Testing nonperturbative Ansätze for the QCD field strength correlator

27 April 2000

Physics Letters B 479 Ž2000. 387–394

Testing nonperturbative Ansatze ¨ for the QCD field strength correlator Dmitri Antonov

1

INFN-Sezione di Pisa, UniÕersita´ degli studi di Pisa, Dipartimento di Fisica, Via Buonarroti, 2 - Ed. B - 56127 Pisa, Italy Received 24 January 2000; received in revised form 13 March 2000; accepted 20 March 2000 Editor: P.V. Landshoff

Abstract A test for the Gaussian and exponential Ansatze ¨ for the nonperturbative parts of the coefficient functions, D nonpert. and which parametrize the gauge-invariant bilocal correlator of the field strength tensors in the stochastic vacuum model of QCD, is proposed. It is based on the evaluation of the heavy-quark condensate within this model by making use of the world-line formalism and equating the obtained result to the one following directly from the QCD Lagrangian. This yields a certain relation between D nonpert. Ž0. and D1nonpert. Ž0., which is further compared with an analogous relation between these quantities known from the existing lattice data. Such a comparison leads to the conclusion that at the distances smaller than the correlation length of the vacuum, Gaussian Ansatz is more suitable than the exponential one. q 2000 Elsevier Science B.V. All rights reserved. D1nonpert.,

PACS: 12.38.Aw; 12.40.Ee; 12.38.Lg Keywords: Quantum chromodynamics; Gluons; QCD vacuum

Among presently existing various phenomenological approaches to QCD, stochastic vacuum model ŽSVM. w1–4x is recognized to be one of the most promising for the investigation of both perturbative and nonperturbative phenomena. However, despite obvious achievements of this model, surveyed in Refs. w3,4x, its field-theoretical status requires further justifications. In fact, an exact derivation of field

1 Permanent address: Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, RU-117 218 Moscow, Russia, Tel.: q39 050 844 536; fax: q39 050 844 538; E-mail: [email protected]

correlators, which are up to now used in this model as a dynamical input, from the QCD Lagrangian is extremely desirable. Once being performed, such a calculation would shed a new light on an interpolation between the perturbative and nonperturbative effects in QCD. That was the main reason for various authors to address this problem analytically both in QCD w5x and other models possessing the confining phase w6x. However, despite these efforts, the exact form of the coordinate dependence of the bilocal gauge-invariant correlator of field strength tensors in QCD Žwhich plays the major role ˆ in SVM. remains unknown. The most reliable result known from evaluations of the bilocal correlator on the

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 3 3 7 - 3

D. AntonoÕr Physics Letters B 479 (2000) 387–394

388

lattice w7–9x Žsee Refs. w10,11x for a review of the recent progress in the lattice calculations of field strength correlators. is that the nonperturbative parts of the two coefficient functions parametrizing this correlator, D nonpert. and D1nonpert., rapidly decrease at the distance, which is usually referred to as the correlation length of the vacuum, Tg . The latter one is equal to 0.13 fm for the SUŽ2.-case w7x and is about 0.22 fm for the SUŽ3.-case w8x. The two standard nonperturbative Ansatze for ¨ D nonpert. and D1nonpert., used both in lattice investigations and in phenomenological applications of SVM to the evaluation of various QCD processes, are the Gaussian and the exponential ones. In the present Letter, we propose a nontrivial analytical test of these Ansatze. It is based on the evaluation of the ¨ heavy-quark condensate by making use of the world-line formalism w12x Žsee Ref. w13x for a recent review. 2 and further comparison of the obtained result with the one, which follows directly from the QCD Lagrangian w15x. Such a comparison then leads to some relations between D nonpert. Ž0 . and D1nonpert. Ž0. for both Ansatze. Among those, as we ¨ shall eventually see, the one obtained for the Gaussian Ansatz satisfies the existing lattice data w8–11x concerning such a relation much better than the other one. Then, since the typical sizes of the heavy-quark trajectories in the problem under study are of the order of Tg , this leads to the conclusion that at the distances smaller than Tg , Gaussian Ansatz is more suitable than the exponential one. This result is important for future lattice simulations as well as for the applications of SVM to the high-energy hadron scattering w16x. Our method of derivation of the quark condensate is based on the well known formula

² cc :s y

1 E V Em

²G

Ama

:A

a m

,

Ž 1.

