A Monte Carlo study of the [beta]-function of the SU(3) Wilson action

Nuclear Physics B252 (1985) 189-196 189 North-Holland, Amsterdam

A MONTE CARLO STUDY OF THE B-FUNCTION OF THE SU(3) WILSON ACTION F. Karsch*, CERN, Theory Division, 1211 Geneva 23, Switzerland

We discuss the behaviour of the B-function of the standard SU(3) Wilson action at intermediate couplings. Results obtained from measurements of the deconfinement temperature and string tension on large lattices are compared with those obtained from a systematically optimized Monte Carlo Renormalization Group method.

i. INTRODUCTION

In the large cut-off limit of renormalizable theories it is possible to tune the cut-off and coupling(s) in such a way that the physical content of the theory remains unchanged. The functional relation between the

coupllng(s) and the cut-off is given by the B-function(s) of the theory. In an SU(N) lattice gauge theory the B-function describes the way the bare coupling g(a) has to be changed when the lattice spacing a is varried in order to leave all physical predictions unchanged: B(g)=-adg(a)/da.

However, for a generic value of the cut-off the function g(a) depends on the specific quantity which is kept fixed - there is no way to keep all physical predictions unchanged. It is only in the large cut-off limit that a unique B-function can be defined. But also in this case it still depends on the renormalization scheme choosen. In particular, the B-function depends on the lattice action choosen, only the two leading terms in its perturbative expansion are universal:

B(g) = -b o g3 _ bl g5 + O(g7)

with (i)

b o = IIN/(48~2); b I = ~ (N/16n2) 2.

Early exploratory studies of both SU(2) and SU(3) gauge theories indicated that already at moderate correlation length (i.e. at intermediate coupling constant values) physical quantities seem to scale according to the above perturbative SU(N) B-function [ I ] . However, recent detailed studies on large lattices (which allow to study the theories at smaller couplings)

Address after October 1,1984: Department of Physics, University of Illinois at Urbana-Champalgn, III0 West Green Street, Urbana, Ii 61801, USA

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190

~howeM D ~ D ~ii~sf~nr~al Mpvi~tions From t h ~ nn~zers~l "asvmntotic ~e~lfn~"

behaviour are still present at intermediate couplings [2-5]. This led to sometimes confusing interpretations of q u a n t i t a t i v e results for physical observables. For a, correct i n t e r p r e t a t i o n of results obtained at inter- m e d i a t e couplings and their e x t r a p o l a t i o n to the c o n t i n u u m limit it is thus basically important to reveal and u n d e r s t a n d the q u a n t i t a t i v e structure of the ~-function.

In the following we will discuss the d e t e r m i n a t i o n of the SU(3) B - f u n c t i o n by using the d e c o n f i n e m e n t temperature as a physical observable which is held fixed under changes of the cut-off. In section 3 we will discuss a MCRG a p p r o a c h to d e t e r m i n e the B - f u n c t i o n [6] and compare the results with those of s e c t i o n 2 and recent m e a s u r e m e n t s of the string tension [4,7].

Section 4 contains our conclusions.

2. ~ T R A C T I N G A B-PUNCTION FROM THE SU(3) D E C O N F I N ~ E N T T ~ P E R A T U R E At finite temperature the SU(3) gauge theory exhibits a first order d e c o n f i n i n g phase t r a n s i t i o n [8,9]. On a lattice of size N B x N 3 (N o >>

o NB) the order parameter for this transition, the thermal Wilson line, is d i s c o n t i n o u s at the critical coupling B C (NB)= 6/g~ (a). This jump in the order parameter provides a very clear signal for the critical temperature

Tcl= NBg(g2c). (2)

D e m a n d i n g that T c remains u n c h a n g e d when the cut-off a is varried allows to d e t e r m i n e a d i s c r e t i z e d v e r s i o n of the B-function: Changing the temporal extend of the lattice by a factor n corresponds to a decrease of the lattice spacing by the same factor in order to keep T c u n c h a n g e d

-I (g2cl) (g2c2)

T c = Ns,la = NB,2a

w i t h (3)

( 2

a gc2 ) NB, I a( 2

gel ) N3,2

In the following we will restrict ourselves to scale changes by a factor n=2. The change AB in the critical c o u p l i n g s 8 c (N 8) = 6/g2c,

AB =

B e

Bc(N B)

F. Karsch / The/3-function o/the SU(3) Wilson action

Is re]ateN to the B - F u n c t i o n through B

dx 21n2 (5)

