The String Tension in Gauge Theories: a

arXiv:hep-lat/9210021v1 19 Oct 1992The String Tension in Gauge Theories: aSuggestion for a New Measurement MethodEnzo MARINARI(1,2), Maria Luigia PACIELLO(3),Giorgio PARISI(1,4) and Bruno TAGLIENTI(3)(1): Dipartimento di Fisica and Infn,Universit`a di Roma Tor Vergata,Viale della Ricerca Scientifica, 00133 Roma (Italy)(2): Physics Department and NPAC,Syracuse University, Syracuse, NY 13244 (USA)(3): Infn Sezione di RomaUniversit`a di Roma La Sapienza,Piazzale Aldo Moro, 2 - 00185 Roma (Italy)marinari, paciello, parisi, taglienti @roma1.infn.itNovember 19, 2018(4): Address after November 1st: Dipartimento di Fisica and Infn - Universit`a di RomaLa Sapienza, Piazzale Aldo Moro, 2 - 00185 Roma (Italy)ROMA 92 - 903hep-lat/92100211

AbstractWe discuss a new method for testing confinement and measuringthe string tension (in the Coulomb gauge). Our numerical simulationsdemonstrate that the problems related to Gribov copies are not harm-ful and that the method is effective in the case of pure gauge Q.C.D..We discuss the relevance of the method for studying gauge theoriescoupled to fermionic matter.2

In this note we introduce a new method (following an en passant remarkof ref. [1]) for measuring the string tension σ2 in gauge theories, and toestablish a criterion for confinement.

We show numerically the validity ofthis approach, and we discuss its relevance toward the simulation of fullycoupled Q.C.D..Our approach will be based on the use of Coulomb gauge; we will deal withthe gauge fixed lattice theory, in a gauge that is smooth at fixed euclideantime, i.e. where the spatial gauge fields are brought, by gauge transforma-tions, as close as possible to the identity.We will consider in the following an SU(3) gauge theory defined on alattice of volume L3 and time extent T, with periodic boundary conditionsin the 4 dimensions.The gaugeon of length n is defined byGn(⃗x, t0) ≡t0+nYt=t0[Ut(⃗x, t)] ,(1)and we integrate over a spatial 2-plane (going to zero 2-momentum) bysettingGn(xα, t0) ≡1L2X⃗xβ,⃗xγGn(⃗x, t0) ,(2)where α, β, γ = 1, 2, 3, α ̸= β ̸= γ.

We compute the zero 2-momentumcorrelation functions by definingCn(d) ≡13LTXδ=1,2,3Xt0=1,TXxα=1,L⟨Gn(xδ, t0)G†n(xδ + d, t0)⟩. (3)Let us introduce the point by elaborating, in the form discussed in ref.

[1],a point originally developed by Ferrari and Picasso[2]. The argument hintsthe relevance of measuring, in the Coulomb gauge, correlation functions oftime-like gauge fields at the same time (and different spatial points), of thetype (3).The point of view suggests that the photon can be seen as the Goldstoneboson of the gauge symmetry, and that the instantaneous potential 1x (theCoulomb potential in Coulomb gauge) can bee seen as originated from theexchange of a Goldstone boson.

For understanding the point let us consider3

our lattice theory, which has an invariance SU(3)L3T.This is the gaugesymmetry of the lattice theory, and the gauge group is somehow too large tobe broken from Goldstone bosons.We can anyhow gauge fix our theory, and reduce the symmetry. Let usfix Coulomb gauge, by maximizing the expressionXi3Xµ=1Re{Tr[Uµ(x)]} .

(4)We can get rid of Gribov copies in many different ways. We can definethe expectation values by averaging over all copies with equal probability,or we can choose, with lot of work, the global maximum (which is generi-cally unique).

We could also assign to each copy a different weight which isproportional to its basin of attraction in the algorithm we are using to fixthe gauge. We note that for similar gauge fixing algorithms the basins ofattraction are quite similar.Independently from the method we use to deal with Gribov copies, thecrucial point is that now there is a residual symmetry.

Gauge transformationswhich only depend on time but are space independent leave the quantity (4)invariant. This symmetry is a global SU(3) for each time slice, i.e.

the totalresidual symmetry is SU(3)T. But on a fixed time slice (a sensible entity toconsider in Coulomb gauge) now we have a global symmetry, which in theV →∞can by broken generating a Goldstone boson. In the V →∞limitthe symmetry will be indeed broken in the Coulomb phase (where we willhave a Goldstone boson for each time slice, and the expected propagator),while it will be preserved in the confined phase.

C1(d) will tend to a constantfor d →∞in the Coulomb case, while it will decay exponentially in theconfined phase.This physical picture leads us to suggest to use Cn(d) in order to measurethe string tension. We expect that for n and d large enough Cn(d) will decay,in the confined phase, with a behavioure−σ2nd .

(5)There are two ways which are usually employed to measure the string ten-sion σ2 and to distinguish between the confined and the deconfined phase.One is based on the measurement of large Wilson loops (the original Creutzratios), while the other is based on measuring correlation functions of Polyakov4

loops. Statistical improvements, like for example the use of smeared looppyobservables, turn out to be crucial (related to the fact we are working in acritical limit, where a correlation length is diverging).In a pure gauge theory the expectation value of a Wilson loop of areaA = B ×H behaves as e−A if the theory is confined, and as e−L if the theoryis deconfined.

