Monte Carlo study of SU(3) gauge theory with next-to-nearest neighbour interactions on a 124 lattice

Volume 143B, number 4, 5, 6

PHYSICS LETTERS

16 August 1984

M O N T E C A R L O S T U D Y O F SU(3) G A U G E T H E O R Y W I T H N E X T - T O - N E A R E S T N E I G H B O U R I N T E R A C T I O N S O N A 12'; L A T T I C E Ph. de F O R C R A N D Ecole Polytechnique, CPHT, Palaiseau, France

and C. R O I E S N E L CERN. Geneva, Switzerland

Received 2 April 1984

We report results of a Monte Carlo simulation of SU(3) pure gauge theory on a 124 lattice with action involving planar 1 x 1 and 2x 1 Wilson loops. We study the string tension and glueball spectrum and compare with our previous results on a 64 lattice.

In a previous p a p e r [1] we carried out a n u m e r i cal simulation of SU(3) lattice gauge theory on a 64 lattice with the tree-level i m p r o v e d (TI) action [21: S=5~(1-~

R e T r W~)

/3 i~2Y'~ (1 - ½ Re T r WEre),

(1)

where W[] a n d W ~ are the W i l s o n loops associa t e d to the 1 x 1 a n d 2 x 1 p l a q u e t t e s respectively. W e d i d observe a satisfactory scaling b e h a v i o u r for the Creutz ratios X e x t r a c t e d from our largest (3 × 3) loops over a r a t h e r b r o a d range of /3: 3.6 ~
with action (1). T h e r e f o r e it b e c a m e necessary to r e p e a t our m e a s u r e m e n t s of the string tension a n d the glueball s p e c t r u m on a larger lattice in o r d e r to m e a s u r e larger l o o p s in lattice units, a n d to o b t a i n reliable results at larger fl, further from the p e a k in the specific heat located, in our case, a r o u n d / 3 = 3.9. In this letter we r e p o r t on a M o n t e C a r l o simulation of SU(3) lattice gauge theory d o n e on a 124 lattice with action (1). W e selected four/3 values in the scaling w i n d o w observed on the 64 lattice, n a m e l y fl = 3.6, 3.9, 4.2 a n d 4.5 a n d p e r f o r m e d respectively runs of 1500, 2100, 1900 a n d 1200 iterations. C o n f i g u r a t i o n s were generated using the p s e u d o - h e a t b a t h m e t h o d of C a b b i b o a n d M a r i n a r i [5], with three SU(2) subgroups, for better decorrelation. To speed up convergence t o w a r d s equilibrium, we started from thermalized 64 configurations that were c o p i e d 16 times into the larger lattice. Details a b o u t the p r o g r a m are given in a n o t h e r p u b l i c a t i o n [6]. In o r d e r to extract the string tension we measured the v a c u u m expectation values of all rectangular W i l s o n l o o p s up to length 6 x 6. W e also m e a s u r e d all correlations b e t w e e n these p l a n a r

0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

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loops up to distance 2 to get masses of glueballs with positive parity at a cost of only a few percent in computer time. After a suitable number of s w e e p s ( 3 0 0 o r so) t o r e a c h e q u i l i b r i u m o n t h e l a r g e r l a t t i c e , w e a n a l y z e d 120, 240, 330 a n d 330 c o n f i g u r a t i o n s r e s p e c t i v e l y a t fl = 3.6, 3.9, 4.2 a n d 4.5. M e a s u r e m e n t s w e r e g r o u p e d i n t o b i n s o f 50 i t e r a t i o n s ( o n a v e r a g e ) f o r e s t i m a t i n g all s t a t i s t i c a l e r r o r s . T o give a n i d e a o f t h e e f f i c i e n c y o f o u r program, one Monte Carlo sweep through the w h o l e l a t t i c e t o o k 12 s a n d o n e c o m p l e t e m e a s u r e m e n t o n o n e c o n f i g u r a t i o n t o o k 26 s. We first discuss results on the string tension. V a c u u m e x p e c t a t i o n v a l u e s , n o r m a l i z e d to u n i t y , o f all r e c t a n g u l a r W i l s o n l o o p s t h a t w e c o u l d m e a s u r e u p t o size 6 × 6 a r e l i s t e d i n t a b l e 1. A l l v a l u e s u p t o size 3 × 3 a r e i n a g r e e m e n t , w i t h i n one standard deviation, with those obtained on the 6 4 l a t t i c e f o r fl = 3.6, 3.9 a n d 4.2 [1]. H o w e v e r , t h e r e is a s i g n i f i c a n t d i s c r e p a n c y a t fl = 4.5 ( 2 0 f o r W ( 1 , 1) a n d 3 a f o r I4,'(3, 3)). S u c h a n e f f e c t is t o be expected, given our previous measurements of