~-AB x 3 / 2 B f u n c t ( ( 6 / x ) i / 2 ) / 6

From the p e r t u r b a t i v e B-function, eq.(1), one finds that in the limit g2 + 0 AB a p p r o a c h e s a constant value:

= 132 i n 2 / ( 1 6 ~ 2) = 0 . 5 7 9 (6) (A6) 2

g ÷ o

At finite g2 the inclusion of the two-loop p e r t u r b a t i v e c o r r e c t i o n s leads to somewhat larger values for AB. For i n s t a n c e in the c o u p l i n g r e g i o n B " 6.0 one would expect to find A6 = 0.61, if the p e r t u r b a t i v e B - f u n c t i o n is still valid in this i n t e r m e d i a t e coupling regime. The c r i t i c a l t e m p e r a t u r e T c has by now been d e t e r m i n e d on lattices of various t e m p o r a l extend N 8 [2,5,8,9].

The a v a i l a b l e results are s u m m a r i z e d in table I. As can be seen the

c r i t i c a l t e m p e r a t u r e does not stay constant w h e n one a s s u m e s the v a l i d i t y of the two-loop p e r t u r b a t i v e result for the 8-function.

N2

3 10 4 10

5 12 6116

18o .

6/g z Tc/A L Ap

5,11 ¢0.D1 78~1 5.097=0.001 5.55~0.01 86±1 5.70~0.01 76=1

0.59~0.02 5 696±0.004

5.78-5.82 685~1

592-594 66.5~1 0.38±0.02

5.877:0.006 62:.3 0~33±0.01

6.00:0.02 53:1.5 0.30=0.02

622~0.07 55:4 0.41±0,09

Table I: S u m m a r y of critical couplings of the SU(3) d e c o n f i n e m e n t t r a n s l - 3 N8" The rows w i t h N a ~ refer to the

tion on lattices of size N a x =

e x t r a p o l a t e d data of ref. [5]. The other data are taken from ref. [2, 8].

Also g i v e n are the critical t e m p e r a t u r e s Tc/A o b t a i n e d by a s s u m i n g the v a l i d i t y of eq. (I) and the AB o b t a i n e d from eq. 4.

The last column of table I shows the r e s u l t i n g values for A8 w h e n T c is kept fixed. They clearly show that the a s y m p t o t i c s c a l i n g regime has not been reached b e l o w 8 = 6.0. Indeed they are still d e c r e a s i n g b e t w e e n 8 " 5.5 and 6.0.

192 F. Karsch

/

To extract A8 from measurements of T c for larKer values of B much lar~er lattices are required. In addition T c is just one physical observable and more observables have to be analysed in order to check whether the B-function is unique for all these observables in the ~ range considered. In the following we will discuss a MCRG study of the SU(3) B-function which analyses a large set of different physical observables and allows to make contact with the perturbative regime.

3. THE RATIO METHOD

The basic idea of the ratio method [i0] is to extract the ~ -function from ratios of Wilson loop expectation values which are combined in such a way that the self mass and corner contributions cancel. These ratios satisfy the homogeneous renormalization group equation, thus by comparing ratios of loops calculated at some value of ~ with corresponding once formed from half as large loops at an appropriate 8' the change of the couplings AS=B-8' neccessary to achieve matching between these ratios can be determined.

There are two problems, however. First, ratios composed of small loops are contaminated by lattice artifacts resulting in a systematic error which increases linearly with ~ [6,11]. Second, the matching prediction is distorted by finite size effects if the correlation length is comparable or larger than the lattice size. While the last problem can easily be handled by measuring Wilson loops at ~ and 8' on lattices of size L 4 and (L/2) 4 respectively, the first problem requires a selection of systematically improved observables, which are free of lattice artifacts. These are constructed as follows: First the basic ratios are formed as

W (il, i 2) W i 4 )

f(i I, i2; i 3, i 4) = (i3, ' if+ i2= i3+ i 4

(7) W(i I, i 2) W(i 3, i 4)

g(i I, i2; i 3, i4; i 5, i6; i 7, i 8) = W(is,i6) W(i7,i 8) '

if+ i2+ i3+ i 4 = i5+ i6+ i7+ i 8.

and so on. Here W(il,i 2) is the expectation value of a planar Wilson loop of size il,i 2. Apart from lattice artifacts these loops satisfy the RG- equation

F. Karsch / The #-function o f the SU(3) Wilson action

( 2 f l , 2 t 2 ; 2 t 3 , 2 f 4 ; 8 , L ~ = f ( t l , t 2 ; i ' 3 , t 4 ; 8 ' , L / 2 3 ( R 3

and similar equations for g, .... Any linear combination of the functions f,g,.., defined in eq.(7) satisfy eq.(8) also. In the improved ratio method the mixing coefficients are determined by the requirement of cancelling the lattice artifact corrections to eq.(8) systematically order by order in perturbation theory [6,12]. The improvement procedure is illustrated in table 2.

