The complete cancellation of Gribov copies and the Neuberger 0/0 problem of
lattice BRST can be avoided in modified lattice Landau gauge. In compact U(1),
where the problem is a lattice artifact, there remain to be Gribov copies but
their number is exponentially reduced. Moreover, there is no cancellation of
copies there as the sign of the Faddeev-Popov determinant is positive. Applied
to the maximal Abelian subgroup this avoids the perfect cancellation amongst
the remaining Gribov copies for SU(N) also. In addition, based on a definition
of gauge fields on the lattice as stereographically-projected link variables,
it provides a framework for gauge fixed Monte-Carlo simulations. This will
include all Gribov copies in the spirit of BRST. Their average is not zero, as
demonstrated explicitly in simple models. This might resolve present
discrepancies between gauge-fixed lattice and continuum studies of QCD Green's
functions.Comment: 5 pages, talk presented at the 8th Conference Quark Confinement and
  the Hadron Spectrum, September 1-6, 2008, Mainz, German

André Sternbeck

Dhagash Mehta

Er Jorkowski

Lorenz Von Smekal

English

arXiv

Submitted to Cornell University’s online archive www.arXiv.org in 2008 by Lorenz von Smekal.The complete cancellation of Gribov copies and the Neuberger 0=0 problem of lattice BRST can be avoided in modified lattice Landau gauge. In compact U(1), where the problem is a lattice artifact, there remain to be Gribov copies but their number is exponentially reduced. Moreover, there is no cancellation of copies there as the sign of the Faddeev-Popov determinant is positive. Applied to the maximal Abelian subgroup this avoids the perfect cancellation amongst the remaining Gribov copies for SU(N) also. In addition, based on a definition of gauge fields on the lattice as stereographically-projected link variables, it provides a framework for gauge fixed Monte-Carlo simulations. This will include all Gribov copies in the spirit of BRST. Their average is not zero, as demonstrated explicitly in simple models. This might resolve present discrepancies between gauge-fixed lattice and continuum studies of QCD Green’s functions.Lorenz von Smekal, Alexander Jorkowski, Dhagash Mehta and André Sternbeckhttp://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=7

Von Smekal, L.

Jorkowski, A.

Mehta, D.

Sternbeck, A.

