Short-distance expansion of Wilson loops, gluon condensation and Monte Carlo lattice results

Nuclear Physics B210 [FS6] (1982) 567-582 © North-Holland Publishing Company

S H O R T - D I S T A N C E E X P A N S I O N OF W I L S O N L O O P S , G L U O N CONDENSATION AND MONTE CARLO LATTICE RESULTS R. KIRSCHNER, J. KRIPFGANZ, J. RANFT and A. SCHILLER

Sektion Physik, Karl-Marx-Universit?itLeipzig, DDR Received 11 May 1982 The short-distance expansion of Wilson loops in terms of composite gluon operators (gluon condensate) is studied. The 2-loop perturbative contribution to Wilson loop rations is evaluated. Monte Carlo lattice results for SU(2) gluodynamics are analyzed in terms of this expansion. Consistent results for the gluon condensate are found.

1. Introduction The Monte Carlo approach to lattice gauge theories allows one to study nonperturbative quantities not accessible otherwise. For non-abelian gauge theories a non-vanishing string tension is found [1]. It shows a dependence on the bare coupling constant as predicted by asymptotic freedom and the renormalization group. This numerical result provides strong evidence for confinement. More recently gauge theories with fermions have also been studied. Results for lowestlying meson masses are quite satisfactory, indicating that chiral symmetry is indeed spontaneously broken [2]. A n o t h e r rather interesting non-perturbative p h e n o m e n o n is gluon condensation. It manifests itself through the appearance of non-vanishing vacuum expectation values of composite gluon operators, like (G~G",~). These operators show up as higher twist contributions in the short-distance expansion of various current correlation functions. Through sum rules they can be related to parameters of low-lying resonances in the corresponding channel [3]. This approach yields a phenomenological value 1 /as

a

~\

N c ~--~ G ~ G ~ ) ~

0.004 GeW 4 ,

(Nc = 3),

(1)

for the gluon condensate. The gluon condensate has been studied along similar lines in the lattice Monte Carlo approach [4]. H e r e the relevant correlation function is the Wilson loop. The gluon condensate again appears as a contribution of higher dimensions and may be identified due to its characteristic (exponential) dependence on the lattice coupling constant. 567

568

R. Kirschner et al. / Wilson loops

Previous numerical studies [4-6] of the gluon condensate mostly refer to lattice gauge theories with gauge group SU(2). Few lattice data are available for the gauge group SU(3). They have been analyzed in ref. [7]. The results are consistent with the phenomenological value (1) although the numerical uncertainties are large, and a direct comparison is not really possible since fermions have not been included in the corresponding Monte Carlo calculations. To our knowledge, there are no new results as far as SU(3) is concerned, and we have nothing to add to ref. [7]. Our main objective is rather a study of the consistency of the approach. This can equally well be done for SU(2) gauge theory. A major difficulty in determining the gluon condensate is the separation of logarithmic (perturbative) and power-like contributions. Perturbative terms are typically larger than the non-perturbative signal one is interested in. If one works with elementary plaquettes [5, 6] this signal-to-background ratio is less than 10 -2 • • • 10 -3. This is one of the reasons why we prefer to work with larger loops [4]. In this case non-perturbative contributions b e c o m e more prominent, and the continuum limit could be studied in a systematic way. Nevertheless, it is important to know the perturbative background as accurately as possible. We have therefore worked out the two-loop perturbative contribution to rectangular Wilson loops, or actually to certain Wilson loop ratios which have a continuum limit. This calculation is discussed in sect. 2. Some technical details are presented in the appendix. High precision Monte Carlo data on SU(2) Wilson loops became available recently [8]. H e r e we also present some of our own data. In sect. 3 they are analyzed in terms of a gluon condensate contribution. Consistent results are found if the perturbative background is handled appropriately, i.e. if the expansion in the lattice coupling constant is partially resummed. In sect. 4 we provide a s u m m a r y of our results.

