Search for Effective Lattice Action of Pure QCD

TL;DR

This paper investigates the renormalization group flow of pure QCD gauge systems using Monte Carlo methods, compares four different actions, and assesses how they restore rotational symmetry near the renormalized trajectory.

Contribution

It provides a comparative analysis of four lattice actions and identifies how their proximity to the renormalized trajectory improves rotational symmetry restoration.

Findings

The renormalized trajectory is roughly located in two coupling space.

Rotational symmetry restoration improves as actions approach the renormalized trajectory.

Different actions show varying effectiveness in symmetry restoration.

Abstract

We study a coupling flow of pure QCD gauge system by using the Monte Carlo Renormalization Group method. A rough location of the renormalized trajectory in two coupling space is obtained. Also we compare 4 different actions; (a)standard Wilson, (b)Symanzik's, (c)Iwasaki's and (d)QCDTARO's. The rotational symmetry is restored better as an action gets close to the renormalized trajectory.

Tables

Table 1. Table 1: couplings in each action

	${\gamma }_{12}$	${\beta }_{11}$	${\beta }_{12}$
(a)Wilson	0
(b)Symanzik	-0.05
(c)Iwasaki	-0.091
(d)QCDTARO		6.1564(53)	-0.6241(23)
		7.986(13)	-0.9169(41)

	Wilson	Symanzik	Iwasaki	QCDTARO
$\chi$	.302	.228	.092	.076

Equations

$$S$$

$$U_{plaq}=\begin{picture}(20.0,20.0){\put(0.0,0.0){\vector(1,0){20.0}} \put(20.0,0.0){\vector(0,1){20.0}} \put(20.0,20.0){\vector(-1,0){20.0}} \put(0.0,20.0){\vector(0,-1){20.0}} } \end{picture}\hskip 14.22636pt,\hskip 14.22636ptU_{rect}=\begin{picture}(40.0,20.0){\put(0.0,0.0){\vector(1,0){20.0}} \put(20.0,0.0){\vector(1,0){20.0}} \put(20.0,0.0){\circle*{2.0}} \put(40.0,0.0){\vector(0,1){20.0}} \put(40.0,20.0){\vector(-1,0){20.0}} \put(20.0,20.0){\vector(-1,0){20.0}} \put(20.0,20.0){\circle*{2.0}} \put(0.0,20.0){\vector(0,-1){20.0}} }. \end{picture}$$

$$({\beta }_{11},{\beta }_{12})\to ({\beta }_{11}^{\prime },{\beta }_{12}^{\prime })$$

$$F=\prod _{\bar{l}}\int d{U}_{\bar{l}}\mathrm{Im}\mathrm{Tr}({\lambda }^a{U}_l{G}_l^{\alpha })e^{2\mathrm{ReTr}(U_l{G}_l)},$$

$$\frac{dF}{dϵ}$$

$$\frac{8}{3}\mathrm{Re}<\mathrm{Tr}(U_l{G}_l^{\alpha })>=$$

$$\sum _{\gamma }\frac{\beta }{6}\{-\mathrm{Re}<\mathrm{Tr}(U_l{G}_l^{\alpha }{U}_l{G}_l^{\gamma })>$$

$$+\mathrm{Re}<\mathrm{Tr}(G_l^{\alpha }(G_l^{\gamma }{)}^{†})>$$

$$-\frac{1}{3}\mathrm{Re}<\mathrm{ImTr}(U_l{G}_l^{\alpha })\mathrm{ImTr}(U_l{G}_l^{\gamma })>\}.$$

$${\chi }^2\equiv \sum _i(V(R_i)-f(R_i){)}^2$$

Full text

1 Introduction

In order to achieve the continuum physics in lattice gauge theory, a lattice spacing $a$ should be small enough. However the scaling violation of Wilson action at presently accessible $a$ is clearly seen [1, 2, 3]. Since the cost of a Monte Carlo Simulation of QCD can estimated as $\propto a^{-6}$ [3], it seems difficult to adopt smaller $a$ without a surprising computer power-up or improvement of the algorithm.

Recently ”improved” actions have been extensively applied to resolve the scaling problems [4]. Although in these studies, several types of actions are proposed and used. What is the most optimal choice or what is a ”perfect” action is still an open question.

In this paper, we analyze the coupling flow of pure SU(3) gauge theory by the Monte Carlo Renormalization Group (MCRG) method. And we compare several actions, which are widely used in actual simulations, by investigating the rotational symmetry.