Adelaide Research &amp; Scholarship

PoS(Confinement8)048Lattice Landau gauge via Stereographic ProjectionLorenz von Smekal∗, Alexander Jorkowski, Dhagash Mehta and André SternbeckCentre for the Subatomic Structure of Matter, School of Chemistry and Physics,The University of Adelaide, SA 5005, AustraliaE-mail: lorenz.smekal@adelaide.edu.au,alexander.jorkowski@student.adelaide.edu.au,dhagash.mehta@adelaide.edu.au, andre.sternbeck@adelaide.edu.auThe complete cancellation of Gribov copies and the Neuberger 0/0 problem of lattice BRST canbe avoided in modified lattice Landau gauge. In compact U(1), where the problem is a latticeartifact, there remain to be Gribov copies but their number is exponentially reduced. Moreover,there is no cancellation of copies there as the sign of the Faddeev-Popov determinant is posi-tive. Applied to the maximal Abelian subgroup this avoids the perfect cancellation amongst theremaining Gribov copies for SU(N) also. In addition, based on a definition of gauge fields onthe lattice as stereographically-projected link variables, it provides a framework for gauge fixedMonte-Carlo simulations. This will include all Gribov copies in the spirit of BRST. Their averageis not zero, as demonstrated explicitly in simple models. This might resolve present discrepanciesbetween gauge-fixed lattice and continuum studies of QCD Green’s functions.8th Conference Quark Confinement and the Hadron SpectrumSeptember 1-6, 2008Mainz, GermanySupported by the Australian Research Council.∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(Confinement8)048Lattice Landau gauge via Stereographic Projection Lorenz von Smekal1. IntroductionThe Green’s functions of QCD provide a basis for hadron phenomenology [1]. Their infraredbehaviour is also known to contain information about the realisation of confinement in Landaugauge QCD. Dyson-Schwinger equation (DSE) studies [2] have established that the gluon propa-gator alone does not provide long-range interactions of a strength sufficient to confine quarks, andthat the infrared dominant correlations are instead mediated by the Faddeev-Popov ghosts of thisformulation, whose propagator was found to be infrared enhanced. This can be understood in termsof confinement in QCD [1, 3, 4], as a consequence of the Kugo-Ojima criterion which is based onthe realisation of the unfixed global gauge symmetries of the covariant continuum formulation. Inorder to distinguish confinement from Coulomb and Higgs phases it requires: (a) The massless sin-gle particle singularity in the transverse gluon correlations of perturbation theory must be screenednon-perturbatively to avoid long-range fields and charged superselection sectors as in QED. (b) Theglobal gauge charges must remain well-defined and unbroken to avoid the Higgs mechanism. InLandau gauge, in which the (Euclidean) gluon and ghost propagators are parametrised by the twoinvariant functions Z and G, respectively, this entails that(a): limp2→0Z(p2)/p2 < ∞ ; (b): limp2→0G−1(p2) = 0 . (1.1)The translation of (b) into the infrared enhancement of the ghost propagator thereby rests on theghost/anti-ghost symmetry of the Landau gauge or the symmetric Curci-Ferrari gauges. It repre-sents an additional boundary condition on DSE solutions which then lead to the prediction of aconformal infrared behaviour for the gluonic correlations in Landau gauge QCD [4]. In fact, thisbehaviour is directly tied to the validity and applicability of the framework of local quantum fieldtheory for non-Abelian gauge theories beyond perturbation theory. The subsequent verificationof this infrared behaviour with a variety of different functional methods in the continuum meanta remarkable success. These methods which all lead to the same prediction include studies oftheir Dyson-Schwinger Equations (DSEs) [4], Stochastic Quantisation [5], and of the FunctionalRenormalisation Group Equations (FRGEs) [6]. This prediction amounts to infrared asymptoticformsZ(p2) ∼ (p2/Λ2QCD)2κZ , and G(p2) ∼ (p2/Λ2QCD)−κG , (1.2)for p2→ 0, which are both determined by a unique critical infrared exponent κZ = κG ≡ κ , with0.5 < κ < 1. Under a mild regularity assumption on the ghost-gluon vertex [4], the value of thisexponent is furthermore obtained as κ = (93−√1201)/98 ≈ 0.595 [4, 5].The conformal nature of this infrared behaviour in the pure Yang-Mills sector of Landau gaugeQCD is evident in the generalisation to arbitrary gluonic correlations [7] which has furthermorebeen shown to represent a unique scaling solution [8]. In particular, the ghost-gluon vertex is theninfrared finite and the non-perturbative running coupling of [2] approaches an infrared fixed-point,αS(p2) =g24piZ(p2)G2(p2) → αc for p2→ 0 . (1.3)If the ghost-gluon vertex is regular at p2 = 0, its value is maximised and given by αc ≈ 8.9/Nc [4].However, the uniqueness of infrared scaling does not rule out solutions with infrared finite gluonpropagator and a ghost propagator with a free massless-particle singularity, i.e., Z(p2) ∼ p2/M2 ,and G(p2) ∼ const., for p2 → 0. This solution corresponds to κZ = 1/2 and κG = 0. It doesnot satisfy the scaling relation κZ = κG in (1.2) because transverse gluons decouple for momenta2PoS(Confinement8)048Lattice Landau gauge via Stereographic Projection Lorenz von Smekalp2  M2, and it is therefore called the decoupling solution [9]. The interpretation of (1.3) as arunning coupling does not make sense in the infrared in this case, in which there is no infraredfixed-point and no conformal infrared