Monte Carlo Renorrnalization Group
Study of the SU(3) Lattice Gauge
Theory(Non-perturbative lattice /3-function and Perfect action search)

T. Takaishi
Department of Physics, Hiroshima University,
Kagamiyama, Higashi-Hiroshima, 724 Japan

January 13, 1995

Contents

1

2

Introduction

3

1.1

Problem on Scaling in the Lattice Gauge Theories

3

1.2

Perfect Action .

. . . . . . . . . . .

10

MCRG Analysis of the lattice (3- function

15

2.1

Monte Carlo Renormalization Group

15

2.2

Blocking Transformation

19

2.3

Matching Method . . .

21

2.3.1

De.nnit.ion of ~(3

21

2.3.2

~(3 from Matching Methou

22

2.4

3

Numerical Simulation . .

26

2.4.1

Update ProgralTI

26

2.4.2

Configurations ..

26

2.5

Measurement of Wilson loop

27

2.6

Results of ~(3

....

28

2.6.1

Optimal value of q parameter

28

2.6.2

~(3

29

..

2.7

Improved couplings

31

2.8

Scaling Test

36..

Perfect Action Search

39

3.1

39

Effective Action on the blocked trajectory
1

CONTENTS

2

4

3.2

The Demon Method

40

3.3

Testing the demon method

42

3.4

The Demon Method on Blocked Configurations

45

Conclusion

49

A Blocking Scheme

55

B Correction factor to the 2 loop formula

59

C Relation between the coupling and the demon energy

61

Chapter 1
Introduction
1.1

Problem on Scaling in the Lattice Gauge
Theories

In 1974, Wilson[l] has proposed a lattice regularization for Quantum Chromo
Dynamics (QCD). The lattice regularization seems the most natural way for analyzing a continuum field theory. This formulation allow us not only a strong
coupling expansion but also a Monte Carlo simulation. In particular, the later
fascinates people who want to know non-perturbative aspects of QCD. The Monte
Carlo simulation for SD (2) lattice gauge theory, first, has been proposed and implemented by Creutz[2] in 1980. Since then, much efforts, improvelnents of simulation technique and computer developments, have been poured into the Monte
Carlo simulation of the lattice gauge theory. Especially it might be worthwhile
to remind that physicist's demand on COlTIputer went over the supply from thecommerce. Some people who are not satisfied with the performance of commercial computers have been starting to construct much faster computers, such asparallel computers[3] and the efforts in this direction are still going on. Recently atypical lattice size for a silTIulation of pure SD (3) gauge sector (without fern1ions)has become 244 rv 32 4 or more[4] owing to developments of computer.
3

CIIAPTER 1. INTRODUCTION4

Why do we need to go to such a large lattice? The answer is that we Inustobtain values of physical quantities in the continuum lilnit. Mainly observablesfrom the lattice calculations are suffered from two systematic errors!, the finite
lattice spacing and the finite volume effects. Naively we expect that as the latticesize becomes large and the lattice spacing become small we approach to the continuum limit. As for the finite volume effect, it is said that we are already in theregion where the finite volume effect is under control by such as an extrapolation
to the infinite volun1e limit[4], and the finite volume effect itself is already small
on a lattice currently used in Monte Carlo simulation. On the other hand, thefinite lattice spacing effect is still a serious problem.
Now we discuss the finite lattice spacing effect and the scaling violation problem. Let us consider an SU(3) lattice gauge theory. An expectation value ofgauge invariant physical quantity Q is defined by Euclidean path integral withrespect to gauge field UJ.L (n):< Q >= ~JD[U]Q[U]exp( -S[U])(l.1 )

where S[U] is an action, Z is a normalization factor and D[U] is a Baal' measureof SU(3). The standard choice of the action for the SU(3) gauge theory is a simple
plaquette Wilson action:

S(U) = {3L ReTr(l - W11)/3

(l.2)

Wll

where {3 = 6/g 2 and WII is a plaquette. This action is very popular in the Monte
Carlo simulations since it is easy to implement the Monte Carlo simulation on
cOlnputers. We call this action the standard Wilson action from now on and referto this action if we do not say anything about the form of the action.
In the Monte Carlo simulation, < Q > is evaluated as an average in sequential
configuration {U = UI , U2 , U3 , ..• , UN} generated in Markov chain with a proba1 Herewe consider a pure gauge theory . In the dynamical case ( with fermion ), there are

two more systemat ice error, the quenching effect and the finite quark mass effect.

1.1. PROBLE!v! ON SCALING IN THE LATTICE GAUGE THEORTES

5

bility distribution:

P[U] ex exp( -S[U]),(1.3)

that is, the important sampled average[5] and a value of < Q > is obtained by:
1 N<Q>~<Q>= N~Q[Ui]'

(1.4)

t

Statistical error of < Q > is propotional tok. This error can be reduced bysampling a number of Q[Ui]. In the limit of N - - - t 00, < Q > coincides with

< Q >. Let us take a physical quantity M with mass dimension as an example. Corresponding to M, Q is constructed of some lattice operators. Q has no
dimension and it may be written as:<Q>=m(g)(1.5 )

where we indicate the coupling 9 explicitly in order to show the m can be dependent on it. Therefore the m is also given as a non-dimensional quantity andrelated to the physical observable M wi th mass dimension by:M(g, a) = m(g)
a

(1.6)

or alternatively,M(g, a)a = m(g).

(1. 7)In the lattice gauge theory, scaling means that dimensional physical quantities
are independent on the lattice spacing a. This requires that the coupling 9 should
be a function of a and for infinitesimal a it is written as:

a dM(g(a), a) = 0
da

(1.8)

for all the quantities lVJ(g(a), a) with mass dimension. At finite lattice spacing,however, there is a 0 ( a 2 ) correct ion:

(1.9)

CHAPTER 1. INTRODUCTION

6
Assuming e.q.(l.8), we obtain:

8g8m
= O.
8a 8g

1?l-a--(1.10)

This equation means that scaling behavior is controlled by the ;3-function 2 which
is defined by:

8g;3j(g) = -a 8a'(1.11)

Using e.q.(l.II), e.q.(l.10) is rewritten as:(l.12)In the perturbative region, the ;3-function can be expressed by an expansion:(l.13)
The first two coefficients, bo and b1 are universal( regularization independent )and well known by the two loop calculation[6]:b = lIN - 2njo
3 x 167r 2(l.14)

b = 34N 2 - 10Nnj - 3nj(N 2 - I)/N1
3 x (167r 2)2(l.15)

where N is the number of colors and n j is the number of quark flavors. The ;3function with the first two terms is called the 2 loop ;3-function. As we recognize
from the definition of the ;3-function (e.q.(l.II)) the ;3-function can give us arelation between the coupling 9 and the lattice spacing a. The important propertyon the ;3 function is that the coupling 9 goes to 0 as the lattice spacing a goes too since the coefficient bo is positive provided that 11 N > 2n j. Therefore there isa ultraviolet fixed point at 9 = 0 where the continuum limit of the lattice gauge
theory is defined. Taking only the universal first two coefficients, the ;3 function
2Do not confuse the coupling {3 and the {3-fundion. In order to avoid the confusion, we
denote {3J for the fJ-function.

1.1. PROBLEM ON SCALING IN TIlE LATTICE GAUGE TIIEORIES

7

can be easily integrated and the lattice AL parameter with mass dimension isdefined as:(1.16)
This tells us the relation between the lattice spacing a and the coupling g. Taking
a ratio of e.q.(1.7) and e.q.(1.16), we obtain:m(g(a))

M(g(a),a)

aA L

AL

(1.1 7)In the continuum limit, physical quantities should be constant. Hence e.q.(1.17)
should also be constant in the continuum limit:

M(g(a), a)
AL

M

(LIS)

= AL = constant.

If e.q.(1.1S) holds for all physical quantities, we call it asymptotic scaling.

In order to obtain physical quantities in the continuum limit, it is importantto establish e.q.(1.1S), that is, the asymptotic scaling for all quantities from thelattice Monte Carlo simulation. How can we check the asymptotic scaling? Thiscan be done by analyzing non-perturbatively the IJ-function on the lattice since
the asymptotic scaling is based on the two loop IJ-function with the first two
coefficients in e.q.(1.13) and the IJ-function calculated non-perturbatively on thelattice can show whether or not the two loop IJ-function is valid for the region of
the coupling IJ where we are employing the Monte Carlo simulation. Although
it is difficult to get the IJ-function itself directly, it is possible to obtain the IJfunction in terms of ~(3 ( this (3 Ineans the coupling (3 = 3N/ g2, for SU (N) gauge
theory) which is a coupling shift when the lattice spacing a is changed by somefactor.

~IJ can give us inforn1ation of the original IJ-function.

Here we give a

relation between 6IJ and the two loop IJ function. Using the definition of the
IJ-function, e.q.(1.13), we integrate:j,(3(a)-6(3

{3(a)

dg

lba

IJj(g)

a

-- =

dIna

(1.19)

CHAPTER 1. INTRODUCTION

8

where (3(a) = 6/g 2 (a) for SU(3) gauge theory and we take a scale factor b, that
is, a ~ ba. Using the 2 loop (3-function, we obtain:(1.20)
where we use values of the two loop coefficients for SU (3) gauge theory without
ferrrUons, N= 3 and nj = O. E.q.(l.20) is our starting point for analyzing

the asymptotic scaling of the SU(3) lattice gauge theory. Comparisons between!:::,.{32Loop of e.q.(l.20) and !:::"{3 from the lattice study tell us whether or not we arein the asymptotic scaling region.
There are two popular methods to analyze !:::,.{3, the ratio method[7] and theoperator n1atching rnethod by the Monte Carlo Renormalization Group ( MCRG
) technique[24]. The ratio method is based on the fact that the ratio of Wilson
loops:

R(· . k I) = W(i,j)
l,]"W(k, I)(l.21)

satisfies an approximate homogeneous renormalization group equation:

R(2i, 2j, 2k, 21,g(a), 2L) = R(i,j, k, l,g(2a), L)(l.22)

where i + j = k + t. Using values of Wilson loops frOlTI the Monte Carlo sinlttlation on 2 lattices of size 2L and L, the couplings g(2a) and g( a) which satisfies
e.q.(1.22) is searched out. This gives us !:::"{3 we want as:!:::"{3 = _6_ _

g2(a)

6
g2(2a)(l.23)

Due to the lattice artifacts, however, e.q.(l.22) is exact for Wilson loops of infinite
size, that is, i,j, k, I ----+ 00 . Therefore in order to obtain reliable results of !:::,.{3,one has to prove the convergence of the value of !:::"{3 as a function of loop size.
However measurements of large Wilson loops are practically difficult since weneed a large lattice and much statistics. ~10reover it is known that from SU(2)Monte Carlo simulations, the convergence with respect to loop size is very slow.

