aa r X i v : . [ h e p - l a t ] O c t On gauge fixing
Axel Maas ∗ † Karl-Franzens-University Graz, Universitätsplatz 5, A-8010 Graz, AustriaE-mail: [email protected]
Gauge fixing is a useful tool to simplify calculations. It is also valuable to combine differentmethods, in particular lattice and continuum methods. However, beyond perturbation theory theGribov-Singer ambiguity requires further gauge conditions for a well-defined gauge-fixing pre-scription. Different additional conditions can, in principle, lead to different results for gauge-dependent correlation functions, as will be discussed for the example of Landau gauge. Also therelation of lattice and continuum gauge fixing beyond perturbation theory will be briefly outlined.
The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14-19, 2010Villasimius, Italy ∗ Speaker. † Supported by the FWF under grant number M1099-N16. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ n gauge fixing
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1. Why gauge fixing?
Gauge fixing is essentially a choice of coordinates in the field configuration space, and thusnothing else than choosing a suitable coordinate system for calculations. In fact, to calculate ob-servables, it is intrinsically not necessary to fix a gauge. However, as often, it is much more simplerto select a useful coordinate system than not. Hence, almost all current calculations, from perturba-tion theory to lattice calculations of the hadron spectrum, use at one time or another gauge-fixing.One application of gauge fixing has been calculations beyond perturbation theory when com-bining different methods. A particular useful example is the the combination of lattice and con-tinuum functional methods [1]. In this case, the lattice calculations are providing gauge-fixedcorrelation functions in the domain where they are rather reliable. Functional methods then extendthese calculations to regions where lattice calculations are not feasible, like finite density, the chirallimit, disparate energy scales, and so on.Furthermore, all non-perturbative methods require at some point systematically uncontrolledapproximations. For functional methods, these are truncations. For lattice calculations, these arefinite volume and discretization, for which no analytical tools are available, and it is necessary torefer to extrapolations. Thus a comparison of different non-perturbative methods helps to controlthese systematic errors, in particular when it comes to predictions for experiments. Comparing thenthe simplest, and thus in a gauge theory gauge-dependent, correlation functions helps to establishsystematic coherence. However, gauge fixing becomes itself a problem beyond perturbation theory.
2. Gauge fixing as a selection process
The original idea of a gauge theory is to trade a redundant set of coordinates for locality. As aconsequence, it is possible to describe, e. g., Yang-Mills theory using only local interactions . Theprice paid is that not every field configuration describes a different physical process. In continuumgauge theories, there is an denumerable infinite degeneracy in terms of physical observables for thelocal variables, the gauge fields: Each physical situation is represented by a continuous gauge orbitof gauge-equivalent fields, i. e., field configurations which can be transformed into each other by agauge transformation. In the case of Yang-Mills theory L = − F a mn F mn a F a mn = ¶ m A a n − ¶ n A a m + g f abc A b m A c n an infinitesimal move along this gauge orbit is given by A a m → A a m + D ab m f b D ab m = d ab ¶ m + g f abc A c m . with A a m the gauge fields, g the gauge coupling, f abc the structure constants of the associated gaugealgebra, and the f a are arbitrary functions. It is in principle possible to go back to a formulation without redundant coordinates, like in the loop-formalism oflattice gauge theory. However, the variables and interactions then become inherently non-local. n gauge fixing Axel Maas
When performing a path integral quantization, the problem is how to ensure that each gaugeorbit is counted the same number of times. This can be ensured by either of two ways [2]. Oneway is that an average over all possible representatives of the gauge orbit is performed, which isthe basic idea of gauges like the Feynman gauge. The second possibility is to select a uniquelycharacterized representative for each gauge orbit. This is done, e. g., in Landau gauge, whichchooses perturbatively the one representative which satisfies ¶ m A a m = d -function, respectively [2]. In the continuum, such prescriptions are usually not evaluatedas a selection procedure on Gribov copies, but are rewritten with the help of auxiliary fields, theghost fields. Lattice calculations, on the other hand, have the advantage to be able to really selectgauge copies for the calculation of the path integral. Thus, in lattice calculations a literal imple-mentation of a gauge condition is possible.
