Mass gap and finite-size effects in finite temperature SU (2) lattice gauge theory

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Mass gap and finite-size effects

in finite temperature SU (2) lattice gauge theory

J . Engels and V.K. Mitrjushkin L

Fakultiit J~r Physik, Universitiit Bielefeld, W-4800 Bielefeld, FRG Received 14 January 1992; revised manuscript received 1 March 1992

This letter is devoted to the investigation of the point-point Polyakov loop correlators in SU (2) lattice gauge theory on 4N 3 lattices with Ns=8, 12, 18 and 26. We use an analytic expression for point-point correlators provided by the transfer matrix formalism to study the temperature dependence of the mass gap/Lm.g, and the corresponding matrix element v near the critical point in a finite volume. The finite-size scaling analysis of the values #m.d.(r; Ns) obtained gives the possibility to extract the critical value tic, the critical exponent v and the surface tension as.t:

1. Introduction

As is well k n o w n in S U ( N ) gauge theories the con- f i n e m e n t - d e e o n f i n e m e n t phase t r a n s i t i o n c o n n e c t e d with the s p o n t a n e o u s b r e a k i n g o f the global ZN sym- m e t r y occurs at s o m e n o n z e r o t e m p e r a t u r e 0c [ 1,2 ].

The n u m e r i c a l study o f t r a n s i t i o n p h e n o m e n a is a r a t h e r delicate p r o b l e m because o f strong finite vol- u m e effects n e a r the t r a n s i t i o n point. O n a finite lattice the s y m m e t r y can never be s p o n t a n e o u s l y b r o - ken, a n d the t u n n e l i n g between different m i n i m a o f the effective p o t e n t i a l at 0 > 0c lifts the d e g e n e r a t i o n o f the v a c u u m in the d e c o n f i n e m e n t phase m a k i n g the results sensible to the size o f the lattice n e a r 0~.

I n s t e a d o f two degenerate v a c u a 10+ > a n d 10_ ) (for the SU ( 2 ) g r o u p ) one rather has on finite lattices two n o n d e g e n e r a t e " m i x e d " states 10s) a n d 10a) a b o v e 0c. In this letter we study the influence o f the tunneling p h e n o m e n a on the p o i n t - p o i n t P o l y a k o v loop correlators in the S U ( 2 ) lattice gauge theory. T h e s t a n d a r d W i l s o n a c t i o n is

Sw(U~)=flY', (1-½Tr Up), (1)

t2

Work supported by the Deutsche Forschungsgemeinschaft under research grant En 164/2-3.

Permanent adress: Joint Institute for Nuclear Research, Dubna, Russia, USSR.

where r = 4/g2are a n d Up e SU ( 2 ) are p l a q u e t t e vari- ables. O n a lattice N c N ~ the t e m p e r a t u r e is d e f i n e d as 0 = 1 / a N , where a = a (fl) is the lattice spacing. In w h a t follows we shall put the spacing equal to unity, measuring, therefore, all d i s t a n c e s a n d energies in units o f a a n d a - ~ , respectively. T h e average o f the P o l y a k o v l o o p ~ ( x ) is d e f i n e d in a s t a n d a r d way:

< ~ > - Z - ' f I~ dU~ ~(x) exp[-Sw(U~)], (2)

links

where

~ ( x ) - ½Tr

U4 ( x , ~" ) , (3) a n d periodic b o u n d a r y conditions in all directions are assumed.

In the infinite v o l u m e l i m i t the average value o f the P o l y a k o v loop < ~ > is zero below the critical p o i n t a n d differs f r o m zero at 0 > Oc(fl> tic) because o f s p o n t a n e o u s breaking o f the global Z2 symmetry.

