a detailed explanations see, e.g., n ) . Indeed, the entropy factor is given now by the number Ni of monopole trajectories of the same length L. This number can be evaluated only upon introduction of discretization of the space-time. For a hyper-cubic lattice the number is 12 : NL = exp(ln7-L/a)

.

(9)

Thus, the probability to find a monopole trajectory of length L is proportional to:

WL ~ e x p ( - 4 + "ln7")--

(10)

K

ez ' a where we put In 7 in quotation marks since Eq (9) does not account for neighbors (numerically, though, the effect of neighbors is small 1 2 ). The probability (10) is a function of the electric charge alone. In particular, if e 2 is equal to its critical value,

then any length L is allowed and the monopoles condense. Eq. (10) demonstrates also that in the limit a —» 0, generally speaking, monopoles are either very rare or too common, depending on the sign of the difference in the exponential. Only a very narrow band of values of e 2 , elrit ~ fh-ae2phVs


0, see (19). This means that in this limit we are exactly at the point of the phase transition to the monopole percolation. In other words, the tuning of the entropy and action factors is exact. 4.2. Fine tuning

of vortices

on the

lattice

To define vortices, or two-dimensional surfaces one projects further the U(l) fields A^ into the closest Z^ fields, i.e. onto the matrices ±1. The surfaces are then unification of all the negative plaquettes in terms of the projected Zi fields. By definition these surfaces are infinitely thin and closed. Their relevance to confinement has been intensely investigated, see review 2 and

28

references therein. We are interested in the entropy and non-Abelian action associated with the surfaces. The results of the measurements 4 are reproduced in Figs. 2,3. At first sight, there is nothing dramatic: in both cases we have only weak dependence on a. The 'drama' is in units: the total area (per volume / m 4 ) is approximately constant in physical units while the action density is a constant in lattice units. 1

'

1

'

I

'

1

'

I -

-

C/2 C/D

1

Hill

0.7

0.5

1

• all plaquettes on P-vortices • plaquettes near monopoles • excluding plaquettes near monopoles A'side plaquettes' T'closest plaquettes'

0.3

0.1 t -0.1 0.00

1

0.02

1

1

0.04

1

M 1

1

0.06

,

0.08

1

0.10

,

1

.

0.12

1

,

0.14

0.16

a(fm) Figure 2. Excess of the non-Abelian action associated with the vortices. The excess of non-Abelian action is measured separately on the average over the vortex and on the plaquettes which simultaneously belong to monopoles. On the neighboring plaquettes (geometrically, there two different types of them) there is no excess of the action.

Thus, the excess of the action associated with the surface is approximately « 0.5^ ,

(20)

where A is area and a is the lattice spacing. While for the total area of the percolating surfaces one has: Avortex - 4(fm)

2

Vi ,

(21)

29

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

a (fin) Figure 3.

Scaling of the total area of the vortex

where V4 is the volume of the lattice. Thus, one can say that coexistence of the infrared and ultraviolet scales in case of the surfaces is seen directly on the lattice.

5. Monopoles and short-distance physics Taken at face value, the data on monopoles bring us to an amusing conclusion that monopoles make sense at short distance and can be treated as point-like particles. Indeed, the results on the monopole action and density can be summarized by saying that the lattice monopoles behave themselves as the Dirac monopoles whose action is fine tuned to the entropy. The paradox is that it seems granted that only gluons are point-like and we are not 'allowed' to introduce extra elementary particles, let them be called monopoles or else. Trying to resolve this paradox would bring us to further predictions 9 . And to further poisoned questions. There is no yet end in sight to this process and we are in position to describe only a few first steps along the road.