1.1. PROBLEM ON SCALING IN THE LATTICE GAUGE THEORIES

9

On the other hand, the operator matching method by the MCRG techniqueseems to be more efficient. There are two main groups who have analyzed ~{3 using the MCRG technique. Bowler et al.[8] have obtained ~{3 up to {3 = 6.6 on 16 4
lattices using the MCRG with a scale factor 2 blocking. They have reconfirmedthat there is a pronounced dip around {3 = 6.0 which has already observed in the
study by the ratio method[7] and showed that ~{3 approaches to the predictionof the 2 loop {3-function as {3 increases and reaches it at {3 = 6.6. In their further
research[9]' however, they have observed the difference from the prediction of the
2 loop {3-function at higher {3, 6.9 and 7.2. On the other hand, Gupta et al.[10, 11]analyzed ~{3 on 94 lattices up to {3 = 7.5 and they have claimed that ~{3 reaches
the prediction of the 2 loop {3-function already at {3 = 6.75, and concluded that
there is the asymptotic scaling above (3 = 6.75. Those results are plotted in Fig.l.
The results from b = J3 blocking are rescaled to b = 2 blocking according toe.q.(1.20) although this rescale procedure is not accurate if the results are not in
the asymptotic scaling region. Hoek[12] have also analyzed ~{3 using the MCRG
technique and he has found the slow approach to the asymptotic scaling region
using phenomenological fit to the {3-function[13]. This is a controversy and we can
not conclude from these results of ~{3 where the asymptotic scaling is. Chapter 2
is devoted to solve this controversy. We carry out the MCRG study with a scale
factor 2 blocking on both 16 4 and 32 4 lattices. A feature of our study is that
we use a large lattice, a 32 4 one and do high statistics Monte Carlo simulations.
Typically we update gauge configurations 20,000-100,000 sweeps and measure observables every 10th sweep. Therefore We have 2,000-10,000 configurations. This
statistics is much higher than others (They have used about 100 configurations.).
We consider that the controversy comes from 2 effects, the finite temperaturephase transition and the poor matching condition. On a small lattice the latticesystem goes through the finite temperature phase transition as {3 increases, for example at {3 ~ 6.4 on a 16 4 lattice. The finite temperature phase transition causes
the difficulty of determination of ~{3. If we employ the work on a 32 4 lattice, weCl-IAPTER 1. INTRODUCTION

10remain in the confinement region up to {3 ~ 7.0. This is one of advantages ofusing a 32 4 lattice. The second advantage is that on a large lattice we can makethe matching condition better since for instance on a 32 4 lattice we can do thescale factor 2 blocking one more time than on a 16 4 lattice and this ensures thatthe blocked trajectory approaches to the renormalized trajectory more.
The analysis of 6.{3 using the MCRG technique is a main theme in Chapter
2 and we solve the controversy and discuss the scaling behavior[14, 15,16,17,18,
19,20,21,22].

1.2

Perfect Action

As we will see later in Chapter 2, the standard Wilson action has a poor scaling
property. This can not be completely compensated with some remedies such as
improved couplings. Do we need to go to higher (3 and a larger lattice in order
to confirm the scaling? The answer is not necessarily YES. We have anotherway, improvements of actions.

The standard Wilson action is widely used in

the current Mont Carlo simulations because of its simplicity. We can choose
other actions in such a way that the the cut-off dependence near the continuum
limit decreases.

Such a systematic procedure to improve the action has been

proposed by Symanzik[23]. Let us see Symanzik's improvement program briefly.
The standard Wilson action has O( a 2 ) correction to the con tinuurn action in thevicinity of the continuum limit. E.q.(1.2) has a form in the limit of the latticespacing a goes to 0:(1.24)
O( a 6 ) terms and other higher power of a terms disappear in the continuum limit.
However O( a 6 ) tenns can be renloved out of the continuum lilni t by adding 6
link loop terms. lIence it is considered that such an action for which O( a 6 ) tenTIS
are removed approaches to the continuum limit fast. The action including 6 link

1.2. PERFECT ACTION

11loop terms is written as:

S(U)6 =

L coTr(l - Wll) + L cI(l - W12)
+ L c2(1 - W chair) + L c3(1 - Wtwist)
Wll(1.25 )

WI2

Wchair

Wtwistwhere shapes of 6 link loops are displayed in Fig.2. After some calculations, wehave the following:(1.26)a4-2(cO + SCI + 16c2 + SC3) L Tr(F~v(n))
n,J.i.,v+a (C2 + 3)
6

~

C3

.G

n,J.i.,V,A

Tr(DJ.i.FJ.i.A(n)DvFv>Jn))a6+~(co + 20CI + 4C2 - 4C3)
1~

+a6~

L Tr(DJ.i.FJ.i.v(n))2n"v,,...,

L Tr((DJ.i.FvA (n))2) + O(a B )n,J-.L,V,A

The conditions to remove O( a 6 ) terms are:( 1.27)(1.2S)(1.29)
From these conditions, we have coefficients for the so-called Symanzik improved
action:

5

Co =-

3

1
12

CI = - -(1.30 )
(l.31)an d C2 = C3 = o.
Other improved actions have been proposed by Wilson[24] and Iwasaki[25].
They have detennined coefficients of 6 link loop action using the blocking transformation. The coefficients they have obtained are:Co= 4.376(1.32)

CHAPTER 1. INTRODUCTION

12Cl= -0.252(1.33)=0(1.34)= -0.17(1.35)

C2

C3

for Wilson's improved action for the SU(2) gauge theory andCl= -0.331(1.36)(1.37)(l.38)
for Iwasaki's improved action for the SU(3) gauge theory. Using these improved
actions some Monte Carlo simulations have been performed but drastic improvements have not been fulfilled yet[26, 27, 28].
Recently other program to improve the action has been advocated by P.Hasenfratz
and Neidermayer[29]. They have determined the classical perfect action 3 ( or fixed
point action) by solving the saddle point equation for 0(3) non-linear a model.
The con tinu urn limi t is defined on the critical surface ( the correlation length isinfinity). The continuum limit means that there is no lattice artifact and thecontinuum physics is obtained there . There is a fixed point on the critical surface.
At the fixed point, the fonn of the action S* is unchanged by the Renormalization
Group transformation :S* = RS*(1.39)

where R indicates the operation of the Renormalization Group transformation.
There is a one dimensional flow, the Renonnalized trajectory, which flows fromthe fixed point according to the Renorn1alization Group transformation . On the
Renormalized trajectory, there is also no lattice artifact. They approximated theRenormalization trajectory with the fixed point action 131 S*( See Fig.3 ).
This program successfully works for 0(3) non-linear a model and restorationof the rotational invariance is shown. The program on SU(3) gauge theory is also
3The perfect action means that its action is completely free from lattice artifacts.

1.2. PERFECT ACTION

13

going on[30]. However the action obtained by such a program is not a complete
perfect action. Even if we get a classical perfect action, it is not insured thatthe action works at moderate finite coupling since the renormalization effects are
non-trivially dependent on the blocking scheme at finite j3( See Fig.3 ). Namely
at moderate finite coupling, the approximation of the fixed point action is notenough.
The MCRG technique can be used for the perfect action search since theblocked trajectory goes to the renormalized trajectory and flows along the renormalized trajectory lastly. In principle, if we can obtain an effective action after
enough blocking steps, it will be one of perfect actions. However it was very difficult to obtain such an effective action since we did not have a practical method
to obtain it. Recently, however, quite an efficient method, the canonical demon
method, has been proposed and applied for 0(3) non-linear (J' model[51]. It has
been shown that the method works well for 0(3) non-linear (J'model. In Chapter
3, we report an application for physically interesting case, SU(3) gauge theory
and try to obtain an effective action on blocked configurations[52].

14

CHAPTER 1. INTRODUCTION

Chapter 2
MCRG Analysis of the lattice (3function
2.1

Monte Carlo Renormalization GroupThe idea of the Monte Carlo Renormalization Group (MCRG), which combines
the Renormalization Group with the Monte Carlo simulation, first, was given
by Ma[31] and developed by Swendsen[32] in spin systems and by Wilson[24] inlattice gauge theories. In the vicinity of the critical point, the correlation length
diverges and all the length of scales contribute the dynamics of the system. In
other words, infinitely many degrees of freedom arise in the system. The standard
Monte Carlo simulations face with hard problelTIs at or near critical point. The
physical size of the system, in Monte Carlo simulations, should be larger thanthe correlation length, at least a few times, in order to obtain meaningful results.
As we approach to the critical point, computer time used for the Monte Carlosimulation increases inevi tablely since a large lat tice is needed to keep size of thesystem larger than the correlation length . There also exits the critical slowing
down problem which makes the Monte Carlo silTIulation much difficult.
The Renormalization Group method sheds light for us to handle the problems
In the vicinity of the critical point. General concepts of the Renormalization
15

16

CHAPTER 2. MCRC ANALYSIS OF TJ-IE LATTICE (3- FUNCTION

Group is an integration over the irrelevant degrees of freedoll1 keeping those which
are relevant for physical quantities of the system. In the MCRG technique, theRenormalization Group transformation is performed for configurations generated
by the Monte Carlo silTIulation[31]. To illustrate what the MCRG is like, let us
consider the 2-dimensional Ising system which described by a Hamiltonian:H = -1(1 2: SiSj

(2.1 )<i,j>

where spin Si takes ±1 and < i, j > means that the summation is taken for thenearest neighbor pairs. The partition function of the system is written as:

Z = 2: e-H= 2: e K1S1

(2.2)

S1

where we use the brief expression, 51 = L<i,j> SiSj, and configurations are generated with the probability distribution:(2.3)
The Renormalization Group transformation acts on spIn configuration Si andtransforms them to new transformed spin configuration s~. The transformation
procedure is often called blocking . Let us take a scale factor 2 blocking. Spinsin a block a are mapped onto ± 1 according to a transformation rule, such as
majority rule(See Fig.4). This procedure is symbolically written as:(2.4)
where i E block a.

