The string tension from continuum QCD

Volume

152, number

PHYSICS

1,2

28 February

LETTERS

1985

THE STRING TENSION FROM CONTINUUM QCD

E. BAGAN ’ , J.I. LATORRE and R. TARRACH Department Received

of Theoretical Physics, University of Barcelona, Dhgonal647,

19 December

QCD and including dynamic quark degrees of freedom within a sum rule of the vacuum condensates. The output is &= 0.5 f 0.05 GeV and is

The QCD sum rules approach [I] allows one to compute meson and baryon properties in terms of a few parameters which characterize the physical ground state of the theory. Within continuum QCD there is no reliable way in which these vacuum condensates can be calculated. They are therefore determined from experiment. In lattice QCD however, vacuum condensates can be computed [2]. But there is of course more interest in the computation of quantities of more immediate physical significance. One outstanding instance is the string tension [3]. The aim of this note is to compute the string tension in continuum QCD within an approach based on the operator product expansion. The comparison of the large distance string tension with the vacuum condensates is performed in the manner we have learned from the QCD sum rules approach. The results are quite accurate and show that both approaches lead to compatible results. They also show that quarks do not seem to contribute very much to the string tension. The neglect of the dynamic quark degrees of freedom in lattice computations seems thus numerically justified. The Wilson locip, in euclidean space-time, is defined as $,4;(x)

:X@ dx,

C

Its perturbative ’ Universitat

expansion

Autonoma

Barcelona, Spain

1984

The string tension is computed in continuum like approach. The inputsare the standardvalues very insensitive to quarks.

W, = NC 1 Tr P exp(-ig)

08028

is well understood

. >

(1)

[4]. To

de Barcelona.

0370-2693/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

one loop order and in dimensional regularization there is only one type of singularity. It comes from the cusps of the contour. The result can be taken from ref. [S J. It is w, = 1 + C&2/27?) X {e-l

t In&L) t 1 +f(L/T)

t L ++T} ,

(2)

for a rectangular L times T Wilson loop. Here E = 4 -d, /J is the renormalization scale, Cr: = (N; - l)/ 2Nc and f’(x)=xarctgx-

i ln(1 +x2).

(3)

We have computed the dominant and subdominant nonperturbative contributions to the Wilson loop. We have followed technically Shifman’s work [6]. The result is

- (8/48)fab,(G~vG~,G~,)}.

(4)

It should be noted that the last term has the opposite sign compared to Shifman’s result. It is easy to see that the,problem lies in the continuation from minkowskian to euclidean space-time. We have therefore performed the computation twice, one time in minkowskian space-time and the result then continued to euclidean space-time and another time directly in euclidean space-time. 113

Volume 152, number 1,2

PHYSICS LETTERS

28 February 1985

The condensates are in euclidean s p a c e - t i m e . Their relation with the standard minkowskian condensates [1] is Q a odGuvGuv) = a(G 2) = (0.04 + 0.01) GeV 4 ,

([Do~, G a ] [Dr, Gvael) = I-sg ' 2J V F I J v2c - 1)(~-q)2 . . . . .

(~-q) = ( - 0 . 2 5 GeV) 3 ,

(5)

. a b c = -g3(fG3) ~-- - 1 .la(G 2) GeV 2, g 3Jabc(GuvGv~Gau) where the rhs expressions are minkowskian. The values for (G 2) and (~-q) are the standard ones and the value for (G 3) has been taken from ref. [7], which is perfectly compatible with the value given in ref. [1 ]. For the second condensate of eq. (5) both factorization and flavour independence for N F flavours have been assumed. Once we have the small contour expansion of the Wilson loop, it is our aim to compare it with its wellknown large contour behaviour,

W = Ce - a L T - a ( L + T ) .