16 August 1984

t h e c r i t i c a l t e m p e r a t u r e f o r N t = 6 [1]. I n d e e d we a l s o m e a s u r e d o n t h e 6 4 l a t t i c e tic = 4.35 + 0.1. In table 2 are listed the derived values for the Creutz ratios

X(I,j)=_Ln

W(I,J)×W(I-1,J-1) (2) W ( I , J - 1) × W ( I - 1, J ) "

Convergence of x(I, J) for loops of increasing a r e a t o w a r d s a l i m i t i n g v a l u e is n o t c l e a r ( e x c e p t at fixed I or J) so we can only extract an upper b o u n d o n t h e r a t i o v ~ / A L w h e r e K is t h e s t r i n g tension. From a weighted average of the X(I, J) w i t h I , J >~ 3 a t fl = 3.6, 3.9 a n d I, J >~ 4 a t fl = 4.2, 4.5 w e get r e s p e c t i v e l y : 25.1+0.8,

25.5+0.5,

21.9+1.3

(3) These values show a significant violation of a s y m p t o t i c s c a l i n g b e t w e e n fl = 3.9 a n d fl = 4.2. H o w e v e r it h a s b e e n e m p h a s i z e d r e c e n t l y [7,8] that the phenomenological procedure of sub-

Table 1 Measured values ( x l 0 5 ) of Wilson loops W(1, J) at fl = 3.6, 3.9, 4.2 and 4.5.

W(I, J) W(1, 1) W(2, 1) W(2, 2) W(3, 1) W(3, 2) W(3, 3) 14z(4, 1) W(4, 2) 14,'(4, 3) W(4, 4) W(5, 1) W(5, 2) w(5, 3) w(5, 4) w(5, 5) w(6, 1) W(6, 2) W(6, 3) W(6, 4) W(6, 5) W(6, 6)

454

B 3.6

3.9

4.2

4.5

46516 + 11 20864 4-12 4163 + 7 9387 4- 9 855 -+ 5 87-+ 4 4231 + 5 178--+ 3 10+ 4

54089 -+ 19 29546 + 29 9670 ___27 16365 4- 26 3391 -+ 15 810-+ 8 9096 + 21 1212-+ 7 204_+ 5 30_+ 6 5054 _+17 430 -+ 5 44-+ 4 11_ 5

59918_+11 37257 + 18 16614_+26 23628 _+21 8051 4- 24 3243 _+20 15052_+21 3989 4-19 1364 -+ 13 489 -+ 7 9565 _+15 1962 + 10 570 -+ 6 180-+ 4 58-+ 4 6093 + 12 977 -+ 6 240 4- 4 66-+ 4 18+ 3

63565 + 7 42222 _ 12 21587_+20 28613 _+14 119614-18 5840 + 18 19480 + 13 6763 -+ 15 2973 _ 15 1388_+11 13271 _+12 3843 _+12 1528 _+10 666 _+ 7 302 _+ 6 9046 _+10 2190 _+10 793 4- 7 320-+ 5 134+ 4 59_+ 5

1915 + 6 39-+ 4

2811 + 11 151 -+ 5 9+ 3

22.3+1.2.

16 August 1984

PHYSICS LETTERS

Volume 143B, number 4, 5, 6

Table 2 Measured values of Creutz ratios X(I, J) at fl = 3.6,3.9,4.2and4.5.

