The two-loop average plaquette on the BCH lattice

Volume 200, n u m b e r 1,2

PHYSICS LETTERS B

7 January 1988

T H E T W O - L O O P AVERAGE PLAQUETTE O N T H E BCH LATTICE Michele M A G G I O R E a,b and Haralambos PANAGOPOULOS c,b a Scuola Normale Superiore, 1-56100 Pisa, Italy b INFN, Sezione di Pisa, 1-56100 Pisa, Italy c Dipartimento di Fisica, Universita di Pisa, 1-56100 Pisa, Italy Received 15 September 1987

We perform a two-loop calculation of the average triangular plaquene on the body-centered hypercubic (BCH) lattice. Combining our result with Monte Carlo SU (2) data on this lattice, we estimate the value of the gluon condensate.

1. The Wilson [ 1 ] formulation of lattice gauge theories allows for the computation of non-perturbative quantities such as hadron masses, string tension or the deconfinement temperature [ 2 ]. Another interesting quantity which can be computed is the gluon condensate [ 3,4]. Most of these studies have been performed on a hypercubic lattice with the Wilson action. A motivation for alternative formulations of lattice gauge theories, in which one changes the form of the lattice action [5] or the geometry of the lattice [6,7], is to verify that physical results are independent of the regularization procedure, i.e. that universality holds. Besides, among the various formulations belonging to the same universality class, some may be more convenient than others from the point of view of Monte Carlo simulation. Particularly interesting is the body-centered hypercubic (BCH) lattice [ 7 ]. In ref. [ 8 ] the ratio between the A parameters of the BCH and hypercubic lattice was extracted from a measurement of the string tension. The same ratio has been computed perturbatively in ref. [9] and the two values are in agreement. Ref. [10] computes the deconfinement temperature which, using the value for the ratio of the A parameters found through the string tension, is in agreement with the value found on the hypercubic lattice. In this paper we show the result of the perturbative calculation of the average triangular plaquette up to two loops, in quarkless QCD. This result al-

lows us to extract the value of the gluon condensate from the existing Monte Carlo data [ 11 ] for the average plaquette. However, since these data cover only a small range of values for fl, the determination of the gluon condensate is only an order of magnitude estimate. 2. For a more detailed explanation of perturbation theory on the BCH lattice we refer to ref. [9]; here we sketch the basic ideas. The action on the BCH lattice is S = 2g-1S ~ tr(1 - Ua) ,

(1)

where the sum runs over the triangular plaquettes of the lattice, with both orientations. On the BCH lattice there are 12 links joining a site with its nearest neighbours, instead of 4 as in the hypercubic lattice. To compute the average plaquette we parametrize the 12 matrices associated with site n as

Un

exp(igy~,) ,

(2)

where v runs over the 12 links. To interpret this theory as QCD 7~.= v~A, + qvf~J(,O with j ( v ) = l ..... 12,

q,.=0,

j = l .... , 4 ,

~/u=l,

j = 5 .... , 1 2 .

we write

(3)

Then it can be shown (see refs. [9,12 ]) that in the

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continuum limit the field A u propagates as the gluon field, while the 8 fields 0; decouple from Au and acquire an infinite mass, and therefore do not propagate. The perturbative calculation o f the plaquette proceeds along the same lines as in the case of the simple hypercubic lattice [4]; the main difference is that the propagator and the vertices are now matrices of dimension 12. Defining, for S U ( N ) , 1 - P = 1 - (l/N) Tr U~ and fl=2N/g 2, we write for the weak coupling perturbative expansion

( 1 - P ) =c~/fl+c2/fl 2 "~ C3/j~ 3 "~- . . . .

7 January 1988

C) C) C-) CK) a

b

c

d

e

f

(4)

In a functional integral representation we have (l-P)-

1 f[dU] exp(-S) S 16fl~ f[dU]exp(-S) '

(5)

where N~ is the number of lattice sites; a compensating factor o f N~ will come out o f the functional integral, so that the limit N ~ - , ~ can then be taken. To proceed with the weak coupling expansion we must decompactify the measure [dU] to [dT]. The jacobian of this change of variables contributes an extra term S m e a s to the action, which to two-loop order equals Smea s -

g2N 24 Z

~a Y~'(n+kv)y~(n+lv) •

(6)

In addition, one must as usual introduce a gauge fixing and a ghost term to the action; these are the same as in ref. [9]. The expression for the average plaquette now reads (I-P)

1(f

= 16~----~

o°

2 :

.....

%

V /N

'° 1

o.°

C)::

°.

°..°* °°*. . . . . . .

°. °°"

..-"

g

h

Fig. 1. One- and two-loop diagrams contributing to the average plaquette. Diagram a is O(g2); diagrams b-g are O(g2). The solid line denotes the gluon propagator, the dashed line denotes the ghost propagator, the line with a solid square denotes the contribution of Sm.... the lines with crosses denote the quadratic part of S (inverse propagator without the gauge-fixingpart).

c, = ~ 2 ( N 2 - 1) , [d~,] [dcl [de]

c2 = ( N 2 - 1 ) × [N2(0.0382 + 0.0001) - 121/(64)<32)].

×exp[ - (S+S

X(f

....

-~-Sgf"l-Sgh)]S)

In particular, for S U ( 2 ) , c1=~,

[dY] [dc] [dc] \ --I

)< exp[ - ( S + S

....

