Vistas in Astronomy, Vol. 37, pp. 305-308, 1993 Printed in Great Britain. All rights rescued.

0083-6656/93 $24.00 © 1993 Pergamon Press Ltd

NUMERICAL STUDY OF GAUGE FIXING AMBIGUITY

Atsushi Nakamura* and Masashi Mizutani

Department of Physics and Science and Engineering Research Laboratory, Waseda University, Tokyo 169, Japan

Abs t rac t After reviewing how Gribov copies are found in lattice gauge theories, we propose a stochastic

gauge fixing for the standard compact lattice gauge theories. Gauge fixed fields remain within the Gribov region in the algorithm.

1. In t roduc t ion Gauge theories are, needless to say, the most successful and beautiful tool to describe the nature.

There are many good text books for them, and we are apt to think that they had been completed. Within the perturbative calculation, quantum gauge theories are satisfactorily formulated; And yet beyond the perturbative regime there is still a skeleton in the closet: we do not know how to fix gauge uniquely in a proper manner.

This problem was pointed out more than ten years ago by Gribov [Gribov 1978]. Mathe- maticians have then proved generally that the gauge fixing is impossible [Singer 1978, Killingback 1984]. Many examples were presented under special assumptions for gauge configurations. But after several years it was forgotten, or we pretend to forget it. Recent observations of Gribov copies on the lattice have called us our attention again on this problem. In this brief report, we shall first briefly review how Gribov copies are seen on the lattice. Then we will formulate a lattice version of the stochastic gauge fixing first proposed by Zwanziger [Zwanziger 1981] and show that gauge fields simulated numerically remain within the Gribov region.

2. Latt ice gauge fixing In this section, we briefly describe the gauge fixing procedure in lattice gauge theories. Gauge

potentials A,(x) appears as link variables, U~(x),

U~,(x) = P e i f : +' .4,d~, ,., e~aA~(~), (2.1)

where a is the lattice distance, x +/2 is the nearest neighbor point along the direction ~. The gauge transformation is performed through

U~(x) --, wt(x)U~,(x)w(x + ~), (2.2)

*Address after April,1993: Faculty of Education, Yamagata University, Yamagata 990, Japan

306 A. Nakamura and M. Mizutani

for both Abelian and non-Abellan gauge fields, where w(x) are gauge transformation fields. Here we consider the Landau gauge as an example, i.e.,

4

a~A~(x) : 0, for all x. (2.3) p : l

Lattice version corresponding to Eq.3.1 is

4

A'(x) -- ~ 2 ImTr f f (U , (x ) - Up(x -/~)} = 0 for all x. (2.4) pffil

Wilson and Mandula and Ogilvie [Wilson 1980,Mandula 1987] proposed and investigated the following condition as a lattice gauge fixing procedure:

Maximize o, I -- ~.~., ~';. ReTr Up(x), (2.5) z p

i.e., for fixed gauge configuration U~(x), we tune the gauge transformation matrix w(x) so that I is maximum. When this condition is satisfied, the derivative of I with respect to the gauge transformation is zero,

~Z = 0. (2.6)

But of course the condition (2.6) does not mean the condition (2.5). They are equivalent to each other only when no local maximum/minimum point exists except the global maximum. In other words, if there are other local maximum/minimum points, the lattice gauge fixing condition (2.4) is satisfied also there. They are copies.

Our task is now to mazdmiT.e I which can be written as,

I = ~_, Re Tr wt(x)U~(x)w(x + fi), (2.7) ~,P

by changing w(x) for fixed Up. It has the form of the spin model with random interaction Up. This is a well-known difficult problem. We use therefore the following strategy: we gauge transform U~ with randomly chosen w(x) and then try to maximize I using Mandula's iterative algorithm. We repeat this procedure many times for a fixed Up. If we find two configurations U(~°)(x) and U(b)(X), we calculate a gauge transformation w(~b)(x) which transforms one to another.

uco) (x) = - Cob)t(x)UCb)( ) Cob)Cx + (2.8)

and if w ('b) is diiferent from the global gauge transformation, then they are copies to each other. In this way several cases were investigated and found copies for

• compact U(1) in 2, 3 and 4 dimension with Landau gauge condition [Forcrand 1991, Gupta 1992, Nal~mura 1991]

• SU(2) in 4 dimension with Landau and Coulomb conditions [Forcrand 1991]

Gauge Fixing Ambiguity 307

The compact U(1) model has the Coulomb phase at small coupling region where the photon becomes massless, but we get masaless photon propagators only when we take the maximum of I [N,k~mura 1991].

