A preliminary study of the Gribov ambiguity in lattice SU(3) Coulomb gauge

Physics Letters B 268 ( 1991 ) 236-240 North-Holland

PHYSICS LETTERS B

A preliminary study of the Gribov ambiguity in lattice SU ( 3 ) Coulomb gauge C l a u d i o P a r r i n e l l o a, S i l v a n o P e t r a r c a b a n d A n a s t a s s i o s V l a d i k a s ° Physics Department, New York University, 4 Washington Place, New York, NY 10003, USA b DipartimentodiFisica, Universit&diRoma "'LaSapienza", andlNFN, SezionediRoma "'LaSapienza'; Piazzale Aldo Moro 2, 1-00185 Rome, Italy c Dipartimento di Fisica, Universit& di Roma "Tor Vergata'; andINFN, Sezione di Roma "Tor Vergata'; Via E. Carnevale, 1-00173 Rome, Italy Received 18 July 1991

We report on simulations of pure SU (3) gauge theory on a 103X 20 lattice at fl= 6.0 in the Coulomb gauge, from which the Gribov ambiguity appears to be maximal, in the sense that the gauge-fixing process is highly unstable with respect to variations of the starting configuration via random gauge transformations. We give a heuristic explanation of the larger number of Gribov copies in such a gauge with respect to the Landau gauge.

1. Introduction

v o l u m e V and inverse coupling fl, nor a c o m p a r a t i v e Study between the two gauges.

Recent studies o f the G r i b o v p r o b l e m [ 1 ] ~1 on the lattice have d e m o n s t r a t e d the existence o f gauge-fixing ambiguities for both abelian [ 3 ] a n d n o n a b e l i a n [4], theories on small lattices in the L a n d a u gauge. In order to d e t e r m i n e the relevance o f the G r i b o v ambiguity for Q C D simulations involving gauge-dep e n d e n t operators, one must first investigate a n d establish its existence in the scaling region on a large lattice, for the gauge o f interest. In this letter we present the first results o f such a program for pure S U ( 3 ) theory in the C o u l o m b gauge. This gauge condition, together with the Landau gauge, is the most used both in those simulations related to the study o f gauge-dependent quantities [5,6], as well as to Q C D p h e n o m e n o l o g y simulations where the composite propagators are s m e a r e d gauge-dependent objects (see for example refs. [7, 81). We stress the p r e l i m i n a r y nature o f this work in the sense that we d i d not a t t e m p t a systematic study o f the dependence o f the G r i b o v copies on the lattice ~t For a review of the Gribov ambiguity in the continuum see for example ref. [2]. 236

2. Lattice gauge fixing and Gribov copies Let us recall how the C o u l o m b gauge can be implem e n t e d on the lattice by means o f a m i n i m i z a t i o n process [ 7,8 ]: for a given t h e r m a l i z e d link configuration { U}, generated by some gauge-invariant Monte Carlo algorithm at a fixed value offl, consider a generic configuration { U g} o b t a i n e d applying some local gauge transformation {g} on { U}: ugu(n) = g ( n ) U ~ , ( n ) g t ( n + [ t ) .

( 1)

We indicate with Uu the link variable emanating from site n in the positive d i r e c t i o n # ( # = 1, ..., 4). At each lattice site n we can evaluate the following functional o f the spatial links U k ( n ) and g ( n ) : f[ Ug(n) ] 3

=

Z ReTr[U~(n)+V~(n-~)].

(2)

k= i

Recalling the lattice definition o f the gauge Potential A , ( n ) =- [ U z ( n ) - U ~ ( n ) ]t .... tess/2ia, # = 1, ..., 4, it

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can be shown that the condition of stationarity at each site n o f f [ Ug(n) ] with respect to small gauge transformations g(n) implies that the Coulomb gauge condition Z 3~= 1 0kA g = 0 in its discretized form 3

A(n)- •

[Agk(n)--Agk(n--I~)]=O

(3)

k=l

holds at each site. In order to obtain from a generic link configuration { U} a gauge-related configuration { Ug} satisfying (3) at each site, one can consider an iterative algorithm for the minimization of