2. Short-distance expansion of Wilson loops Wilson loops have been studied in perturbation theory by several groups [9-12]. Their renormalization is now well understood. There are two new types of singularities characterizing these non-local objects. First of all, there are linear divergences (poles at d = 3 in dimensional regularization) which depend only on the perimeter of the loop, and which exponentiate. They may be interpreted as mass renormalization of the heavy test particle and are multiplicatively renormalizable. The second type of singularity appears for contours with cusps. This new logarithmic singularity (pole at d = 4) is also multiplicatively renormalizable, with a renormalization constant depending on all the cusp angles but not on the size of the contour [12]. One may avoid this multiplicative renormalization by studying suitable ratios of Wilson loops. Such ratios still contain logarithmic divergences. They are removed by the usual charge renormalization, i.e. the ratio is finite in the limit when the

R. Kirschner et al. / Wilson loops

cut-off is removed and renormlized coupling g 2 tions to the Wilson loop The relation between dimensions is

569

the bare coupling g2 is varied in such a way that the is kept fixed. W e have calculated the perturbative contriburatio in dimensional regularization in configuration space. the renormalized and the bare coupling in d = 4 - e

with flo = ½ ( 1 1 N c - 2 n 0 .

f(e, g2)

is any function of e with the property f(0, g2) = 1. Each different continuation of g2 to d dimensions leads to different finite parts in the two-loop contributions. In particular we absorb those finite pieces connected with the dimensional regularization into the definition of the effective coupling constant. Such a MS scheme in configuration space corresponds to the choice

f(e, g2) =

1 + ~e (log ~" + ' r E ) ,

YE = 0.5772.

(3)

Note the numerical difference to the usual MS scheme in m o m e n t u m space. With this choice we obtain g =g~

/1

16~r z

e+21ogLotz

))

,

(4)

where 1

tz = t2"f-~ exp (~yE) •

(5)

For convenience, we c h o o s e / z = 1/Lo, where L0 denotes the characteristic length of a reference loop. For the renormalized ratio of rectangular Wilson loops we obtain up to two loops

W(L1, L2) = l + g ~ ( L o ) W ~) W(Lo, Lo) a ~

4

H e r e I ~ ) denotes the l - l o o p contribution

g4(Lo) (1

1,~,~))

(16zrZ) 2 \ ~ . ( I ' ~ ) 2 +

(CF = [N~

"

(6)

- 1]/2Nc), (7)

and lgz~) the non-abelian 2-loop contribution (C2 = 9~(31Nc - 10n0), = 8CF/[/30 log2 L1 + 2(/30 + Nc(a + 1.644)) log L---L tl_ Lo L0

+2 0rt )

L,

L1

570

R. Kirschner et al. / Wilson loops

/ L1 / L I \ _ f ( 1 ) \) + C2~l°g-~oo+f ~ ) + Nc(C(~2)-C(I))]+[LI<->L2]}.

(8)

Various functions appearing in eq. (8) are given in the appendix. Using this result we can calculate the perturbative contribution to the often used quantity

W(tl, L2) W(L1 - AL, L 2 - A L ) x(L1, L2, AL)=- - l o g W(L1 - A L , L2)W(L1, L 2 - A L )

(9)

'

as xP(L 1, Le, AL) = X~ ) (L1, L2, AL)g2R (Lo)+ X~ ~(L~, LE, AL; Lo)g 4 (Lo)+ . . . .

(lO) We note the explicit dependence of the 2-loop contribution on the renormalization length. The l-loop term X~ ) is easily evaluated: CF [ /L1-AL\

L1

/L2-AL\

<._.>L21

(11) The 2-loop term can be read off from eqs. (6), (8) and (9). A priori, the continuum limit (10) is applicable to a lattice calculation of X P only if both L1, L2 and AL are much larger than the lattice spacing:

L1, Lz, AL >>a. It turns out, however, that the expression (10) still gives a very good description for AL = a, L1 = I a , L2 = J a , I, J/>2. To show this we compare our results with the same xV(L, L, AL) obtained on the lattice [13] (for pure Yang-Mills case, rtf = 0 ) . We express our perturbative result in terms of the coupling ga of the theory regularized by means of a lattice with lattice spacing a. The relation between our g2(Lo) and g2a is given by

11

gR2

~

ga- ~-

log

(~a L0)_~_ /31 -~-

1

g2

/3016rr2 log ~ =

2

0, (12)

/31 = ~(34Nc - 38nf), to 2-loop accuracy. The ratio of A/Aa can be easily found using the known relations AMoM/Aa [14] and A~-'-S/AMOM [15]. Using A 1 = ~ exp (YO, A~-s

(13)

571

R . Kirschner et al. / Wilson loops

we obtain in the Feynman gauge for SU(2) A/A, = 17.6,

(14)

A/ha = 25.7.