3. The Gribov-Singer ambiguity
The problem of gauge fixing beyond perturbation theory is that it becomes complicated tospecify gauge conditions [3]. The reason is that the gauge algebras of Yang-Mills theory are simpleLie algebras [4]. As a consequence there is no single coordinate system which is able to cover thewhole field configuration space. Hence, it is not possible to give a local (and thus single coordinatesystem) prescription to characterize the selection criterion for representatives of the gauge orbits.This problem is known in the context of covariant gauges as the Gribov-Singer ambiguity, but itoccurs in one disguise or the other for all gauges studied so far, as long as a continuum formulationis desired. In case of the Landau gauge this problem boils down to the fact that beyond perturbationtheory there is more than one solution to the equation (2.1), and it is not possible to impose purelylocal constraints to single out one and only one representative for each gauge orbit.However, it should be remarked that there is no conceptual difference between Gribov copies,i. e. the multiple solutions of (2.1), and ordinary gauge copies. All of them are just regular repre-sentatives of a gauge orbit. The only difference is that two Gribov copies are not infinitesimallyclose to each other, as otherwise a local distinction would be possible. They are therefore separatedby non-infinitesimal gauge transformations. Still, they are just ordinary gauge copies, and theirremoval is in no way different from the removal of other gauge copies. It is just that because of thegeometrical structure of the gauge orbits it is not possible to do so by local conditions.It is of course possible, instead of trying to specify with non-local gauge condition such asingle representative, to just enlarge the averaging procedure of perturbation theory such as toencompass also the Gribov copies. This case has additional challenges, known as the Neubergerproblem [5]. However, these can be overcome [6], and such gauges can indeed be constructed.Irrespective of which possibility is chosen there is no reason to suspect a-priori that there existsa unique way of how to extend the selection process. To discuss this in more detail, the setting willbe simplified by first reduce the gauge orbits to Landau gauge i. e., all remaining representatives ofthe gauge orbit, the residual gauge orbit [7], fulfill (2.1).3 n gauge fixing
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4. The example of Landau gauge(s)
The residual gauge orbits can then be partitioned in Gribov regions [3], which give the orbitspace a shell-structure, with the shell surrounding the origin also including perturbation theory. Itcan then be shown that each shell contains at least one representative of each gauge orbit, but thatthere exists also for at least some gauge orbits multiple representatives in each shell [8]. Thus, eachshell contains all physical information. The shells are called Gribov regions, their boundaries arecalled Gribov horizons.Among all possibilities how to treat the residual gauge orbits, two have received most attention.One averages over all Gribov regions [5]. This construction can be shown to harbor a fullnon-perturbatively well-defined version of BRST symmetry [5, 6], which has the same algebraicstructure as the perturbative one. However, in any calculation this requires to take into account thesigned measure of each Gribov copy, which so far cannot be turned into a simple expression.The second one reduces the residual gauge orbit to the innermost Gribov region. It then re-mains to specify further how to deal with the remaining representatives. Three different, equallyvalid, choices have been especially pursued in lattice gauge theory recently: • Minimal Landau gauge [7, 9, 10]. In this case for each orbit a random representative ischosen among the possible ones. Results indicate that for correlation functions this may beequivalent to average over all representatives of each residual gauge orbit. • Absolute Landau gauge [7, 8, 11]. In this case the representative is selected which minimizesa certain non-local functional (the integral of the trace of the gluon propagator [7]) absolutely,though this has still a minor problem with topological identifications of certain gauge copies.It has been conjectured that for correlation functions made from a finite polynominal of thegauge fields this should yield the same correlation function as the minimal Landau gauge [8],which is supported by available results [11]. • Landau- B gauges [10]. In this case a representative is chosen in which a certain non-localquantity (the ghost propagator at zero momentum) is agreeing best with a predefined value(called B ). These gauges have shown the largest variability of correlation functions yet,though they tend to agree for finite polynomials of the fields also with