where V is the four-volume of observation and m is the quark mass. Next, the average ² . . . :Ama on the R.H.S. of Eq. Ž1. is implied w.r.t. the gluodynamics 2 action in the Euclidean space-time, 14 Hd 4 x Ž Fmna . ,

with a s 1, . . . , Nc2 y 1 and Fmna s Em Ana y En Ama q gf ab cAmb Anc being the Yang-Mills field strength tensor. Finally in Eq. Ž1., ² G Ama :A a is the averaged m one-loop quark effective action Ži.e. one-loop quark self-energy. defined as w12,13x

²G

Ama

:A

a m

q`

The combination of the SVM with the world-line formalism has been studied in various different contexts in Ref. w14x.

eym

T

T

H0 dt ž

=exp y

=

1

2

T

HP D x HA Dc m

1 4

T

/

H0 dt Ž A

tr P exp yig

Nc

m

x˙m2 q 12 cm c˙m

½ ¦

˙

m xm

; 5

ycm cn Fmn .

Ama

y1 .

Ž 2.

Here, the subscripts P and A denote the periodicity properties of the respective path-integrals, cm’s are antiperiodic Grassmann functions, and Am ' Ama T a with T a ’s standing for the generators of the SUŽ Nc .-group in the fundamental representation. We have also adopted the standard normalization conditions ² G w 0 x :Ama s 0 and ²W Ž 0 . :Ama s 1, where 1

²W Ž C . :Am ' a

Nc

¦

ž

T

H0 dt A

tr P exp yig

˙

m xm

/;

Ama

Ž 3.

is just the Wilson loop, which is defined at the contour C parametrized by the vector xmŽt .. It turns out that the gauge-field dependence of Eq. Ž2. can be reduced to that of the Wilson loop Ž3. only. That is because, as it has been demonstrated in Ref. w17x, the spin part of the world-line action can be rewritten by means of the operator of the area derivative of the Wilson loop and becomes separated from the average ² . . . :Ama : 1 Nc

¦

T

H0 dt Ž A

tr P exp yig

ž

m xm y cm cn

˙

d

T

H0 dtc c ds

s exp y2 2

dT

H0

s y2

m n

mn

Ž x Žt . .

Fmn .

/

;

Ama

²W Ž C . :Am . a

Ž 4.

D. AntonoÕr Physics Letters B 479 (2000) 387–394

Within the SVM, the Wilson loop Ž3. can further we rewritten by virtue of the non-Abelian Stokes theorem and the cumulant expansion as follows w1–4x

group invariant coefficient functions D and D1 as follows

¦² Fmn Ž z . F Ž z , zX . Fl r Ž zX . F Ž zX , z . :;A

²W Ž C . :Am a

s 1 , Nc

½

tr exp y

=

HS w C x

d sl r Ž zX .

X

X

¦² Fmn Ž z . F Ž z , z . Fl r Ž z . F Ž z , z . :;A

a m

5

.

Ž 5. Here ²² O O X ::Ama ' ² O O X :Ama y ² O :Ama² O X :Ama , and S w C x is a certain surface bounded by the contour C and parametrized by the vector zmŽ j . with j s Ž j 1 , j 2 . standing for the 2D-coordinate. This surface is usually chosen to be the one of the minimal area for a given contour C. We have also denoted for brevity z ' z Ž j ., zX ' z Ž j X . and introduced the notation F Ž z, zX . for the parallel transporter factor taken along the straight line joining the points zX and z:

Nc

ž

z

P exp yig

Hz A Ž u . du X

m

m

/

.

Next, in a derivation of Eq. Ž5., the so-called bilocal approximation has been employed, according to which the irreducible gauge-invariant bilocal field strength correlator Ž cumulant. ² Fmn Ž z . F Ž z , zX . Fl r Ž zX . F Ž zX , z . : Ama dominates over all the cumulants of higher orders Žsee e.g. Ref. w18x for details of a derivation of Eq. Ž5. and discussion of related approximations.. Finally, it is worth commenting that the factor 1r2! on the R.H.S. of Eq. Ž5. is simply due to the cumulant expansion, whereas the factor 1r4 is due to the Žnon-Abelian. Stokes theorem with the usual agreement on the summation over all the indices Žnot only over those, among which the first one is smaller than the second, used in Refs. w1–4,18x.. Let us further parametrize the bilocal cumulant according to the SVM by the two renormalization-

¦

2 E zm

E q

X

1

E

1 q

HS w C x

N

Nc

2! 4

=

F Ž z , zX . '

1ˆ Nc=Nc

1 g2

d smn Ž z .