F[Ug]=-l~f[ug(.)],

(4)

where Vis the four dimensional lattice volume. If such an algorithm converges (this is not mathematically guaranteed and must be checked in practice), its fixed point will be a link configuration { C } - { U ~} [where c(n) is the local gauge transformation generated by the algorithm], satisfying the lattice Coulomb gauge condition. The lattice Landau gauge can be implemented via the minimization of a quantity which is defined analogously to Fwith the addition of the links in the time direction. I f F[ Ug] has local minima, these correspond to gauge-related link configurations in the Coulomb gauge, i.e., Gribov copies. Should these minima be "numerous", the fixed point of our algorithm may strongly depend on the initial condition and on the minimization algorithm. In order to study the approach to the Coulomb gauge we monitor two quantities [ 10,4 ]. The first is F[ Ug] itself, which decreases and then reaches a plateau. The other, denoted by 0, is defined as follows:

o - l ~ o(n)== l ~ Tr[A(n)A* (n) ] .

(5)

This quantity behaves like a kind of first derivative of F[U g] during the gauge-fixing process: it decreases (not strictly monotonically) reaching zero when F[ Ug] gets constant. Moreover, one can verify that F[ Ug] is a lattice analogue of Z 3k=1Trfd4x(A~) 2, while 0 corresponds to Trfd4x(O~A~) 2. Actually it is 0 that provides evidence that the gauge condition is fulfilled. The basic strategy for the search of Gribov copies has been the following [4 ]: we take the original link

10 October 1991

configuration { U} and transform it in a Coulomb gauge-fixed configuration { U c°} by minimizing F[ Ug ] while monitoring the approach to zero of 0. Then we go back to the original configuration { U} and perform a random gauge transformation on it, obtaining { Ug'}. This configuration will be the starting point of a new gauge-fixing process that produces the configuration { Uc!}); after that we try a different random gauge transformation on {U}, producing { ugh}, gauge-fix it to { UC2},and so on. In this way, for a fixed thermalized { U}, we can generate an "ensemble" of gauge-fixed configurations { UCi}, i = 0, ..., N. The next step consists of the analysis of the gaugefixed ensemble, in order to understand whether all the members of the ensemble are the same configuration, or true Gribov copies have appeared. We note in passing that two configurations, say {U Ci} and {Uq}, are defined to be gauge equivalent (i.e., the "same" configuration) if they are related to each other by some global, site-independent gauge transformation. The analysis starts by evaluating the values of F[C~], which is invariant under global gauge transformations, so that for two configurations { U c~} and

{ uc~}

8F c,'cj- I F [ C , ] - F [ C A I ¢ 0

(6)

is a sufficient condition for them to be distinct Gribov copies. In principle 8FC~'CJ=O being satisfied does not exclude the possibility that the two configurations be distinct Gribov copies. In this case, one may examine other gauge-dependent operators capable to distinguish among different copies, like 8TC~,CJ[U~(n)]=[Tr[UC'(n)-UCJ(n)][ 2. If Tci, cj [ Uk(n) ] is zero for all spatial links U~i(n) and U~(n) of the two configurations, then we have evidence that { Uc'} and { U ~j} are gauge equivalent.

3. Results

We have considered a pure SU(3) gauge theory defined by the standard Wilson action on a 103 × 20 lattice at r = 6. We have used three thermalized configurations generated by a Cray Y-MP. All our runs have been performed on such a machine, so that we always work in double precision (64 bit). The total amount of CPU time used in our work is approximately 50 Cray Y-MP hours. 237

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The gauge-fixing code i m p l e m e n t s an efficient iterative m i n i m i z a t i o n algorithm for F, which updates the S U ( 3 ) matrices v i a S U ( 2 ) subgroups, as proposed in ref. [ 11 ]. The r a n d o m local gauge transform a t i o n s that we have a p p l i e d to the thermalized configurations in o r d e r to produce the ensembles as described in the previous section are generated according to the S U ( 3 ) p a r a m e t r i z a t i o n p r o p o s e d in ref. [12]. We have m o n i t o r e d the quantities F and 0 after every lattice sweep o f the gauge-fixing algorithm; in such a way we have been able to decide the quality o f gauge fixing. We m e n t i o n in passing that after each gauge-fixing sweep we reorthogonalize the link variables, to make sure that they are not driven off the S U ( 3 ) group. In fig. 1 we plot r F - ( F [ C ] - F ) / F [ C] and 0 as functions o f the n u m b e r o f gauge-fixing sweeps for a typical configuration; in this case we have reduced 0 up to 10 - 15. We observe that near the end o f the curve rF, instead of decreasing monotonically, starts to fluctuate. As a consequence, our d a t a show that the m a x i m u m precision on rF that we can get from our gauge fixing algorithm is rF~ 1 0 - 12. While the order o f m a g n i t u d e o f this kind o f error is the same for every configuration that we have produced, the n u m b e r o f sweeps required to reduce 0 to a fixed value depends on the

. . . .