1

0W 0W

or

1 02W

W OTOL '

(7)

and assume, in the spirit o f the sum rules approach, that there exist some intermediate values of L and T for which it is meaningful to compare this quantity as computed from the small contour expressions of eqs. (2) and (4) with the same as computed from eq. (6). This leads to (N C = 3, N F = 3) o = (8a/37r) [T - 2 arctg(L/T) + L - 2 arctg(T/L)

+ 1/LT] + }rrLT(oeG 2) --~--4LT(L 2 + T 2) X [~

n2a2(~q) 2 + i~g3(fG3)] .

(8)

Finally we will renormalization group improve our results. This means that the a which appears in the perturbative contribution will be substituted b y the running coupling constant

-~(LT)/rr = - 4 / 9 ln(L T A 2 ) , where [5]

114

,I i[ L

Fig. 1. x ~ a s a function of L and T, both varying from near 0 up to 3.5 GeV -1 .

(6)

In order to get rid o f C, a and/~ we will consider the quantity

W20L

"~

(9)

A / A f f g = 1~ evE,

7E = 0 . 5 7 7 ....

(10)

Recall that (aG 2) is renormalization group invariant. We will not bother about renormalization group improving the other condensates because the contribution of the quark condensate is very small and the contribution o f g 3 ( ) G 3) is uncertain and eventually not very important either. Numerically the sum rule (8) is very well behaved. It has an extensive plateau in both variables L and T as shown in fig. 1. In this large region of stability which starts at values around L = T ~--2.7 GeV 1 we find for the string tension x / ~ = 0.5 + 0.05 GeV. The error takes into account the uncertainties coming from the estimation of the condensates and from the value of AM--g (which has been varied over the region 0.10 to 0.15 GeV). The region of stability as well as the dependence of o on A are shown in fig. 2 along the line L = T. Notice that for small surfaces the perturbative part dominates and there is no area law. For large surfaces higher order non-perturbative contributions become dominant and the approach is meaningless. Nevertheless our results show an intermediate region o f stability where we think they can be trusted. The non-perturbative contribution to o is strongly

Volume 152, number 1,2

PHYSICS LETTERS

28

February 1985

~o- (GeV) [ 4.

[-

3.5 3,

2.5

2. 1.5 I.

0.5

O.

O.

I

I

I

I

I

I

I

0.5

1

1,5

2.

2.5

3.

3.5

Fig. 2. ~/~along the line L = Tfor Am = 0.13 GeV (solid line), AM-"S dashed line). dominated by (G2>. Neither the nor the <~-q>2 terms contribute significantly to its value. More specifically contributes less than 1(~o than what 2 not more than 1%. All this make us believe that second order perturbative corrections, perturbative corrections to (G2), renormalization group improvement of the and <~q>2 contributions, inclusion of m(~-q> and <6"4> nonperturbative terms, etc. would not alter substantially our result and in any case suffer, after very long computations, from uncertainties due to the lack of knowledge of the values of the higher order condensates and due to the error with which we know the value of (G2). We conclude that the sum rule approach, which allows to estimate the string tension from Continuum QCD, confirms the commonly accepted numerical value of o and its gluonic nature.

=

L:T (GeV")

0.10 GeV (dashed line) and AM-~= 0.15 GeV (dot-

This work has been supported financially by the CAICYT project number 0435.

References [1 ] M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B147 (1979) 385,448. [2] J. Kripfganz, Phys. Lett. 101B (1981) 169. [3] M. Creutz, Phys. Rev. Lett.45 (1980) 313; N.A. Campbell, C. Michael and P.E.L. Rakow, Phys. Lett. 139B (1984) 288. [41 A.M. Polyakov, Nuel. Phys. B164 (1980) 171; R.A. Brandt et al., Phys. Rev. D24 (1981) 879. [5] R. Kirschner, J. Kripfganz, J. Ranft and A. Schiller, Nucl. Phys. B210 (1982) 567. [6] M. Shifman, Nucl. Phys. B173 (1980) 13. [7] A. Di Giacomo, K. Fabricius and G. Paffuti, Phys. Lett. l18B (1982) 128.

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