BASIC RATIOS W(3,3) RI =

W(2,4) W(1,1)W(3,3)

R 2 -

W(1,2)W(2,3) W(1,2)W{2,3) Ra =

W(1,3)W{1,3)

WEAK COUPLING 613

- o . i ~

-0.057[]

0.046p

1 - LOOP IMPROVED RATIOS

R12~ = R2 + 0.0279T7 R2 + 0.702917 R, 0+579

Table 2: lllustration of the improvement procedure for three basic

ratios. Tree level and one-loop improved ratios are formed fromm the three basic ratios RI,R 2 and R 3. The mixing leads to a systematic improvement of the weak coupling behaviour by canceling the O(B -I) (tree level) and 0(8 ° ) (one-loop level) lattice artifacts for the observables considered. The last column shows the shift A~ for the listed ratios obtained in the weak coupling limit.

A large number of systematically improved ratios can be obtained this way. These mixed ratios have been used in a MC analysis to determine AB

[6]. Ratios of Wilson loops measured on a 16 4 lattice at several values of [4] have been compared with those on a 8 4 lattice at B'. The results obtained for A8 are shown in fig. i together with those deduced from the deconfinement temperature and measurements of the string tension [4,7].

TREE LEVEL IMPROVED RATIOS

R,~ = R2 + 0.E88298 R, 0.582

R=3 = R~ * 0.523659 R~ 0.492

1 9 4 F. Karsch

/

Ai3

0.5

0.1

I , I

5.6 6.0

o 1-1oop i m p r o v e d r a t i o t e s t x String t e n s i o n

• T c

r I i

6.~ I~

Figure I: The a v e r a g e shift AB as a f u n c t i o n of B o b t a i n e d from the a n a l y s i s of o n e - l o o p i m p r o v e d ratios (squares). (At B = 5.8 the basic ratios are used.) The error bars refer to s t a t i s t i c a l f l u c t u a t i o n s (thin bars) and the a v e r a g e f l u c t u a t i o n s (thick bars) of the m a t c h i n g p r e d i c t i o n s o b t a i n e d from d i f f e r e n t ratios. Also shown are the p r e d i c t i o n s for AB o b t a i n e d from the string t e n s i o n (crosses) and the critical t e m p e r a t u r e (full points).

As can be seen the m a t c h i n g p r e d i c t i o n s start a p p r o a c h i n g the p e r t u r b a t i q e result a b o v e B = 6.0. In p a r t i c u l a r the result at B=6.6

~B (B = 6.6) = 0.56 + 0.06 (9)

shows that there are only small d e v i a t i o n s from a s y m p t o t i c s c a l i n g for

B ~ 6 . 0 .

4. OONCLUSIONS

The a v a i l a b l e i n f o r m a t i o n on the SU(3) ~ - f u n c t i o n shows that there are s u b s t a n t i a l d e v i a t i o n s from a s y m p t o t i c s c a l i n g in the i n t e r m e d i a t e c o u p l i n g

F. Karsch / The [3-function o f the SU(3) Wilson action

r ~ m ~ 8 ~ ~.~ - 6.N . ~ w e v e r , shove 8 = 6.N the H e v ~ t ~ o n s From a s y m p t o t i c scaling seem to he small. The fact that quite dlffernt

approaches and physical observables lead to compatible results for A8 seems to indicate that already in the n o n - a s y m p t o t i c regime a u n i q u e E - f u n c t i o n exists. This is supported by recent MCRG calculations [[3,14] w h e r e block- spin transformations have been used to extract AS. Thus a consistent q u a n t i t a t i v e u n d e r s t a n d i n g of the rather non-trivial way the standard W i l s o n action approaches c o n t i n u u m seems to be emerging.

A C K N O W L E D G E M E N T S

This talk is based on work done in c o l l a b o r a t i o n with A. Hasenfratz, P. Hasenfratz and U. Heller. I would like to thank them for their support and many valuable discussions.

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t( Karsch / The O-function o f the SU(3) Wilson action

~3) K.C. Bowler, R.D. Kenway, G.S. Pawley, D.J. Wallace, A. Hasenfratz, P. Hasenfratz, U. Heller, F. Karsch and I. Montvay, CERN preprint, Ref. TH.-3952-CERN (1984).

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