;

389

E zn

½

a m

Ž dml dnr y dm r dnl . D Ž Ž z y zX . 2 .

Ž Ž z y zX . l dnr y Ž z y zX . r dnl .

Ž Ž z y zX . r dm l y Ž z y zX . l dm r .

= D1 Ž Ž z y zX .

2

.

5

.

Ž 6.

Here, 1ˆ Nc=Nc is the unity matrix, and we have used the standard normalization condition for the functions D and D1 w3x with a mn

N'

¦Ž F

Ž 0. .

2

;

Ama

24 Ž D Ž 0 . q D1 Ž 0 . .

.

By virtue of the Žusual. Stokes theorem, the contribution to the double surface integral standing in the argument of the exponent on the R.H.S. of Eq. Ž5., which emerges from the D1-part of the cumulant Ž6., can be rewritten as a boundary term. After that, we eventually arrive at the following expression for the Wilson loop within the SVM

²W Ž C . :Am a

½

s exp y

g2 8 Nc

HS w C xd s

N 2

= D Ž Ž z y zX . q dxm dxmX

2

mn

Ž z.H

d smn Ž zX .

S wC x

. q`

EC EC HŽ xyx . dt D Ž t . X 2

1

5

.

Ž 7.

Next, one can extract the volume factor V from the heavy-quark self-energy Ž2. by splitting the coordinate xmŽt . into the center-of-mass and the relative coordinate w13x, xmŽt . s xm q ymŽt ., where the center-of-mass is defined as follows xm s T1 H0Tdt xmŽt .. The empty integration over x then obviously yields the factor V. After that, owing to the fact that d xmŽt . s d ymŽt ., we can also replace drdsmn Ž x Žt ..

D. AntonoÕr Physics Letters B 479 (2000) 387–394

390

in Eq. Ž4. by drdsmn Ž y Žt .., which is possible due to the formula w19x

d dsmn Ž x Ž t . . q0

s

(

d

X X

Hy0 dt t

1

ž

d xm t q

2

2

t X d xn t y

/ ž

1 2

.

tX

/

All the above mentioned considerations lead to the following intermediate expression for the heavyquark self-energy Ž2.

²G

Ama

:A

a m

dT

q`

H0

s y2V

T

eym

T

H0 dt ž

=exp y

½ ž ½

=exp y

2

T

HP D y HA Dc m

y˙ q 12 cm c˙m

m n

g2 8 Nc

HS w G xd s

mn

q dym dymX

/

d

T

mn

=

m

1 2 4 m

H0 dtc c ds

= exp y2

Ž yŽt . .

HS w G xd s

N 2

mn

/

Ž w.

2 Ž wX . D Ž Ž w y wX . .

q`

EG EG HŽ yyy . dt D Ž t . 1

X 2

5 5

y1 .

Ž 8.

Here, the contour G is parametrized by the vector ymŽt ., and the surface S w G x, parametrized by the vector wmŽ j ., is bounded by this contour. Next, it can straightforwardly be shown that in the heavy-quark limit under study the typical quark trajectories are small. Indeed, let us consider the free part of the bosonic sector of the world-line action standing in the exponent on the R.H.S. of Eq. Ž8., 1 Sfree s

4 1

s

T

2 m

H0 dt y˙ T

reparametrization transformations t ™ s Ž t ., d srdt G 0, it is convenient to choose the proper-time parametrization t s srm. Here, s is the proper length of the contour G so that T s Lrm, where L ' Lw G x s HG ds ' Hssinfin d s y˙m2 Ž s . is just the length of G . Within this parametrization, H0Tdty˙m2 Ž t . s mHG ds = Ž dymŽ s .rds . 2 s mL, since Ž dymŽ s .rds . 2 s 1 by the definition of the proper time. Therefore Sfree s mL, which means that the typical quark trajectories are so that L F 1rm, i.e. they are really small in the heavy-quark limit. However, according to the general concepts of the SVM, the distances smaller than the correlation length of the vacuum, Tg , are forbidden for a test quark, i.e. L cannot be arbitrarily small. That is because SVM is an effective low-energy theory of QCD with Tgy1 playing the role ˆ of the UV momentum cutoff. Therefore, infinitely small contours C Žwhose contribution to the Wilson loop Ž7. would obviously dominate if they are allowed. are forbidden within the SVM. Moreover, C’s should be so that not only their lengths obey the inequality L G Tg , but even among such contours those which lie inside the sphere ST g r2 Ž x . are forbidden. We conclude that within the SVM, typical heavy-quark trajectories are located from outside to the circle of the radius Tgr2 and with the center placed at x, nearby to it Žin the sense that their lengths L’s obey the inequality L F 1rm.. Therefore, the correlation length of the vacuum is related to the area S inside such a trajectory according to the formula

2 m

H dty˙ 2 0

Ž t . q m2 T

Ž t. q

m2 T 2

.