(15)

and for SU(3) Expanding X p in terms of g] we have instead of eq. (10) X

P

2 __ (2) 4 =Xa(1) ga±X,, ga+''',

(1) = X ~ )

(16)

I'

(17)

x~

This result is c o m p a r e d in table 1 with lattice calculations for xP(L, L, AL) of ref. [13]. Similar results are already found for very small loops. Deviations are of the order of 10 percent. Therefore, our continuum results can be used for all loops for which no explicit lattice calculations are available. In table 2 we have given l - l o o p and 2-loop contributions to X P expanded in the renormalized coupling at length L0 g2R(L0) for different interesting values of L 1/AL, Lz/AL. Two possible choices for L0 are considered:

Lo=LI,

Lo=x/L1L2.

When folded with the coupling constant calculated at the corresponding value of L0 differences in the relevant coupling range are of the order of 10%. This characterizes the uncertainty due to the choice of the renormalization point for which two scales (L1 and Le) are available. The results of our perturbative calculations can be c o m p a r e d with results obtained by Fischler [16] for the potential of a heavy quark-antiquark system at short distances. Consider the quantity

1 W(T,R)W(T-6T, R -6R) F = -STS-----Rlog ~ ' ~ V - ~ , R - S R ) "

(18)

TABLE 1

Comparison of the perturbative expansion of XI'(L, L, A L ) for lattice and continuum calculations (I = L a = L A L )

2

3

4

(1) X~

0.0543 0.0496

0.0194 0.0164

0.0091 0.00816

SU(2)

lattice [13] continuum

Xa(2)

0.0158 0.0143

0.0066 0.00559

0.0034 0.00305

SU(2)

lattice [13] continuum

x~(2)

0.0472 0.0428

0.0194 0.0164

0.0098 0.0089

SU(3)

lattice [13] continuum

R. Kirschner et al. / Wilson loops

572

TABLE 2 One- and two-loop contributions to xP(LI, L2, ~ L ) [eq. (10)] in pure SU(N) gluodynamics for two renormalization points Lo = L 1 and Lo = x / L - - ~

2 2 2 2 2 3 3 3 3 4 5 6 7 8 9

2 3 4 5 6 3 4 5 6 4 5 6 7 8 9

0.0661 0.0459 0.0421 0.0409 0.0404 0.0218 0.0167 0.0150 0.0142 0.0109 0.00652 0.00435 0.00310 0.00233 0.00181

-0.00139 -0.000568 -0.000438 -0.000407 -0.000396 -0.000296 -0.000119 -0.0000643 -0.0000436 -0.000116 -0.0000592 -0.0000352 -0.0000230 -0.0000161 -0.0000119

-0.00139 -0.00100 -0.00111 -0.00128 -0.00143 -0.000296 -0.000230 -0.000242 -0.000272 -0.000116 -0.0000592 -0.0000352 -0.0000230 -0.0000161 -0.0000119

xP([AL, JAL, AL ) = g2 (Lo)CFX~) (I, J) + g4 (Lo)CF NcX(R2)(I, J ; ~L ) in the limit 6R, a T <
g2 CF (

g2R (~Uc-~nf-[3o(Z-logR2122-1og4zr))),

4~ R z 1 +1--~ 2

(19)

which agrees with Fischler's result. Non-perturbative contributions due to the non-trivial vacuum structure of gluodynamics are power-like suppressed with respect to the loop size. Without taking quantum corrections into account those effects have been studied by Shifman [17]. For the Wilson loop ratio (9) one finds x(LI,

L2,

AL) =

xP(L1, L2,

AL) + (2L1 l(a~

× lrr2(AL)2 ~ c

AL)(2L2 - A L )

G2~G~ ) +''',

(20)

plus terms of still higher dimension. Quantum corrections also add to the coefficient function of the gluon condensate (i.e. ( G ~ v G ~ ) ) term. These contributions have not yet been studied and will be ignored in the following.

