Copyright © 2002. Submitted to Cornell University’s online archive www.arXiv.org in 2002 by Anthony G. Williams. Post-print sourced from www.arxiv.org. Copyright © 2002 World Scientific Publishing Co. All rights reservedThe standard definition of perturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leads to ghosts and the local BRST invariance of the gauge-fixed perturbative QCD action. In the nonperturbative regime, there appears to be a choice of using nonlocal Gribov-copy free gauges (e.g. Laplacian gauge) or of attempting to maintain local BRST invariance at the expense of admitting Gribov copies and somehow summing or averaging over them. It should be recognized that the standard implementation of lattice QCD corresponds to the former choice even when only physical (i.e. colour singlet) observables are being calculated. These issues are introduced and briefly explained

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Williams, A.

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LATTICE QCD, GAUGE FIXING AND THE TRANSITION TO THE PERTURBATIVE REGIME

Adelaide Research &amp; Scholarship

arXiv:hep-lat/0212038v1  26 Dec 2002February 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings GargnanoLATTICE QCD, GAUGE FIXING AND THE TRANSITIONTO THE PERTURBATIVE REGIMEA. G. WILLIAMSSpecial Research Centre for the Subatomic Structure of Matter (CSSM),University of Adelaide, SA 5005, AustraliaE-mail: Anthony.Williams@adelaide.edu.auPerturbative QCD uses the Faddeev-Popov gauge-fixing procedure, which leadsto ghosts and the local BRST invariance of the gauge-fixed perturbative QCDaction. In the asymptotic regime, where perturbative QCD is relevant, Gribovcopies can be neglected. In the nonperturbative regime, one must adopt either anonlocal Gribov-copy free gauge (e.g., Laplacian gauge) or attempt to maintainlocal BRST invariance at the expense of admitting Gribov copies. These issuesare explored and discussed. In addition, the relationship between recent Dyson-Schwinger based calculations of the infrared behavior of QCD Green’s functionsand the lattice calculation of these quantities is examined.1. IntroductionPerturbative quantum chromodynamics (QCD) is formulated using theFaddeev-Popov gauge-fixing procedure, which introduces ghost fields andleads to the local BRST invariance of the gauge-fixed perturbative QCDaction. These perturbative gauge fixing schemes include, e.g., the standardchoices of covariant, Coulomb and axial gauge fixing. These are entirelyadequate for the purpose of studying perturbative QCD, however, they failin the nonperturbative regime due to the presence of Gribov copies. Per-turbative QCD works because in doing a weak-field expansion around theAµ = 0 configuration these Gribov copies are not encountered1.One could define nonperturbative QCD by imposing a non-local Gribov-copy free gauge fixing (such as Laplacian gauge) or, alternatively, one couldattempt to maintain local BRST invariance at the cost of admitting Gribovcopies. One of the well-known difficulties for the latter option is the prob-lem of pairs of Gribov copies with opposite sign giving a vanishing pathintegral2,3,4,5. Whether or not a local BRST invariance for QCD can bemaintained in the nonperturbative regime remains an open problem.The standard lattice definition of QCD is equivalent to the choice of a1February 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings Gargnano2Gribov copy free gauge-fixing. There is a negligible chance of selecting twogauge-equivalent configurations (strictly zero except for numerical round-off error). Calculations of physical observables