X(I,J)

fl 3.6

3.9

4.2

4.5

X(2, 2) X(3, 2) X(3, 3) X(4, 2) X(4, 3) X(4, 4) X(5, 2) X(5, 3) X(5, 4) x(5, 5) X(6, 2) X(6, 3) X(6, 4) X(6, 5)

0.810 + 0.002 0.784 + 0.006 0.702 _ 0.047 0.772 + 0.018 0.594+0.403

0.512 + 0.003 0.457 + 0.006 0.384 ± 0.014 0.442 ± 0.008 0.350+0.027

0.332 + 0.002 0.269 +_0.003 0.185 4- 0.007 0.251 ± 0.006 0.164 + 0.013 0.160 5:0.025 0.258 ± 0.007 0.163 ± 0.014 0.127 5:0.030 0.133 _+0.083 0.246 -+_0.008 0.168 + 0.021 0.138 +__0.067

0.262 ± 0.001 0.201 ± 0.002 0.126 ± 0.004 0.186 _ 0.003 0.105 ± 0.007 0.087 _+0.013 0.181 ± 0.004 0.100 ± 0.009 0.069 + 0.016 0.057 _+0.030 0.179 + 0.006 0.094 ± 0.012 0.077 ± 0.022 0.080 ± 0.041

-

-

0.726 ± 0.104 -

0.449 ± 0.014 0.498 ± 0.095 0.460 ± 0.035 -

-

t r a c t i n g the p e r t u r b a t i v e c o n t r i b u t i o n f r o m X ( I , J ) i m p r o v e s t h e c o n v e r g e n c e to K a 2. T o get a n app r o x i m a t e e s t i m a t e of the p e r t u r b a t i v e c o n t r i b u t i o n x P ( I , J ) w e c a n use the c o n t i n u u m results o f ref. [9]. T h e n if w e d e n o t e K ( I , J ) = X ( I , J ) xP(I, J), we do observe that the quantities K ( I , J ) , for p l a q u e t t e s of d i f f e r e n t sizes, a g r e e m u c h b e t t e r a m o n g t h e m s e l v e s a n d are c o n s i s t e n t w i t h a f i x e d string t e n s i o n at e a c h fl value. T h e c o r r e s p o n d i n g r a t i o s v ~ / A L at fl = 3.6, 3.9, 4.2 a n d 4.5 are s i g n i f i c a n t l y l o w e r t h a n in eq. (3), respectively:

g a t e d p l a q u e t t e s to try a n d d e t e r m i n e )t a n d # at fl = 4.2 a n d 4.5, w i t h the results

2 2 . 2 _ 0.2,

17.9___ 0.4.

1 / ~ / ( A L ) T I -~ 19.9 _+ 0.8

( f l = 4.2),

(8a)

(4)

x / K / ( A L ) T ~ = 17.9 + 1.8

(fl = 4.5),

(8b)

21.6 + 0.2,

18.2 4- 0.3,

X = 0.26 _+ 0.02,

t~=0.12+0.01

(fl--4.2), (7a)

X=0.25-t-0.01,

~t= 0.05 _+ 0.01

(fl=4.5). (7b)

T h e c o u l o m b i c t e r m t u r n s o u t to b e v e r y close to t h e v a l u e X - - ~ r / 1 2 [10], a l t h o u g h n o e f f o r t has b e e n m a d e to s e p a r a t e l o n g d i s t a n c e f r o m s h o r t d i s t a n c e effects. T h e l i n e a r t e r m yields

T h e r e m i g h t be a s i z e a b l e s y s t e m a t i c e r r o r in the n u m b e r s o f eq. (7) d u e to t h e a p p r o x i m a t i o n s i n v o l v e d in their d e r i v a t i o n . H o w e v e r a s y m p t o t i c s c a l i n g v i o l a t i o n b e t w e e n fl = 3.9 a n d 4.2 is still there. A n o t h e r c h e c k of t h e v a l u e o f the scale ( A L)TI c o n s i s t s in fitting the w h o l e lattice p o t e n t i a l

w h i c h c o r r e s p o n d s to AMO M = 350 M e V ( t a k i n g ~/K = 420 M e V a n d i g n o r i n g f e r m i o n loops). T h e lattice s p a c i n g of o u r b o x is t h e n a = 0.16 f m at f l = 4 . 2 a n d - - 0 . 1 1 f m at fl---4.5. O u r v a l u e at fl = 4.5 c a n b e c o m p a r e d to the o n e o b t a i n e d in ref. [11] o n a 163 x 32 lattice for fl >~ 6.0 w i t h the s t a n d a r d W i l s o n action, u s i n g t h e s a m e m e t h o d :

VL(R)+

( A L ) w = 9.6 × 10 . 3 ~ .

lim Ln[W(R,T)/W(R,T-1)] T~oo

(5)

O n e gets

w i t h a l i n e a r p l u s a C o u l o m b t e r m for e a c h fl: V L ( R ) = Vo - X / ( R / a )

+ ~(R/a).