Here, in the last equality we have performed a rescaling t s 2t , T old s T new r2. Among all the

Tg2 ,

4

p

S

Ž 9.

with a good accuracy. We also see that for any two points ymŽt . and ymŽt X ., the following inequality holds < y Ž t . y y Ž t X . < F Tg .

Ž 10 .

Note in particular that since typical sizes of Wilson loops in the problem under study are approximately equal to Tg , our final conclusion on the advantage of the Gaussian Ansatz w.r.t. the exponential one will be valid at the distances < x < F Tg . Let us now evaluate heavy-quark condensate Ž1. with the Gaussian and exponential Ansatze ¨ for the

D. AntonoÕr Physics Letters B 479 (2000) 387–394

nonperturbative parts D nonpert. and D1nonpert. of the functions D and D1. The first of these Ansatze ¨ 2 2 reads D Ž x 2 . s D Ž0.eyx r T g , D1 Ž x 2 . s D1Ž0. = 2 2 eyx r T g , where from now on we shall denote for brevity D nonpert. by D and D1nonpert. by D1. By making use of the result of Ref. w20x for the string tension Žsee also Ref. w18x for a detailed derivation., we get due to Eq. Ž9. the following leading D-function contribution

HS w G xd s

2

mn

X 2

X

Ž w.H

SwG x

d smn Ž w . D Ž Ž w y w .

X m

.

2

s D1 Ž 0 .

Ž yy yX . 2

EG EG 2

, D1 Ž 0 . Tg2 P

Tg2

T g2 X X m yn yn

EGdy EGdy m

2 s 2 D1 Ž 0 . Smn .

This leads to the following expression for the Wilson loop standing on the R.H.S. of Eq. Ž8.

½

8 Nc

HS w G xd s

N 2

HS w G xd s

=

mn

mn

Ž w.

2 Ž wX . D Ž Ž w y wX . .

q dym dymX

X 2

dt D1 Ž t .

5

2

AGa uss Smn , eyA ,

where AGauss

s 2 D1 Ž 0 .

Tg2

, 2 D1 Ž 0 .

Tg2

q` y

X 2

EGdy EG

dymX

EGdy EG

dymX

m

m

ž

ž

1q

1y

't Tg

< y y yX < Tg

< yyyX <

/

Ž y y yX .

y

e

Tg

2

/

Tg2

Here in the evaluation of the D- and D1-dependent parts, Eqs. Ž9. and Ž10., respectively, have been applied. We see that for the exponential Ansatz, Eq. Ž11. remains valid with the replacement AGauss ™ Aexp s 2 AGauss . In order to proceed with the evaluation of the path-integral Ž8., it is useful to linearize the quadratic Smn-dependence of Eq. Ž11. by introducing the integration over an auxiliary antisymmetric tensor field. Clearly, since Smn depends on the contour G as a whole, this field will be space-time independent. Introducing for brevity the common notation A for both AGauss and Aexp , we have

ž

A exp yA

m , n s1

1 4 D Ž 0 . q D1 Ž 0 .

24 Nc D Ž 0 . q D1 Ž 0 .

P

as

a mn

Ž 0. .

ž

3

dB Hy` mŁ -n

mn

2

;

Ama

Ž 11 .

ž

=exp y

2 Bmn

8A

y iBmn Smn

Ý m- n

q`

Ž 8p A .

¦Ž F p

/

Smn2 sexp y2 A

Ý

s

p '

m

4

q`

EG EG HŽ yyy . 2

X m

EGdy EGdy HŽ yyy . dt e

2 s 4 D1 Ž 0 . Smn .