Let me give a comment on the U(1) case. The reader may wonder why there are copies in the Abelian theory; Gribov copies is peculiar to non-Abelian theories and there is no copy in QED. There are two essential differences between QED and the compact U(1) theory treated here. In the latter case we work on the finite lattice space and impose the periodic boundary condition. That means the theory is defined on the toms. In order to see more detail, let us write w as

o~(x) ---- e/A("),

The gauge transformation (2.2) then reads

A.(x) A.Cx) + {ACx + - ACx)}/a, (2.9)

We require the periodic boundary condition for w, but not for A. Therefore

a (x + L) = a(x) (rood 2~r), (2.10)

where L is the lattice size.

3. Stochast ic gauge fixing The stochastic gauge fixing was proposed by Zwanziger [Zwanziger 1981]: he added a force

tangent to the gauge orbits and has shown that (i) the force decreases HAIl 2 in the Langevin steps and (ii) the Gribov region where the Faddeev-Popov operator is positive and Eq.(2.3) is satisfied is an attractor for points near it. The condition (i) corresponds Eq.(2.5) on the lattice if we are near the continuum limit. Its non-compact lattice version was investigated by Seiler, Stamatescu and Zwanziger [Seiler 1984]. Here we propose a stochastic gauge fixing for the standard compact lattice gauge theories. Our Langevin equation is

U~,(z; r + At) = exp(if*t*)[P~(U~(x;'w)) l, (3.1)

where r is the Langevin fictitious time, t* are color generators, and f is the standard Langevin force,

y* = -as/aA ar + v*J

An operation P~ in Eq.(3.1) is defined in Eq.(2.8) and gauge rotations oJ is given as

o~ = exp(i~A*t*Ar/a). (3.2)

A" is defined in Eq.(2.4) and a is a gauge parameter. We can easily show that Eq.(3.1) becomes Zwanziger's form in the continuum limit. We can

also prove that I in (2.5) is increasing in this algorithm and that the Gribov region is an attractor. Preliminary analysis reveals that the compact U(1) model simulated by Eq.(3.1) has the correct massless photon propagators in the Coulomb phase.

In Fig.1 we show I in (2.5) as a function of the Langevin step for U(1) case. Around 4400th step and 5300th step it jumps to the higher values, i.e., the configuration is going beyond the

308 A. Nakamura and M. Mizutani

Gribov region, and is pulled back. This can be seen in Fig.2 where < A s > is displayed. The value remains small, i.e., the configurations stay near the hyperplane where the condition (2.4) is satisfied. But during they are pulled back to the Gribov region, they leave the hyperplane.

O.

D.d

O.

).3

O.

).~

O,

0,!

O.

0,@

@ 2C~0 20000 30000 40000 50000

!

I II I

ij !

0 20000 ~ ~ 40000

Fig.l ! Fig.2 < AS >

References

Ph.deForcrand, J.E.Hetrick, A.Nskamura and M.Plewnia, Nucl.Phys. B(Proc.SuppL)20(1991)194-198.

V.N.Gribov, Nucl.Phys. B139(1978)1-19.

S.Gupta and A.NAkamura work in progress.

J.E.Mandula and M.Ogilvie, Phys.Lett., B185 (1987) 127-132.

A.Nakamura and M.Plewnia, Phys.Lett. B255 (1991)274.

E.Seiler, I.O.Stamatescu and D.Zwanziger, Nucl.Phys. B239 (1984) 177-200.

I.M.Singer, Commun.math.Phys.60( 1978)7-12.

T.P.Killingback, Phys.Lett. B138(1984)87-90.

K.G.Wilson, in Recent Developments in Gauge theories, ed. G.t'Hooi~ (Plenum Press, New York, 1980) p.363.

D.Zwanziger, Nucl.Phys. 192 (1981) 259-269.