T h e p o i n t - p o i n t c o r r e l a t o r F ( x ) o f the P o l y a k o v loops defines the color averaged p o t e n t i a l for a static qCl p a i r s e p a r a t e d by a distance Ix l

(' )

F(x)=<~(x)~(O)>~exp

-~v~q(x;O/ . (4)

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The temperature dependence of this potential is of considerable interest for a quantitative understand- ing ofcolour screening and its possible relation to the deconfinement transition. One expects that below the phase transition point 0~ (or tic) the potential is line- arly rising at large distances

(1/O)Vq~(r;

0 ) = /~m.g.(0)r+... where /tm.g. is the so-called mass gap, while above the phase transition point/Zm.g. = 0 and Debye screening of static color charges presumably takes place.

For the investigation of the long distance proper- ties of the plasma phase the correct functional form of the potential Vq~(x; 0) is of principal importance, especially in view of speculations about the possible breakdown of simple Debye screened Coulomb behaviour [3 ]. The usual choice for the parametrization ofVq~ (x; 0) above the phase transition point is

Vqq(r;

O)=c(O)(~d

exp(--/lr)

, )

+ ( N ~ - r ) dexp[

-It(N,-r) ] , ⁽⁵⁾

with an arbitrary power d a n d / t defines the Debye screening. Because of the tunneling in finite volumes /tm.g. is not zero at 0> 0c, which influences strongly the functional form of

Vqq(r)

in the critical region.

The parametrization as in eq. (5) is not correct in a finite volume at temperatures near the critical one 0~0c.

Monte Carlo studies [4-8] of the heavy quark potential so far gave no arguments in favour of perturbative behaviour at large distances. It is possible that nonperturbative modes play an important role in the large distance behaviour of the chromoplasma even at high temperatures [ 9,10 ]. Moreover, the very validity of the perturbative approach is now under question. At large enough distances r this expansion is divergent for all (nonzero) couplings g~ and therefore not even the lowest-order calculations are reliable [ 11 ]. The determination of the heavy quark potential appears thus to be a nonperturbative prob- lem even at very high temperatures (see, however, refs. [12,13]).

As a first step in the investigation of the heavy quark potential we want to deduce the value of the mass gap, Pm.g. (or for fl < tic the inverse of the corre-

lation length ~_ ) from the full correlator

F(x).

To that end we use data from 4N 3 lattices, i.e. with cubic geometry, which were taken in the course of a general finite size scaling analysis [ 14 ] with the in- tention to extract the correlation length and the potential Vq~ (r). We shall show that full correlator data may be used just as well as zero momentum correlator data to evaluate the mass gap.

The dependence of the mass gap/lm.~ on fl was already studied for SU (2) on lattices with cylindrical geometry

Nt.NZ.Nz

(Nz>>Ns) [15-17] using zero momentum correlators. The cylindrical geometry has the obvious advantage that the highest excitation level of the transfer matrix may be projected out more re- liably than in the cubic case for the same Ns. We are aware of this fact and will therefore, when we estimate numbers as the critical coupling and the surface tension, leave the results from our smallest (Ns = 8) lattice out of consideration and additionally check the other results. As we shall see in the following the mass gap and matrix element values which we obtain from our cubic lattices are nevertheless well in accord with the expectations from finite size scaling and universality.

2. Correlators of Polyakov loops in finite volumes We define the zero momentum operators ) (z) as follows:

~(z)= ~

1

~(x).

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In the transfer matrix formalism the zero momentum correlators P ( z ) are given by

P(z)=<~(z).~(O))

= Z - ~ ( n l ~ ( 0 ) I m ) 2

n , m

xexp(--flmZ) e x p [ - - f l , ( N s - Z ) ]

= vZ{exp( - Zpm.g.) + e x p [ -- (Ns - Z)P-m.s. ] }

+v2{exp(--z/al)+exp[--(Ns--z)#,]} ...

⁽⁷⁾

where v 2 -- Z - ~ < 0s I ~ (0) I 0a > 2 and v~, ~q .... corre- spond to higher exitations of the spectrum of the transfer matrix.

It is worthwhile to note here that in a finite volume

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at fl-~ tic rotational invariance is broken:

F(x, y, z) - F(x) ¢ F (

Ix1 ) and only invariance with respect to p e r m u t a t i o n s x, y, z survives. Because in our data Po- lyakov loops are separated by distance r along one o f the three axes we shall use the notation

F(r) =-F(x).