1

X 2

s D1 Ž 0 . Tg2 dym dymX ey

g2

1

X 2

q`

EGdy EGdy HŽ yyy . dt D Ž t .

exp y

q`

X m

m

For further purposes let us rewrite this result as 2 8 D Ž0. Smn , where Smn ' Smn w G x s Smn w C x ' EG ym dyn is the tensor area associated with the contour G Žor C .. Here, we have used the fact that for contours under study, which slightly deviate from the 2 circle, S 2 s 12 Smn . By virtue of Eq. Ž10., we also have m

with a s ' g 2r4p . Note that the behaviour of small 2 Wilson loops of the type eyc onst.S was for the first time obtained in Ref. w2x. In the same way one can treat the exponential 2 Ansatz, D Ž x 2 . s D Ž0.ey 'x r T g , D1 Ž x 2 . s D1Ž0. 2 =ey 'x r T g . In that case we have: 2HS w G x d smn Ž w . = 2 2 HS w G x d smn Ž wX . D Ž Ž w y wX . . , 16 D Ž0. Smn ,

EGdy EGdy HŽ yyy . dt D Ž t .

, 4Tg2 D Ž 0 . d 2 teyt P S , 16 D Ž 0 . S 2 .

H

391

/

Smn2

/

D. AntonoÕr Physics Letters B 479 (2000) 387–394

392

1

q`

s

Ž 8p A .

2 Bmn

ž

Hy` mŁ -n

3

ž

=exp y

dBmn ey 8 A

spinor QED, i.e. one-loop electron effective action in a constant background field Žsee e.g. w13x., we have 1 2 1 T D ym Dcm exp y dt y˙ q cm c˙m 4 m 2 P A 0

/

H

4

i 2

/

Bmn Smn .

Ý m , n s1

ž

d

H0 dtc c ds

exp y2

m n

mn

T

žH

s exp i

Ž yŽt . .

/

Ama

:A

q`

Ž 8p A . q`

=

H0

3

dT T

2 Bmn

ž

y mn e 8 A

dB Hy` mŁ -n

eym

2

1 s

Ž 4p T .

/

Here, the standard notations were adopted:

(

2

T

H0 dt

yiBmn cm cn

/

ž

1

Ama

:A

4

1 2

1 y

D

Ž 4p T .

cm c˙m y

¶ • ß.

,

2V

q`

sy i 2

Bmn ym y˙n

P

1 3

3

q`

H0

2 Bmn

ž

2 y mn Bmn e 8 A

dTT

eym

2

/

T D

.

2

Next, the pole at D s 4 emerging in the integral over proper time,

2

Taking now into account the known one-loop expression for the Euler-Heisenberg Lagrangian in

,

dB Hy` mŁ -n

Ž 4p T . Ž 12 .

2

a m

m

A

y˙m2 q

2

with E ' i Ž B14 , B24 , B34 . , H ' Ž B23 ,y B13 , B12 . . 2 Next, due to the factor eym T in Eq. Ž12., in the heavy-quark limit under study only small values of the proper time T are sufficient, and the R.H.S. of Eq. Ž13. can be expanded in powers of T. Then, the leading term of this expansion, which in the expansion of the fermionic determinant corresponds to the diagram with two external legs of the Bmn-field, 2 reads T 2 abcotŽ aT .cothŽ bT . y 1 s T3 Ž b 2 y a 2 . q 2 O Ž T 4 Ž E P H . . . This leads to the following expression for the heavy-quark self-energy Ž12.

Ž 8p A . =exp y

2

a 2 s 12 E 2 y H 2 q Ž E 2 y H 2 . q 4 Ž E P H .

T

m

2

Ž 13 .

²G

° ~H D y H Dc ¢ P

2

T 2 abcot Ž aT . coth Ž bT . y 1 .

D

(

a m

2V

D

q Ž E 2 y H 2 . q 4Ž E P H .

/

sy

y

Ž 4p T .

ey 2 Bmn Smn

we get upon the insertion of Eq. Ž11. into Eq. Ž8. the following expression

²G

1

/

b 2 s 12 y Ž E 2 y H 2 .

i

dt Bmn cm cn ,

0

ž

H

i y Bmn ym y˙n y iBmn cm cn 2

Note that only in this equation we have emphasized explicitly the summation over indices in order to avoid possible misleadings. In what follows, in the X we shall as usual expressions of the type Omn Omn assume the summation over all the values of indices. The one-loop heavy-quark self-energy Ž8. has now reduced to that in the constant field Bmn , which should eventually be averaged over. Indeed, since due to the Stokes theorem, Bmn Smn s HS w G x d smn Ž w . =Bmn Ž w ., the following equality holds Žcf. Ref. w17x. T

H

q`

H0

dTT

eym

2

T

m Dy4 D

Ž 4p T .

