Gribov copy effects in maximally abelian gauge

N.H

ELSEVIER

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 49 (1996) 25(~261

Gribov copy effects in maximally abelian gauge. G.S. Bali ~, V. Bornyakov b*, M. Mfiller-Preussker ¢ and K. Schilling d ~Physics Department, The University, Southampton SO17 1B J, UK b Department of Physics, Kanazawa University, Kanazawa 920-11, J a p a n ¢Institut fiir Physik, Humboldt Universitht zu Berlin, 10099 Berlin, Germany dFachbereich Physik, Bergische Universitht, Gesamthochschule, 42097 Wuppertal, Germany and HLRZ c/o KFA, D-52425 Jiilich, Germany We study the effect of maximally abelian gauge Gribov copies in the context of the dual superconductor scenario of confinement. We present a new approach to estimate the systematic uncertainties from incomplete gauge fixing. Numerical studies have been performed, in SU(2) lattice gauge theory, based on the overrelaxed simulated annealing gauge fixing algorithm. As a result, we obtain the ratio 0.92(4) between abelian and nonabelian string tensions at fl = 2.5115.

1. I N T R O D U C T I O N Understanding how confinement arises from QCD is a central problem of strong interaction physics. G. 't Hooft and S. Mandelstam proposed the QCD vacuum state to behave like a magnetic superconductor [1,2]. A dual Meissner effect is believed to be responsible for the formation of thin string-like chromo-electric flux tubes between quarks in SU(N) Yang-Mills theories. The application of this idea to non-abelian gauge theories is based on the abelian projection [3], reducing the non-abelian SU(N) gauge symmetry to the maximal abelian (Cartan) subgroup U(1) g - 1 by gauge fixing the off-diagonal components of the gauge field. Then the theory can be regarded as an abelian gauge theory with magnetic monopoles and charged matter fields: quarks and off-diagonal gluons. The dual superconductor idea is realized if these abelian monopoles condense. In this scenario large distance (low m o m e n t u m ) properties of QCD are carried by the abelian degrees of freedom (abelian dominance). Nonperturbative investigations of this scenario became possible after formulating the abelian projection for lattice gauge theories [4]. In previ* P e r m a n e n t address: I H E P , 142284 P r o t v i n o , R u s s i a

ous numerical studies (see reviews [5,6] and references therein), it has been demonstrated that the maximally abefian (MA) gauge was a very suitable candidate for lattice investigations of the 't Hooft-Mandelstam confinement scenario, being the only known renormalizable abefian gauge. These studies provided strong evidence for the QCD vacuum to act like a dual superconductor. At this stage it appears to be important to address these issues on a quantitative level. This requires a careful study of the problem of gauge (or Gribov) ambiguities [7], and the resulting biases on abelian observables. In the present paper we develop a numerical procedure to estimate gauge fixing biases. This enables us to carry out high precision measurements of the abelian string tension and other abefian observables, with control over systematic errors. The results have been obtained on a 324 lattice a t / 3 = 2.5115 in SU(2) gauge theory. Fixing MA gauge on the lattice [4] amounts to maximizing the functional (V = Nsites) ,

F(U) = 1 ~_Tr(crsUn,~cr3U?n,~)

(1)

with respect to local gauge transformations , Un,/~

0920-5632/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved.

PII: S0920-5632(96)00342-8

,

*'l. J . t

V~,~ ----g,~ n,vg,~+~,

(2)

G.S. Bali et al./Nuclear Physics B (Proc. Suppl.) 49 (1996) 256 261

Condition Eq. (1) fixes (besides other possible degeneracies) g.~ only up to multiplications g,. v~g~ with v. = e x p ( i a . r s ) , r3 = ~3/2, - 2 r <_ a . < 21r, i . e . g . C S U ( 2 ) / U ( 1 ) . It has been shown that the corresponding continuum gauge, defined by the bilinear differential gauge condition, (O. ± .•g A ts. ) A 4. =0

,

A .4- = A u1 ± iA2u

,

(3)

is renormafizable [8], a feature, that is crucial in order to allow for a continuum interpretation of lattice results. After a configuration has been transformed to satisfy the MA gauge condition, a eoset deeomposition is performed: U . , . = C . , . V m . , where V.,. = exp(iC.,.r3), -2~r < ¢ . , . < 2a-, transforms like a (neutral) gauge field and C m . like a charged matter field with respect to transformations within the residual abelian subgroup, ? V.,. ~ vnVn,.vn+.