The probability of observing a new configuration [s~] isproportional to exp( -H'[s~]) which is written as:
e-II'[s~] =2: II 8(s~ - !(si)liEcJe-H[sd.
lSi]

(2.5)

C\'

After the blocking, the Renormalization Group transformation may produces
many couplings. Hence the form of 11' can be complicated. For instance, we can2.1. MONTE CARLO RENORMALIZATION GROUP

17

generalize the Hamiltonian H' by writing:

-H'

(2.6)
<i,j>

«i,j»

i,j,k,l

(2.7)

31 =

L:= SiSj

(2.8)

<i,j>

32 =

L:=

SiSj

(2.9)

SiSjSkSt(2.10)«i,j»

53 = Li,j,k,l

and«

i,j »

the next-to-nearest neighbor pair and i,j,k,l is the plaquetteinteraction pair.
Note that the partition function of the transformed system equals that of theinitial system provided that the Kadanoif constraint is satisfied:

L:= IT 8(s~ - f(si)liEex) = 1.
[s~](2.11)

ex

Using this condition, it is easily checked that:z = L:= e-H[siJ = L:= e-H[s~] = Z'
[siJ

(2.12)[s~]

It is also considered that the Renormalization Group transformation acts oncoupling space.

(2.13)
where Rex(3 is a transfonnation nlatrix of the couplings. Let us denote the couplingat the fixed point I(~. At the fixed point, I(~ is not changed as:(2.14)In the vicinity of the fixed point, the coupling I( ex is expressed by:(2.15)

18

CIIAPTER 2. MCRG ANALYSIS OF TIlE LATTICE (3- FUNCTION

Therefore we obtain:

1(* + 81('

1('

Cl'

Cl'

(2.16)

Cl'(2.17)
where the transformation matrix is:(2.18)
Generalizing e.q.(2.18) to n-th blocking, we obtain:

81( n81(i- 1IK •81( n 8 < sn >---8 < s~ > 81({3n-l .
Cl'

Cl'

(2.19)(2.20)

This can be calculated using the following identities[32]:

8 < s~ >snsn-lsn
sn-l81(n-l =<{3> - <>< {3 >
Cl'Cl'

(2.21)

Cl'

8 < s~ >
sn sn
sn
sn
8I(n
=<
{3 > - <>< (3 >
Cl'Cl'

(2.22)

Cl'

where < s~ > are the expectation values on the n-th blocked lattice. The criticalexponent v is obtained by the leading eigenvalue ,\ of RCl'{3:

In b
In ,\

v=-

(2.23)

where b is the scale factor of the blocking.
In the lattice gauge theory, blocking is son1ehow complicated since the variable
is defined on the link. The blocked link is constructed from a sum of paths:(2.24)
In general, this sum is not an SU(N) matrix and the new blocked link is selectedwith the probability distribution:(2.25)

2.2. BLOC](ING TRANSFORMATION

19

where p is a parameter to optimize the blocking.
Several ways to construct the paths of e.q.(2.24) have been proposed since
there is no unique blocking. Wilson[24] has proposed a scale facto b = 2 blocking
whose geometry of the blocking is shown in Fig.5. 8 links are summed up( inFig.5 only 4 of 8 links are shown in 3-dimension.). This blocking requires thegauge fixing since the end of links are not connected. The gauge fixing procedure
is very costly in the Monte Carlo simulations. Hence this blocking transformation
has not been pursued since Wilson proposed and tried to obtain values of 6.{3.
Swendsen[33] has proposed the gauge invariant scale factor b = 2 blocking, which
does not require the gauge fixing. In this blocking step, paths which connects 2
sites are summed up(Fig.6). Therefore all the paths are connected at the blockedsites. Other blockings with a different scale factor, based on Swendsen blocking,have also been proposed, scale factor b = -13"[34] and b = V2[35]. These are
specific to the gauge theories in 4-dimension.

2.2

Blocking Transformation

We follow Swendsen's scale factor 2 blocking[33]. The blocking procedure we takeis as follows: Path ordered links connecting two lattice sites separated with twolattice spacings are sumn1ed up( Fig.6 ):Qt-t(n) = aUt-t(n)Ut-t(n+ll)+b

L Uv(n)Ut-t(n+l/)Ut-t(n+Il+l/)U~(n+21l) (2.26)

vf.t-twhere we take a = 1 and b = 1/2. The matrix Qt-t(n) is not an SU(3) matrix. So
we have to extract an SU(3) matrix from it. In the case of SU(2) gauge theory,
such a sum is in proportion to an SU(2) matrix because of the speciality of SU(2)matrix. Therefore it can be easily normalized to an SU (2) matrix. On the other
hand, the SU(3) matrix does not have such property. We have to employ otherprocedure to project a non-SU(3) matrix onto one of SU(3) group . We definesuch a projection by a ITlaxilTlization of a function ReTr(Qt-t(n)PZ(n)), where an

20

CHAPTER 2. MCRC ANALYSIS OF Tl-IE LATTICE (3- FUNCTION

SU(3) matrix PJ-£(n) is taken so that ReTr(QJ-£(n)P1(n)) takes a maximum:(2.27)
This definition preserves local gauge invariance. The maximization can be doneby the polar decon1position[36, 10, 11]. In the polar decomposition step, QJL(n)is decomposed as:(2.28)
where D is a positive definite diagonal matrix and H and V are SU(3) matrixes.
Using this decomposition, we can maximize:

ReTr(De- i ¢ X)

(2.29)X = VtHtpV.

(2.30)

where X is given by:

In the polar decomposition, 11 is given by:(2.31)
where a= arg(det(Q{L(n))). It is known that for {3 > 6.0 the phase ¢ is ap-

proximately 0 and Hand P are approximately same. At {3 = 6.0 the difference
between Hand P by comparing average values of Wilson loops is already small( about 1 ~ 2%) and the difference decreases as (3 increases. Therefore we use II
as an SU(3) matrix mapped from QJ-£(n), that is, PJ-£(n) = II.
We introduce a paralneter q in order to optimize the blocking. The meaningof the optimization will be clarified later. A blocked link U~b) (n) is given as a
product of PJ-£(n) and a random gaussian SU(3) matrix:(2.32)H Urandom-lJauss.

(2.33)The random gaussian SU(3) matrix is given by:

co,\o)i?= T
8

Urandom_gauss= exp (

1=1

(2.34 )

2.3. MATCHING METIIOD

21

where Ai, i = 1, ... ,8 are SU(3) generators and the Ci are real numbers generated
with the distribution:(2.35)
In our Monte Carlo simulations, the blocking is repeatedly performed until thesize of the lattice becomes 22.
Our blocking scheme is also written in the form of e.q.(2.5) (See AppendixA):
exp( -S'[U'])![dU]exp(-S[U] + N(Q)+p

L ReTr(U~b)(Qt(n)QJ-L(n)) -;1 e-iex 3/ Qt(n))(2.36)

n

where
2

1

p=--q 2(2.37)

det[QJ-L(n)] = rexp(ia)(2.38)and the normalization factor N (Q) is:N( Q) =

2
1
L
-In[exp(3( - - "2 )(2q7r )4].
n,J-L
q

2.3

Matching Method

2.3.1

Definition of 6.(3(2.39)

Here we give a definition of 6.f3. 6.f3 is defined as a coupling shift when the latticespacing a is changed to 2 x a:

6.f3 = 6.f3(f3(a)) = f3(a) - f3(2a)(2.40)

where a is a lattice spacing and f3(a) = 1/g 2 (a).
Inversely we can consider the lattice spacing as a function of the coupling f3.
Starting from a coupling f3, we search a coupling f3' which satisfies:

2a(f3) = a(f3').

(2.41)

CHAPTER 2. MCRC ANALYSIS OF THE LATTICE (3 - FUNCTION

22

6(3 is obtained by:~(3((3)2.3.2= (3 - (3'.

(2.42)

6.f3 from Matching Method

How can we obtain a value of ~(3 in the blocking? We can utilize the operator
matching ( or two lattice matching) method in order to obtain ~(3. This method
has been originally advocated by Wilson[24] and he tried to obtain ~(3 on rather
srnalllattices, 88 ones.
To see what the operator matching method is, let us do the blocking n times.
After n-th blocking step, the lattice spacing becomes 2n times bigger than theone on the initial lattice. The lattice spacing can be written as:(2.43 )
where (3 indicates that the blocking is done on the configuration generated at(3 = g~' We would like to find (3' which satisfies e.q. (2.41). In order to obtain arecipe to find (3', first let us assume that e.q.(2,41) is satisfied and do the blocking(n - 1) times at (3'. We obtain a relation:(2.44)
Using e.q.(2.41) and e.q.(2.43) we find the following relation:(2.45)
This equation is telling us that (3' can be determined by searching a value ofthe coupling at which the lattice spacing after (n - 1 )-th blocking step coincide
wi th the one after n- th blocking step at (3.

Here we can ask ourselves "Howcan we recognize such a coincidence of the lattice spacing in e.q.(2,45)?".

It

is not so easy to find the coincidence by directly measuring the lattice spacinga.

Fortunately the operator matching method can give us a criterion of thecoincidence. Therefore we follow the operator matching method and hereafter wecall it simply the matching method .

2.3. MATCHING MET~HOD

23

Let us assume that after n-th blocking step, the blocked trajectory reaches therenormalized trajectory c0111pletely(See also Fig.7). We define a value of Wilson
loop operator Ok there as:(2.46 )In terms of the operator, e.q.(2.45) means:(2.4 7)
In e.q.(2.4 7) we put the suffix k on {3' in order to indicate that we are searching {3'for each Wilson loop operator. However if both of the blocked trajectories from

{3 and {3~ are already on the renormalized trajectory, a value of each {3~ should be
the same value. Namely for all the Wilson loops the next equation holds:(2.48)
Therefore {3' can be obtained by comparing values of Wilson loops on n-th blocked
lattice and on (n -1)-th blocked one, and searching (3' which matches e.q.(2.48).
This is the reason that the method is called operator matching method. The
measurements of Wilson loop should be done on a lattice with the same physicalvolume to avoid the finite volume effect since we are employing on the finite
lattice. This means that when Ok(a(n)({3)) is measured on a blocked lattice fro111
a N 4 lattice, Oi(a(n-l)({3:)) should be measured on a blocked lattice fron1 a (N/2)4
lattice in order to keep the physical volume same.
So far we have assumed that the blocked trajectory after n-th blocking step
already reaches the renormalized trajectory. If not the case, e.q.(2.48) does nothold for all the Wilson loops.

Hence 6.{3 which we obtain from each Wilson

loop does not give us the same value. How can we make SLIre that the blocked
trajectory are on the renol'111alized trajectory? It is not obvious that how many
blocking step we should do in order to reach the renormalized trajectory.

