Lattice QCD predicts a phase transition from Hadrons gaz (HG) to quark-gluon plasma (QGP) at deconfinement temperature, T 170 MeV. It is believed that QGP has been produced in relativistic heavy ions collision [1] -[4] where in the initial pre-equilibrium stage of QGP about half the total center-of-mass energy, , goes into the production of a semi-classical gluon field [5] -[17] . Therefeore, to study the production of a QGP from a classical chromo field, it is necessary to know how quarks and gluons are formed from the latter. The production rate of quark-antiquark from a given constant chromo-electric field has been derived in Ref. [18] and the integrated distribution has been obtained in [19] -[22] (for a review see [23] ).

In this short technical note, we will extend the results of Ref. [18] to a general constant background field. The method presented here may simplify the complexity found in the Non-perturbative QCD calculations. Also, the obtained distribution for quark (antiquark) production can be used in the analysis of the experimental results at the RHIC and the LHC colliders.

The paper is organized as follows: in the next section, we will calculate the one loop effective action needed in the evaluation of the distribution of the quark (antiquark) production. In Section 3 the distribution is presented. Finally, in Section 4, an application to heavy ion collision is given.

As described in the above section, we will evaluate here the one loop effective action in the presence of a constant chromo-field. For this purpose, we start from the QCD Lagrangian density for a quark in a non-abelian background field which is given by

Then the vacuum to vacuum transition amplitude is given by

And the one loop effective action can be written in this form

Thus, using the invariance of trace under transposition and the following relation

we obtain the following expression1

The quickest way to calculate the effective action is to work in a basis that are the eigenstates of defined by:

which is a part of the one loop effective action of Equation (5).

As an application to this idea, we first consider the case of a constant electric field in the direction (direction of the beam in the heavy ion collision). In this case, we choose a gauge such that we can take. Thus the second part of Equation (6) can be written in this form

The Hamiltonian becomes

After a straightforward algebra one can find the following eigenvalues of the Hamiltonian

where are the eigenvalues over the Dirac matrices such that and. And, with, are the eigenvalue for over the group space and are given by [18] .

with given by

where

Using the obtained eigenvalues of the Hamiltonian, the effective action becomes

Performing the i and n summations we found

which is the same results as Ref. [18] . Clearly, the one loop magnetic effective action can be found upon the following substitution. Therefore

Now, in the same manner as in Ref. [18] we may derive the non-perturbative quarks (antiquarks) production per unit time, per unit volume and per unit transverse momentum from a given constant chromo-electric field. Thus as done in Ref. [18] we can find that

where is the effective mass of the quark and the eigenvalues are given above.

Let's consider the situation of two relativistic heavy nuclei colliding and leaving behind a semi-classical gluon field which then non-perturbatively produces gluon and quark-antiquark pairs via the Schwinger mechanism [19] . As estimated in Ref. [24] for Au-Au collision at RHIC collider with fm and center-of-mass energy GeV per nucleon, the initial energy density is GeV⁴ and GeV⁴. For our analysis we take which can be justified by the sensitivity check that has been made in Ref. [24] where it has been found that the production rate is not very sensitive to.

In Figure 1 we plot the rate of quark production as a function of the transverse momentum for two values of (used in [25] ) and with initial energy density GeV⁴. Clearly seen from this figure that the production rate decrease with and becomes negligible at GeV. The obtained distribution for quark (antiquark) production can be used to fix the initial conditions for the QGP in heavy ion collision at the RHIC and the LHC colliders.

In this note we have proposed a method for calculating the vacuum to vacuum transition amplitude in the presence of the non-abelian background field. The method can be applied to a general background field and it can be updated to study the non-perturbative soft gluon production [26] . Also, we have evaluated the rate for

quark (antiquark) production in a constant chromo-electric field. These results are used to determine the quark (antiquark) production rate in heavy ion collision.