are unaffected by arbitrarygauge transformations on the configurations in the ideal gauge-fixed en-semble. A lattice QCD calculation using an ideal gauge-fixed ensemble willgive a result for a gauge-invariant (i.e., physical) quantity which is identi-cal to doing no gauge fixing at all, i.e., equivalent to the standard latticecalculation of physical quantities.We begin by reviewing the standard arguments for constructing QCDperturbation theory, which use the Faddeev-Popov gauge fixing procedureto construct the perturbative QCD gauge-fixed Lagrangian density. Thenaive Lagrangian density of QCD is LQCD = −14FµνFµν +∑f q¯f (iD/ −mf )qf , where the index f corresponds to the quark flavours. The naiveLagrangian is neither gauge-fixed nor renormalized, however it is invariantunder local SU(3)c gauge transformations g(x). For arbitrary, small ωa(x)we have g(x) ≡ exp {−igs (λa/2)ωa(x)} ∈ SU(3), where the λa/2 ≡ ta arethe generators of the gauge group SU(3) and the index a runs over theeight generator labels a = 1, 2, ..., 8.Consider some gauge-invariant Green’s function (for the time be-ing we shall concern ourselves only with gluons) 〈Ω| T (Oˆ[A]) |Ω〉 =∫DA O[A] eiS[A]/∫DA eiS[A] , where O[A] is some gauge-independentquantity depending on the gauge field, Aµ(x). We see that the gauge-independence of O[A] and S[A] gives rise to an infinite quantity in both thenumerator and denominator, which must be eliminated by gauge-fixing.The Minkowski-space Green’s functions are defined as the Wick-rotatedversions of the Euclidean ones.The gauge orbit for some configuration Aµ is defined to be the set of allof its gauge-equivalent configurations. Each point Agµ on the gauge orbit isobtained by acting upon Aµ with the gauge transformation g. By definitionthe action, S[A], is gauge invariant and so all configurations on the gaugeorbit have the same action, e.g., see the illustration in Fig. 1.2. Gribov Copies and the Faddeev-Popov DeterminantAny gauge-fixing procedure defines a surface in gauge-field configurationspace. Fig. 2 is a depiction of these surfaces represented as dashed linesintersecting the gauge orbits within this configuration space. Of course, ingeneral, the gauge orbits are hypersurfaces as are the gauge-fixing surfaces.Any gauge-fixing surface must, by definition, only intersect the gauge orbitsFebruary 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings Gargnano3A orbitµA µgggaugeFigure 1. Illustration of the gauge orbit containing Aµ and indicating the effect ofacting on Aµ with the gauge transformation g. The action S[A] is constant around theorbit.at distinct isolated points in field configuration space. For this reason, itis sufficient to use lines for the simple illustration of the concepts here. Anideal (or complete) gauge-fixing condition, F [A] = 0, defines a surface calledthe Fundamental Modular Region (FMR) that intersects each gauge orbitonce and only once and typically where possible contains the trivial config-uration Aµ = 0. A non-ideal gauge-fixing condition, F′[A] = 0, defines asurface or surfaces which intersect the gauge orbit more than once. Thesemultiple intersections of the non-ideal gauge fixing surface(s) with the gaugeorbit are referred to as Gribov copies 2,3,4,5 Lorentz gauge (∂µAµ(x) = 0)for example, has many Gribov copies per gauge orbit. By definition anideal gauge fixing is free from Gribov copies. The ideal gauge-fixing surfaceF [A] = 0 specifies the FMR for that gauge choice. Typically the gaugefixing condition depends on a space-time coordinate, (e.g., Lorentz gauge,axial gauge, etc.), and so we write the gauge fixing condition more generallyas F ([A];x) = 0.