2 Improved Action

In this paper we consider 2-coupling space in which the $1\times 1$ simple plaquette and $1\times 2$ rectangle loop are included. The action $S$ is,

[TABLE]

where

[TABLE]

Although there are several arguments about the necessity of other loops as possible candidates of an action [3, 5], actions with two couplings are widely used as improved ones in actual lattice QCD simulations [4]. Here, we restrict ourselves on a 2 coupling space and our purpose of this paper is to propose an optimal choice of action in 2 coupling space. Inclusion of other couplings is now in progress and will be presented elsewhere.

For actions with 2 couplings, we consider 4 candidates which are widely used; (a) Wilson [6], (b) Symanzik [7], (c) Iwasaki [8], (d) QCDTARO [5].

$\beta$s (or the ratio $\gamma_{12}\equiv\beta_{12}/\beta_{11}$) in these actions are listed in Table 1.

3 Coupling Flow by MCRG

First we follow the coupling flow by MCRG. As MCRG method, here we adopt Swendsen’s factor 2 blocking scheme [9]. For details of this blocking, see [2, 5].

3.1 Schwinger-Dyson method for determination of effective action

From the original configurations $\{U\}$ which is generated by an action $S$ with $(\beta_{11},\beta_{12})$ in eq.(2) we obtain the blocked ones $\{U^{\prime}\}$ after the above blocking procedure. When the blocked configurations $\{U^{\prime}\}$ can be considered as generated configurations by an action $S^{\prime}$ with $(\beta^{\prime}_{11},\beta^{\prime}_{12})$ in eq.(2), an action $S^{\prime}$ is an effective action of the blocked configurations, and

[TABLE]

can be regarded as the coupling flow associated with this blocking.

For the determination of an effective action, here we use a Schwinger-Dyson method [10].

This method is simply based on the following identity. For a link $U_{l}$, we consider a quantity;

[TABLE]

where $\bar{l}$ means link except for $l$. $G_{l}$ is a sum of staples $G_{l}^{\alpha}$ for link $l$ as $G_{l}=\sum_{\alpha}{\beta\over 6}G_{l}^{\alpha}$. For infinitesimal transformation; $U_{l}\rightarrow(1+i\epsilon\lambda^{\alpha})U_{l},$ invariance of $F$ leads to

[TABLE]

Using a well-known formula of $\lambda$-matrix and summation over $a$ and integral over link $U_{l}$ , we have

[TABLE]

We then apply this equation to the blocked configurations.

3.2 Lattice Simulation and Results

We use $8^{4}$ lattice size and about 2000 configurations separated by every 10 sweeps are used. For the starting point $(\beta_{11},\beta_{12})$, we try (a)-(d) coupling sets in Table 1 and other points are also tested to search the renormalized trajectory(RT) in detail. All simulations have been done on CRAY J90 at Information Processing Center, Hiroshima University and on VPP500 at KEK (National Laboratory for High Energy Physics).

In Fig.1, the result of coupling flows in $(\beta_{11},\beta_{12})$ plane is shown. Arrow denotes the measured coupling flow in eq.(2).

Ending points seem lie on the universal line which can be thought as the RT of this blocking scheme. For comparison, we prepare 2 sets of (a)-(d) points at almost the same scale; $a^{2}\sigma\simeq 1.51(6)$ and $a^{2}\sigma\simeq 0.82(6)$. Among (a)-(d) at almost the same scale, (d) is the closest to this RT and (c) is the next.

4 Rotational Invariance

In order to see the improvement of the action, here we check the rotational invariance by measuring the heavy quark potential. In this paper, we compare 4 actions in Table 1 at $a^{2}\sigma\simeq 1.51(6)$.

In Fig.2, the result is shown. Since the lattice spacing at this scale is rather large, rotational symmetry can not be expected to be restored on the standard Wilson action, which is clearly seen in Fig.2(a). For Symanzik action in Fig.2(b), the situation is similar to Wilson case. On the other hand for Iwasaki and QCDTARO cases in Figs.2(c) and (d), the rotational invariance can be seen. It is also noted that errors become small as the action gets close to RT.

In order to compare quantitatively, we fit the measured potential $V(R_{i})$ ($R_{i}$ is the measured point) with the linear plus coulomb term as $f(R)=a+bR+c/R$ and evaluate the effectiveness of this fit by

[TABLE]

From these results, rotational symmetry can be restored better as the action gets close to RT.

AKNOWLEDGEMENTS

To attend this conference, S.H is supported by the Grant-in-Aid for Scientific Research of the Ministry of Education No.08640379.