behaviour.Because of the inevitable finite-volume effects, early lattice studies of the gluon and ghostpropagators could have been consistent with both, the scaling solution or the decoupling solution.Recently, the finite-volume effects have been analysed carefully in the Dyson-Schwinger equationsto demonstrate how the scaling solution is approached in the infinite volume limit there [10]. Thisis clearly not what is being observed, however, in more recent SU(2) lattice data on impressivelylarge lattices [11, 12]. Present lattice data is fully consistent with the decoupling solution whichposes the question whether we are perhaps comparing apples with oranges when comparing theminimal lattice Landau gauge correlations with those of local quantum field theory in the infrared?The latter is based on a cohomology construction of a physical Hilbert space over the indefinitemetric spaces of covariant gauge theory from the representations of the Becchi-Rouet-Stora-Tyutin(BRST) symmetry. But do we have a non-perturbative definition of a BRST charge in presence ofGribov copies? In the most direct translation of BRST symmetry on the lattice, there is a perfectcancellation among these gauge copies which gives rise to the famous Neuberger 0/0 problem.It asserts that the expectation value of any gauge invariant (and thus physical) observable in alattice BRST formulation will always be of the indefinite form 0/0 [13] and therefore preventedsuch formulations for more than 20 years now. In present lattice implementations of the Landaugauge this problem is avoided because the numerical procedures are based on minimisations of agauge-fixing potential w.r.t. gauge transformations. To find absolute minima is not feasible on largelattices as this is a non-polynomially hard computational problem. One therefore settles for localminima which in one way or another, depending on the algorithm, samples gauge copies of the firstGribov region among which there is no cancellation. For the same reason, however, this is not aBRST formulation. The emergence of the decoupling solution can thus not be used to dismiss theKugo-Ojima criterion of covariant gauge theory in the continuum.2. Lattice BRST and the Neuberger 0/0 ProblemIn principle, a BRST symmetry could be implemented on the lattice by inserting the partitionfunction of a topological model with BRST exact action into the gauge invariant lattice measure.Because of its topological nature, this gauge-fixing partition function ZGF will be independent ofgauge orbit and gauge parameter. The problem is that in the standard formulation this partitionfunction calculates the Euler characteristic χ of the lattice gauge group which vanishes [14],ZGF = χ(SU(N)#sites) = χ(SU(N))#sites = 0#sites . (2.1)Neuberger’s 0/0 problem of lattice BRST arises because we have then inserted zero instead ofunity (according to the Faddeev-Popov prescription) into the measure of lattice gauge theory. Ona finite lattice, such a topological model is equivalent to a problem of supersymmetric quantummechanics with Witten index ZGF, except that for gauge-fixing we need a model with non-vanishingWitten index to avoid the Neuberger 0/0 problem. Then however, just as the supersymmetry of thecorresponding quantum mechanical model, such a lattice BRST cannot break.In Landau gauge the Neuberger zero, ZGF = 0, arises from the perfect cancellation of Gribovcopies via the Poincaré-Hopf theorem. The gauge-fixing potential for a generic link configuration3PoS(Confinement8)048Lattice Landau gauge via Stereographic Projection Lorenz von Smekalthereby plays the role of a Morse potential for gauge transformations and the Gribov copies are itscritical points (the global gauge transformations need to remain unfixed so that there are strictlyspeaking only (#sites−1) factors of χ(SU(N)) = 0 in (2.1)). The Morse inequalities then immedi-ately imply that there are at least 2(N−1)(#sites−1) such copies in SU(N) on the lattice, or 2#sites−1 incompact U(1), and equally many with either sign of the Faddeev-Popov determinant.The topological origin of the zero originally observed by Neuberger in a certain parameterlimit due to uncompensated Grassmann ghost integrations in standard Faddeev-Popov theory [13]becomes particularly evident in the ghost/anti-ghost symmetric Curci-Ferrari gauge with its quarticghost self-interactions. In this gauge the same parameter limit leads to computing the zero in (2.1)from a product of independent Gauss-Bonnet integral expressions, for each site of the lattice [15],corresponding to the Gauss-Bonnet limit of the equivalent supersymmetric quantum mechanicsmodel in which only constant paths contribute [16]. The indeterminate form of physical observ-ables as a consequence of (2.1) can be regulated by a Curci-Ferrari mass term. While such a massm decontracts the double BRST/anti-BRST algebra, which is known to result in a loss of unitarity,observables can then be meaningfully defined in the limit m→ 0 via l’Hospital’s rule [15].3. Lattice Landau Gauge from Stereographic ProjectionThe 0/0 problem due to the vanishing Euler characteristic of SU(N) is avoided when fixingthe gauge only up to the maximal Abelian subgroup U(1)N−1 because the Euler characteristic ofthe coset manifold is non-zero. The corresponding lattice BRST has been explicitly constructedfor SU(2) [14], where the coset manifold is the 2-sphere and χ(SU(2)/U(1)) = χ(S2) = 2. Thisindicates that the Neuberger problem might be solved when that of compact U(1) is, where thesame cancellation