With the definition o f the zero m o m e n t u m opera- tor ) ( z ) as in eq. (6) the connection between the zero m o m e n t u m correlator P ( z ) and the p o i n t - p o i n t correlator

F(x±; z)

on a finite lattice is

P(z)-

( ~ ( z ) ~ ( 0 ) ) = }2

r(x±; z).

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x ± s

The Fourier-transforms o f the correlators

F(x)

and

P(z)

are

F ( p ) = Y~ e x p ( i x - p )

F ( x ) ,

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x

F(Pz) = ~ e x p ( i z p . ) / ~ ( z )

2

- Y~ exp(izpz)

F ( x ) ,

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x

where

p , = ~ k , ,

2rr k , = 0 , + 1 ... + ( ½ N ~ - 1), ½N~ (11) for even values o f N~. Therefore the correlators F and P are connected by the evident relation

P(pz)=-F(p±

= 0 , p ~ ) . (12) Discarding higher exitations in eq. (7) (# = #m.g ) one obtains for the Fourier-transform F ( p . )

F(Pz)

= v2'2¢t [ 1 - e x p ( - N ~ / t ) ] G(p~; N~,/.t), (13) where

G(pz;

N~, ~) - 1 - e x p ( - 2 # ) 2/~ { [ 1 - e x p ( - / ~ ) ] 2 + e x p ( - - ~ ) . 4 sin2 (½p_~) } -~ (14) To derive the correlator

F(p)

we use the following substitution in eq. (14):

3

4 sin 2 (½p=) --,D (p) --- ~ 4 sin 2 (½Pi)

, ( 15 )

i = l

where

D(p)

is just the lattice laplacian in 3D m o - m e n t u m space. One cannot exclude in principle that in the d e n o m i n a t o r o f G (p; N , / t ) , say, cross-terms ~

sin2(lpi ) sin2(½pi) m a y appear. So, our ansatz eq.

(15) is based on the a s s u m p t i o n that these cross- t e r m s play a negligible (if any) role. As a result we find

F(p)=v2.21L[1-exp(-Nslz)]G(p;Ns,l ~) ,

(16) where

1 - e x p ( - 2 # ) ( [ l _ e x p ( _ / z ) ]2

G(p; Ns, It) =- 2l t

- - l

+ e x p ( - / t ) i=, ~ 4 s i n 2 ( l p / ) ) (17) Performing the inverse Fourier-transform one ar- rives at the following expression for the correlator F ( r ) :

F(r)

= v2.2/~ [ 1 - e x p ( - N~/t) ] N3 1 27r N~

X Z e x p ( - i r p z ) G = ~ - k ; s , # ) . (18) In the limit o f small/~ (/~ << N ~- l ) the correlator F ( r ) will tend to a constant value. At/~Ns >> 1 the correlator F ( r ) is equivalent to the superposition o f two Yu- kawa-type potentials:

Fvuk (r) ⁼ ^{V 2}~ [ 1 - - e x p ( - - # N ~ ) ]

x(lexp(--I.tr)+N@_reXp[--I~(Ns--r)l).

(19) Fig. 1 shows the behaviour o f the ratio

R ( r ) - F(r)/F(r)

Yuk as a function o f r at different values o f

#Ns. We observe that R (r) -~ 1 only for/.t >> N ~- l and, therefore, the correlator

F(r)

can be represented in the f o r m o f a superposition o f two Yukawa-type potentials only far f r o m the phase transition point. At f l ~ tic, w h e r e / ~ ~ N ~ -l finite-volume corrections are too strong and this is not possible.

Defining the average o f the " s q u a r e d magnetization".

2

z r(x)= z , (20)

we get for ( .~2 ) f r o m eq. ( 18 )

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3.0

2.0

1.0

0.5

' ' ' 1 . . . . I ' ' ' ' 1 ' ' ' ' 1 ' ' ' '

V (r)/V (r) ruk.