(9)

(6)

W e h a v e e n o u g h a c c u r a t e d a t a p o i n t s for e l o n -

w = 5.8 _ 0.6,

(lO)

s e e m s in a g r e e m e n t

w i t h the t h e o r e t i c a l

(AL)Tt/(AL) which

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value predicted from one-loop perturbation

theory

(AL)TI/(aL)W= Comparison

5.29 + 0.01.

among

(11)

Creutz ratios would yield the

same result. a

standard

variational

masses

momentum

correlations,

from

method

[13-15]

correlations

between

to

get zero-

o p e r a t o r s . T h e set o f o p e r a t o r s selected

in the m i n i m i z a t i o n the rectangular

p r o c e s s w a s q u i t e n a t u r a l l y all

Wilson loops

6 × 6 that we had

W(I, J)

to measure

the string tension. We measured

u p t o size

in order to extract

because

to o n - d i a g -

they give a minimum

m a s s w h i c h is q u i t e s t a b l e w i t h i n o u r e r r o r s :

C,(t) = (O,(t)O,(O)) The corresponding

We now turn to the glueball spectrum. We used glueball

it is s u f f i c i e n t t o r e s t r i c t m i n i m i z a t i o n onal

[121:

16 August 1984

( O , ( 0 ) ) 2.

(12)

effective glue ba ll m a s s at dis-

t a n c e t is u s u a l l y d e f i n e d a s

M,(t)a Our table

[C,(t)/C~(t-

= -Ln

1)].

(13)

d a t a f o r t h e 0 ÷+ g l u e b a l l s t a t e a r e l i s t e d in 3. T h e

error bars

at dis ta nc e

2 are rather

large. A reasonably precise estimate of the glueball mass can be obtained from

both on-diagonal

a n d o f f - d i a g o n a l c o r r e l a t i o n s u p t o d i s t a n c e 2, b u t

Mga

= min {i)

- ½ Ln

[Ci(2)/C~(O)],

Table 3 0 ++ glueball masses M,(t)a from diagonal correlations at /3 = 3.6, 3.9., 4.2 and 4.5 between rectangular Wilson loops distance t = 1 (first entry) and 2 (second entry).

w(1, 1) 14/(2, 1) w(2, 2) w(3, 1)

3.6

3.9

4.2

4.5

2.13 ± 0.12 1.47 ± 0.63 2.13±0.11 1.62 ± 0.75 3.11 ± 0.30 2.45 ± 0.18

1.78 ± 0.11 1.59 _ 0.55 1.54+0.10 1.20 __+0.40 1.77 ± 0.11 1.30 ± 0.36 1.57 ± 0.11 1.14 ± 0.34 2.07_+0.12 1.06 ± 0.44 3.36 ± 0.27

2.15 ± 0.13 1.98+0.14 1.34 ± 0.54 1.92 ± 0.12 1.15 ± 0.33 2.09 ± 0.16 1.03 ± 0.48 1.83±0.15 0.78 -+ 0.34 2.46 -+0.17 0.82 ± 0.35 2.29 5:0.20 0.94 _+0.54 2.02 ± 0.17 0.93 ± 0.44 2.46 -+ 0.19 0.75 -+ 0.41 2.56 ± 0.25 1.10 _+0.67 2.55 ± 0.15 0.99±0.51 3.06 ± 0.27 0.74 _+0.46 3.10 _+0.42 2.88±0.19

2.66 ± 0.13 2.31 ±0.10

-

w(3, 2)

-

w(3, 3) w(4, 1) w(4, 2)

-

w(4, 3) w(5, 1) w(5, 2) w(5, 3)