R. Kirschner et al. / Wilson loops

573

3. Discussion of some lattice Monte Carlo results

As pointed out in sect. 2, only suitably defined ratios of Wilson loops have a continuum limit. Most commonly used is the quantity x(L1, L2, A L ) defined by eq. (9). In order to obtain reliable information on the continuum theory one would have to study large loops such that the relevant length scale is much larger than the lattice spacing a: L1, L2, A L >>a. In practice, however, one is restricted to rather small loops, and in particular we will have to choose A L = a. This might introduce systematic errors which could show up as an apparent loop size dependence of this gluon condensate, or as a possible dependence on the lattice action chosen. The range of available Monte Carlo data is still too limited to study these questions comprehensively. We shall investigate loop size effects for the Wilson action as few as this is possible. We introduce the short notation x ( L J ) = - x ( I a , Ja, a). The short-distance expansion of X is given by eq. (20). Monte Carlo data will be analyzed in terms of this expansion. As scale p a r a m e t e r we use the string tension or instead of the h parameter. This is convenient because o- (a2(r) is directly measured in the Monte Carlo procedure. The most accurate available result for the SU(2) string tension is [18] a2o-=(0.011+0.002)

2(fiog2(a)) ~l/~?~exp(-1/(fiog2(a)))

(21)

with rio =/3o/16"/7" 2 ,

fil = f11/(167r2) 2 •

When translating into units of G e V we further use tr = 1 / 2 7 r a ' ,

(22)

with a ' - ~ 1 G e V z. We shall represent results for the gluon condensate in reference to the I T E P value (1), and define (l INc)((asl zr)G ~G ~v>lpureSU(2) r=--- (llNc)((aslzr)G~vG~)liTEp

(23)

This ITEP value is of course of no immediate reference for SU(2) pure gauge theory but should still indicate the order of magnitude one expects for the gluon condensate, r should therefore be of order I. This is indeed found. We analyze the Monte Carlo data of refs. [8, 19], and present some of our own results. Our data are obtained on a 84 lattice. The Metropolis method is used for the updating, and the gauge group SU(2) is approximated by its 120 element icosahedral subgroup. Measurements are done after 5 sweeps through the lattice in order to reduce correlations between the following measurements. For the error

574

R. Kirschner et al. / Wilson loops TABLE 3 Monte Carlo results (this work)

4/g2(a)

a'(2, 2)

2.3 2.375 2.45 2.6

0.299+0.031 0.258±0.023 0.235±0.023 0.190±0.012

e s t i m a t e , 12 m e a s u r e m e n t s a r e always g r o u p e d t o g e t h e r . 7 such g r o u p s of m e a s u r e m e n t s h a v e b e e n t a k e n for e a c h fl = 2Nc/g~ value. T h e results for X(2, 2) a r e given in t a b l e 3. T h e v a r i o u s M o n t e C a r l o d a t a , t o g e t h e r with the results of o u r analysis are s h o w n in figs. 1, 2. F i r s t of all, we o b s e r v e t h a t a l r e a d y for r a t h e r small l o o p s ( I = 3, c o m p a r e fig. 1) a m u c h b e t t e r d e s c r i p t i o n of t h e p e r t u r b a t i v e b a c k g r o u n d is o b t a i n e d if o n e e x p a n d s in t e r m s of t h e r e n o r m a l i z e d c o u p l i n g c o n s t a n t i n s t e a d of t h e b a r e one. U s i n g t h e s h o r t - d i s t a n c e e x p a n s i o n (20) including t h e gluon c o n d e n s a t e we o b t a i n g o o d fits to t h e d a t a for I = 2 a n d I = 3, with p a r a m e t e r s •=2:

r = 0.34+0.38,

A~=34+7, (A)

•=3:

r = 0.33+0.13,

--=24+5. Aa

x (3, 3)

..... i

2.2

-----s-_ __i

2.6

J

i

i

3'.0

Fig. 1. Comparison of X(3, 3) [see eqs. (9), (20)] in SU(2) with Monte-Carlo results of ref. [8]. Dashed line: 2-1oop perturbative contribution (bare coupling expansion); Dashed-dotted line: 2-loop perturbative contribution (expansion in terms of renormalized coupling); solid line: fit (A) including gluon condensate.