30

5.1. Monopole

percolation

Monopoles (in the confining phase) can be treated as a percolating system. This is the simplest realization of the idea that the monopoles are pointlike on the scale a 8 . We will review here a few points on the monopole percolation. In case of percolation (see, e.g., 15) the basic quantity is the probability p for a given link to be 'occupied'. In our case, this is the probability for a link to belong to a monopole trajectory. At p = pc there is a phase transition so that at p > pc there always exists an infinite percolating cluster. Moreover, if (p — pc) <S 1 the percolating cluster has low density and the probability 8(p) (see Eq (19)) is small: 9(p) ~ (p-Pc)a,

0< a 0 corresponds to the point p = pc and the fine tuning is perfect in this limit. Also, one can discusses properties of finite clusters as function of p — pc. In our case, these are the finite clusters extending to the infrared scale, ^-QCD- There are no much data on such clusters. To the infinite and finite clusters, which are commonly discussed in connection with percolation we would add "short clusters" which are not specifically sensitive to the infrared scale but are determined by physics at short distances 8 . Properties of such clusters can readily be predicted. Indeed, according to the idea of fine tuning monopoles at short distances correspond to free particles. At the next step, one can add Coulomb-like interaction between the monopoles. The simplest vacuum graph for free monopoles is just a closed loop without self-intersections. This graph corresponds to the the following spectrum of the clusters in their length L:

N{L) ~ - 1 ,

(23)

as can be understood by inspecting, e.g., equations in Ref. u . The spectrum (23) remains true with account of the Coulomb-like interaction as well 8 . Moreover, the radius of the cluster, R should satisfy the relation:

R ~ VI .

(24)

Both predictions (23) and (24) are in perfect agreement with the data 16,6 . Thus, we can say that the simplest vacuum loop corresponding to the monopole field is directly observed on the lattice.

31

5.2.

Vacuum condensate

of the magnetically

charged

field

In view of this apparent relevance of the free field theory to properties of the monopole clusters it is worth to discuss the vacuum expectation value < I^MI 2 > where <J>M is a hypothetical magnetically charged field. The point is that we can directly relate < \(J>M\2 > to the monopole density 7 . To this end, let us go back to the polymer representation of the field theory mentioned in Sect. 3.2. By differentiating the partition function with respect to the mass M(a) introduced in the polymer representation (see (5)) we get for the average length of the monopole trajectory:

< L>

= mXnZ

(25)

•

Moreover, using the relation between the mass M(a) and the standard field theoretical mass mpr0p we find:



= ;4^ l n 2 •

M\2 a n d the derivative in the r.h.s. of Eq (26) is related to < \• Unifying all these simple equations we get: M|2|0>

=

^(Pperc +

Pfin)

•

(27)

Note that the percolating cluster produces condensate of order < |^M|2 > ~ (a • AQC , £))AQ C £ ) and vanishes in the limit a —> 0. Nevertheless the whole of the confinement in the monopole-dominated picture is due to the percolating cluster and pperc ~ const is sufficient to maintain the confining force for external heavy quarks in the limit o —• 0. We conclude this subsection with a remark on gauge invariance of the condensate (27). The point is that gauge invariance of the monopole density-discussed above- apparently implies gauge invariance of the condensate (27). In terms of the original gauge fields gauge invariant condensate of dimension d = 2 was introduced rather recently, see 17 . It would be natural to identify (27) with the non-perturbative part of the condensates discussed in 17 .

32

5.3. Association

of the monopoles

with the

vortices

The contribution of the percolating cluster to the < \<J>M\2 >< see Eq. (27), can be viewed as the classical part while the piece proportional to pfin corresponds to quantum fluctuations. Moreover, we mentioned in the previous subsection that some properties of the short monopole clusters are reproduced by free field theory. If so, we should have reproduced also the standard ultraviolet divergence in the condensate, < \<J>M\2 > ~ 1/a2On the other hand, one can argue that ultraviolet divergences are allowed only for operators constructed on the gluonic fields. This conjecture is a generalization of the constraint that the magnetically charged field cannot change the /3-function. Thus, at first sight, we have a contradiction. However, let us turn to the data. The full monopole density is fitted as 14,6,

+ c2A2QCDa-1

,

(28)

where ci,2 do not depend on a. Substituting (28) into (27) we find

~ &QCD >

(29)

which is very reasonable. Geometrically (28) implies that the monopole trajectory are in fact associated with a two-dimensional submanifold of the whole d=4 space. One could have predicted this by imposing the constraint that < |