NBI-HE-94-30 March 1994

Scaling with a modified Wilson action which suppresses Z_{2} artifacts in SU(2) lattice gauge theories

J. Ambjørn and G. Thorleifsson

The Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark

Abstract

A modified Wilson action which suppresses plaquettes which take negative values is used to study the scaling behavior of the string tension. The use of the β_{E} scheme gives good agreement with asymptotic two loop results.

Contents

1 Introduction 3 2 Numerical method 4 3 Scaling of the string tension 5 4 Conclusions 6

1 Introduction

The simplest and most popular choice of a gauge lattice action is the one proposed by Wilson, which for the group SU(2) reads:

S_{W} (U_{l}) = β ∑_{□}(1 − \frac{1}{2}Tr U_{□}),  (1)

where β = 4/g^{2}, U_{l} ∈ SU(2) is the link variable defined on the link l ≡ (x, μ), while □ ≡ (x, μν) refers to the location and orientation of the corresponding plaquette. U_{□} is the standard plaquette variable:

U_{□} ≡ U_{x,μν} = U_{x,μ}U_{x+μ,ν}U_{x+ν,μ}^{†}U_{x,ν}^{†}.  (2)

In any Monte Carlo simulation the crucial question is whether the field configurations generated provide an adequate representation of the continuum physics. The naive continuum limit can only be obtained for \frac{1}{2}Tr U_{□} ≈ 1. Especially any lattice configurations which locally have some U_{□}’s where \frac{1}{2}Tr U_{□} ≈ −1 are lattice artifacts which a priori have nothing to do with continuum configurations. A complete suppression of such configurations should not change any continuum physics. For the values of β where one performs the Monte Carlo simulations such local Z_{2} fluctuations are not at all rare if S_{W} is used. It is thus very important to make sure that these fluctuations have no impact on continuum observables. The easiest way to do so is to modify the lattice action in such a way that these small-scale fluctuations are suppressed, without influencing plaquettes where Tr U_{□} > 0. This can be done by modifying the Wilson action as follows [1]:

S = S_{W} + S_{λ},  (3)

where S_{W} is the standard Wilson action (1), while S_{λ} is

S_{λ} = λ ∑_{□}[1 − sgn(Tr U_{□})].  (4)

The action (3) was studied for λ = ∞ in [2] while the phase structure in the (β, λ) plane was studied in [1]. For a fixed λ one would expect the same continuum limit for β → ∞ and also the same identification β = 4/g^{2}. However, as shown in [1] there is a marked difference between, say, Creutz ratios, measured for λ = 0 (the Wilson action) and for λ = 0.5 in the case of β = 2.5. One would expect the difference to be even more pronounced with increasing λ. This illustrates that plaquettes with negative values may play an important role for the range of β’s where the scaling of pure SU(2) lattice gauge theories is usually studied, and one could be worried about

β No. of sweeps No. of measurements
1.5  55250  11050
1.6  65250  13050
1.7  63750  12750
1.8  88750  17750
1.9  45000  9000

Table 1: Statistics collected at various β. In all measurements λ = 10

the relation to continuum physics. The fact that one gets acceptable agreement with continuum scaling relations could be fortuitous since one would expect the action S to reproduce continuum physics better for large λ. One purpose of this article is to show that we indeed get the correct scaling relations for large λ and that there is no reason to expect the plaquettes with negative values which appear in the Wilson action to play any important role for β ≥ 2.2. Another purpose is to test whether the λ modification of the action may improve the approach to scaling for the reasons mentioned above.

2 Numerical method

The numerical simulations were performed using a standard Metropolis algorithm to update the gauge fields, combined with an overrelaxation algorithm to decrease the autocorrelations. For every Metropolis sweep we performed 4 overrelaxation steps. Lattice size 12^{4} with periodic boundary conditions was used and we measured all Wilson loops up to the size 6 × 6. In order to improve the statistics we modified the configurations using the Parisi trick [6] before measuring. Unlike in the case of λ = 0 the integration involved in the Parisi trick had to be performed numerically and was thus costly in computer time. But the gain in the statistics did more than compensate for that.

The choice of parameters was λ = 10, which was sufficient to suppress all negative plaquettes, and β ∈ [1.0, 2.5]. We found a scaling window for β ∈ [1.5, 1.9] and concentrated the simulations in that interval. Usually 1000 sweeps where used for thermalization and the number of sweeps used for measuring, for various β, is shown in table 1. Measurements were performed every fifth sweep and the errors were estimated by using the jackknife method.

3 Scaling of the string tension

The β-values in the scaling window are even smaller than the ones used for the standard Wilson action. If one wants to test scaling by comparing with the perturbative two-loop result using the identification β = 4/g^{2} it is doomed to fail. We clearly need a suitable effective coupling which we can use instead of g^{2}. Here we will use the so-called β_{E} scheme [3, 4, 5]. This scheme is simple and well suited to deal with “perturbations” of the Wilson action, like the ones given by (3)-(4). Let us define

β_{E} = \frac{3}{4} \frac{1}{1 − ⟨\frac{1}{2}Tr U_{□}⟩}.  (5)

From weak coupling expansions (β → ∞, λ fixed), it is seen that

β_{E} → β for β → ∞.  (6)

It is consistent to use the g_{E} = 4/β_{E} in the two loop β-function since g_{E} is a function of g and the two first coefficients of the β-function are invariant under a non-singular coupling constant redefinition. The use of β_{E} in the two loop β-function is known to diminish scaling violations in a variety of situations: The string tension in SU(2) and SU(3) gauge theories, the mass gap in the SU(2) and SU(3) chiral models and the deconfining transition temperature T_{c}, again for SU(2) and SU(3) gauge theories, just to mention some. We will now show that the method works well for the modified Wilson action with λ = 10.