. . . . ~ , ~ : ~ : ~ - , ~ . ~ . . . . ..."

l / u = ~ = ~ ... 4 - - - -

2 . . . 8 - - . - - p

r l { I t l r l , r l t i i B , , I . . . .

5 1 0 15 20 25

Fig. 1. The behaviour of the ratio F(r)/F(r)vuk as a function of rat different values of~ on an Ns=26 lattice.

N 3 ( ~ 2 >

=V2[I

- - e x p ( --/.lm.g. 2~s) ]

× 1 +exp(--/~m.g.) (21)

1 -- exp ( - ¢tm.g. ) "

It is worthwhile to note here the nontrivial role o f the factor with the explicit size dependence

1 - exp ( - #m.g.Ns) in eq. (21 ). Far below the transi-

tion point it tends to unity with increasing lattice size Ns. But at ~ > tic, where #m.g.N~ << 1 it gives after con- traction with #m.~. from the d e n o m i n a t o r an addi- tional power o f N~.

In the t h e r m o d y n a m i c limit below the phase transition point, fl< tic, the correlator F ( x ) decays expo- nentially at large distances,

( ( ~ ) N~--,oo, r > > l (22) F ( x ) ~ e x p ~_ ,

which entails

V ( x . ; z) < o e . (23)

x ±

F r o m eq. (23) we conclude that for a finite lattice size all matrix elements are independent o f Ns in the large volume limit so that

2 0

v ~ N ~ , N s ~ o o . (24)

Above the transition point, fl> tic, the large distance

behaviour o f the correlator F ( x ) in the thermody- namic limit is

F ( x ) - ( ~ 2 ) ~ e x p ( - - - N~-,oo, r>> 1 ,

and at large but finite N~

Z F ( x ± ; z ) ~ U ~ .

x ±

+

(25)

(26) The matrix element has therefore the following dependence on the lattice size:

2 2

v ~ N ~ , N ~ o o . (27)

3 . D a t a a n a l y s i s

The data we analyse were produced on lattices with N t = 4 and N s = 8 , 12, 18, 26. Part o f the data were already evaluated and described in ref. [14]. The correlators F ( r ) were measured every 10th update in these runs, so that between 10 000 and, close to 0c, upto 45 000 measurements per r-value were available.

To extract values o f the mass gap/~m.g. (fl; Ns) and the matrix element v(fl; Ns) we made simultaneous fits o f our data for the correlators F(r) and the squared magnetization ( ~ 2 ) using eq. ( 18 ) and eq.

(21 ). The errors Of#m.g. and the matrix element v were determined from the X 2 o f the fits such as to include possible deviations with a probability o f 75%. Sys- tematic errors due to the neglect o f the higher levels have not been taken into account. Exploratory fits in- cluding more levels in formula (18) show that the mass gap values resulting from the fit close to and above the critical point are changed to slightly lower values. A corresponding effect is found if we discard in the fitting procedure the r = 1 or the r = 1 and 2 correlator data. Then/Zm.g.-values near to tic are low- ered by about 10% (12%) for the bigger lattices and by 40% (60%) for the Ns = 8 lattice. In general, however, we find from our full correlator formula, eq.

( 18 ), mass gap values which are o f comparable size to those deduced from zero m o m e n t u m correlators (see, e.g. ref. [ 17] ).

The dependence

of ]Am.g.(fl;

Ns) on fl for different Ns is shown in fig. 2a. F o r f l > 2.27/~m.g. it shows strong

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1.O

0.5

0.0

2.5

2.0

t..5

~ . . 0

I I , , , , I , , , , I , J , ,

t~ ~m. g.