Itcompletely depends on what blocking scheme we take. However even if we take aslow blocking scheme(bad blocking scheme), enough many blocking steps can lead

24

CHAPTER 2. l\i[CRG ANALYSIS OF TIlE LATTICE (3 - FUNCTION

the blocking flow to the renormalized trajectory since it is fated that the blocked
trajectory becomes to flow along the renormalized trajectory lastly. Therefore to
do many blocking steps is an advantage in a sense of reaching the renormalized
trajectory and making sure that we are on the renormalized trajectory. This is
one of reasons why we use a 32 4 lattice twice bigger than a 16 4 lattice as mentioned
briefly in Sec. 1. 1. We can do the blocking one more time.
There is a freedom to optimaize the blocking in such a way that the blocked
trajectory reaches the renorrnalized trajectory fast. Actually, it is very importantto take such an optimal blocking scheme since the number of blocking step we
can do is limited within a few blocking steps. In Sec.2.2 we have introduced the qparameter in the blocking transformation. This parameter can be tuned so thatthe blocked trajectory reaches the renormalized trajectory fast. In principle, we
can find a value of the optimal parameter changing a value of q continuously in a
Monte Carlo run. However such a process to find the optilnal parameter is very
costly. In practice we can employ at most at several q points.
Fortunately dependence of 6.(3k on q is moderate and we can approximate it
by a linear function in terms of q:(2.49 )In the Monte Carlo run, as mentioned above, we run at several q points ( ql = qo,
ql, q3, ... ).

Hence we obtain 6.(3k(qL) ± 86.(3k(qL) for ql, where 86.(3k(QI) is anerror of /:}.(3k(ql ). Using this set, e.q.(2.49) is obtained by a fitting. Note thatthe cri terion that we are on the renormalized trajectory is that for all the Wilson
loops we obtain the same value of 6.(3. Therefore we can define the optimal q
parameter where all Jk (q) ill tersect each other. OUf proced ure to fi nd the optimal
q parameter is as follows:An error of the fitted line is defined as:(2.50)

2.3. MATCHING METHOD

25

The optimal value of q (= qopt) is defined at the best intersection poin t where thefunction S( q) takes a minimum:S(qopt) = min S(q),(2.51)(2.52)
where f(q) is an average of all the fk(q) at q, which is defined by:(2.53)
Using qopt, we obtain ~{3 as:(2.54)In order to know accuracy of obtained ~{3, we define the error of ~{3 as follows:First we define an error of qopt as:(2.55)

Here the derivative in e.q.(2.55) can be evaluated by a numerical calculation.
Using e.q.(2.55), the error of ~{3 is defined by:(2.56)
Alternatively ~{3 can be considered as a function of the ~{3k(qd. Therefore theerror of ~{3 is also defined as:(2.57)The derivatives in e.q.(2.57) is also calculated nUlnerically. We have checked thetwo definitions of 8~{3 and it turned out that the two definitions give the almost
same result. This indicates the validity of our procedures.

26

CHAPTER 2. MCRC ANALYSIS OF THE LATTICE (3 - FUNCTION

2.4

Numerical Simulation

2.4.1

Update Program

Our update program consists of two algorithms, the pseudo-heat bath[37] and theover-relaxed[3S] algorithm. The pseudo-heat bath algorithm is a standard method
widely used for generation of SU(3) gauge configurations and it is more efficientthan the Metropolis algorithm[39]. The over-relaxed algorithm is combined withthe pseudo-heat bath algorithm so that decorrelation between configurations is
accelerated. The decorrelation rate in terms of the autocorrelation time has beenstudied using blocked Wilson 100ps[14, IS, 19]. Due to the difficulty of measuring
the autocorrelation till1e, however, a drastic ilnprovelnent of decorrelation has notbeen seen yet. Even if so, there is a n1erit in using the over-relaxed algorithm.
It is faster to update a lattice system than pure pseudo-heat bath algorithm[22].
Our update speed for the pseudo-heat bath algorithm is 2.6 flsec/link. On theother hand, one for the over-relaxed algorithm is 2.0 flsec/link. In updating, we
use the pseudo-heat bath and the over-relaxed algorithm stochastically in a ratioof 1:9.
Our programs are developed on the parallel computer AP1000 in Fujitsu
laboratory[21, 22]. This machine has 512 floating processors and can be extended
to a 1024 processors version at maximum. The total speed of AP 1000 with 512
processors is 4.3Gflops. The sustained speed for our program is about 60% of
the total speed. Each processor has l6Mbytes n1en10ry. I-Ience APlOOO with 521
processors has SGbytes memory. The huge memory became a great advantage todevelop flexibly a program for the blocking[22].

2.4.2

Configurations

We generate three sizes of a lattice, 32\ 16 4 and s4, for our purpose. All theconfigurations are made at cold start, that is, all links are initially set to the unit
matrix U = I. We update configurations typically up to 20,000-100,000th sweep.

2.5. MEASUREMENT OF WILSON LOOP

27

Fist 2,000-10,000 sweeps are used for therl11alization. The independence between
configurations can be monitored by a measurement of the autocorrelation time.
The blocked Wilson loops are also used for operators for the autocorrelation
measurement since it contains a long range property of the lattice systeI11[14]. It
turned out that the blocked Wilson loop is useful to see a occurrence of a finite
temperature phase transition of the system. This study has been done[18, 19, 20]
and it has been showed that the autocorrelation time has a pronounced peak at
finite cross over or finite temperature phase transition point. This means thatat the finite cross over point it is very hard to obtain statistically independentconfigurations. Actually, we generated configurations of 30,000 sweeps on a 32 4
lattice at {3 = 7.0, where the systel11 is considered to be at or near the deconfinement phase transition point. However no lneaningful results could obtain due tothe large autocorrelation. Then we decided not to use there configurations forthe analysis.

2.5

Measurement of Wilson loop

As mentioned in Sec.2.3.2, 6.{3 can be determined by the matching method. At
first a value of 6.{3 is determined for each Wilson loop and each q parameter. And
then 6.{3 is extrapolated as a linear function of q. We have chosen three points,
q = {O.O, 0.002,0.004}. We measure 6 Wilson loops for purpose of the matching

method. The shape of those Wilson loops are displayed in Fig.l and we call themWll, W12, W22, Wchair, Wtwist and Wsofa respectively.
What we have to do in the matching method is evaluate (3' which satisfies
e.q.(2.47) by comparing values of Wilson loops. For this purpose, We have done
Monte Carlo run at a nUl11ber of {3 points. We have simulated at 96 points, from

{3 = 5.3 to {3 = 9.5 on a 8 4 lattice and at 33 points, from {3 = 5.75 to {3 = 7.7
on a 16 4 lattice. Values of Wl1 operator for each level are plotted in Fig.8 as an
example.

28

CHAPTER 2. MCRG ANALYSIS OF TIlE LATTICE {3- FUNCTION

2.6

Results of 6(32.6.1

Optimal value of q parameter~(3

is determined for each Wilson loop operator, each q parameter and eachblocking level. So we can write 6.{3 as:(2.58)
where i indicates the Wilson loop operator and I means the number of the blocking
steps and we call it Level. As mentioned in Sec.2.3.2., we express 6.{3?\qm) asa linear function of q and determine ~(3 at an optimal q point where those lines
intersect each other maximally.
Fig.9( a)-( d) shows how the optinlal q's are detennined or how values of 6.{3are obtained. Those data are from matching of the 32 4 lattice at {3 = 6.8 andthe 16 4 ones. It is seen that a functional dependence of 6.{3 is nicely expressedby a linear function of q. We can see some features of q dependence of ~(3. At
levell, the q dependence of 6.{3 is strong, that is, the slope of the line is very
steep. And we can see no good intersection for those lines. This indicates that theblocking scheme we took can not be optimized by only the q parameter such thatthe blocked trajectory can reach the renormalized trajectory after one blocking
step . On the other hand, it is seen that the intersection is getting good as we go
to high blocking level. It is noted that the data are presented in finer scale inFig.9( d) . So matching at Level 4 is not worse in comparison with those at Level
3. The q dependence of ~{3 becomes weak as blocking goes on. This trend seems
natural due to the following reason. As mentioned in Sec.2.3 .2, this means thatthe blocking flow comes close to the renormalized trajectory after enough many
blocking steps even if we start with a bad blocking schelne, that is, a blocking
scheme with a non-optimal q parameter. Hence if we employ further blocking
step at the level 4, we will see an approximate non-q-dependence of ~{3.
We plot values of optimal q for each level in Fig .10. The values of qopt at Level
2 and 3 are almost same. This indicates that after 2nd blocking step, the blocked2.6. RESULTS OF !:1(329

trajectory is already close to the renormalized trajectory. On the other hand,that of qopt at Levell is very different, almost constant. This also means that we
can not optimize the blocked trajectory tuning only the q parameter so that theblocked trajectory reaches to the renormalized trajectory by one blocking step.
Even if the optimal value for Levell in Fig.IO is used, the blocked trajectory is
still not close to the renormalized trajectory. Therefore values of !:1(3 at Level 1
can not be accurate.

2.6.2

6(3

In Fig.II we plot !:1(3 obtained at each level on 32 4 lattices as a function of Level.
For all (3, we can see the same functional dependence, that is, !:1(3 decreases as
Level increases and seems to converge to a constant value after Level 3, that is,
3rd blocking step. Although at Level 1 the value of !:1(3 is very different fromothers, at level 2 the value of !:1(3 is close to the stable value at Level 3 and 4.
This behavior supports the interpretation for the approach to the renormalized
trajectory as mentioned in Sec.2.6.1. Namely we can not optimize the blocking
scheme by one blocking step. Therefore again we conclude that at Levell thevalue of !:1(3 is not accurate. At Level 2 the blocked trajectory is already close to
the renormalized trajectory but not sufficient to obtain a reliable value. Therefore
we can see a slight but clear difference between Level 2 and Level 3, 4.
Here note that situation on the approach to the renormalized trajectory isdependent on the blocking scheme we have taken. In this sense, good blocking
scheme is the one which come close to the renormalized trajectory within first a
few steps. Although we have taken the q paralneter as an optimization paralneter,
there is another choice of an optimization parameter. For instance, it is possible
to use the parameter a and b in e.q.(2.26) as an optirnization parameter andthere exists a work in which the parameter a and b are used as optilnization
parameters [12] .
In Fig.12 we plot !:1(3 horrl 16 4 lattices for each level. As n1entioned above,

30

CHAPTER 2. MCRC ANALYSIS OF THE LATTICE {3- FUNCTION~{3 at Levell are not accurate and very different fron1 6.{3 at Level 2 and 3.

Forlevel 2 and 3, 6.{3 are very similar each other but for {3 ~ 6.4, values of 6.{3 atLevel 3 are systematically small. One of the possibilities of interpretation for thisbehavior is that 6.{3 is affected by the finite temperature phase transition since
the transition point is about {3 ~ 6.4 on a 16 4 lattice and for {3 ~ 6.4 the systemof a 164 lattice is in the deconfinement region. As already mentioned in Sec.l.1
one of the reasons we use a 32 4 lattice is we can remain in the confinement region
till about (3 ~ 7.0 and we can avoid the systematic error coming from the phasetransition. We plot 6.{3 at the deepest level from 16 4 and 32 4 lattice together in
Fig.13. These values are summarized in Table 2.6.2.