,

C . , . ~ vnCn,.v'tn

. (4)

Quark fields are also charged with respect to such U(1) transformations. The abelian lattice gauge fields Vm. constitute an abelian projected configuration. The abelian Wilson loop for charge one static sources is defined as 2

Vt

= Re

uz

,

(5)

257

Magnetic monopole currents k . , . , residing on the links of the dual lattice, are defined as 1

k . , . = ~-~r~.~..a~0,~,.~

,

(7)

where a lattice forward derivative is used, O v f . = f . + ~ - f . . It is easy to see, that k . , . is a conserved current: ~ . a . k n , . = O. 2. B I A S E S

FROM

GRIBOV

COPIES

We used overrelaxed simulated annealing (OSA) algorithm for gauge fixing procedure. Details of our implementation were described in [10,11]. In order to estimate systematic errors induced by an incomplete gauge fixing procedure, we have generated N = 20 random gauge copies on each of Nc = 30 equilibrium gauge configurations. Subsequently, these copies have been fixed to the maximally abelian gauge. Each abelian configuration (i.e. each gauge copy) Cj(i) = {0(J'i)(n) : # = 1 , . . . , 4 , n E V} is labeled in the following way: j runs from one to the number of gauge copies, N, while i runs from 1 to the number of SU(2) gauge configurations, No. On eaeh of these copies, abelian quantities ((smeared) Wilson loop, plaquette or monopole density) A ( C j ( i ) ) and the gauge fixing functional F ( C j ( i ) ) are measured. All gauge copies (on a given configuration) are sorted by the value of this functional: F(C1) < F ( C : ) <_... <_ F ( C N )

where un,t, = exp(iO.,.), On,. = ½¢n,.. C denotes a closed contour. The Re symbol can be omitted from the definition of the abelian Wilson loop since Im(I-Izcc ul) = 0. We define magnetic monopoles with respect to the residual U(1) gauge group in the way proposed in Ref. [9] for U(1) lattice gauge theory. Abelian plaquette variables, On,.~ = On,. + O.+#,v - On+o,. - On,v can be decomposed into a periodic (regular) part, - r < 0.~,.~ < a', and a singular part, m.~,.~ = 0, + l , ±2:

0,~,.~ =

0,~,.~

+ 2rrn~,.~

(8)

We are now prepared to investigate, which value of A we would have obtained on the "best" out of m < N copies. To answer this question one has to select m random copies out of the N copies that have been generated in total and subsequently, extract A from the copy with largest functional F, i.e. largest index i. Averaging over all possible choices yields

m

x

j-a

j=m

j--m

)

A(Cj)

(9)

(6)

0,~,.~ describes the U(1)-invariant 'electromagnetic' flux through the plaquette and mn,.u is the number of Dirac strings passing through it.

as the "average best copy" expectation of A on subsets of size rn where the gauge configuration index has been omitted. As expected, the above formula corresponds to the average over all copies

258

G.S. Bali et al./Nuclear Physics B (Proc. Suppl.) 49 (1996) 256-261

for the special case m = 1 while for m = N one obtains the value of A on the "best" copy. Quantities like the abelian potential can be computed from averages of these Ams over the Arc configurations. Our interest lies in determining how large the deviation of (Am) = 1 ~N~I Am(i) from the result ( A ~ ) on an absolute m a x i m u m is. To estimate this effect we make the following assumptions:

where ( A ~ ) , ct and dl are free parameters. The statistical errors on (Am) imply a large uncertainty on ( A ~ ) . However, due to strong correlations among the data, the differences A A can be obtained quite accurately. Because of our limited statistics for the study of gauge fixing ambiguities (Arc = 30) we have not applied full correlated fits. Nonetheless, some of the correlations have been taken into account by fitting the differences 0.00020

• The absolute m a x i m u m is unique. • The algorithm is in principle able to reach the absolute m a x i m u m Fglobal, i.e. F ~ = Fglobal. • In the neighborhood of the absolute maximum, (Am) approaches ( A ~ ) as a monotonous function of ( F ~ - Fro). The first assumption is supported by Zwanziger's proof of the non-degeneracy of absolute maxima within the interior of the fundamental modular region in Landau gauge on the lattice [12] while the third assumption is supported by numerical evidence as all our observables exhibit strong correlations with the average value of the functional. Absolute m a x i m u m is to be understood modulo the trivial degenerations due to constant gauge transformations, gauge transformations within the unfixed diagonal U(1) subgroup and the 16 degenerate m a x i m a due to Z2 center group transformations within hyperplanes perpendicular to the four possible lattice orientations, which only effect expectation values of Polyakov-line like operators but have no impact onto the spectrum. We call the difference, A A : (Ao~ - Am), the bias on A, implied by incomplete gauge fixing. The statistical uncertainty in estimating this bias constitutes the systematic error on our final result. We start from the ansatz AAm : ct e x p ( - d t m ) + . . .