-

W(6, 1) W(6, 2)

456

-

1.76 -+0.11 1.03 _+0.32 3.41 ± 0.44 2.07 ± 0.10 1.05 ± 0.41 2.59 ± 0.16 1.06 ± 0.62

2.23 ± 0.13 1.86 ± 0.57 2.26 _.+0.08 2.47 ± 0.68 2.01 ±0.10 1.81 ± 0.46 2.09 ± 0.06 1.35 ± 0.30 2.37 ± 0.08 2.12 ± 0.55 1.99 ± 0.09 1.97 ± 0.46 2.02 -+0.06 1.42 + 0.36 2.62 ± 0.06 2.22 _+0.08 2.37 _+0.06 2.92 _+0.12 2.58 ±0.12

(14)

W(I,

J), at

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Then, a s s u m i n g a s y m p t o t i c scaling, we get:

M(O++)/(AL)x,

= 54_+ 9, 53 + 6, 74 +_ 1 3 , 1 3 4 + 12,

(15)

respectively at 13 = 3~6, 3.9, 4.2 a n d 4.5. These n u m b e r s c o n f i r m our results on the 64 lattice [1]. A s y m p t o t i c scaling a p p e a r s to be satisfied a t / 3 = 3.6 a n d 3.9, b u t such a result should be i n t e r p r e t e d with caution, due to the vicinity of the p e a k in the specific heat. D e p a r t u r e from scaling at higher 13 is p r e s u m a b l y mostly due to spin-wave effects. W e w o u l d need better statistics at d i s t a n c e 2 to check scaling up to 13 = 4.2 with a r e a s o n a b l e accuracy. If we use eq. (11) the 0 ++ glueball mass in physical units is M ( 0 + + ) = 1200 MeV. W e can also get a r o u g h e s t i m a t e of the 0 ++ glueball size f r o m the average ( I ) of the eigenvector of the c o r r e l a t i o n m a t r i x built from the ( I x 1) W i l s o n loops, b o t h at d i s t a n c e 0 a n d 1. Results so o b t a i n e d a p p e a r i n d e p e n d e n t of d i s t a n c e a n d consistent with scaling a t / 3 = 3.6 a n d 3.9, respectively 1.9 a and 2.8 a, which c o r r e s p o n d s to a physical size o f a b o u t 0.45 fm. W e have also l o o k e d for excited glueball states with positive parity. As with the 64 lattice, we f o u n d no sign of scaling except for the 2 ++ state at d i s t a n c e 1, where results are consistent with a s y m p t o t i c scaling at /3 = 3.6 a n d 3.9, n a m e l y M ( 2 + + ) / ( A L ) v l = 122 _+ 13 a n d 133 _+ 8. Of course these n u m b e r s should be c o n s i d e r e d o n l y as upper bounds. A s a final check of the scaling b e h a v i o u r of the T I action, we tried to c o m p a r e our results for X(2, 2), X(3, 2), X(3, 3) o b t a i n e d on a 64 lattice with spacing a, with the c o r r e s p o n d i n g expressions c a l c u l a t e d on a 124 lattice with spacing a/2. F i n i t e size effects are m i n i m i z e d in such a c o m p a r i s o n so that the shift in 13 necessary to m a t c h the 2 sets of values c o r r e s p o n d s rather precisely to a change of scale b y a factor 2 [16,17]. A s y m p t o t i c scaling w o u l d require a shift A/3 ----0.64. N o w the corres p o n d e n c e seems to be the following: /3(124) = 4.5 ---,/3(64 ) = 4.04,

(16a)

/3(124) = 4.2 - o / 3 ( 6 . 4 ) = 3.74,

(16b)

/3(124) = 3.9 ---, 13(6 4) < 3.

