Estimate of the gluon condensate from Monte Carlo calculations

Nuclear Physics B190 [FS3] (1981) 692-698 © North-Holland Publishing Company

ESTIMATE OF THE GLUON CONDENSATE FROM MONTE CARLO CALCULATIONS* T. BANKS 1 Tel Aviv University, Ramat Aviv, Israel

R. HORSLEY, H.R. RUBINSTEIN and Ulli WOLFF2 Physics Department, Weizmann Institute, Rehovot, Israel

Received 23 February 1981 Using existing Monte Carlo data we estimate the value of the gluon condensate tb = (O[(~(g)/g)F2v ]0). Given the limitations of the method and the available data we find reasonable agreement in both sign and magnitude with the value needed in QCD sum rule calculations.

1. Introduction T h e Q C D v a c u u m is u n d o u b t e d l y v e r y different f r o m t h e state d e s c r i b e d by r e n o r m a l i z e d p e r t u r b a t i o n t h e o r y : s t r o n g i n f r a r e d v a c u u m fluctuations a r e r e s p o n sible for the forces t h a t i m p l e m e n t q u a r k c o n f i n e m e n t a n d the chiral o r d e r p a r a m e t e r (~to) has a finite v a c u u m e x p e c t a t i o n v a l u e e v e n for a t h e o r y of massless q u a r k s . S e v e r a l y e a r s ago S h i f m a n , V a i n s h t e i n a n d Z a k h a r o v [1] a r g u e d t h a t a n o t h e r n o n - p e r t u r b a t i v e f e a t u r e of t h e Q C D v a c u u m was a n o n - z e r o e x p e c t a t i o n v a l u e for t h e t r a c e of the e n e r g y - m o m e n t u m t e n s o r (in t h e massless q u a r k theory). I n c o r p o r a t i n g this e x p e c t a t i o n value into a d u a l i t y s c h e m e b a s e d on t h e o p e r a t o r p r o d u c t e x p a n s i o n , t h e y s u g g e s t e d that it was r e s p o n s i b l e for the large splitting b e t w e e n ~c a n d tO, a n d o t h e r i m p o r t a n t n o n - p e r t u r b a t i v e effects. T h e i r s c h e m e a n d e x t e n s i o n s b y R ¢ i n d e r s , R u b i n s t e i n a n d Y a z a k i [2] h a v e b e e n q u i t e successful p h e n o m e n o l o g i c a l l y for systems of h e a v y q u a r k s , b o u n d states of h e a v y a n d light q u a r k s [3], a n d e v e n p u r e light q u a r k systems [4], w h e r e its t h e o r e t i c a l justification is m o s t suspect. It is i m p o r t a n t to p o i n t o u t t h a t t h e gluon c o n d e n s a t e

a p p e a r s m u l t i p l i e d b y different W i l s o n coefficients in t h e v a r i o u s a p p l i c a t i o n s of t h e m e t h o d , a n d t h a t t h e p o w e r c o r r e c t i o n s r e q u i r e d by t h e p h e n o m e n o l o g y a r e t h o s e e x p e c t e d f r o m t h e d i m e n s i o n s of this o p e r a t o r . * This work was partially supported by the United States-Israel Binational Science Foundation. 1 Partially supported by the Israel Commission for Basic Research. 2 On leave of absence from the Max-Planck-Institut, Miinchen, Germany. 692

T. Banks et al. / Gluon condensate

693

The theoretical basis of SVZ's use of 4, is a matter of some debate [5]. Although connected matrix elements of the operator (/3(g)/g)F]~ are finite in perturbation theory, the vacuum expectation value is infinite. SVZ suggest that the appropriate definition of the operator is to "normal order it in the perturbative vacuum", i.e. 4 , = ([3B(gO)go
(B=bare),

(2)