of lattice Gribov copies arises because χ(S1) = 0. A surprisingly simple solutionto this problem is possible, however, by stereographically projecting the circle S1→ R which canbe achieved by a modification of the minimising potential [17]. The resulting potential is convexto the above and leads to a positive definite Faddeev-Popov operator for compact U(1) where thereis thus no cancellation of Gribov copies, but ZU(1)GF = NGC, for NGC Gribov copies, which followsfrom a simple example of a Nicolai map [16]. As compared to the standard lattice Landau gaugethe number of copies is furthermore exponentially reduced. This is easily verified explicitly in lowdimensional models. While NGC grows exponentially with the number of sites in the standard caseas expected, the stereographically projected version has only NGC = Nx copies on a periodic chainof length Nx, and lnNGC ∼ Nt lnNx on a 2D lattice of size Nt ×Nx in Coulomb gauge, for example,and in both cases their number is verified to be independent of the gauge orbit.Applying the same techniques to the maximal Abelian subgroup U(1)N−1, the generalisationto SU(N) lattice gauge theories is possible when the odd-dimensional spheres S2n+1, n= 1, . . .N−1,of its parameter space are stereographically projected toR×RP(2n). In absence of the cancellationof the lattice artifact Gribov copies along the U(1) circles, the remaining cancellations betweencopies of either sign in SU(N), which will persist in the continuum limit, are then necessarilyincomplete, however, because χ(RP(2n)) = 1. For SU(2) this program is straightforward. Startingfrom a modified gauge-fixing potential [17] one defines stereographically-projected gauge fieldson the lattice (see [18]),Axµ =12ia(U˜xµ −U˜†xµ), where U˜xµ ≡ 2Uxµ1+ 12 TrUxµ,4PoS(Confinement8)048Lattice Landau gauge via Stereographic Projection Lorenz von Smekalsuch that the gauge-fixing condition is given by the usual lattice divergence but in terms of thesestereographically-projected gauge fields. A particular advantage of the non-compact gauge fieldsis that they allow to resolve the gauge condition of the stereographically-projected lattice Landaugauge by Hodge decomposition. This provides a framework for gauge-fixed Monte-Carlo simula-tions which is currently being developed for SU(2) in 2 dimensions [19]. In the low-dimensionalmodels mentioned above it can furthermore be verified explicitly that the corresponding gauge-fixing partition function is indeed given by ZSU(2)GF = ZU(1)GF 6= 0, as expected from χ(RP(2)) = 1.4. Conclusions and OutlookComparisons of the infrared behaviour of QCD Green’s functions as obtained from lattice Lan-dau gauge implementations based on minimisations of a gauge-fixing potential and from continuumstudies based on BRST symmetry have to be taken with a grain of salt. Evidence of the asymptoticconformal behaviour predicted by the latter is seen in the strong coupling limit of lattice Landaugauge where such a behaviour can be observed at large lattice momenta a2 p2 1 [18]. Observeddeviations from scaling at small momenta in the strong-coupling limit are not finite-volume effects,but discretisation dependent and hint at a breakdown of BRST symmetry arguments beyond pertur-bation theory in this approach. Non-perturbative lattice BRST has been plagued by the Neuberger0/0 problem, but its improved topological understanding provides ways to overcome this problem.The most promising one at this point rests on stereographic projection to define gauge fields on thelattice together with a modified lattice Landau gauge. This new definition has the appealing featurethat it will allow gauge-fixed Monte-Carlo simulations in close analogy to the continuum BRSTmethods which it will thereby elevate to a non-perturbative level.References[1] R. Alkofer and L. von Smekal, Phys. Rept. 353 (2001) 281.[2] L. von Smekal, A. Hauck and R. Alkofer, Phys. Rev. Lett. 79 (1997) 3591; Annals Phys. 267 (1998) 1.[3] R. Alkofer, L. von Smekal and P. Watson, arXiv:hep-ph/0105142.[4] Ch. Lerche and L. von Smekal, Phys. Rev. D 65 (2002) 125006.[5] D. Zwanziger, Phys. Rev. D 65 (2002) 094039.[6] J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett. 93 (2004) 152002.[7] R. Alkofer, C. S. Fischer and F. J. Llanes-Estrada, Phys. Lett. B 611 (2005) 279.[8] C. S. Fischer and J. M. Pawlowski, Phys. Rev. D 75 (2007) 025012.[9] C. S. Fischer, A. Maas and J. M. Pawlowski, arXiv:0810.1987 [hep-ph].[10] C. S. Fischer, A. Maas, J. M. Pawlowski and L. von Smekal, Annals Phys. 322 (2007) 2916.[11] A. Sternbeck, L. von Smekal, D. B. Leinweber and A. G. Williams, PoS LAT2007 (2007) 340.[12] A. Cucchieri and T. Mendes, PoS LAT2007 (2007) 297.[13] H. Neuberger, Phys. Lett. B 175 (1986) 69; ibid. 183 (1987) 337.[14] M. Schaden, Phys. Rev. D59 (1998) 014508.[15] L. von Smekal, M. Ghiotti and A. G. Williams, Phys. Rev. D 78 (2008) 085016.[16] D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Phys. Rept. 209 (1991) 129.[17] L. von Smekal, D. Mehta, A. Sternbeck, A. G. Williams, PoS LAT2007 (2007) 382.[18] A. Sternbeck and L. von Smekal, PoS LATTICE2008 (2008) 267; and arXiv:0811.4300 [hep-lat];see also A. Sternbeck’s talk at this conference.[19] A. Jorkowski, Honours thesis, University of Adelaide, November 2008;A. Jorkowski, A. Sternbeck and L. von Smekal, in preparation.5

Lattice Landau Gauge via Stereographic Projection

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Lattice Landau gauge via Stereographic Projection

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