575

R. Kirschner et al. / Wilson loops

~(2,2)

i

z~

i

2.6

t

I

3.0

|

i1

I~

Fig. 2. Comparisonof X(2, 2) with Monte Carlo results of ref. [19] and this work. Fit (A) is used.

X 2 is about one per data point. We have used Ae~ as a second fit parameter (besides of the gluon condensate). For the renormalization prescriptions used in this paper one has A/Aa = 17.6 [compare eq. (14)]. By allowing A to be an effective parameter Aef~-higher (3-loops) effects are simulated to a certain extent. Aen/Aa is close enough to the calculated value for A/Aa. The perturbative background therefore seems to be essentially under control, especially for larger loops where this is to be expected. We have also tried the bare coupling expansion as background prescription. In this case we use the unknown 3-loop expansion coefficient X~ ~ as second fit parameter. For I = 2 we find I=2: r = 0.68+0.38,

X~ ) = 0 . 0 1 7 + 0 . 0 0 4 .

(B)

This fit is not as good as fit (A) but still reasonable. Using a bare coupling expansion the gluon condensate tends to come out somewhat larger. If one tries to fit the I = 1 (plaquette) data with only one free (3-loop) parameter in the background contribution, one obtains rather large values for the gluon condensate (r = 1 . . . 2). The result is also quite sensitive to whether one fits the logarithm or the plaquette itself. In any case, the quality of the fit is very bad 0( 2 ~ 130 per data point). This was also found recently by Di Giacomo and Paffuti [20]. In analyzing plaquette data these authors have to include 3 free background parameters (corresponding to perturbative contributions up to 5 loops) in order to obtain a good fit. Of course, one cannot hope to carry the perturbative calculations to such an accuracy in the near future. In such a situation the only conclusion

576

R. Kirschner et al. / Wilson loops

should probably be that in the case of the elementary plaquette the perturbative background is badly understood. Wilson loops may also be studied in the adjoint representation. The large-distance behaviour of adjoint loops is expected to be quite different from those in the fundamental representation. An adjoint loop corresponds to an external test charge with the quantum numbers of a gluon. Therefore, it can be screened by dynamical gluons, and no area-law behaviour is expected. On the other hand, the short-distance expansions of loops in the fundamental or adjoint representation are closely related. For SU(2), these loops are given by W ( L , T) ----½(Trl/2 U(C)),

(24)

W"a(L, T)½(Tr~ U(C))

(25)

resp., where the traces (characters) of the fundamental (I = 21) and adjoint (I = 1) representation are related by Trl (U) = (Trl/2 U) 2 - 1.

(26)

Defining ratios of adjoint loops in analogy to eq. (9) one easily verifies

xad(L J) =8X(L J) .

(27)

This simple relation is true for the l-loop and 2-loop perturbative contributions as well as the gluon condensate term. Thus, the gluon condensate is neither enhanced nor suppressed compared to the perturbative background. However, deviations from relation (27) will tell us something about the importance of still higher loop contributions and/or higher order composite operators. Monte Carlo results for adjoint loops have been obtained by Bernard [21]. In fig. 3 we show these results together with the prediction based on eq. (27) and our fits to the ratios of loops in the fundamental representation. Evidently the agreement is quite good. We take this as an important consistency check of our approach.

4. Summary One of our results is the calculation of the 2-loop perturbative contribution to Wilson loop ratios defined through eq. (9). These ratios have a finite continuum limit in terms of the renormalized coupling constant, i.e. the corresponding coefficients do not depend on which cut-off is used in the calculation. We have used dimensional regularization which simplifies the calculation considerably compared to lattice regularization. Strictly speaking, our results should be equivalent to explicit lattice calculations only in the limit of loop sizes much larger than the lattice spacing. However, a comparison of the l-loop terms shows that both

577

R. Kirschner et al. / Wilson loops

.8

.6

•

I=2

*

I=3

i,

.7

m

2.2

2.6

3.0

p

Fig. 3. Wilson loop ratios in adjoint representation. Data of ref. [21] and predictions of fit (A) based on eq. (27).