=0F[A]=0F’[A]=0AµFigure 2. Ideal, F [A], and non-ideal, F ′[A], gauge-fixing.Let us denote one arbitrary gauge configuration per gauge orbit, A0µ, asthe origin for that gauge orbit, i.e., corresponding to g = 0 on that orbit.Then each gauge orbit can be labelled by A0µ and the set of all such A0µ isequivalent to one particular, complete specification of the gauge. Under agauge transformation, g, we move from the origin of the gauge orbit to theconfiguration, Agµ, where by definition A0µg−→ Agµ = gA0µg†−(i/gs)(∂µg)g†.February 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings Gargnano4Let us denote for each gauge orbit the gauge transformation, g˜ ≡ g˜[A0],as the transformation which takes us from the origin of that orbit, A0µ,to the corresponding configuration on the FMR, AFMRµ ≡ Ag˜µ, which isspecified by the ideal gauge fixing condition F ([Ag];x) = 0. In other words,an ideal gauge fixing has a unique g˜ which satisfies F ([Ag];x)|g˜ = 0 andhence specifies the FMR as Ag˜ ≡ AFMRµ ∈ FMR. Note then that we have∫DA =∫DA0∫Dg =∫DAFMR∫D(g − g˜) .The inverse Faddeev-Popov determinant is defined as the integral overthe gauge group of the gauge-fixing condition, i.e.,∆−1F [AFMR] ≡∫Dg δ[F [A]] =∫Dg δ(g − g˜)∣∣∣∣det(δF ([A];x)δg(y))∣∣∣∣−1(1)Let us define the matrix MF [A] as MF ([A];x, y)ab ≡ δF a([A];x)/δgb(y) .Then the Faddeev-Popov determinant for an arbitrary configuration Aµcan be defined as ∆F [A] ≡ |detMF [A]|. (The reason for the name is nowclear). Note that we have consistency, since ∆−1F [AFMR] ≡ ∆−1F [Ag˜] =∫Dg δ(g − g˜)∆−1F [A].We have 1 =∫Dg ∆F [A] δ[F [A]] =∫D(g − g˜) ∆F [A] δ[F [A]] by defi-nition and hence∫DAFMR≡∫DAFMR∫D(g − g˜) ∆F [A]δ[F [A]]=∫DA ∆F [A]δ[F [A]] (2)Since for an ideal gauge-fixing there is one and only one g˜ per gauge orbit,such that F ([A];x)|g˜ = 0, then |detMF [A]| is non-zero on the FMR. Itfollows that since there is at least one smooth path between any two con-figurations in the FMR and since the determinant cannot be zero on theFMR, then it cannot change sign on the FMR. The first Gribov horizon isdefined to be those configurations with detMF [A] = 0 which lie closest tothe FMR. By definition the determinant can change sign on or outside thishorizon. Clearly, the FMR is contained within the first Gribov horizon andfor an ideal gauge fixing, since the sign of the determinant cannot change,we can replace | detMF | with detMF , [i.e., the overall sign of the functionalintegral is normalized away in the ratio of functional integrals].These results are generalizations of results from ordinary calculus, where|det (∂fi/∂xj)|−1~f=0=∫dx1 · · · dxn δ(n)(~f(~x)) and if there is one and onlyone ~x which is a solution of ~f(~x) = 0 then the matrix Mij ≡ ∂fi/∂xjis invertible (i.e., non-singular) on the hypersurface ~f(~x) = 0 and hencedetM 6= 0.February 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings Gargnano53. Generalized Faddeev-Popov TechniqueLet us now assume that we have a family of ideal gauge fixings F ([A];x) =f([A];x) − c(x) for any Lorentz scalar c(x) and for f([A];x) being someLorentz scalar function, (e.g., ∂µAµ(x) or nµAµ(x) or similar or any non-local generalizations of these). Therefore, using the fact that we remainin the FMR and can drop the modulus on the determinant, we have∫DAFMR =∫DA detMF [A] δ[f [A] − c] . Since c(x) is an arbitrary func-tion, we can define a new “gauge” as the Gaussian weighted average overc(x), i.e.,∫DAFMR ∝∫Dc exp{−i2ξ∫d4xc(x)2}∫DA detMF [A] δ[f [A]− c]∝∫DA detMF [A]exp{−i2ξ∫d4xf([A];x)2}∝∫DADχDχ¯ exp{−i∫d4xd4y χ¯(x)MF ([A];x, y)χ(y)}×exp{−i2ξ∫d4xf([A];x)2}, (3)where we have introduced the anti-commuting ghost fields χ and χ¯. Notethat this kind of ideal gauge fixing does not choose just one gauge config-uration on the gauge orbit, but rather is some Gaussian weighted averageover gauge fields on the gauge orbit. We then obtain〈Ω| T (Oˆ[...]) |Ω〉 =∫DqDq¯DADχDχ¯ O[...] eiSξ[...]