"l-Sgf-~-Sgh) ]

)

(7)

(c is the ghost field). The diagrams arising up to two loops are drawn in fig. 1. We evaluate these diagrams and obtain (cf. eq. (4)~ 130

c2=0.281+0.001.

(8)

We also verify, as a check, that our result is gauge invariant. 3. The strategy to extract the gluon condensate from the lattice has been explained in ref. [4]; defining

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7 January 1988

Fig. 2 shows ( P ) from the Monte Carlo calculation of ref. [ 1 ]; fig. 3 shows on a logarithmic scale the difference

a,]w

one expects, for a ~ 0 G ~ (c~/a 4) exp( -2~flog 2) = c o A 4 .

(10)

In the naive continuum limit on the BCH lattice ( 1-P)

~ Ga4tr2/128N

+ operators depending on the 0 fields +operators of higher dimension.

(11 )

To obtain the gluon condensate we must subtract from the Monte Carlo data the perturbative expansion until ( 1 - P ) shows an exponential behaviour; the exponent should be given in terms of the beta function as specified by the renormalization group. We have used the Monte Carlo data reported in ref. [ 11 ], obtained on a 144 BCH lattice, for SU(2), with values offl ranging from 2.40 to 4.00. Since this range offl is quite small, we cannot expect a precise determination of the condensate but we can only obtain an order of magnitude estimate. As in ref. [4] subtracting the two loops expansion does not leave with an exponential behaviour, and we have to subtract also a term cslfl 3, to be determined by the fit.

(12)

- (33/32fl+O.281/fl 2 +c3/fl 3) ,

with c3 determined by the fit to be c3= 1.09+0.07. A best square fit then gives for c~, defined in eq. (10), the value c o = 1.22× l0 II

(13)

.

Due to the incertitude in the fit caused by the small range o f / / w e do not put any error on this number. Using for the A parameter of the simple hypercubic lattice the same value as ref. [4], that is A~ = (1/2~) e x p ( - 12n2/11) GeV 2 ,

(14)

and for Abch/Ah the value of ref. [9],

5.00

4.50

-InA

4.00

0.70 0.65 3.5C 0.60

i

0.55 3.00

0.50

' ti

0.45 0.40

M

r

2.5

°

I

3.0

I

3.5

[

4.0

Fig. 2. Monte Carlo versus theoretical value of ( P ) . 33 -I f~ (fl) : 1 -- ~fl , A(fl) =f~ (fl) --0.281fl -2, f 3 ( f l ) = f 2 ( f l ) -1.088/~-3,AtP) =A(/~) - (1.22 × 10' ') :~s~%xp( - ::~-p). ,2

2.50

2140 2'.50

2'.60

2170

2180

Fig. 3. The value of A extracted from a fit of the exponential tail. The angular coefficient of the straight line is determined from scaling.

131

Volume 200, number 1,2

Abch/A h = 0.28948 ,

PHYSICS LETTERS B

(1 5)

we get for t h e v a l u e o f the g l u o n c o n d e n s a t e ext r a c t e d o n the B C H lattice G=0.01 GeV 4 ,

(16)

to be c o n f r o n t e d w i t h t h e v a l u e o f ref. [4] G=0.015 +0.002 GeV 4

(17)

We stress that, while the a b s o l u t e v a l u e o f the c o n d e n s a t e s d e p e n d s o n t h e v a l u e a s s u m e d for An, t h e i r ratio is i n d e p e n d e n t . T a k i n g i n t o a c c o u n t the unc e r t a i n t i e s in the fit, the a g r e e m e n t is satisfying. We w o u l d like to t h a n k P r o f e s s o r A. D i G i a c o m o for m a n y useful d i s c u s s i o n s a n d suggestions.

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J. Engels, F. Karsch, H. Satz and I. Montvay, Phys. Lett. B 101 (1981) 89; G. Bhanot and C. Rebbi, Nucl. Phys. B 180 [FS2] (1981) 469; H. Hamber and G. Parisi, Phys. Rev. Lett. 47 ( 1981 ) 1792; E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795. [3] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385,448, 519. [4] A. Di Giacomo and G.C. Rossi, Phys. Lett. B 100 (1981) 418; A. Di Giacomo and G. Paffuti, Phys. Lett. B 108 (1982) 327; Nucl. Phys. B 205 [FS5] (1982) 313. [5] J.M. Drouffe, Phys. Lett. B 105 (1981) 46; P. Menotti and E. Onofri, Nucl. Phys. B 190 [FS3 ] (1981 ) 288; N.S. Manton, Phys. Lett. B 96 (1980) 328. [ 6 ] J.M. Drouffe and K.J.M. Moriarty, Nucl. Phys. B 220 [ FS8 ] (1983) 253. [7] W. Celmaster, Phys. Rev. D 26 (1982) 2955. [8] W. Celmaster, Phys. Rev. Lett. 52 (1984) 403. [9] A. Di Giacomo, M. Maggiore and H. Panagopoulos, Phys. Rev. D, to be published. [ 10 ] W. Celmaster, E. Kovacs, F. Green and R. Gupta, Phys. Rev. D 33 (1986) 3022. [ 11 ] W. Celmaster and K.J.M. Moriarty, Phys. Rev. D 33 (1986) 3718. [12] M. Maggiore, Master thesis, Universit~t di Pisa (986), unpublished.