regularization schemes yield quite similar results for rather small loops already. Deviations are of the order of 10%. This is confirmed by the comparison of our 2-loop results with some explicit lattice calculations [13] which became available when our work was essentially finished. We conclude that dimensional regularization can be used for estimating perturbative contributions to lattice quantities whenever explicit lattice results are not available. We have used the improved knowledge of the perturbative contribution to Wilson loop ratios in order to reanalyze lattice Monte Carlo results in terms of a gluon condensate. Some new Monte Carlo data have also been presented. For loops which are not too small (i.e. ! ~>3) we find that the perturbative tail of the lattice data can be reasonably well understood by the 2-loop result expressed in terms of the renormalized coupling. Expanding in terms of the lattice coupling constant does not lead to acceptable results, neither for the plaquette nor for larger loops. Fits clearly show the presence of a condensate contribution. The effective value for the gluon condensate given from the analysis of the average plaquette is inconsistent with results obtained for larger loops. This may be partly due to the fact that the background estimate is particularly difficult in the case of the plaquette. Some dependence on the loop size is also to be expected and should disappear only for large loops. Our results are consistent with such a behaviour although it would be desirable to demonstrate this better by extending the analysis to larger loops. Our results are also consistent in comparing loops in the fundamental and adjoint representation.

R. Kirschner et al. / Wilson loops

578

Appendix

Here we outline the calculation scheme for the perturbative expansion of the rectangular Wilson loop including two-loop corrections, and collect our results. In configuration space the used Feynman rules are given by (A.1)

"-')----- 0 ( 0 " 2 - - O ' 1 ) o- 1 o- 2

describing the propagation of a test particle along the contour between the points xl = x (0-1) and x2 = x (0-2) and the vertex i

cr

j

=tgAijx (o')

(A.2)

The a ~- are colour matrices with tSqCF = 6ii A iakA~j. Our calculations are performed using dimensional regularization technique (d = 4 - e ) . The regularized gluon propagator in the coordinate representation including one-loop corrections is given by (in the Feynman gauge) 2-e

D~,~(x,e) =D~(x, t 1, e), + gl ~P.2 ~,-.2 x , e ), ,

(A.3)

with (£ is the gauge parameter) 1

{1+~:

r(1-1e)

~r(2-1e)'~,

D~(x, ~, e)-4~r 2 ~/2 ~--~--g, ~)-f-z-~ ~-(1-~)x"x (x2)2_~/2] 2

1 /

(A.4)

10--3e\ 2

D~,v(x, e)= B (2-21e, 2 - ~ e ) ~ - 4 n f + 2 N c ~ - e 1 {(1 1 +e)D,~(x, 2+e

--

e

1, 2e)+2Dl~(x, - 1 , 2e)}

(A.5)

The unrenormalized Wilson loop has the perturbative expansion in 4 - e dimensions 2

W(L1, L2, e ) = 1 + g~16~r2W(1)(L1, L2, e) 4

+ g__L__

(167rZ)2(l (w(1)(L1, L2, e))E+w(z)(L1, Lz, e)).

(A.6)

In eq. (A.6) the well-known exponentiation of the one-loop quasi-abelian contribution is explictly taken into account. The general form of the one-loop result (diagrams la, lb) can be written as follows: 1

W(I)(I-,1, L2, e) = --4CF~ ~

1

Io Io do'20(o'2-o'l)xlD,~(x2-xl, ~, e)x2. dcq

(A.7)

579

R. Kirschneret al. / Wilsonloops In the Feynman gauge we obtain

fo60"20(O"2--O"1)

1

1

W(1)(LI'L2'E)=-4CF(t~2"rr)~/2F(1-18) fo do-1 2(1-e/2) •

X (21 • X 2 ) / ( X 2 - - X 1 )

(A.8)

Due to the scalar product (21" 22), diagrams with gluons connecting orthogonal parts of the contour vanish. The two-loop contribution resulting from the gluon propagator correction is obtained using the propagator (5) and the gauge independence of the one-loop result 1

W]2)

1

do-1

fo d0-20(0-2-°l)2~D2v(x2-x1'g)2~"

(A.9)

The structure of diagrams 2b-e is given by 1

1

1

1

W(z2)=-aNcCv(122rr)eF2(1-~e)fo dO'l Io d0-2Io d0-3fo do'4 (21 " 22)

(2C3 " 24)

(X2_TXl)2(1_e/2) (X4_X3)2(l_E/2).