Table 2.6.2(a) 16 4 lattice

{3

6.{3

e1Tor

{3

~{3

error

{3

~{3

error

5.85

0.323

0.007

6.25

0.430

0.009

6.60

0.545

0.045

5.90

0.317

0.013

6.28

0.422

0.011

6.65

0.507

0.013

5.95

0.325

0.010

6.30

0.432

0.010

6.70

0.527

0.030

5.98

0.334

0.004

6.35

0.448

0.012

6.75

0.500

0.010

6.00

0.338

0.007

6.38

0.457

0.012

6.80

0.535

0.027

6.03

0.343

0.007

6.40

0.451

0.013

6.85

0.511

0.025

6.05

0.351

0.005

6.43

0.460

0.011

6.90

0.531

0.028

6.10

0.366

0.006

6.45

0.485

0.006

7.00

0.539

0.017

6.12

0.372

0.013

6.47

0.468

0.011

7.20

0.546

0.041

6.15

0.390

0.003

6.50

0.461

0.005

7.35

0.568

0.048

6.17

0.387

0.015

6.53

0.478

0.016

7.50

0.557

0.019

6.20

0.387

0.007

6.55

0.521

0.020

7.60

0.579

0.078

6.23

0.418

0.009

2.7. IMPROVED COUPLINGS

31

Table 2.6.2(b) 32 4 lattice
{3

~{3

error

6.35

0.433

0.004

6.55

0.481

0.014

6.65

0.500

0.015

6.80

0.536

0.010

All the ~{3 frorn 32 4 lattice are results in the confinement region. We can notsee significant difference between ~{3 from 16 4 and one from 32 4 lattices even if~{3 from 16 4 lattices are from the deconfinement region. However ~{3 from thedeconfinement region have large error and fluctuate very lTIuch. In Fig.13, we alsoline the prediction of 6.{3 from the 2 loop (3 function, e.q.(1.20), which is indicatedas bare g2. There is a clear discrepancy between 6.{3 from MCRG and one fromthe 2 loop prediction till about {3 ~ 7.5 Although the discrepancy decreases as{3 increases, there is still a 10 % discrepancy at {3 = 7.0. From this, we may saythat there is no asymptotic scaling below {3 = 7.0. This result is very serious forMonte Carlo sin1ulation since the current simulation has been typically carriedout at about {3 = 6.5 on the largest lattice such as 32 4 lattice and at {3 = 6.5
there is a 20 % discrepancy. As we will see in the next section, the asymptotic
scaling violation can be reduced by some remedies.

2.7

Improved couplings

As we have seen above, the large deviation from the prediction with the 2-loop{3-function still remains above {3 ~ 7.0. However this situation can be improvedby changing the schen1e of the coupling to a mean field scheme[40] or an effective
coupling scheme[41, 42] in stead of using 'poor' bare coupling.
Lepage and Mackenzie[40] have showed that the choice of the lattice bar coupling g2 is not practical. For instance, let us take a look at the plaquette. In the

32

CHAPTER 2. MCRG ANALYSIS OF THE LATTICE (3- FUNCTION

perturbative expansion, the plaquette value is expressed by[44, 43]:
46
< Uplaq >pt= 1 - c}g 2- C2g- C3g + ...

(2.59)

where the coefficients c}, C2 and C3 are(2.60)

1

2

C2

C3

1

= (N -1)(0.0204277 - 32N2)4(2.61 )= (N 2 _ 1)N (0.0066599 _ 0.020411 + 0.0343399) ~
N2

N4

6

(2.62)

where N is the number of colors. At (3 = 6.0, e.q.(2.59) with only the first two
terms gives us:< Uplnq >pt~ 0.667.

(2.63)On the other hand, from the Monte Carlo silTIulation, we obtain:< Uplnq > MC~ 0.597.

(2.64)

There is still 11 % discrepancy between the perturbative calculation and the Monte
Carlo calculation although the coupling in tern1S of ex = g2 /47r is already small (ex ~ 0.08 ). Therefore we may conclude that the bare lattice coupling g2 fails to
reproduce values of the plaquette correctly. In Ref.[40], the more continuum like
coupling scheme or the mean field scheme has been proposed:
1 ( /)< Uplnq >-27r a = 2( / ) + 0.008204N
9 MS
9 7r a(2.65)

where the new coupling is denoted as g~s since this coupling exactly corresponds
to the modified minimal subtraction (M S) scheme in the continuum limit (a --+0).

The relation between g~s and g2 coupling is known by the perturbative

calculation[45]:

N 2 -1
8N + 0.008204N.

(2.66)

2.7. IMPROVED COUPLINGS

33

Using e.q.(2.59) it is easily verified that e.q.(2.65) goes to e.q.(2.66) in the continuum limit. Therefore the scale parameter of the scheme defined by e.q.(2.65)
takes the same value of A MS ' Assuming the 2 loop ,B-function (e.q.(1.13)) for thiscoupling, and following the definition of Ref. [46] we obtain A MS as:-MSa(x)AMS == 12(x) = 7r

2b
( x + bl/b O
) bl/ 5 X,b

exp ( -o2bJ

(2.67)

where x = -}- for the later convenience. Note that this definition is a littlegMS

different from the definition of e.q.(1.16). Here we write it again for the later
convenIence:a(x)AL == h(x) = (~r/2b6 exp ( -2:J .

(2.68)

Using e.q.(2.67) and e.q.(2.68) the relation between A MS and AL in the continuum
limit is easily obtained:

AMS = I'1m /fS(l/gk) = 7rexp (_ A/?b0) "'"- 28 .81
ALa-+O
12 ( 1/ 9 2)"-'

(2.69)

N 2 -1
8N + 0.008204N.

(2.70)

Ll

where
~ = -

The ratio of A MS and AL is very large.
The other in1proved coupling, the effective coupling, is proposed by Karschand Petronzio[42] motivated by Parisi[41]. They have proposed theg; coupling

as follows:2ge =1- < Up1aq >(2.71 )

where Cl is the value of (2.60). Using the perturbative expansion of e.q.(2.59),
we obtain:(2.72)
or(2.73)

CHAPTER 2. MCRC ANALYSIS OF T~fIE LATTICE (3- FUNCTION

34

Assuming the 2 loop (3-function for this coupling and using e.q.(2.68) as a defini tion of the scale parameter Ae for this coupling:(2.74 )
the relation between Ae and AL is obtained by:where x = ~,
ge

gn

Ae
. 12(1/-A = hm f ( / 2) = exp
La-+O 2 1 9(-bC2 )
~ 2.0756
2Cl 0(2.75)

and we also obtain the ratio of A MS and Ae:

A MS

2.0756AL = 13.88(2.76)

Bali and Schilling[43] have used the similar coupling, g;2' defined by:(2.77)
where Cl and C2 are taken frOITI (2.60) and (2.61). Using the perturbative expansion e.q.(2.59), we obtain the inverse of g;2:
1
-2

ge2

1

2

= 2" + O(g ).

(2.78)

9

Therefore the scale parameter for the coupling g;2 is unchanged :(2.79)The qualitative behavior of the coupling g;2 is similar with that of the couplingg;[43] .
Using these ilTIproved couplings, we compare ~(3 from the MeRG study with
one from the 2 loop (3-function with the improved couplings. The curve M in
Fig .14 shows the coupling shift 6.(3 from the g~s coupling. This scheme partiallyexplains our results of 6.(3. However the large deviation is still seen at low (3(f3 :::; 6.6 ). Similar analysis is also implemented for the effective coupling scheme,g;. The curve E in Fig.14 is 6.(3 from the effective coupling scheme. This scheme
also partially accounts for our results of 6.(3.

2.7. IMPROVED COUPLINGS

35

Now we consider the possibility to obtain further improvements. Let us introduce a next order correction in the ,B-function and define a new effective coupling
2.
gu'

dxu
b1
b'
2d in a = bo + -Xu + x2u(2.80)where1

1= 2 = - 2 - - Xo·(2.81)
gu
9 MS
Here b' and Xo are not known and these values are determined so that this couplingXu

scheme accounts for 6.,B fonn the r..1CRG study. Assun1ing the correction tern1 b'
is small, the integration of e.q.(2.80) leads to (See Appendix B):-b'a(xu)i\u == f(x u ) = f2(xu) exp( - 2b ).

2 OXu(2.82)

where the function f2(Xu) comes from e.q.(2.68).
Using Xo and b' as free parameters, we attempted two fits for the data of ~,Babove ,B = 6.0:

(A) introduces only b'.

(B) introduces both Xo and b'.
The case of (A) allows to add tbe correction ternl O(9~). The results areshown in Fig.14 as curve A. Unfortunately this case gives only a poor fit andthe coefficient b' is relatively large against bo : b' /b o = -0.085(0.0013) (Note thatb1 / bo = 0.0587).
The case of (B) allows xo and b' as free paralneters. This case has a quit wellfit for all the data of 6.,B( curve B in Fig.14). The values of parameters obtainedby the fit are:= 0.442(0.004)(2.83)

b' /b o = -0.0119(0.0008)(2.84)Xo

The coefficient h' is srnall bowever the shift Xo is non-negligible. In this schemewith the shift Xo, the relation to the (},;{ S) scheme in the continUUlTI limit IS
obtained by:

Au = exp(;:J

A MS = 23.95A MS(2.85 )

36

CHAPTER 2. MCRC ANALYSIS OF THE LATTICE (3 - FUNCTION

Therefore, Au is O(GeV) and much larger than A_.
Although we have betterMSscaling in this scheme, this fact needs a explanation. At present, however , wehave no reasonable explanation yet.

2.8

Scaling Test

Based on various improved couplings introduced above, we exam the scaling of
the physical quantities in the unit of A~s where (0) means we are employing thequenched case, n f = O. First we exam the string tension, one of popular physicalobservables in the Monte Carlo simulation. The data of string tensions comefrom Ref. [43]. Using various couplings, we plot the ratio of the string tension
and the A~s parameter in Fig.I5. As shown in e.q.(l.I8), those values should
be constant in the asymptotic scaling region. The rightest value corresponds
to one from (3 = 5.7 and the leftest value is from (3 = 6.8.

We can see thelarge asymptotic scaling violation for the lattice bare coupling g2. The improved
couplings
Fig .I5.

givts and g; can reduce the asymptotic scaling violation as we see in

However the visible deviation still remains.