I0 o

I

. . . .

I

. . . .

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sample; empirically we see that O (2000) sweeps are required to reduce 0 to ~ 10 -8, which corresponds to a satisfactory i m p l e m e n t a t i o n o f the gauge c o n d i t i o n in order to disentangle G r i b o v copies. On each sample we have checked that gauge-invariant operators like the Polyakov loop a n d the average plaquette r e m a i n constant u n d e r gauge fixing, within the precision given above. In table 1 we show the values o f F [ Ci] for the three gauge-fixed ensembles generated from our thermalized configurations. F o r the ensembles A a n d B in most o f the cases we have performed the m i n i m i z a tion up to 0 = 10- i 5, and in a few cases up to 0 = 10-12. T h e latter precision on the zero o f 0, which corresponds to the m a x i m u m precision on rF allowed by rounding, does not seem necessary to get a stable value o f F , so that for the ensemble C we have stopped the algorithm when 0 = 1 0 - 8. F r o m table 1 one finds that in all the ensembles ~F cl,cs is several orders o f m a g n i t u d e greater than rF, for every pair o f configurations, so that we have a sharp evidence that each gauge-fixed configuration corresponds to a different G r i b o v copy. This is a somewhat surprising result, i f c o m p a r e d to the results for the L a n d a u gauge [ 4 ], in which most o f the times the same G r i b o v copy was r e o b t a i n e d starting from different, r a n d o m l y gauge-related configurations.

I

Table 1 The values found for F[Ci] in each of the three gauge-fixed ensembles.

10-4 I0-8

Ensemble

F[ Ci]

i0-I~:

A i0-16 i0 o 10-3 I0-6

15.75804504 - 15.75809753 -15.75776950 - 15.75790330 15.75682942 15.75812351 - 15.75723626 -

10-9

-15.77112661 - 15.76791702 • -15.77061331

10-12 0

500

i0 0 0

1500

2000

Number of gauge fixing sweeps

Fig. 1.0 and rr= (F[C] -F)IF[C] versus number of iterations. F[C] is the minimum value of F obtained in the gauge-fixing process. 238

C

-

15.77253345 15.77064109 15.77219401 15.77308875

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In addition, we r e m a r k that some m e m b e r s in the gauge-fixed ensembles have been generated by applying to the thermalized configurations a r a n d o m global gauge transformation, and then gauge fixing, Even this p r o c e d u r e has generated new copies. Again, this was not expected a priori, since when classifying G r i b o v copies there is a natural equivalence relation a m o n g t h e m with respect to global gauge transformations that leave F i n v a r i a n t . Nonetheless, there is no reason for this equivalence to hold during the gauge-fixing process. We also tried in two cases gauge fixing after performing gauge t r a n s f o r m a t i o n s close to the identity, still obtaining new G r i b o v copies. Since our goal was to establish the existence o f copies on our lattice in the C o u l o m b gauge, without att e m p t i n g at this stage a statistical analysis o f such p h e n o m e n o n , we d i d not produce large gauge-fixed ensembles. Finally, we want to stress that the a b u n d a n c e o f copies in the C o u l o m b gauge should not be naively c o m p a r e d with the results that we m e n t i o n e d for the L a n d a u gauge [ 4 ]. In fact, while L a n d a u gauge fixing is o b t a i n e d by m i n i m i z i n g a quantity that is invariant only u n d e r global gauge transformations, i.e., p r o v i d e d by the same group element g on the whole lattice, in the C o u l o m b case F is invariant u n d e r global gauge transformations a n d u n d e r transformations that are constant only within each timeslice, i.e., space-independent, t i m e - d e p e n d e n t transformations. In other words, in the C o u l o m b gauge one is still free to " r o t a t e " each timeslice with respect to others. This means that C o u l o m b gauge fixing is imp l e m e n t e d i n d e p e n d e n t l y on each timeslice, so that it makes sense to investigate gauge-fixing ambiguities within each timeslice that contains its own G r i b o v copies. In fact, we can see from table 2 that i f we single out corresponding timeslices in each m e m b e r { U c'} o f a gauge-fixed ensemble a n d c o m p a r e the values o f