X 0(O-2--0-1)0(O'4--0-3)

(A.10)

The three-gluon contribution can be evaluated using the form oo

1

1

1

W(32)=-8NcCFI22ef_~ d4 ~Xfo d0-1Io d0-2 fo 60"30(O'2--O'1)0(O'3--O'2) 1 1 1 . t * . v ,-r ×Dgu'(XIX, 1, 8)D~.,(x2-x, 1, e)DT~'(X3--X, 1, 8)XlX2X3

o X[g•'v'(0Xl

o

+

~2")'r'

o

o

+

o o gv"'(0-X2 0--~3)u' g*'"'(OX3 0~l) v']"

(A.11)

The only non-vanishing type of contribution is shown in table a. We obtain W(32) =

8CFNc{ A

(/A'L1)2E +

e

B(LI]

\~]+LI"c-~L2J ~ '

(A.12)

with c~

A

1

y____

y

dx ;o d, (1 + y 2)(1 + x 2 + Y2) 41---~x arctg 41--~x x

+

A = 1.644,

arct

,

(A.13)

R. Kirschneret al. / Wilson loops

580

TABLE 4 Non-vanishing 1- and 2-loop perturbative contributions to the Wilson loop of size Lx x L2 Diagrams

Result

la

8CF{(tZLel)~+(IxL1)E+LI.~-},L2}

lb

8Cv{f(~)(l +e log (L2p-)) + -e h /LX\+LI<.._ ~L2} i--] 2 2 \L21

2a

W(1}(L1,L2, 2e)(-;2 C1q-C2), L

2e

8CFNc{2(P'el2) +5 (lz'L1)2~:+Ll~"'~L2}e

2b

2c

C1 = 31(2nr- 5Nc)

ILl

L1 1 8CFNc{4f(~)('~+ +4

dzarctgzlog

~o L1

2d +2

/T

z

-z

+2h(LZ]+L,~"~L21 \L21 J

L, 2+ 1) arcfg2 ~2

LI I.,/1/L2dz arctgzf(~z-z)+

Lz

2e

L1 2 log (L2/x)+ 1 - log ~ )

L2

8Cr:Nc{A (/xLz)2~ /LI\

2h(~)+Lx','~L2}

R. Kirschneret al. / Wilson loops

( )io f ,fo

B L1

7r2

dx _

x arctg

581

dzx/1--.~y ( l + z 2 ) ( l + y 2 + z 2 ) ( l + z 2 + ( y + x ) 2

)

2z ~/1-+-Y2 (-1-~ l+Y2+z2+YX xx/i~-z2 ( L l x'~ 2 x ~/1----~-~z arctg 1 + y 2 + z 2 + y x -). l+Y2--z2+\~ ] (A.14)

The results for all contributions are collected in table 4. All diagrams of the same type are summed over. Factors which can be reabsorbed after renormalization into the definition of the renormalization charge are not explicitly given. Constant contour-independent pieces are omitted. With the notations (A.6), (A.8)-(A. 10) we get the final results for the perturbative expansion of the unrenormalized Wilson loop:

1 / L I \ +L 1~__>L2}

(A.15) (IxL1)2~

W(2)(L1,L2, e) = 8CF{/30 (/zL~)2~ +e (2t3o+½(Nc+A + C2)) - + 2flof (~-~21)(1+ 2 log (Le/x))+ 2floh(~) + C2f (L~)+ NcC(-~2 ) + LI<--->L2}.

(A.16)

The following notations are introduced:

f(x) = x arctg x -21 log (1 +x2), h(x)=x

x dz l~--j21og(l+z2)-~log2(l+x2),

fo

CIx)=B x)+ ix)16 41ogx Il+x2)arctg2x +2

dz a r c t g z ( 2 1 o g ( z ( x - z ) ) + f ( x - z ) ) ,

C2 = 1(3 1 N c - 10nf).

(A.17)

582

R. Kirschner et al. / Wilson loops

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