I ' ' ' ' I ' '

(a)

0 26ax4 [] 18ax4

~ k X x x 12ax4

,~ 8ax4

~zr~

2.15 2.20 2.25 2.30 /~

' ' (b)

~

. ~ . ' ~ ' ~ N~m" g "

<) 26aX4 . ~ =

X 12ax4 ~

a 8ax4 <~

0 . 5 I I I

2.2B 2.29 2.30 2.31

Fig. 2. (a) The dependence of//m.g, on flfor different hrs. (b) The dependence o f Nd&..g on fl for different Ns near the transition point tic. The straight lines are linear least square fits to the points in the neighbourhood of tic.

finite-volume effects as expected. Also the mass gap is tending to zero with increasing N~, as it should be near to and above the critical point.

To scrutinize the temperature and finite-volume dependence o f the mass gap we used the finite-size scaling (FSS) technique [ 18,19 ]. Below the critical point the mass gap can be identified with the inverse correlation length (see, e.g. ref. [ 19 ] ):

~-(fl; Ns)=-ltm.~.(fl;Ns), fl<~flc.

(28) According to FSS theory any observable O with critical b e h a v i o u r is supposed to have the following form;

O=NC/"fo(xN]/"), Ns-~OO,

⁽²⁹⁾

for fixed small x-= ( f l - tic)/tic a n d p is the critical ex- p o n e n t o f the observable O. Since the critical exponent o f the correlation length is again v we expect for the mass gap

-~sfu(xN]/'), x~O

(30) /tm.g.(fl; N)s =

and at the critical point

/lm.g. (tic, Ns) ~ N s ' , (31)

d-flUm.g. (tic ; d

Ns) ~ N z '+t/v.

(32)

Below the transition point the susceptibility X(fl;

N~) is defined as follows:

Z(fl;Ns)=N3 ( ~2), fl<~flc.

(33) The critical exponent o f the susceptibility is 7 and eq.

(29) then leads to

( ~2)=Nu3+~/" f~,(xN~/"), x~O.

(34) C o m b i n i n g eqs. (21), (30) and (34) we finally obtain

v2(fl)=Ns'+~/Vfv(xNls/v), x~O.

(35) The following should be m e n t i o n e d here. Eq. (28) and eq. (33) are valid only at//~<~¢ (x~<0). Never- theless, because o f the analytic dependence on fl (or x ) on

afinite

lattice we expect that the corresponding FSS equations (30), (34) and (35 ) will be valid also above the critical point, i.e., at x>~ 0. This is really the case as can be seen below.

We used eq. (31 ) to estimate the transition point tic. T h o u g h observables, which are directly connected to the measured data (i.e. not through a fit) like the cumulant g~ (see ref. [ 14 ] ) or ( ~2 ) are better suited to that purpose, this gives us a check on the consistency o f our results for the mass gap and a compari- son to the zero m o m e n t u m results. In fig. 2b we show the dependence o f Ns~m.g. o n fl for different values o f N~ in a narrow region near tic. The straight lines cor-

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respond to linear least square fits of Ns///m.g" n e a r the transition point. We have chosen the intervals where these fits were done much smaller (by a factor three or more) than in ref. [17]. This was necessary because, as can be seen in fig. 2b, with increasing N, the curvature of Ns#m.g. increases and it was possible since we have enough points close to the critical point.

The intersection point fl(Ns, N's ) of the two lines corresponding to the lattice sizes Ns and N's gives an estimate of the critical point. At large enough values of N,, N'~ the intersection point must tend to tic. We obtain for fl(Ns, N's )

fl(12, 8 ) = 2 . 3 0 1 3 + 0 . 0 0 1 2 , fl( 18, 8) = 2.3005_ 0.0006, //(18, 12)=2.3001 ___0.0009, //(26, 8)=2.2999_+0.0006, //(26, 12)=2.2997_+0.0006,

//(26, 18)=2.2994-+0.0009. (36)

With increasing, Ns, N'~ the intersection point fl(Ns, N~ ) approaches the critical point from above, and //(26, 18) in eq. (36) is in agreement with the value obtained in ref. [ 14 ],//c = 2.2985 _ 0.0006. The slight discrepancy with the value//c obtained in ref. [ 17 ] can be naturally explained by the fact that the lattice volumes used in that paper are not large enough.