(16c)

16 August 1984

Because of the difficulty of m e a s u r i n g large l o o p s at low r , o n l y 2 a n d 1 x - r a t i o s could be used as i n p u t at fl(124) = 4.2 a n d 3.9 respectively, thus increasing the error at low ft. Nonetheless, the m e a s u r e d shifts Aft p r o v i d e an i n d e p e n d e n t indic a t i o n of a sizeable a s y m p t o t i c scaling violation in the m i d d l e of our scaling window, with a crossover to strong c o u p l i n g between fl = 3.3 a n d 3.9. In conclusion, this study of the SU(3) p u r e gauge theory on a 124 lattice with action (1) confirms the picture f o u n d in our previous work. Sizeable scaling violations have been o b s e r v e d between /3 = 3.9 a n d 4.2 for all physical quantities we could extract, i.e. string tension a n d deconfinem e n t temperature. F o r higher /3, our results are n o t accurate e n o u g h to decide whether scaling violations increase [18], decrease or even d i s a p pear. F o r lower/3 a s y m p t o t i c scaling a p p e a r s to be satisfied a m o n g o u r 2 d a t a p o i n t s (/3 = 3.6 a n d 3.9) for the string tension a n d the glueball mass: this is m o s t likely an i n d i c a t i o n that we enter the crossover region to strong coupling shortly below fl = 3.9. O u r m e a s u r e m e n t of the glueball mass is therefore s o m e w h a t inconclusive. W e can, however, check universality b y evaluating, at /3 = 3.9, the string tension, the glueball mass a n d the critical t e m p e r a t u r e ; they are in the ratios : M0++ : Tc --0 1 : 2.1 + 0.3 : 0.58 4- 0.03.

(17)

These results are quite consistent with universality. F i n a l l y it is rather d i s t u r b i n g that very similar scaling b e h a v i o u r s of v ~ a n d Tc are o b s e r v e d with a c t i o n (1) a n d with W i l s o n ' s action [3,19]. W i t h b o t h actions a s y m p t o t i c scaling is violated b y m u c h m o r e than is expected from higher o r d e r terms in the C a l l a n - S y m a n z i k / 3 - f u n c t i o n [20]. W e gratefully a c k n o w l e d g e the g r a n t i n g of c o m p u t e r time on the C R A Y - 1 S of C C V R . The C P U time used for this w o r k was a p p r o x i m a t e l y 35 h. W e t h a n k R. Petronzio for r e a d i n g the m a n u s c r i p t .

References [1] Ph. de Forcrand and C. Roiesnel, Phys. Lett. 137B (1984) 213. [2] P. Weisz, Nucl. Phys. B212 (1983) 1. 457

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[3] F. Gutbrod, P. Hasenfratz, Z. Kunszt and I. Montvay, Phys. Lett. 128B (1983) 415. [4] M. Fukugita, T. Kaneko, T. Niuya and A. Ukawa, University of Tokyo preprint INS-REP-473 (1983). [5] N. Cabbibo and E. Marinari, Phys. Lett. l19B (1982) 387. [6] Ph. de Forcrand, D. Lellouch and C. Roiesnel, in preparation. [7] G. Curci and R. Petronzio, Phys. Lett. 132B (1983) 133. [8] S. Belforte, G. Curci, P. Menotti and G. Paffuti, Phys. Lett. 137B (1984) 207. [9] R. Kirschner, J. Kripfganz, J. Ranft and A. Schiller, Nucl. Phys. B210 (1982) 567. [10] M. Luscher, K. Symanzik and P. Weisz, Nucl. Phys. B173 (1980) 365. [11] D. Barkai, K.J.M. Moriarty and C. Rebbi, BNL preprint BNL-34462 (1984).

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[12] w . Bernreuther and W. Wetzel, Phys. Lett. 132B (1983) 382; P. Weisz and B. Wohlent, unpublished. [13] M. Falcioni et al, Phys. Lett. ll0B (1982) 295. [14] K. Ishikawa, G. Schierholz and M. Teper, Phys. Lett. ll0B (1982) 399. [15] B. Berg and A. Billoire, Nucl. Phys. B221 (1983) 109. [16] M. Creutz, Phys. Rev D23 (1981) 1815. [17] A. Hasenfratz, P. Hasenfratz, U. Heller and F. Karsch, Phys. Lett. 140B (1984) 76. [18] F. Gutbrod and I. Montvay, Phys. Lett. 136B (1984) 411. [19] T. t~elik, J. Engels and H. Satz, Bielefeld preprint BI-TP83/23. [20] R.K. Ellis and G. Martinelli, LNF preprint LNF-84/l(P).