If we couple the theory to fermions we can close a fermionloop only paying a price proportional to the loop length, and we get again ane−L decay. So the Wilson loop ceases to be a good indicator when we dealwith the full theory.The Polyakov loop correlation function at distance d behaves for larged as e−σ2Ld in a confined pure gauge theory, and gets a non-zero connectedpart when the pure theory deconfines.

Also in this case the fully coupledtheory does not acknowledge a difference between the two phases, since alsoin the confined phase the fermion loops give a non zero expectation valueto the loop-loop correlation function. The two most popular ways used todetermine the string tension σ2 and to distinguish between the two phasesare not effective in the case of the theory coupled to fermions.On the contrary we expect the gaugeon-gaugeon correlation functionsCn(d) behave asymptotically (for large d) as e−f(n)d both in the pure gaugetheory and in the theory coupled to fermions in the confined phase.

In thecase of the pure gauge theory we expect f(n) to coincide (for large n) withσ2n. Here indeed the U cannot take an expectation value if the symmetryis unbroken.

The method can be used both in the pure gauge and in thefermionic theory, and is likely to be a very effective method in both cases. Inthe following we will discuss a pure gauge numerical simulation in which wedemonstrate its effectiveness.We have analyzed 100 configurations on a 103×20 lattice and 412 configu-rations on a 103×6 lattice, both at β = 5.8.

They have been separated (after2000 thermalization sweeps) of 500 sweeps of an 8 hit Metropolis updatingscheme. Coulomb gauge has been fixed by using an over-relaxed procedure.On each independent gauge configuration we have gauge fixed ten times,starting from 10 different randomly gauge transformed samples.

We wereinterested to check if Gribov copies can have an influence of such a quantity(since it is computed in a gauge fixed environment). So we have averagedseparately the configurations which turned out to have a maximum value of(4), the medium ones and the minimum ones.

We have independently com-puted the rate of the 3 decays, and in the limit of our statistical error we5

have not seen any difference between them.In fig. 1 we show the effective mass estimator, as defined from the loga-rithm of the ratio of two Cn(d) (with the corrections needed from the presenceof periodic boundary conditions) for our largest lattice.

We have points forthe ration of distance 1 and 2, 2 and 3 and 3 and 4. The lines give our bestfit, of the formσ2(n) = σ2(∞) + c1n ,(6)which turns out to be perfect in all cases.

Here measuring directly an estima-tor for the string tension (also at distance 1 and 2, which is however highlybiased, since we are used local, non-smeared Wilson loops) is impossible,since the lattice is too large (the time asymptotic result is of order 0.1, seefor example [3]). In order to stress the very good linearity of our data as afunction of the gaugeon size n we plot the effective mass as a function of nin Fig.

2.In Figs. 3 and 4 we plot the same data for the 103 lattice.

Here we cancompute the true estimator for σ2effective at distance 1 over 2, and we find thatthere is a small systematic difference from the curve extrapolated by usingthe non-gauge invariant gaugeons. We expect such a small systematic effect,which will tend to zero in the continuum limit, since this is a gauge invariantmeasurement.

This additional systematic error has to be kept under control,but does not seem to be a dangerous effect, already at a quite low value ofβ.This result is very satisfactory, as far as the n-dependence of the effectivemass estimator at fixed distance seems to be under very good control. In thelimit of our statistical precision The presence of Gribov copies seems to beirrelevant as far as our results do not depend on criterion we have used tochoose the copies.

Obviously in a practical implementation of the methodone will limit is search to a single gauge fixed copy for each independentstarting gauge configuration (where, at β = 5.8, the decorrelation time forlocal observables will be surely smaller than the one we used here to be sureto get rid of all non-local correlations).As far as we have been able (by using non-smeared Wilson loops) tomeasure the effective mass only up to distance 3 over 4 the extrapolation ofthe effective mass estimator at large distance is problematic. We can onlynotice that the estimator from times 3 over 4 (extrapolated at large n) is6

about the 50% higher than the asymptotic value of the string tension: thisis the best upper bound to the value of σ2 we have been able to obtain andis a quite reasonable result. We expect the use of smeared operators to bevery effective in decreasing the error.

Checking how effective the method iswhen dealing with smeared operators seems to be the next important step.7

References[1] G. Parisi, A Short Introduction to Numerical Simulations of LatticeGauge Theories, in Ph´enom`enes Critiques, Syst`emes Al´eatories, The-ories de Jauge, p. 87, edited by K. Osterwalder and R. Stora (North-Holland, Amsterdam, The Netherlands 1986). [2] R. Ferrari and R. Picasso, Nucl.

Phys. B31 (1971) 316.

[3] E. Marinari, Nucl. Phys.

B (Proc. Suppl.) 9 (1989) 209.8

1Figure CaptionsFig. 1 Distance dependent effective mass estimator as a function of the gau-geon size n. 103 × 20 lattice.

Continuous lines are the best fits.Fig. 2 As in fig.

1, but as a function of n−1.Fig. 3 As in fig.

1, but 103 × 6 lattice.Fig. 4 As in fig.

1, but 103 × 6 lattice and as a function of n−1.9

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