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TRAN HUU PHAT Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, VN . NGUYEN TUAN ANH

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Fifth International Conference

ON FLAVOR PHYSICS

24 - 30 Sept. 2009

______________________________________________________________________

ON THE ANSATZ FOR

GLUON PROPAGATOR

TRAN HUU PHAT

Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, VN.

NGUYEN TUAN ANH

Institute for Nuclear Science and Technique, 179 Hoang Quoc Viet, Hanoi, VN.

PHAN HONG LIEN

Military Academy of Technology, 100 Hoang Quoc Viet, Hanoi, VN.

I. Introduction.

II. Minimum of Effective Potential.

III. Conclusion and Discussion.

We consider the commonly accepted propagators, which behave like

and

in IR region by means of the Cornwall-Jackiw-Tomboulis (CJT) effective action.

It is shown that the minimum of the effective potential corresponds to μ = 0, υ≠0.

This implies that these two modeled gluon propagators are ruled out.

The perturbative calculations of QCD is a powerful tool for considering the strong interaction at high energies.

However, low-lying hadron properties emerge in the low

energy region, which is dominated by nonperturbative effects of QCD.

In this connection, great attempts were made for searching a low energy models which incorporate both relevant symmetries and confinement of QCD.

The NJL model and its extensions share a lot of conceptually important features of QCD, but do not involve confinement [1-4].

The confinement mechanism is assumed to be generated by the IR behavior of the gluon propagator G(k).

I. INTRODUCTION

Two forms for G(k) in IR region are usually accepted so far,

a) (1.1)

b) (1.2)

The form (1.1) has been considered for the first time by Pagels [5] and, subsequently, by many others [6-8].

The form (1.2) is treated to be the regularization of (1.1) and it leads to the desired confinement mechanism [9] and, as a consequence, is the main issue for a lot of considerations [10, 11].

In a series of papers [10], Roberts and his collaborators developed the so-called global color symmetry (GCS) model approximate to QCD, in which the Euclidean generating functional reads

where

is the part of the effective action corresponding to quark interactions via gluon exchange with the nonperturbative gluon propagator

taken in the Feynman gauge.

We showed that [12], for Ggiven by (1.2), although

the confinement mechanism is produced, the minimum

of the effective potential corresponds to μ = 0.

In an effort to extend the NJL model including confinement, Bel’kov, Ebert and Emelyanenko [13] propose a new form for the gluon propagator:

(1.3)

starting from the assumption that the condensation of constant gluon field is not gauge-invariant; here μ and υ are two parameters, θis the step function and Λ is an UV cutoff.

If (1.3) is a priori postulated as an ansatz for the IR behavior of the gluon propagator; it clearly expresses the simplest generalization of the NJL model that contains confinement, too.

In the present paper μ and υ are treated to be two variation parameters of effective potential V . Their values, corresponding to the minimum of V , would probably be meaningful for the whole model.

The effective potential in the two-loop approximation for the GSC model reads

(2.1)

in which S(p) is the quark propagator represented

in the form

(2.2)

Inserting (1.3) and (2.2) into (2.1) yields

(2.3)

where .

From (2.2) the Schwinger-Dyson equations are derived simply

(2.4)

Here, for convenience, we introduce

(2.5)

It is know that for A and B satisfying Eqs. (2.4),

V [A,B] is just the energy density of vacuum state.

Now let us determine the minimum of the effective potential. Solving the system of algebraic equations (2.4) yields four solutions depending upon and : two solutions are real and positive, and other two are complex. Here we do not write them down due to their complicated expressions.

We select only two real solutions and insert them respectively into (2.3). Then the corresponding effective potentials turn out to be functions of two parameters and , and their values at the minimum of effective potentials are given by

(2.6)

where V1 and V2 correspond, respectively, to the first and second solutions of (2.4).

Setting Λ = 700MeV and m0 = 14MeV as input values

the − dependence of and

are shown, respectively, in Fig. 1 and Fig. 2,

which indicate that the minimum of V1 no longer occurs and the minimum of V2 locates at

This result proves one again that the confinement mechanism generated by (1.2) is not accepted, and the situation is not improved even by ansatz (1.3).

Fig. 1

Fig. 2

We have demonstrated that the simplest version of the QCD-motivated NJL model, involving confinement, does not succeed: the confinement mechanism does not correspond to the minimization of the effective potential.

In this respect, it is required that an ansatz for the IR behavior of gluon propagator has simultaneously to satisfy two conditions:

• it generates the confinement and

• it is compatible with the minimization of the effective potential.

In recent years several attempts are devoted either to studying exact infrared properties of gluon propagator [14-19] or to modeling gluon propagator [20].

To testify whether or not these results fulfill two preceding conditions is of interest.

[1] D. Ebert and H. Reinhardt, Nucl. Phys. B271 (1986) 88.

[2] M. Wakamatsu and W. Weise, Z. Phys. A331 (1991) 50.

[3] R. Ball, Int. J. Mod. Phys. A5 (1990) 4391.

[4] D. Ebert, H, Reinhardt and M. K. Volkov, Prog. Part. Nucl. Phys. 33 (1994) 1.

and refernces therein.

[5] H. Pagels, Phys. Rev. D15 (1997) 2991.

[6] B. A. Arbuzov, Sov. Phys. Part. Nucl. 19 (1998) 1.

[7] L. von Smekal, P. A. Amundsen and R. Alfoker, Nucl. Phys. A529 (1991) 633.

[8] V. Sh. Gogohia, Phys. Rev. D40 (1989) 4157;

V. Sh. Gogohia and B. A. Magradze, Phys. Lett. B217 (1989) 162;

V. Sh. Gogohia, Int. J. Mod. Phys. A9 (1994) 759;

V. Sh. Gogohia, G. Kluge and M. Prisznyak, Phys. Lett. B368 (1996) 221; ibid. B378 (1996) 385.

[9] H. J. Munczeck and A. M. Nemirovsky, Phys. Rev. D28 (1983) 181.

[10] R. T. Cahill and C. D. Roberts, Phys. Rev. D32 (1985) 2419;

R. T. Cahill, C. D. Roberts and J. Praschifka, Phys. Rev. D38 (1987) 2804;

L. C. L. Hollenberg, C. D. Roberts and B. H. J. McKeller, Phys. Rev. D46 (1992) 2057.

[11] A. G. Williams and G. Krein, Ann. Phys. (NY) 210 (1991) 464.

[12] Tran Huu Phat and Nguyen Tuan Anh, Nuo. Cim. A110 (1997) 337.

[13] A. A. Bel’kov, D. Ebert and A. V. Emelyanenko, Nucl. Phys. A552 (1993) 523.

[14] A. Cucchieri and T. Mendes, arXiv: 08123261[hep-lat].

[15] A. Cucchieri and T. Mendes, Phys. Rev. D78 (2008) 094503.

[16] L. L. Bogolubsky, E. M. Ilgenfritz, M. Miller - Preussker, and A. Sternbeck, arXiv:

0901.0736[hep-lat].

[17] O. Oloveira and P. J. Silva, arXiv:0809.0258[hep-lat].

[18] A. Sternbeck, L. von Smekal, D. B. Leinweber and A. G. Williams, arXiv:

0710.1982[hep-lat].

[19] D. Zwanziger, arXiv: 0904.2380[hep-lat].

[20] V. Sauli, arXiv:0902.1195[hep-lat].

Thank you !