This situation needs thefurther interpolation to the continuUln limit in order to obtain the continuum
physics. On the other hand, the g~ coupling based on our MCRG results gives us
a good improved scaling behavior. The values ofva/A ~s are a nearly constantin the whole region (5 .7 < (3 < 6.8) . The estimated value is:va(0) = 2.2(1).

AMS

Assuming(2.86)

va = 440A1 e V as a input value, we can obtain the value of A~s:A~s = 200MeV.

(2.87)

This value is smaller than that obtained by the extrapolation to the continuum
limit using the couplingg; in Ref.[43] where A~s = 233MeV.

Further test for other quantities are attempted . Fig.I6 shows the test for the
charmoniun1 1p-1s splitting[47]. We can see the silnilar scaling behavior. The2.8. SCALING TEST
values from the coupling37

g; are almost constant and we obtain:(2.88)

Substituting Me = 450NJ e V into e.q.(2.88), we obtain:

A~s = 210MeV.

(2.89)

Fig.17 shows the test for glueball 0++ mass. In this case, it seems that the ratioof glueball mass and A~s from the improved couplings increases as f3 increases
even if g~ coupling is used . Hence for glueball 0++ mass, a good scaling behavior
is not seen.
From above scaling tests, we can learn the followings:(1) The improvements of the improved couplings (including g~) are only partial
since in the asymptotic scaling region, all the physical quantities should show theasymptotic scaling behavior.
(2) It is difficult to obtain reliable values in the continuum limit . Careful
extrapolations to the continuum limit should be needed at this stage.

38

CHAPTER 2. MCRC ANALYSIS OF THE LATTICE (3- FUNCTION

Chapter 3
Perfect Action Search
3.1

Effective Action on the blocked trajectory

As we have seen in Chapter 2, the standard Wilson action has a poor scaling
property and even if we use the improved couplings the complete asymptotic
scaling is not seen. As mentioned in Chapter 1, there is a possibility to use an
improved action, such as perfect action. The perfect action, which defined on therenormalized trajectory, contains only a relevant interaction with respect to scaletransformation. Therefore it is expected that the perfect action has a almost
complete scaling behavior.
Our motivation to find an effective action for blocked gauge configuration is
as follows. If we can offer an effective action or renormalized couplings on theblocked trajectory after enough blocking steps, we can obtain a perfect action
since the blocked trajectory flows along the renormalized trajectory lastly .
There exists, however, a difficulty that it is very hard to directly determine
the effective action or renormalized couplings since the effective action can becomplicated enough with ITlany couplings. And if we truncate some couplings it
is also hard to know the effect of the truncation.
There are several ways to find an effective action for a given set of gauge configurations. Swendsen[48] has proposed a minimization procedure to find renor-

39

Cl-IAPTER 3. PERFECT ACTION SEARCII

40

malized couplings, however, this n1ethod requires a good starting point and it
becomes more difficult to give the good starting point as we deal with a complicated action. Creutz[49] has invented the microcanonical demon method and it
has been applied to deter111ine renormalized couplings on an SU(2) configuration
with fundamental and adjoint couplings[50]. In actual applications, however, themethod works poorly and the couplings obtained by the method have large systematic errors due to fini te size effect. These errors are caused by extra degrees
of freedom of the demons which carry a considerable part of the energy in a small
system. Recently an idea, the canonical demon method, which does not cause systematic errors has been advocated and applied for 0(3) non-linear cr-model[51].
Although the idea is rather simple, the results coming from it are highly reliable.
In this chapter we in1plen1ent this 111ethod for a physically interesting case, SU(3)gauge theory and obtain the effective action on blocked configurations[52].

3.2

The Demon Method

Let us consider a configuration which is produced with a probability distribution:

P(U) ex exp( -/3S[U]).

(3.1 )

Now we introduce a demon into the lattice and consider a joint system of thelattice and the demon. On the joint system, the microcanonical partition function
is written as:
ZMic= L I:>5(S[U] + Ed - Einitiad
Ed

(3.2)

U

where Einitial is a total energy of the joint system which is initially determined
and kept constant, and Ed is the demon energy. Notice that we need not a priori
know information of the coupling to do the microcanonical step. After enoughmicrocanonical sweeps, the delTIon energy is distributed with the Boltzmann's
distribution:(3.3)

3.2. THE DEMON !vfETI-IOD

41

In the thermodynan1ic lilnit, the average of the demon energy is given by:(3.4)
where Z is a partition function given by integrating out e.q.(3.3) with respect tothe demon energy. The range of integration is taken suitably . We restrict theenergy to -Emax < Ed < Emax. In this case, e.q.(3.4) becOInes( See AppendixC):
1< Ed >= 73 - Emax/ tanh(/3Emax).

(3.5 )

This equation gives a relation between < Ed > and /3. The coupling /3 can beobtained through the average demon energy by solving the above equation.
The extension to lnany couplings is straightforward. In this case, for eachcoupling we must introduce a corresponding demon. Therefore the corresponding
microcanonical partition function is given by:

ZMic = L L II 8(Si[U] + E~ - E;nitial),
Ed U i(3.6)

and the coupling /3i corresponding to E~ is obtained by:·

1 ·< E~ >= /3i - Emax/ tanh(/3t Emax).

(3.7)

In the microcanonical step, the demon visits one of links on a lattice and tries
to change the link, that is, a new SU(3) matrix is taken fron1 SU(3) group and thelink is replaced with it. This trial change causes a shift of energy on the lattice,

6E, which must be compensated with the demon energy since the Lotal energy
on the system is kept constant during the lnicrocanonical update. After this trialchange, new demon energy is given by:(3.8)
If the new demon energy ren1ains in the allowed region, in the above case -Emax <E;;ew < E max , this trial change is accepted. If not the case, the demon moves toCI-IAPTER 3. PERFECT ACTION SEARCl-I

42a next link keeping its old energy. In order to raise an acceptance ratio of the
n1icrocanonical update, it is possible to do some trial changes before the demon
moves to a next link. If a number of demons are introduced in the system, the trialchange is accepted only when all the demons remain in the allowed region. Afteradequate microcanonical sweeps, the demon moves into another configuration
keeping its energy to make itself canonical[51].

3.3

Testing the demon method

In order to exalnine effectiveness of the Inethod, first, we test the method forSU(3) gauge configurations generated on a 44 lattice with an action with 7 couplings(For those values see Table 3.1.) corresponding to the Wilson loop operators
in Fig.1. We choose rather small lattice, 4 4 lattice, to test how the method works
since it is getting worse to obtain the effective action as the lattice size becomes
small. The action S is written as:
7

S =L L ReTr(W )/3
(Ji

i=l

(3.9)

i

Wi

where vVi is a Wilson loop corresponding to one in Fig.l. Our numerical description is as follows.
Table 3.1: Coupling values used for update of a 44 lattice( the first row" A") andcoupling values obtained by the demon method( the second row "B") .
11

{312

{322

{3chair{3$oj a{3twi$t(34Dtwi3t

A

.5

1.0

-0.1

-0.1

-0.1

-0.1

-0.1

B

.5029(84)1.0004(32)-0.1027(37)-0.0988(18)-0.10032(94)-0 .0988(34)-0.1028(12)We need to introduce 7 demons, one for each of the 7 Wilson loop operators. Let us call the 7 demons "one demon system" . We prepare 100 independent demon systems and 100 configurations and each demon system visits oneof these configurations and the microcanonical update is carried out. Initially,3.3. TESTING THE DEA10N METHOD

43

we set all the demons energy to zero. All the demons energy are restricted to-10 < Ed < 10 during the lnicrocanonical step. After 100 microcanonical sweeps,
each demon system, keeping its demon energy, moves into a second configuration
taken from a next set of 100 configurations such that this second configuration is
statistically independent to the first one. This procedure eliminates the systematic error due to the finite volume of the system. After another 100 microcanonicalsweeps on the second configuration, each demon system again moves into a third
configuration, and so on.
Fig.18 shows the demon energy averaged over the 100 deinon systems for every
sweep. On each configuration used by a demon system, after some microcanonjcal
sweeps, the demon energy seeins to be thermalized. However a clear differenceof the demon energy between the first configuration and the second one can be
seen. Although there also remains a difference between the second one and thethird one, the difference is much smaller and seems to go to zero within thermal
fluctuation. This behavior can be seen more clearly in Fig.19.
Fig.19 shows the average of the demon energy which correspond to the 5 couplings: f322, f3chaiTl f3sofa, f3twist and f34Dtwist, on the first, second, etc configuration
used by a demon system. We averaged over the last 20 lnicrocanonical sweeps(
We are doing 100 microcanonical sweeps on one configuration.) and furthermore
over the 100 den10n systems. Note that the configurations are generated with thesame value( -0.1) for these couplings, hence the average values should be san1e forthese demons. On the first 3 configurations, however, a clear difference is seen,
which might be considered due to the finite volume effect of the micro canonicalupdate. After enough movements of the demon system, in this case 3 movements,
the average value seems to converge to one value.
This elimination of the finite volume effect works in the following way. In theinfinite volume limit, initial values of the demons energy are not so importantsince the demons carry only a vanishingly small percent of the energy of a system.
On the other hand, in our case the system is so small that the initial values ofCl-IAPTER 3. PERFECT ACTION SEARCH

44

the demons energy becolne important. Since we have set all the demons energy
to zero, on the first configuration the demons energy becomes distributed with aBoltzmann's distribution exp( -(3' E~) which is away from the true Boltzmann's
distribution exp( -(3E~) due to the finite volume effect. The demons move into
a second configuration keeping their energy with the wrong Boltzmann's distr'ibution which have ever has a larger overlap with the true one than the initial
distribution(E~ = 0). On the second configuration, the demon energy becomes

distributed with a Boltzmann's distribution which is ever closer to the true one.
By doing further movements, the distribution of the demon energy converges tothe true one. To check this scenario, we set initial values of the demons energy
to values distributed with the true Boltzmann's distributions and did the nucrocanonical update. In this case, we obtained the desired demons energy on thefirst configuration.
The coupling values, which correspond to the demons energy in Fig.19, converted through e.q.(3.4) are shown in Fig.20. Each coupling nicely converges
to the value of -0.1. We proceeded the method to the 22nd configuration andobtained coupling value corresponding to each demon averaging over the configurations except the first 3 configurations as thermalization. The couplings arewell reproduced within a few percent error bar.(See Table 3.1.)
So far, we have restricted the demon energy to -10 < Ed < 10, however, there
is no reason to do so. If the constraint is set to a wide region, the demon carriesa large part of energy of the system. This enhances a systematic error. To see
this, setting all the deIllons to -50 < Ed < 50, we perfonned the ll1icrocanonicalupdate as above. Fig.21 shows the coupling value converted from average energyof the demon corresponding to (311 in the both regions, that is, -10 < Ed < 10
and -50 < Ed < 50. Although for the both cases, each denl0n energy seems toconverge, the convergence of the demon energy in the wide region is very poor.
From this point of view) it lnight be better to restrict the demons in a narrow
region as possible as we can. As a matter of fact we can not take a very narrow3.4. THE DEMON METIIOD ON BLOCI(ED CONFIGURATIONS

45

region, since the acceptance ratio of the microcanonical update goes to zero.
Therefore the constraint to the demons energy should be optimized suitably.