2

s

D

Ž 4p .

2

ž

G 2y

D 2

/

,

D. AntonoÕr Physics Letters B 479 (2000) 387–394

can be subtracted by making use of the MS-prescription ŽClearly, the additional constant, which appears if we use e.g. the MS-prescription, does not contribute to the quark condensate, since the derivative of this constant w.r.t. m in Eq. Ž1. vanishes... This finally yields

²G

Ama

:A

a m

2V sy

L2

1

3 2 Ž 8p A . 3 Ž 4p .

ln

m2 2 Bmn

q`

393

Table 1 Theory

D1Ž0.r D Ž0.

Quenched approximation at the distances 0.1 fm F < x < F1 fm Quenched approximation at the distances 0.4 fm F < x < F1 fm The same as in the previous line, but with the perturbative-like part of the fit fixed to zero Full QCD with the quark mass Žin lattice units. equal to 0.01 The same as in the previous line, but with the quark mass equal to 0.02 Average over the above values

0.22"0.05 0.29"0.13 0.34"0.04

0.13"0.08 0.13"0.07 0.20"0.10

2 y mn Bmn e 8 A

P

dB Hy` mŁ -n 2V

sy

1

3 Ž 8p A . 3 Ž 4p .

L2 ln 2

7

1

3

5

2

2

2

2

m2

P 2 p A P Ž 8p A . P 6

sy

2V A

p

2

L ln

m

with L standing for the UV momentum cutoff. Owing to Eq. Ž1., at Nc s 3 we arrive at the following value of the heavy-quark condensate ² cc :s y2 Arp 2 m, which should be equal to the result following directly from the QCD Lagrangian Ži.e. from the corresponding triangle diagram. w15x

² cc :s y

1 12 m

P

as

a mn

¦Ž F p

Ž 0. .

2

;

Ama

.

For A s AGauss , this yields the following relation between the nonperturbative parts of the functions D and D1 at the origin: D1Ž0. s 12 D Ž0.. Similarly, for A s Aexp , we obtain D1Ž0. s 5 D Ž0.. These relations should be compared with the results of those lattice measurements w8–11x, whose x 2 is minimal. Such results can be summarized, following the data reviewed in Ref. w11x, in the form as done in Table 1. The table indicates that the result obtained within the Gaussian Ansatz is closer to the existing lattice data than the other one. It is worth noting that as it has for the first time been mentioned in Ref. w21x and then confirmed in Ref. w22x, the connection of SVM with the heavy quark effective theory requires that at large distances

the exponential Ansatz is more preferable. However, this does not affect the above statement on the advantage of the Gaussian Ansatz at small distances. One more argument in favour of this statement was presented in Ref. w21x. There, it was demonstrated that the requirement for the operator product expansion to be an asymptotic one demands that at small distances the bilocal correlator should be a quadratic function of the relative distance. In conclusion of the present Letter, we have proposed a nontrivial analytical test for the Gaussian and exponential Ansatze ¨ for the nonperturbative parts of two coefficient functions, which parametrize the gauge-invariant bilocal field strength correlator in QCD within the SVM. It was based on the evaluation of a heavy-quark condensate within this model by making use of the world-line formalism and further comparison of the obtained result with the exact one, following directly from the QCD Lagrangian. The outcome of this investigation suggests that the Gaussian Ansatz satisfies the existing lattice data better than the exponential one. This means that at the distances < x < F Tg , the former Ansatz is more favourable than the latter one. Such a conclusion is important for further applications of the SVM to the evaluation of various QCD processes as well as for the lattice measurements of field correlators.

Acknowledgements The author is indebted to A. Di Giacomo for critical reading the manuscript, useful suggestions

394

D. AntonoÕr Physics Letters B 479 (2000) 387–394

and discussions, and E. Meggiolaro for bringing his attention to Ref. w11x and useful discussions. Besides that, he has benefitted from discussions and correspondence with N. Brambilla, H.G. Dosch, M.G. Schmidt, C. Schubert, Yu.A. Simonov, and K.L. Zarembo. He is also greatful to Prof. A. Di Giacomo and the staff of the Quantum Field Theory Division of the University of Pisa for kind hospitality and INFN for financial support.

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