∫DqDq¯DADχDχ¯ eiSξ[...], (4)whereSξ[q, q¯, A, χ, χ¯]=∫d4x−14F aµνF aµν −12ξ(f([A];x))2+∑fq¯f (iD/−mf )qf+∫d4xd4y χ¯(x)MF ([A];x, y)χ(y) . (5)4. Standard Gauge FixingWe can now recover standard gauge fixing schemes as special cases of thisgeneralized form. First consider standard covariant gauge, which we obtainby taking f([A];x) = ∂µAµ(x) and by neglecting the fact that this leadsto Gribov copies. We need to evaluate MF [A] in the vicinity of the gauge-February 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings Gargnano6fixing surface (specified by g˜):MF ([A];x, y)ab =δF a([A];x)δgb(y)=δ[∂µAaµ(x) − c(x)]δgb(y)= ∂xµδAaµ(x)δgb(y)]. (6)Under an infinitesimal gauge transformation about the FMR, δg ≡ g − g˜,we have (Ag˜)µ → (Ag˜+δg)µ, where(Ag˜+δg)aµ(x) = (Ag˜)aµ(x) + gsfabcωb(x)Acµ(x)− ∂µωa(x) +O(ω2) (7)and hence in the neighbourhood of the gauge fixing surface (i.e., for smallfluctations along the gauge orbit around AFMRµ ), we haveMF ([A];x, y)ab=∂xµδAaµ(x)δ(δωb(y)]∣∣∣∣ω=0(8)= ∂xµ([−∂xµδab + gsfabcAcµ(x)]δ(4)(x− y)).We then recover the standard covariant gauge-fixed form of the QCD actionSξ[q, q¯, A, χ, χ¯] =∫d4x−14F aµνF aµν −12ξ(∂µAµ)2 +∑fq¯f (iD/−mf )qf+(∂µχ¯a)(∂µδab − gfabcAµc )χb . (9)However, this gauge fixing has not removed the Gribov copies and so theformal manipulations which lead to this action are not valid. This Lorentzcovariant set of naive gauges corresponds to a Gaussian weighted averageover generalized Lorentz gauges, where the gauge parameter ξ is the widthof the Gaussian distribution over the configurations on the gauge orbit.Setting ξ = 0 we see that the width vanishes and we obtain Landau gauge(equivalent to Lorentz gauge, ∂µAµ(x) = 0). Choosing ξ = 1 is referred toas “Feynman gauge” and so on. We can similarly derive the QCD actionfor axial gauge.5. Discussion and ConclusionsThere is no known Gribov-copy-free gauge fixing which is a local functionof Aµ(x). In other words, such a gauge fixing cannot be expressed as afunction of Aµ(x) and a finite number of its derivatives, i.e., F ([A];x) 6=F (∂µ, Aµ(x)) for all x. Hence, the ideal gauge-fixed action, Sξ[· · ·], inEq. (5) becomes non-local and gives rise to a nonlocal quantum field theory.Since this action serves as the basis for the proof of the renormalizabilityof QCD, the proof of asymptotic freedom, local BRST invariance, and theFebruary 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings Gargnano7Schwinger-Dyson equations6,7 (to name but a few) the nonlocality of theaction leaves us without a first-principles proof of these features of QCD inthe nonperturbative context.The lattice implementation of Landau gauge finds local minima of thegauge fixing functional, which correspond to configurations lying inside thefirst Gribov horizon. The remaining Gribov copies after this partial gaugefixing then necessarily all have the same sign (positive) for the Faddeev-Popov determinant and hence add coherently in the functional integral.This ensures that the ghost propagator is positive definite7,8. The deriva-tion of the Dyson-Schwinger equations is based on the fact that the in-tegral of a total derivative vanishes6 provided that the surface integral ofthe integrand vanishes when integrated over the boundary of the region.Since the Faddeev-Popov determinant vanishes on the first Gribov horizon,then we can still