3.4

The Demon Method on Blocked Configurations

As we have seen in the previous section, the demon method works very well. In
this section, we implement the demon method for blocked SU(3) gauge configurations. Now the effective action is not known. The blocked configurations whose
size is 44 are produced after twice blockings from 16 4 lattices with the Wilson
action at f3 = 6.2 by QCD_TARO Collaboration[53]. The blocking scheme wetake here is a double sn1eared blocking of a scale factor 2.

With certain op-

timal parameters[30, 53] it does not induce complicated extended interactions
corresponding to large Wilson loops in the classical level. The double smeared
blocking is as follows.
A link UJJ.(n) is smeared twice:

VJJ.(n) = (1 - 6c)UJJ.(n) + C L Uv(n)UJJ.(n + v)U}(n + J-l)v:pJJ.

(3.10)

TJJ.(n) = (1 - 6c)~L(n) + C L VV(n)~L(n + v)V}(n + J-l)v:pJJ.

(3.11)

where TJJ.(n) and VJJ.(n) are once and twice smeared link respectively. The two
adjacent smeared links are connected as:(3.12)
and a blocked link U~b)(n) is obtained according to a probability distribution:(3.13)The parameters, c and K, are tuned so that the effective action becomes local.
Here we take c = 0.077 and K = 10.5[30, 53].

46

CHAPTER 3. PERFECT ACTION SEARCII
Assuming that the effective action has only 7 nterctions as in Fig.2, we dothe same procedure as above. Fig.22 shows the coupling value of 1311 converted
through the demon energy at each configuration. In this case, the demon energy
seems to be already thermalized on the second configuration. The coupling values
obtained by the demon lnethod are summarized in Table 3.2. We can observea coupling decay for those couplings, namely, the values of 8 links Wilson loop
couplings is smaller than the values of 6 links ones. It might be conjectured that
contributions from 1110re than 8 links Wilson loops are smaller.
Table 3.2: Coupling va.lues obtained by the demon method for the double smeared
blocking(D) and Swendsen's blocking(S). The double s111eared blocking is doneat 13 = 6 2 and Swendsen's blocking IS at 13 = 6 0

lBu

f312

f322

f3chair

D l.1402(25)0.1169(16)0 .0204(22)0.20023(94)0.03494(80)0.1935(13)0.07032(89)S ~ . 236( 45)-0.7033(59)0.0653(74)-0.1042(34)0 .0300(30)0.2151(66)0.1023(13)To verify the validity of the effective action obtained in this way we compare
the values of a nurnber of Wilson loop operators. The second row of Table 3.3
shows the values of these Wilson loop operators on the blocked configurations.
The third row shows the ones measured on generated configurations with theeffective action with the couplings of Table 3.2. They agree well within error bars.
However this is not a severe check of the truncation effect since we measured only
Wilson loops which the demons couple to. Unfortunately we did not measured
such Wilson loops which the demons do not couple to.
Table 3.3: Values of the Wilson loop operators on the double smeared blocked
configurations( BL) and on the configurations genera.ted with the couplings obtained by the demon method(GE).
BL p.10119(34)0.01814(21)0.00177( 11)0.02585( 16)0.003253(59)0.02137(15)GE p.10121(15)0.018167(57)0.001722(64)0.025789(53)0.003251(37)0.021528( 43)3.4. THE DEMON METI-IOD ON BLOC](ED CONFIGURATIONS

47

We also measure susceptibilities between Wilson loop operators as anothercheck. We define the susceptibilities a;j as:(3.14)
where < ... > indicates taking an average. The susceptibilities obtained by thisdefinition are summarized in Table 3.4. All the non-diagonal susceptibilities arecorrectly reproduced in sign. As for magnitude of the susceptibilities, most ofthem are correctly reproduced within 2 sigma error bar. There are small but definite deviations for those of W11-W11, Wchair-Wsofa, W12-Wtwist, and WsofaWtwist. Therefore the obtained action is still an approximate one. Complete
reproduction will be obtained by enlarging coupling constant space. However,such fine tuning is too pren1ature for the first trial and we do not come into
further detailed reproduction.
Table 3.4: Susceptibility aij (x 1000). In each row, upper figures from the double smeared blocked configurations and lower figures are from the configurations
genera t e d WI. tl1 tlle coup rIng by th e d emon me th o d .
Wi 1, Wj -+

W ll

W 12

W 22

WchairW"ojaWtwi"tW ll

7.55(13)

3.67( 14)l.57(27)

3.955(85 )l.55( 11)

3.15( 14)8.212(69)3.833(57)l.20(24)4.084(39)l.156(95 )3.172(52)4.73( 10)l.36(23)

2.17(11)l.257(68)1.936(95)

4.727(25)l.37(11)2.088(29)

1.007(82)l.601( 44)5.98(11)0.99(21)0.78(20)l.01(21)6.046(38)0.59(16)0.609(98)0.74(17)3.l83( 40)l.193( 48)

2.522(55 )3.130(19)0.879(45)2.387(29)2.259(28)l.412(82)2.192(19)l.020(28)

W 12

W 22

WchairW"ojaWtwi"t4.125(68)
4.136(36)We also implen1ent the demon 11lethod for blocked configurations generated by

48

CJ-IAPTER 3. PERFECT ACTION SEARCH

a different blocking, Swendsen's scale factor 2 blocking[32, 14, 18J. The blocking
procedure we take is saIne as in Sec.2.1
Blocked configurations are also produced by QCD_TARO Collaboration[14,22], after twice blockings with q = 0 from 32 3 x 64 lattices with the Wilson action
at f3 = 6.0. These configurations have a size of 83 x 16. Therefore it can be
expected that the finite volume effect is much smaller comparing to the case of a
44 lattice. As seen in Fig.10, the optimal q is about 0 at f3 = 6.0. Therefore wehave blocked configurations with q = O. In general, however, this optimal value
does not necessarily mean that the effective action is local.
We introduced only one demon system and used 4 configurations, so far. Although in this case statistics lnight not be enough to obtain a definite result, atleast we could obtain a interesting preliminary result and further investigation
should be needed to improve statistics and verify the result. The result is given
in Table 3.2 together with the double sn1eared blocking case. The interesting
behavior is that although for the double smeared blocking all the couplings takepositive values while for Swendsen's blocking some couplings take negative values
and the f311 coupling is rather increasing. This is an example that the renormalized trajectory is dependent on what blocking scheme we take. Although for
Swendsen's blocking we can also observe a coupling decay for those couplings,
it is rather slow, namely, the f3chair coupling of a 6 links loop and the f34Dtwist
coupling of a 8 links loop are nearly equal. So it might be probable that sometruncation effects are still contained in the effective action which we obtained andthey may be large. A careful analysis should be needed in further investigationin order to see the truncation effect.

Chapter 4
Conclusion
In Chapter 2 we have analyzed scaling property of the standard Wilson action in
tern1S of the coupling shift 6.{3 using the MCRG technique. There was discrepancy
in the previous results of 6.{3 and they were very controversial.
This is partly due to smallness of lattice size which limits blocking step andcauses unwanted finite temperature phase transition at high {3. Another point islack of insufficient statistics.
To solve this, we have worked on 32 4 lattices and done high statistics Monte
Carlo simulations. Our results have showed that there is a large deviation from theprediction of the 2 loop {3-function up to {3 ~ 7.5. From this analysis, we conclude
that the standard Wilson action has not the perturbative scaling property atpresently accessible {3. Therefore in order to obtain values in the continuum limit
using the standard Wilson action, some careful extrapolations to the continuum
limit should be needed or we have to go to higher {3, provided that the performanceof computers is powerful enough .
Using the improved couplings, g~s and g;, however, the deviation can be
reduced but it is only a partial improvement. For {3 :::; 6.5 the large deviation stillremaIns.
We have proposed a new coupling scheme g~ which is determined so that it
accounts for our 6.{3 results. And we have tested the scaling for the string tension,

49

50

CHAPTER 4.

CONCLUSION

the charmonium Ip-ls splitting and the glueball mass using g~ and also bare g2 ,g~s and g;. From this scaling test, we have found that the improved couplingg~s and g; recover the scaling property partially but not completely. On theother hand, We have better scaling for the string tension and the charmonium
Ip-ls splitting by the use of g~ coupling in a region 5.7 < f3 < 6.8. Using theg~ coupling, A MS of the continuum QCD is estimated to be 200M eV from thestring tension and 210MeV from the charmonium Ip=ls splitting. These values
are smaller than that given by

g;, A

MS =

233MeV.

A point to be clarified in that scale paran1eter in the g~ coupling scheIue, Au,
is fairly large, a few GeV. At present, we have no reasonable explanation for thispoint and it is left for future investigations.
For the glueball 0++ mass, we could not see a good scaling even if the couplingg~ is used for them .

We are able to take another approach to the scaling problelu. That is to findan action of lattice QCD which is sitting on the renormalized trajectory and isnamed perfect action. If the perfect action is known and it is a very local action,
it will be a useful and reliable action in the Monte Carlo sin1ulations. As the firststep of the investigation, we study a possibility to find effective action for blocked
gauge configurations.
In Chapter 3 we have developed a tool, the demon method, to find the perfectaction. First the luethod was tested for the known gauge configurations generated
with the 7 couplings. VVe have successfully reproduced the 7 couplings withina few percent error bar.

Based on this study, the method was then used on

the double smeared blocked SU (3) gauge configurations and an effecti ve action
was obtained . To check the validi ty of the effective action we have generated
configurations by this action and measured values of some Wilson loop operators.
Their values turn out to be the san1e as those measured directly on the blocked
configurations within error bars. Susceptibilities were also measured as anothercheck . They showed the salue sign in non-diagonal parts and the san1e magnitude

51
except small deviations on a few susceptibilities. Therefore we conclude that thedemon method works very well to determine effective actions in multi-dimensional
coupling constant space. It is considered that the demon method is a strong tool
to find the perfect action which gives physical prediction in continuum spacetime with good precision. [29] In principle, enough blocking transformations get
the system close to renormalized trajectory even if we start with Wilson action.

If an effective action is determined there, it becomes the perfect action. From
this point of view, searching for the perfect action utilizing the demon method isvery promising and an investigation in this direction is going on [53].