Let us first note that for λ = 10 the β-range 1.4−1.9 is mapped into the β_{E}-range 1.837 − 2.104, which should compared to the corresponding map for the ordinary Wilson action where the β-range 2.2−2.5 is mapped into the β_{E}-range 1.741−2.154. From the measurements of Wilson loops mentioned in the last section we extract the string tension using the simplest method: If W (R, T) denotes a R × T Wilson loop we first form

V_{eff}(R, T) = log[W (R, T)/W (R, T − 1)].  (7)

For T → ∞ V_{eff}(R, T) should go to the ground state static quark potential V_{qq̄}(R). We have available T ≤ 6 and the results agree within error bars for T = 5 − 6. We have used these values as V_{qq̄}(R) for R ≤ 6. Next V_{qq̄}(R) is fitted to the form:

V_{qq̄}(R) = C − \frac{E}{R} + σR  (8)

and the error bars for σ are mainly coming from systematic errors arising from fits with various lower cuts in R. It is clear that with the limited range of R and T ’s available to us nothing will be gained by applying some of the many more elaborate schemes of fitting which are available.

log σ(β_{E}) is finally plotted in fig. 1 against β_{E}. The curve shown is the one obtained from the two-loop β-function, according to which the scaling of the lattice string tension σ(β_{E}) = σ_{cont}a^{2}(β_{E}) is governed by the two-loop scaling of the lattice spacing a(β_{E})^{2}:

a^{2}(β_{E}) = Λ^{−2} (\frac{6π^{2}}{11} β_{E})^{102/121} exp [−\frac{6π^{2}}{11} β_{E}] .  (9)

It is clear that scaling is reasonable well satisfied in the given range of β_{E}.

It is interesting to compare these results with the corresponding ones obtained by using the ordinary Wilson action. The range of β which gives the same range of β_{E} as were used for the modified action will be 2.3 ≤ β ≤ 2.5. In this range it is well known that naive application of scaling does not work particularly well, and to get a fair comparison with the results for the modified action we use again the β_{E} coupling constant transformation. We now follow the same method as in the case of the modified action and extract the string tension^{1}. The result is shown as a function of β_{E} in fig. 1. The use of β_{E} has improved the scaling considerable compared to the use of β as a coupling constant, as already mentioned, and it is apparent from fig.1 that agreement with the two-loop scaling is as good for the Wilson action as for the modified action.

4 Conclusions

The suppression of negative valued plaquettes should not change the continuum limit of pure SU(2) lattice gauge theories since these configurations are pure lattice artifacts. Naively one might even expect a smoother approach to the continuum limit if such a suppression is implemented. The modified Wilson action (3) indeed suppresses negative valued plaquettes for large values of λ and it is possible to approach continuum physics for much smaller values of β. However, for these small values of β the relation between the continuum coupling constant g^{2} and β is not a simple one and in order to extract the scaling behavior one has to use a modified coupling constant closer reflecting the physics of the system. The β_{E}-scheme is such a prescription and using it we have found good agreement with continuum physics. From these results it seems clear that the theory with modified Wilson action belongs to the same universality class as the theory defined by the ordinary Wilson action when β is sufficiently large. The worry mentioned in [1] is thus ruled out. In addition it seems that the approach to scaling is not dramatically improved compared to the situation for ordinary Wilson action. Indirectly this indicates that negative plaquette excitations play no important role in the scaling region for the ordinary Wilson action, at least when we discuss observables like the string tension. However, one would expect that the modified Wilson action is considerable better when it comes to the measurements of topological objects like instantons.

1The data used for β = 2.4 and 2.5 are taken from [7].

Acknowledgement We thank V. Mitrjushkin for discussions and many useful suggestions. Part of the computations were performed at UNI-C and were made possible by a grant from the Danish Ministry of Research and Technology.

References

1 V. G. Bornyakov, M. Creutz and V.K. Mitrjushkin, Phys.Rev. D44 (1991) 3918.

2 G. Mack and E. Pietarinen, Nucl.Phys. B205, (1982) 141.

3 G. Parisi, Proceedings of the XX^{t}h conference on high energy physics, Madison 1980.

4 F. Karsch and R. Petronzio, Phys.Lett. 139B (1984) 403.

5 S. Samuel, O. Martin and K. Moriarty, Phys.Lett. 153B (1985) 87.

6 G. Parisi, R. Petronzio and F. Rapuano, Phys.Lett. 126B (1983) 418.

7 F. Gutbrod, Z.Phys. C37 (1987) 143.

Figure 1: The measured string tension using the modified action (upper curve) and the ordinary Wilson action (lower curve) as functions of β_{E}.