where (F2)p is the vacuum expectation computed in perturbation theory [6]. Non-perturbative contributions to 4, are then supposed to be finite. Checks of this conjecture could presumably be made, for example by computing the higher loop corrections to 4, around an instanton. To our knowledge no such calculations have been carried out. The present paper is an attempt to evaluate (2) by a fit to Monte Carlo simulations of lattice gauge theory. The success of such a fit is a partial confirmation of the conjecture of SVZ on the finiteness of (2). Given the crudity of our methods, our results are quite encouraging. We find that the non-perturbative contribution to 4, has the same sign and order of magnitude as the phenomenological value used by SVZ and RRY. The appearance of the right order of magnitude is non-trivial because it requires that a certain numerical coefficient in our fit be of order 1 0 9 and this agrees with Monte Carlo data. Of course, much more theoretical work is needed before we can conclude that the SVZ conjecture is verified. 2. Connection between lattice theory and continuum Although (2) is well defined in perturbation theory (including non-perturbative semiclassical expansions) it does not make sense when applied to an "exact", i.e. Monte Carlo evaluation of 4,. The point is that the perturbation series is at best asymptotic and it is only defined once a summation method is prescribed. Clearly one possible summation method is just to say that the sum of the series equals the exact value for the function in which case 4, = 0. This is obviously not what was intended by SVZ. We propose instead the following pragmatic definition of (2). Evaluate a certain number of terms in the series and find a range of values of [3 = 4 / g 2 for which the Monte Carlo data deviates from the series by more than the estimated errors in the series (as represented by the highest term calculated). Of course, we restrict ourselves to a range where successive terms in the series are not growing. The difference between the series and the data defines the nonperturbative part of 4,. Clearly we are going to have to work at values of [3 that are not too large if we are to have any hope of separating out the non-perturbative piece. A p r i o r i we have no right to expect that the behaviour of 4, at these values of [3 will be anything like its large [3 asymptote. However, previous Monte Carlo studies of lattice gauge theories appear to show that the functions of the theory take on their asymptotic form as soon as [3 is above the so-called weak-strong transition. We assume that this is also true for 4,. SVZ's conjecture that (2) is finite

694

T. Banks et al. / Gluon condensate

in the continuum limit then implies that the lattice plaquette defined in (5) behaves like W= 1

do

dl

d2

/3 /32 /33

A/32bl/b2o e-t~/2/,o

(3)

even for values of/3 at which we can extract it. This is a renormalization group argument and b0, bl are the first two perturbative coefficients of t h e / 3 function. The constant A is related to the continuum value of ~b: 1.3 × 10 -8 for SU(2) a ( F 2 ) _ -~b =AOMoMA X zr 4zrZb0 8.7 x 10 -9 for SU(3) '

(4)

where AMOM is the Q C D scale p a r a m e t e r in the m o m e n t u m subtraction scheme. To derive (4) we have used the relation between the lattice scale p a r a m e t e r and AMOM in the Feynman gauge calculated by Hasenfratz and Hasenfratz [7]. Note that for values of AMOM of a few hundred M e V the phenomenological value ( a / ' r r ) ( F 2> = 0.012 (GeV) 4 can be reproduced only if A - 108. Notice that for SU(2)

[8] 11 bo = 247r 2 ,

17 bx - 967r 4 ,

1 Alatt -- 5 7 . 5 A M O M ,

while for SU(3) [8] 11 bo - 16~'2 '

51 bl - 128~.~,

Alatt =

1 83.5 AMoM.

The necessity of making our fit in a restricted range of fl is troublesome but it is actually a general feature of Monte Carlo calculations in Q C D . For large /3 [ a c t u a l l y / 3 - 4 - 5 for SU(2)] the C o m p t o n wavelengths of particles in the lattice gauge theory b e c o m e larger than the finite lattice of the Monte Carlo calculation. Even if the Monte Carlo data were exact for the finite lattice they cannot be assumed to give the correct infinite volume limit at values of/3 much larger than this. On the other hand, if /3 is too small, we cannot expect to obtain a good approximation to the continuum (/3 -~ oo) theory. Thus all successful Monte Carlo calculations in lattice gauge theory depend on the existence of a range of intermediate/3 in which important quantities already take on their asymptotic (/3 ~ oo) forms. In our case the problem is slightly exacerbated by the fact that we are calculating a quantity which has a non-trivial perturbation expansion. We must go to small enough/3 to separate the perturbative and non-perturbative effects. Nonetheless, we have been able to find a fit to ~b in the same region in which other authors have fit the string tension. 3. Monte Carlo fit We will now use existing Monte Carlo data to evaluate (2). The first step in the calculation is to find the relation between the plaquette average energy and &. To

695

T. Banks et aL / Gluon condensate

that effect consider a hypercubic lattice with spacing a embedded in a continuum and with axes/z~ = 8.~. For an arbitrary point x, the SU(N) Wilson plaquette, in the/z, p plane adjacent to x, is given by W=--1 R e ( t r Ux,x+a~,Ux+a.,x+a~+a~Ux+a.+ . . . . . . . Ux+.... ). N

(5)

By this we mean the expectation value of the 1 x 1 Wilson loop. In (5), Ux.x +a. = P exp (\ igo

AB.,,

d~

(6)

,

and W has been normalized to belong to [0, 1]. By standard methods we obtain 4

a 1 W = 1 --~- g2 N tr ( F ~ ) + O ( a

5)

(7)

4

=1_4___~ a g2o(F2)+O(a5 ) ,

(8)

where in (8) we have used x independence, and we have averaged over planes and orientations (12 of them). Combining (2) and (8) we obtain, in the continuum limit ~b= lim B B ( g o ) 4 8 N [ w p _ w ] ' a'g°~O go a

(9)

where N is the order of the SU(N) group. Wp is the perturbative part of W. For large/3 the perturbative part Wp is given by do dl d2 Wp=I-/3 /32 / 3 3 ~ - ' " .