Given the critical point//¢ one can determine the critical exponent v using eq. (32). Fig. 3 shows the dependence of In(-dNdlm.g./d//) on InNs at fl=fl¢.

The straight line is a linear least square fit of our data with the inverse slope

u=0.62-+ 0.08. (37)

The error bars in eq. (37) are comparatively large, but the value of v obtained agrees well with u~si~g-~

0.63 for the 3D Ising model (indicated by the dashed line in fig. 3).

To demonstrate the consistency of our data with the universality hypothesis [ 20 ], which predicts equal critical exponents for SU (2) and the 3D Ising model, we show in fig. 4a Ns#m g. and in fig. 4b v2N~ -~+y/" as a function ofy=xN]/", where v and ywere taken from the 3D Ising model. We see that within the error bars all data points in the vicinity of the phase transition lie on the same universal curve, i.e. we have scaling.

I , , , ~ I , , , , I , , r ~ I

2 . 0 2 . 5 3.0

I n N

Fig. 3. The dependence o f l n ( -dNsflm.g.(fl; Ns)/dfl) on InNs at

fl=flc. The line is a linear least square fit to the data, the dashed one corresponds to a fit with v=0.63 as input.

Using our values for v from eq. (37) and flc=fl(26, 18 ) from eq. (36) leads to plots which cannot be dis- tinguished by eye from fig. 4.

The tunneling in finite volumes can be interpreted as the creation of interfaces between domains with different signs o f " s p i n s " ~ (x) separated by domain walls. The associated interface energy density (surface tension) as.,. is defined as

.~,w - ~ ~ w - ~

Oa~.t.(O)= N x ~ - - - ' N ~ (38)

where the free energies ~" and ~w are defined as

~ ( O ) = - O l n Z , ~tw(O)=-OlnZtw, (39) and Ztw is the partition function on the lattice with twisted boundary conditions in the (t, z)-plane [21,22 ]. The surface tension is connected to the mass gap #m.g. through the following relation:

~m.g. ~ e x p [ -as.t.(O)N~] , (40)

where the preexponential factor depends on temperature and can comprise some power dependence

~ N ~ . One may speculate, referring to the analogy with the 3D Ising model, that b = 0 [23,24]. In fig. 5 we show the dependence of - In #m.g. ( 0 ) / N z on 1 / N 2 at fl= 2.35 for Ns = 12, 18, 26. To make sure that

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10

_(a) '~T t'~

^'

i N/Jm. g - I i , '

p=0.63

O 25aX4

X 12ax4

~. 8ax4

~ ; , q D

, ~ , I I I , , ,

0 1

x N 1 / v

I I '

o

v 2N ~ - z / v

× 13 x

x

T=I .24 v = O . 63

- 5 - 1

(b)

0 263x4 0 183X4 X 12ax4

A 8aX4

0

}

& O 0 X

, ~ r I I I ~ K r

-1 0 1

×N 1/u

Fig. 4. (a) The dependence of Ns#m.s.(fl; N,) on xN2/", x-~ (fl- flc) /fl~. (b) The dependence of vZN ~ -y/" on x N ~/~, with the 3D Ising model values for 7 and v as input.

our values for ]Am.g" are as realistic as possible, we have m a d e several fits to the c o r r e l a t o r data: i n c l u d i n g all distances, o m i t t i n g distance r = 1 a n d r = 1 a n d 2. T h e p o i n t s shown in fig. 5 c o r r e s p o n d to the s e c o n d o f these fits. There is essentially no difference to the third fit a n d only a slight one to the first. T h e b r o k e n line in the figure is a l i n e a r least square fit to o u r d a t a --OLs.t.N s ), where with the ansatz IZm.g.=Ca e x p ( 2