(10)

for the (large m) asymptotic behavior, which has to be tested against the data. Starting from this ansatz, we can fit our data to the form ( a m ) = (A~o) - ct e x p ( - d l m ) + . - .

(11)

data , o , fit ...... 0.00015

v

"",., {

0.00010

0.00005

0 5

0

10 m

15

20

Figure 1. The differences, A F ( N , m) = FN -- Fm and an exponential fit.

A A ( N , rn)

= =

= cle - m '

(AN -- Am) - 1)

(12) (13)

instead of the data itself. Such a fit (from m = 5 onwards) is visualized in Fig. 1. From this, the bias on Am within the region of small m can be computed: A ~ = A ~ + A A ( N , m). Results from the above procedure for the functional F and the monopole density Pmon for m = 1, 10, 20 as well as for the extrapolated value (rn = c¢) are compiled in Table 1. We still find biases that are statistically significant when selecting the best out of 10 copies, generated by our OSA algorithm: AFt0 = 0.000055(10) and Apmon, t 0 --~ --0.000015(15). We

G.S. Bali et al./Nuclear Physics B (Proc. Suppl.) 49 (1996) 256-261

Table 1. Dependence of results on the number of gauge copies, m at/3 = 2.5115, V = 324, Nc = 30. m (F) Pmo, K "b 1 0.752641(34) 0.011639(26) 0.0325(11) 10 0.752789(35) 0.011528(30) 0.0311(13) 20 0.752822(36) 0.011515(32) 0.0306(16) cx~ 0.752845(37) 0.011513(33) 0.0303(17)

would like to emphasize that, by computing the biases, we have found a way to extrapolate values, obtained on local maxima, to an absolute maximum. The accuracy of all computations is limited by the statistical error on the biases.

T

0.003

bias fit ...... AKm(Nc= 30) - AK10(Nc=108) .......

]

| I ~

0.001

0 I

0

r

5

10 m

I

I

15

20

Figure 2. Differences between the extrapolated value K~¢b (from the exponential fit) and K ~b, obtained on a finite number of gauge copies, m. The solid line denotes the statistical uncertainty on K ~b from 30 gauge configurations. The horizontal dashed line is the statistical uncertainty on 108 configurations with m = 10.

The abefian potential, vmab(R), as well as the abelian string tension, K,,~, ~b have been computed for various m _< N. We find the abelian string tension, K ~ b to be anticorrelated with the gauge ab fixing functional (K,~b < Kin_l). The systematic bias,

ensemble of 108 configurations, obtained on the best out of 10 gauge copies (horizontal dashed line). We decided to choose m -- 10 for the final simulation since the expected statistical error on the string tension was extrapolated to be of the same size as both, the bias and the systematic uncertainty, i.e. the statistical error of the bias. Contrary to the case of the string tension (Fig. 2), no statistically significant bias is observed on the Coulomb coefficient e and the self energy V0 which are dominated by the short range part of the potential. Results on the abelian string tension from different numbers of gauge copies are included into Table 1 (last column). The bias on 10 gauge copies is A K -- -0.00078(72) while the expected bias on one gauge copy would have been A K = -0.0021(9). 3. P H Y S I C S

0.002

= aKIN,

m) -

is o b t a i n e d

as a

function of m (Fig. 2). The statistical errors on Arc = 30 configurations are indicated in the figure, as well as the error on the final statistical

259

RESULTS

Our main measurements have been performed on the same lattice volume and /3 value as the investigation of systematic gauge fixing errors, presented in the previous section (V -- 324 , /3 = 2.5115), such that the results could be corrected for the estimated biases and systematic uncertainties could be included into the final errors. Abelian projection and measurements have been performed on 108 independent gauge configurations. The following abelian observables (i.e. quantities expressed in terms of abelian gauge fields, O,~,u) have been investigated: charge one and two potentials and string tensions, photon and monopole contributions to the abelian string tension and the abelian monopole density. We have computed the non-abelian potential on 644 configurations to allow for direct comparison of results. Both, abelian and non-abelian potentials have been obtained from Wilson loops with smeared spatial paths in order to enhance the overlap of the QQ creation operator with the QQ ground state [13]. To reduce statistical fluctuations, we have analytically integrated out temporal links of the non-abelian Wilson loops [14]. For large temporal extent, T, the potential can be extracted from the asymptotic expectation, ( W ( R , T)) -- C ( R ) exp ( - V ( R ) T ) + . . . ,

(14)

G.S. Bali et aL /Nuclear Physics B (Proc. Suppl.) 49 (1996) 256 261

260

where C ( R ) denote the ground state overlaps. For finite values of T, we define the following approximants to overlaps and potentials, which will monotonously decrease towards their asymptotic values, VT(R)

=

log ( ( W ( R , T ) )

C (R)

=

exp

"~

(15)

\
(16)

In both cases (abelian and nonabelian), the data for R _> 2 v ~ is well described (for T _> 3) by the parametrization, e

V(R) ----Vo + K R - ~

(17)

By fitting different T-approximants to the potential to this parametrization, we obtain approximants to the string tension, KT. By demanding 0.6

.