the minimal Landaugauge.All of them provide a well-defined prescription how to select a representative. The minimal Landaugauge depends of course on how well the random choice is implemented such that all representa-tives are equally sampled. Though common algorithms seem to perform rather well [10], there isno proof. Fixing the absolute Landau gauge is an NP-hard problem of spin-glass type, and there-fore no guarantee exists whether any existing algorithm attempting to fix to this gauge actuallydoes this, though the results seem to support it [7, 11]. Finally, the Landau- B gauge has not beenshown to be able to differentiate between Gribov copies perfectly, i. e. that each Gribov copy has adifferent B value. However, it just averages in the sense of minimal Landau gauge over remainingcopies [10], which finally makes up for a well-defined prescription in the same sense as minimalLandau gauge. 4 n gauge fixing Axel Maas
5. Scaling and decoupling or why there could be place for more than one solution
To put these constructions into perspective, it is necessary to make a short detour.It is a remarkable fact that both the Dyson-Schwinger equations (DSEs) and the functionalrenormalization group equations (FRGs) are form-equivalent, irrespective of whether they are for-mulated within a single Gribov region or not [12]. Therefore, the solution manifold of these equa-tions has to contain not only the solution inside the first Gribov region, but also in every otherGribov region, and in the whole of the system without specifying the Gribov region. It is fur-thermore quite important that the equations are not closing on themselves alone, but at least oneexternal specification has to be introduced, e. g., the ghost dressing function at zero momentum, atleast for the truncations where this has been investigated so far [1, 13]. It is furthermore intriguingthat the situation in Coulomb gauge is quite similar [14].Using this one parameter, and imposing furthermore that the ghost dressing function shouldbe positive (semi-)definite, an one-parameter family of solutions for these equations are found[1, 13]. One of the end-points of the permitted interval is special, as it corresponds to a qualitativelydifferent kind of solution, the so-called scaling solution, which is characterized by critical behaviorin the infrared [1]. The remaining solutions are characterized by a screening behavior, and thereforethe degrees of freedom largely decouple, except for the photon-like ghost.The natural questions arising are: Assuming that the existence of these solutions is not a puretruncation artifact, to which of the Gribov regions do they belong? And is it possible to reproducethem on the lattice? At least for one of the decoupling-type solutions the latter question can beanswered by yes. It is found equally well in one particular Landau- B gauge [10] and in the minimalLandau gauge [1, 15, 16]. Also, the absolute Landau gauge appears to show a decoupling behavior[11], in contrast to the original expectations [7, 8]. However, it is not yet clear, whether it producesthe same decoupling solution in the infinite volume and continuum limit [7].The interesting question is: Can the other ones be reproduced on the lattice? Given the currentalgorithms, the necessary precondition for a positive answer is, whether they belong to the firstGribov region. Since outside the first Gribov region, the Faddeev-Popov operator acquires moreand more negative eigenvalues, it appears unlikely that the ghost propagator can maintain a singlesign. In 1+1-dimensional Coulomb gauge there is actually a proof that the ghost dressing functionis only positive inside the first Gribov region [17]. This motivates that the whole family of solutionscould be obtainable inside the first Gribov region, and thus with current gauge-fixing algorithms.However, if they are there, then they have necessarily to correspond trivially to a differentselection of Gribov copies than the minimal Landau gauge, because they are different. Thus, ifthey can be found, this implies that they are the solutions obtained in different non-perturbativerealizations of the Landau gauge. At the current time, such a one-parameter-family is found usingLandau- B gauges, but this is only obtained for rather small lattices and coarse discretizations [10].To study this further is mandatory, given the experience with the minimal Landau gauge [1, 15, 16].But it also motivates that the family could be contained in the first Gribov region.A particular case is the scaling version. Particular for a single reason: In favor of its existencein the continuum case it has been embedded [1] in the construction of Kugo and Ojima [18], whichrequired the introduction of a global BRST. As noted, this requires to average over all Gribovregions. Since the functional equations contain the information from all Gribov regions, there