. 0 4 , , , , I , , , , i , , , , - I

• 03

.02

.01

0 0

- l n (~.g.) /N 2

/ / / 0 . I / / i /

/ /

/ / / I /

, I , , , , i , , , I

0 . 0 0 5 0 . 0 1

1 / N 2 Fig. 5. The dependence of-ln#m.g.(fl; N~)/N~ on I/N~ at fl= 2.35. The broken line is a linear least square fit to the data for our three largest lattice volumes.

as.~.(p= 2.35) = 0 . 0 0 2 5 ( 2 ) a n d c , = 0 . 0 7 5 ( 5 ) . I n s i d e the error bars the surface tension o b t a i n e d f r o m re- spective linear fits to the o t h e r two sets o f d a t a is c o m p a t i b l e with this result. A two level fit to the cor- r e l a t o r d a t a also leads to the s a m e ot~.t.-value.

4 . C o n c l u s i o n s

In this p a p e r we s t u d i e d the p o i n t - p o i n t P o l y a k o v l o o p correlators F ( r ) in SU ( 2 ) lattice gauge t h e o r y on 4N~ lattices with N s = 8 , 12, 18 a n d 26. In partic- ular we were interested in the role o f finite v o l u m e effects n e a r the t r a n s i t i o n p o i n t tic.

O u r analysis is b a s e d on an analytic expression for the p o i n t - p o i n t c o r r e l a t o r p r o v i d e d by the transfer m a t r i x f o r m a l i s m a n d the ansatz in eq. ( 15 ). The results o f o u r analysis testify in f a v o u r o f its validity.

We s t u d i e d the fl ( t e m p e r a t u r e ) d e p e n d e n c e o f the mass gap #m.g. a n d the c o r r e s p o n d i n g m a t r i x e l e m e n t v n e a r the critical p o i n t in a finite volume. T h e finite- size scaling analysis o f the values o b t a i n e d f r o m the fit allows to d e t e r m i n e the critical value tic, the critical e x p o n e n t v, as well as the surface t e n s i o n as.t: The value o f the critical p o i n t fl which we o b t a i n e d is in

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a g r e e m e n t w i t h t h e results f r o m t h e a n a l y s i s o f t h e c u m u l a n t [ 14 ] a n d o f t h e z e r o m o m e n t u m c o r r e l a - tors [ 17 ]. T h e v a l u e o f t h e critical e x p o n e n t v is c o n - sistent w i t h t h a t o f t h e 3 D Ising m o d e l .

In t h i s p a p e r we d i d n o t c o n s i d e r h i g h e r - o r d e r c o n - t r i b u t i o n s to t h e c o r r e l a t o r s

F(r)

a n d t h e " s q u a r e d m a g n e t i z a t i o n " ( ~ 2 ) . T h e y c a n p r o d u c e s o m e bias o f t h e v a l u e s o f t h e m a s s gap b u t n o t f o r t h e v a l u e o f t h e t r a n s i t i o n p o i n t a n d t h e c r i t i c a l e x p o n e n t s . T h e s t u d y o f t h e t e m p e r a t u r e a n d f i n i t e v o l u m e d e p e n - d e n c e o f t h e s e h i g h e r - o r d e r c o n t r i b u t i o n s is a n o n - t r i v i a l b u t v e r y i n t e r e s t i n g p r o b l e m e s p e c i a l l y i m p o r - t a n t for t h e i n v e s t i g a t i o n o f t h e h e a v y q u a r k p o t e n t i a l . T h i s will b e a m a i n t o p i c o f a f o r t h c o m i n g p a p e r .

Acknowledgement

W e w o u l d like to t h a n k B. Berg, L. K~irkk/iinen a n d T. N e u h a u s f o r useful discussions. W e a r e i n d e p t e d to J. F i n g b e r g a n d M. W e b e r f o r t h e i r h e l p in t h e m e a s u r e m e n t o f t h e c o r r e l a t i o n data. A l s o o n e o f us ( V . K . M . ) w o u l d like to e x p r e s s his g r a t i t u d e to B i e - l e f e l d U n i v e r s i t y f o r h o s p i t a l i t y .

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Mass gap and finite-size effects in finite temperature SU (2) lattice gauge theory