1 F t [ C / ] --- -~nstim~eslicetf[Ci(rl)]

,

(7)

it appears that not every gauge-fixing o p e r a t i o n has p r o d u c e d a different G r i b o v copy on that timeslice, similarly to what has been o b s e r v e d for the L a n d a u gauge on the whole lattice in ref. [4]. In some sense, when we evaluated the total F [ Ci]

10 October 1991

Table 2 The values of F ~ in and the number of distinct copies for each timeslice in ensemble C of table 1. t

F~ nin

Copies

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-0.7920442922 -0.7875156739 -0.7925935479 -0.7920022878 -0.7886707273 -0.7870507228 -0.7922680233 -0.7905068691 -0.7912612912 -0.7887602549 -0.7842182439 -0.7860181498 -0.7881341084 -0.7865226503 -0.7862809090 -0.7891462155 -0.7864050833 -0.7856798891 -0.7904502057 -0.7881270411

1 1 1 2 1 2 1 1 2 1 2 3 3 2 2 1 3 1 2 1

(cf. table 1 ) we observed a " m i x t u r e " o f independent G r i b o v copies pertaining to different timeslices ~2. In fact, the " p h e n o m e n o l o g y " o f C o u l o m b gauge fixing on a single timeslice a n d L a n d a u gauge fixing on the whole lattice m a y be comparable, since in both cases the gauge-fixing o p e r a t i o n depends on all the link variables.

4. Conclusions We have p e r f o r m e d a p r e l i m i n a r y investigation o f the G r i b o v p r o b l e m in the C o u l o m b gauge on a large lattice in the scaling region. In spite o f a limited n u m ber o f samples, the existence o f G r i b o v copies in this gauge was established. A rough c o m p a r i s o n o f our results to those o b t a i n e d in the L a n d a u gauge [4] suggests that the G r i b o v p r o b l e m is a stronger effect in the C o u l o m b gauge, but we realize that this depends on a mixing effect o f i n d e p e n d e n t G r i b o v copies on each timeslice. The relevance o f such a p h e n o m e n o n from the ~2 This point was clarified in a discussion with D. Zwanziger. 239

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p o i n t o f v i e w o f Q C D s i m u l a t i o n s i n v o l v i n g gauged e p e n d e n t o p e r a t o r s is still n o t clear, a n d we t h i n k that o u r results call for f u r t h e r analysis in this direction.

Acknowledgement We t h a n k N. C a b i b b o , L. M a i a n i a n d D. Z w a n ziger for useful discussions, a n d G. M a r t i n e l l i a n d M. Testa for their suggestions, ideas a n d e n c o u r a g e m e n t t h r o u g h o u t this work. Finally, one o f us (A.V.) wishes to t h a n k the E E C for a research grant.

References [ 1 ] V.N. Gribov, Nucl. Phys. B 139 ( 1978 ) 1.

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[ 21 S. Sciuto, Phys. Rep. 49 (1979) 181, and references therein. [31 A. Nakamura and R. Sinclair, Phys. Lett. B 243 (1990) 396. [41 E. Marinari, C. Parrinello and R. Ricci, Nucl. Phys. B 362 (1991) 487. [5] J.E. Mandula and M. Ogilvie, Phys. Lett. B 185 (1987) 127; P. Coddington, A. Hey, J. Mandula and M. Ogilvie, Phys. Lett. B 197 (1987) 191. [6] C. Bernard, D. Murphy, A. Soni and K. Yee, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 593; C. Bernard, A. Soni and K. Yee, Applications of gauge-fixed correlation functions of quarks and gluons, BNL preprint, Nucl. Phys. B (Proc. Suppl.), to appear. [7] C.R. Allton et al., Nucl. Phys. B 349 ( 1991 ) 548. [8] Ape Collab., P. Bacilieri et al., Nucl. Phys. B 317 (1989) 509;B 343 (1990) 228. [9] J.E. Mandula and M.C. Ogilvie, Phys. Rev. D 41 (1990) 2586. [I0] C.T.H. Davies et al., Phys. Rev. D 37 (1988) 1581. [11] N. Cabibbo and E. Marinari, Phys. Lett. B 119 (1982) 387. [121 M.A.B. Beg and H. Ruegg, J. Math. Phys. 6 ( 1965 ) 677.