is5 n gauge fixing Axel Maas no contradiction. However, if this is correct, and only then, for the question concerning the latticerealization with contemporary algorithms the problem arises if it is possible to select Gribov copiesinside the first Gribov region such that they become equivalent to this average over all Gribovregions. Or, put it in another way, cancel the contribution of all other Gribov regions? If the answeris no, there is no possibility with current gauge-fixing algorithms to verify, or falsify, the existenceof the scaling solution with lattice methods under the assumption that the embedding is correct.Otherwise, the same applies to scaling as applied to decoupling.However, there are number of arguments in favor of this possibility. One is the positivity ofthe ghost dressing function, in particular in view of the 1+1-Coulomb case. The second is thatscaling appears to be realized for all volumes studied so far in two dimensions [15, 19]. Since theKugo-Ojima/BRST construction is the same in two dimensions, at least there such a cancellationappears possible. There is then no a-priori reason that this should not also be possible in higherdimensions, and that it is not only obscured by the problem of finding the right Gribov copies. Thisis also supported by the fact that in three dimensions over some momentum range scaling is seen,before it finally turns decoupling [7, 15]. If this is a problem of finding the right Gribov copies, itis clear that it becomes harder in three and even harder in four dimensions [10].Unfortunately, this is by no means a guarantee that this can work out. Nonetheless, it is suffi-cient motivation to investigate this possibility. Since the Landau- B gauges provide, by construction[10], the most divergent ghost propagator, required for the scaling case, they are the ideal tool forthis search. And while available results increase the motivation [10], only further investigationswill be able to make something close to a statement. Unfortunately, numerical lattice simulationsare never able to verify or falsify such a question like the existence of a solution. Already the expe-rience from solid-state physics teach us that there can always something unexpected happen withthe next order of magnitude of volume or discretization. Thus, the combination with continuumand other methods to obtain a final answer is indispensable.Thus, the question is therefore whether it is possible to impose Landau- B gauges also in thecontinuum as a gauge condition. The formulation of the Landau- B gauges on the lattice suggestthat a property is imposed on the average, and in particular it is not needed to be exactly fulfilledon every gauge orbit. Such constraints can be realized using Lagrange multiplier, suggesting as apossible form for Landau- B gauges for an operator O in the continuum the form < O > = lim x → Z D A m D c D ¯ c O ( A m , c , ¯ c ) q (cid:16) − ¶ m D ab m (cid:17) e − R d x L g exp (cid:18) Z B BV Z d d xd d y ¶ x m ¯ c a ( x ) ¶ y m c a ( y ) (cid:19) , where L g is the Faddeev-Popov Lagrangian, the q -function implements the restriction to the firstGribov region, V is the volume, the integral multiplying B is the ghost dressing function at zeromomentum, and Z B ensures the desired renormalization properties [10]. This is, of course, a non-local object, which explicitly breaks perturbative BRST. This is purely a speculative proposal,guided by the intuition of statistical mechanics. It should be taken only as an incentive for a lineof thought in the future. But if such a realization of the Landau- B gauge is possible, this wouldbe a further step to connect the lattice and the continuum formulation, in particular also to finallyrealize an understanding of what minimal Landau gauge is in the continuum, what is currently farfrom obvious [1, 7, 10]. 6 n gauge fixing Axel Maas
6. Outlook
Gauge fixing is a useful tool to investigate many problems. It has already been quite usefulin perturbation theory, and it is so beyond perturbation theory. However, we have only just begunto explore the possibilities of using the Gribov-Singer ambiguity to design gauges with desirableproperties, and the investigations in Landau gauge will provide us with deep insights into what isthe potential. In particular, it will provide us with an understanding to which extend we can deformthe properties of correlation functions by mere gauge choices. The discovery of multiple solutionsto the continuum equations, and the support of one of them by lattice methods, and algebraicarguments in favor of another only motivates us to understand better what is the role of gauge-fixing in this. But yet lacking analytic control, it is at the current time of significant importance topursue every possibility found to map out the implications of non-perturbative gauge fixing.
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