52

CHAPTER 4.

CONCLUSION

Acknow ledgerne nt s
The study in Chapter 2 was carried out as a project of QCD_TARO collaboration. The author would like to thank to all the member of QCD_TARO collaboration: K.Akemi, Ph de Forcrand, M.Fujisaki, T.Hashimoto, S.Hioki, O.Miyamura,
A.Nakamura, M.Okuda, I.O.Stamatescu and Y.Tago.
The study in Chapter 3 was pursued in part while he was staying at Institut
fur Theoretische Physik, UniversiUit Heidelberg. He also thanks to I.Bender,
J.Holk, M.Plewnia, I.O.Starnatescu and W.Wetzel for their warm hospitality.
Finally he especially thanks to O.Miyamura and M.Yonezawa for their constant encouragement.

53

54

CHAPTER 4. CONCLUSION

Appendix A
Blocking Scheme
Here we give the proof that our blocking scheme is given by e.q.(2.36). Let us
consider a simple case, one variable, and assume e.q.(2.36). A new blocked link

Ub is given according to the probability distribution:

Since Ub is an SU(3) matrix, it can be parametrized as:
(A.2)
where

v = Q( Qt Q) -;1 e- i o:/ 3U' = exp(it

1=1and A, are SU(3) generators and

CI AI )

(A.3)

(A.4)

2

c, are parameters. Using the property of theinvariant measure, we obtain:

dUb = d(VU') = dU'(A.5)

Therefore e.q.( A.l) is rewritten as:

dU' P(U') = dU'

JdU exp [-S[U) + N(Q) + pReTr(U')).
55

(A.6)

56

APPENDIX A. BLOCI(ING SCHEME

This means that U' is taken according to the probability distribution:dU'P(U') ex dU' exp(pReT1'(U'))(A.7)

The ReTT( U') is expanded as:

(A.8)

(A.9)
(A.10)
Assulning C/ are small, e.q.( A. 7) is given by:

dU' P( U') ex dU' exp(p(3 - ~ L cn)
4(A.11)I

If Cl are small, the integration measure dU' is given as:(A.12)
Therefore we obtain:

dU' P( U') ex exp(3p)II exp( - (E4 + ~8 )CndCI.

(A.13)I

This means that the parameter C/ are obtained according to the probability distribution:

p

1

2

P(CI) ex ex p(-(4 + g)c/).

(A.14)

Replacing (p/4 + 1/8) with 1/2q, we finally obtain:(A.15)
Hence U' is a gaussian n1atrix and Ub given by e.q.(A.2) corresponds to the blocked
matrix of e.q.(2.33).

57

N( Q) is a normalization factor and in the Monte Carlo simulation we do notneed to care about it. N(Q) is calculated as follows:21c2
exp(3( - - -))dCI exp( __
I )
q 2
I
2q
2
1
r;;::
exp(3( - - -))( V 2q7r)8
q2
exp( -N(Q)).

J

IT(A.16)(A.17)

Hence we obtain:(A.18)

58

APPENDIX A. BLOCI(ING SCI-IEME

Appendix B
Correction factor to the 2 loop
formulaWe want to integrate the following:J

2d In a =J + ~x +-(B.l)+ A 2 - A 3 + A 4 - A 5 + .... )(B. 2)

b'

~
x

b1

x2

where x = 1/ g~.
The right hand side is expanded as:

dx

dx

b

bo + ~x += -b (1 - A

b'
3"
x

0

where

A=~+~.2
box

(B.3)

box

Arranging the terms in e.q.(B.2), we obtain:

dx
b

b'

bo + 7 + x2

-

dx
b1
b1 2
b1 3
-b { ( 1 - -b + (-b ) - (-b + .... )
0
OX
oX
oX
b'
2b 1 b'b,2
3bi b' 1--b.2 + b2 .3 + (-b
2 - -b3 )-:4 + .... }

J

oX

OX

0

0

X

(B.4 )(B.5)

We can recognize that e.q. (B.4) is an expansion of the contribution of the 2 loop
,B-function. Therefore the correction factor to the 2 loop formula is given by

59

60

APPENDIX B. CORRECTION FACTOR TO T~HE 2 LOOP FORlv!ULA

e.q.(B.5) and this is easily integrated. We obtain the following formula as thecorrection factor up to O( 1 / x 5 ).

(B.6)
where

b1

Cl

= -

C2

= -

bo
b'

bo

(B.7)
(B.8)

and !2( x) is the 2 loop formula. If we assume that the coupling g~ = 1/ x is smallenough, the correction factor is approximately given by the first term, exp( - 2~;x).

Appendix C
Relation bet-ween the coupling
and the demon energy
The partition function of the demon is given by:z =lEma, exp( -f3 E d) dEd
1= [- ~ exp( -(3Ed)]~:~:=(C .l)

E m •n1-~[exp( -(3Emax) - exp( -(3Emin)](C .2)(C.3)

where the integration rage is Emin < Ed < Emax. The expectation value of thedemon energy is given by:

(C.4)
(C.5)
(C.6)
(C.7)
The value of the coupling can be obtained by e.q. (C. 7). Here we show the relations
for several cases .
61

62APPENDIX C. RELATION BETWEEN THE COUPLING AND TIlE DEMON ENERGY(1 )Emin = 0, Emax = 00

(C.S)
(2)Emin = -E, Emax = 00
1

< Ed >= -E +{3(C.g)(3)Emin = 0, Emax = E
1 (

(3E)

< Ed >= -g 1 - exp({3E) _ 1(C.lO)(4)Emin = -E, Emax = E
1< Ed >= -g - E / tanh({3E)(C.l1)

Bibliography[1] K.G.Wilson Phys. Rev. D10 (1974) 2445
[2] M.Creutz Phys. Rev D21 (1980) 2308
[3] See for example, Y.Iwasaki Nuel. Phys. B (Proe. Suppl.) 34 (1994) 78[4] See for example, D.Weingarten Nuel. Phys. B (Proe. Suppl.) 34 (1994) 29
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64

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K.Akemi,Ph.de

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M.Fujisaki,

T.Hashimoto, H.C.Hege, S.Hioki, J.Makino, O.Miyamura, A.Nakamura,
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68

BIBLIOGRAPHY

Figure Captions
Figure 1: 6{3 vs (3, from previous results. The line baTe g is froln
the 2 loop prediction.

Figure 2: Shape of Wilson loop operators. These \Vilson loops
except for W4dtwist are used for the matching method( Chapter 2). In the delllOll method( Chapter 3 ), all the Wilson loops are
considered.

Figure 3: Fixed point action and RenorInalized trajectory.
Figure 4: Blocking in spin system.
Figure 5: Wilson's scale factor b == 2 blocking. 8 links are
summed up in 4 dilllension. Here, 4 of theIll are drawn in 3 dimension. A new blocked link is put between 2 block sites.

Figure 6: Swendsen's scale factor b == 2 blocking.
Figure 7: Blocked trajectory and Renorlnalized trajectory. 2
blocked trajectories Inatch on the Renormalized trajectory.

Figure 8:1 x 1 loop( plaquette ) at each blocking level with
q == O. "Level 0" Illeans no blocking procedure. "Level 1", Level2" and "Level 3" mean blocking level.

Figure 9(a)-(d): Matching on a 32 4 lattce at {3 == 6.8. These69

70

BIBLIOGRAPHY

figures shows the interpolation at each blocking level. It is seenthat the dependence of q becomes weaker at high blocking level.
(N ote that scale of Y axis are different at each blocking level.)
Figure 10: qopt vs (3, on 164 lattices.
Figure 11: 6.(3 vs blocking level, on 32 4 lattices.
Figure 12(a)-(b): 6.(3 at each blocking level on the 16 4 latticevs (3. Filled triangle, squares and filled circles indicate levell, level
2 and level 3 respectively. Fig .12 ( a) is zoomed to Fig .12 (b) .
Figure 13: 6.(3 vs (3 for all data from this study. The solid line
indicated as bare g2 is the prediction from the 2-loop (3 function.
Figure 14: 6.(3 and fitted curves from various coupling schemes.
The curve M is the result of 6.(3 from g~s and the curve E is ones
fromg;. The curve A and B means fitted curves vvith b' only and

one with both b' and Xo respectively.
Figure 15: Scaling test for string tension.
Figure 16: Scaling test for charmonium 1p-1s splitting.
Figure 1 7: Scaling test for 0++ glueball mass.
Figure 18: (a) Delnon energy associating with (311 coupling at
each microcanonical sweep. First, the demon visits a 1st configuration and then moves into a 2nd configuration and so on. (b)Same for (3chair' coupling.
Figure 19: Average demon energy for (322, (3chair, (3sofa, (3lwisland (34Dtwist on each configuration(lst, 2nd, ... , n-th configuration).
Figure 20: Coupling value converted from the demon energy

BIBLIOGRAP IT'll"

71

in Fig.19.

Figure 21: Coupling value of (311 from the demon method inboth regions of -10 < Ed < 10 and -50 < Ed < 50.

Figure 22: Coupling value of (311 from the demon method on
the double smeared blocked configuration.

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Figure 2225

(1) Scaling study of pure gauge lattice QCD by Monte Carlo renornlalization group 111ethod(~~~) K.Ak elni,M.Fujisaki,M.Okuda,Y.Tago,Ph. deForcrand,T.Hashimoto,

S.Hioki, O.Miyalllura, A.Nakamura and LO.Stalllatescu(~;tBlt)

Physical Review Letters 71 3063 (1993)(2 ) Deter111ination of effective actions for SU(3) lattice gauge theory(~;t Bli )Modern Phys ics Letters A (1995) ~:1~ lf&iR:;E

1 Bound for the differential cross-section of elastic hadron scattering(;!~~~) M.Kawasaki and T.Ikellloto

OmuJt~,~~ ) Modern Physics LeLLers A 5 2599 (1990)

2 Break structure of forward peaks on elastic pp and anti-p-p scattering
at high energies(~~~) M.Kawasaki and M.Yonezawa(BrfUJt~,t) Physical Review LeLteres 67 1197 (1991)

3 Generation of shift register random nUl11bers on distributed 111e1110ry
111ultiprocessors(~~~) J.Makino and O.Miyamura(BrfUJt~,t) Computer Physics Cornmunications 70 495 (1992)

4 Does overrelaxed updating In lattice QeD 1111prOVe critical slowing

down?
(~~~) K.Akelni,M.Fujisaki,M.Okuda,Y.Tago,Ph. deForcrand,T.IIashimoto,

S.IIioki, O .lVliyamura, A .Nakan1ura and I.O.Stamatescu(6rfU:Y:J,-&) Physics LeLLers B328 407 (1994)