(10)

In principle these coefficients should be calculated by weak coupling perturbation theory on a finite lattice. This has been done only for do*. However, Lautrup and Nauenberg [9] have fit the coefficients di and d2 to their high accuracy large /3 Monte Carlo data for SU(2). We have performed a similar fit on the (lower statistics) data of Creutz and Pietarinen for SU(3) [10]. The resulting values are do dl d2

SU(2) 3 ~ 0.13 0.29

SU(3) 4

0.8

Our strategy is to attempt to match the Monte Carlo data for W to the formula W = Wp - A / 3 261/b2 e - t 3 / 2 b °

.

* We are presently consideringextendingthis calculationto higher orders.

(11)

T. Banks et al. / Gluon condensate

696 0.80

I

I

I

I

I

I

I

I

1

. I I

1

,1

0.70

f/~l~

~"

1

I

--

o

,

O, 6 0

//.

B

,//";)~/II

i .," t

" r l "" fa Ib

0.50

L 1.5

I ,tl . II 1.9

I1

I I

Ic II

2.3 Beta

2.7

I 3,1

I

I 3.5

Fig. 1. SU(2) plaquette energy versus/3 = 4~go.2 Short dashes: Perturbatwe fit [see (10)] do = 3, dl = 0.13, d2 = 0.29. Dots: Monte Carlo data [9]. Typical errors are below 0.5%. Long dashes: Fit according to (a) A = 108; (b) A = 109; (c) A = 101°.

Then using (9) and the expression

l ( 4 bo)-bJ2b~e-t3/Sbo

(12)

a,.,,= a -~ we find

2

~b=-bo48N(4bo)2b'/b°AA4att

;

(13)

then (4) obtains. For SU(2) the Monte Carlo is very accurate and we obtain a good fit (see fig. 1) down to/3 = 2.1. We can extract the non-perturbative part below/3 = 2.4, and our best estimate for A is 108 t o 10 9. For comparison we note that Bhanot and Rebbi's fit to the string tension (SU(2)) [11] was p e r f o r m e d in the interval 2
T. Banks et al. / Gluon condensate 0.7O

'~

,,,., .¢" ,/~,"S, T

0.60

~-~

Q 0.50

s

. ~r

/ •

OSO

++

/ I r"

Q20 2

I

/'/

aI I I

.,,.+/

+I I I/

/

/

*'= 0.40 m

697

I I

I I I I bl cl I I I

5

I

1 4

l

I 5

Beta

Fig. 2. SU(3) plaquette energy versus/3 = 4/gg. Short dashes: Perturbative fit [see (10)] do = 4, d l = 0.8, d2 = 0. +: Creuz data for Monte Carlo. Dots: Pietarinen data from [10]. Typical errors are below 1%. Long dashes: Fit according to (a) A = 108; (b) A = 3 x 109, (c) A = 10 l°.

To summarize: we have found a fit to the Monte Carlo data for the action density in SU(2) and SU(3) lattice gauge theories which is consistent with the conjecture of S V Z that the perturbatively normal ordered operator ( / 3 ( g ) / g ) F 2~ has a finite non-zero value in continuum Q C D . The deviation from weak coupling perturbation theory on which our fit is based is an order of magnitude above the statistical errors of the Monte Carlo calculation in a significant range of/3 above the weak strong transition. Higher order terms in a perturbation series cannot reproduce this deviation and the exponential behaviour required by the renormalization group is clearly discernible. Our calculation is of course subject to all the usual criticisms of Monte Carlo studies in Q C D (finite lattice effects, absence of fermions etc.). Nonetheless, we believe that certain aspects of our fit are quite clearcut and will survive in more careful studies. These are: (i) The sign of ~b which is obtainable simply from the fact that the data lie below the perturbative curve. This agrees with the sign of the phenomenologically motivated fit to 4'. (ii) The large value of A. This is necessary in order to obtain even order of magnitude agreement with the continuum result. (iii) As for the functional form of W(/3) we do not claim to have proven that it is that given by the renormalization group. However, it is significant that the data do not disagree with the renormalization group fit. Furthermore, there is no doubt that some sort of exponential dependence can be discerned in the range of /3 studied. Polynomial fits are, of course, possible but highly artificial. W e thank B. Lautrup and E. Pietarinen for correspondence and for providing unpublished M o n t e Carlo data, and m a n y colleagues for discussions. U.W. acknowledges a fellowship of the Minerva C o m m i t t e e for I s r a e l i - G e r m a n cooperation in research.

698

T. Banks et al. / Gluon condensate

References [1] [2] [3] [4] [5] [6]

[7] E8] [9] [10] [11]

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