derive the standard Dyson-Schwinger equations from theLandau gauge fixed QCD action even if we restrict the gauge fields to liewithin the first Gribov horizon. This is equivalent to requiring that theghost propagator be positive definite. Thus it is valid to compare latticeLandau-gauge calculations with Dyson-Schwinger based calculations (witha positive definite ghost propagator), since these both consist of consider-ing configurations within the first Gribov horizon. An extensive body oflattice calculations exist for the Landau gauge gluon9 and quark10,12 prop-agators and most recently for the quark-gluon vertex13. Similarly, calcula-tions in Laplacian gauge (an ideal gauge) fixing have also recently becomeavailable14,15.It is well-established that QCD is asymptotically free, i.e., it is weak-coupling at large momenta. In the weak coupling limit the functionalintegral is dominated by small action configurations. As a consequence,momentum-space Green’s functions at large momenta will receive theirdominant contributions in the path integral from configurations near thetrivial gauge orbit, i.e., the orbit containing Aµ = 0, since this orbit min-imizes the action. If we use standard lattice gauge fixing, which neglectsthe fact that Gribov copies are present, then at large momenta∫DA willbe dominated by configurations lying on the gauge-fixed surfaces in theneighbourhood of each of the Gribov copies on the trivial orbit. Since forsmall field fluctuations the Gribov copies cannot be aware of each other, wemerely overcount the contribution by a factor equal to the number of copieson the trivial orbit. This overcounting is normalized away in the ratio offunctional integrals. Thus it is possible to understand why Gribov copiescan be neglected at large momenta and why it is sufficient to use standardFebruary 1, 2008 4:2 WSPC/Trim Size: 9in x 6in for Proceedings Gargnano8gauge fixing schemes as the basis for calculations in perturbative QCD.References1. A. G. Williams, Nucl. Phys. Proc. Suppl. 109, 141 (2002) [arXiv:hep-lat/0202010]; Nucl. Phys. A 680, 204 (2000).2. L. Giusti, M. L. Paciello, C. Parrinello, S. Petrarca and B. Taglienti, Int. J.Mod. Phys. A 16, 3487 (2001) [arXiv:hep-lat/0104012] and references therein.3. P. van Baal, arXiv:hep-th/9711070.4. H. Neuberger, Phys. Lett. B 183, 337 (1987).5. M. Testa, Phys. Lett. B 429, 349 (1998) [arXiv:hep-lat/9803025]; arXiv:hep-lat/9912029.6. C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33, 477 (1994)[arXiv:hep-ph/9403224].7. R. Alkofer and L. von Smekal, Phys. 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Rev.D 64, 074508 (2001) [arXiv:hep-lat/0102013].12. J. B. Zhang, et al., arXiv:hep-lat/0208037; F. D. Bonnet, et al., [CSSM Lat-tice collaboration], Phys. Rev. D 65, 114503 (2002) [arXiv:hep-lat/0202003].13. J. Skullerud, A. Kizilersu and A. G. Williams, Nucl. Phys. Proc. Suppl. 106,841 (2002) [arXiv:hep-lat/0109027]; J. Skullerud, P. Bowman and A. Kizilersu,arXiv:hep-lat/0212011; J. Skullerud and A. Kizilersu, JHEP 0209, 013 (2002)[arXiv:hep-ph/0205318].14. P. O. Bowman, U. M. Heller, D. B. Leinweber and A. G. Williams, Phys. Rev.D 66, 074505 (2002) [arXiv:hep-lat/0206010]. C. Alexandrou, P. De Forcrandand E. Follana, Phys. Rev. D 65, 117502 (2002) [arXiv:hep-lat/0203006]; Phys.Rev. D 65, 114508 (2002) [arXiv:hep-lat/0112043]; Phys. Rev. D 63, 094504(2001) [arXiv:hep-lat/0008012].15. P. O. Bowman, et al., arXiv:hep-lat/0209129; Phys. Rev. D 66, 014505(2002) [arXiv:hep-lat/0203001]; Nucl. Phys. Proc. Suppl. 109, 163 (2002)[arXiv:hep-lat/0112027]; Nucl. Phys. Proc. 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