.

.

.

.

.

-V

0.5

V a'~ v~b ~_~ o

.,~m~

I

0.4 0.3

>

0.2

0.1

-0.1O

s-

f

-o2 [°/ .

2

.

4

.

6

.

.

.

8

10

.

12

14

16

R

Figure 3. The abel]an and non-abelian potentials, V ab and V.

plateaus ofCT(R), VT(R) and KT for T _> Tmln, we find Tmin = 4 and Tmln = 6 for the abelian and non-abelian potentials, respectively. Our results on the fit parameters are in qualitative agreement with previous publications [5]. In particular, the self energy V(~b and Coulomb coe f f i c i e n t eab came out to be by more than a factor two smaller than their non-abelian counterparts while the abelian string tension was found to be close to the non-abelian one (Table 2 and Fig. 3). As pointed out above, all systematic uncertainties are understood and under control in the present

Table 2. Fit parameters for the static potentials in abelian projected SU(2) (corrected by the estimated bias due to incomplete gauge fixing) and the potential of full SU(2). Errors are systematic and statistical. AP SU(2) SU(2) e 0.095(11) 0.252(24) Vo 0.240(4) 0.545(10) K 0.0297(8) 0.0325(12)

investigation, in particular the approach to the T --* oo limit and biases due to gauge fixing ambiguities, which have been ignored in previous studies but turned out to be important as demonstrated above. The value of the abelian string tension comes out to be K ~b -- 0.0305(3) on the best out of 10 OSA gauge copies . By including bias and uncertainty of the bias, we end up with K ab = 0.0297(8), where the error includes the systematic uncertainty. As a result, we find at the given value ofl3 the ratio Kab/K = 0.92(4) (Table 2) where the error includes both, systematic and statistical uncertainties. Though qualitative agreement between the two string tensions is found, the data is sensitive enough to detect a disagreement on the quantitative level. Further measurements at different lattice spacings are required in order to decide whether this ratio approaches unity in the continuum limit, as expected from abelian dominance. The extended discussion of our results including other abelian observables ( charge two abelian potential and corresponding string tension, photon and monopole contributions to charge one abelian string tension, the abelian monopole density) can be found in [11]. 4. S U M M A R Y

AND CONCLUSIONS

Our main conclusions are the following: • The present results have been obtained from a self contained and self consistent analysis on the largest lattice volume that has been studied so far for this kind of simulations with comparatively high statistics and careful checks of possible systematic error sources.

G.S. Bali et al./Nuclear Physics B (Proc. SuppL) 49 (1996) 2 5 6 2 6 1

To obtain reliable results in the abelian projected theory with MA gauge condition one has to investigate and control the uncertainty that is inevitably introduced by the incomplete gauge fixing of numerical practice. Thus, a powerful gauge fixing algorithm turned out to be mandatory. The OSA algorithm has been used for this purpose and a method for estimation of residual uncertainties is suggested. Our conclusion is that K ab is about 10 % smaller than the non-abelian string tension at/3 = 2.5115. We expect this discrepancy to disappear in the continuum limit. We believe that further computations at different /3 values, provided all sources of errors are kept under control as in the present paper, will answer the question whether the abelian projected theory exactly reproduces the large distance behavior of the full theory. 5. A C K N O W L E D G E M E N T S

We thank the DFG for supporting the Wuppertal CM-5 project (grants Sehi 257/1-4 and Schi 257/3-2) and the HLRZ for computing time on the CM-5 at GMD. GB and KS appreciate support by EU contracts SC1"-CT91-0642 and CHRX-CT92-0051. This work was supported in part by the Grant No. NJP000, financed by the International Science Foundation, by the Grant No. NJP300, financed by the International Science Foundation and by the Government of the Russian Federation and by the Grant No. 93-0203609, financed by the Russian Foundation for Fundamental Sciences. During completion of this work GB received funding by EU contract ERBCHBG-CT94-0665. REFERENCES

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