ar X iv : h ep - p h / 99 11 39 8 v 1 1 8 N ov 1 99 9 Gluon Condensate and Non - Perturbative Quark - Photon Vertex ∗

We evaluate the quark-photon vertex non-perturbatively taking into account the gluon condensate at finite temperature. This vertex is related to the previously derived effective quark propagator by a QED like Ward-Takahashi identity. The importance of the effective vertex for the dilepton production rate from a quark-gluon plasma is stressed. PACS numbers: 12.38.Lg Typeset using REVTEX Supported by BMBF, GSI Darmstadt, DFG, and Humboldt foundation Humboldt fellow and on leave of absence from Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Calcutta 700 064, India Heisenberg fellow 1 During the last two decades a substantial amount of activity, both experimentally and theoretically, has been devoted to an improved understanding of the so called quark-gluon plasma (QGP). The prime goal is to find a reliable signature for this new phase, which in turn urgently requires a better theoretical understanding. Some insight can be gained within the framework of thermal field theory [1]. Within the perturbative approach the thermal effects are known to substantially alter soft modes, changing the dispersion relations, and allowing for Landau damping as well as for new modes of propagation for both fermionic and bosonic quasi-particles [2]. However, perturbative calculations based on bare propagators and vertices lead to gauge dependent and infrared divergent results. This inconsistency is due to the fact that at high temperature higher order loop diagrams can contribute to lower order in the coupling constant. This problem has been cured to some extent by the Hard Thermal Loop (HTL) resummation technique developed by Braaten and Pisarski [3], and much progress has also been achieved within HTL resummed perturbation theory in describing dynamical properties of the QGP. In one particular application, the dilepton rate, obtained by using HTL resummed propagators and vertices [4], was e.g. shown to exhibit sharp structures in the low invariant mass regime caused by the presence of collective quark modes in the medium. However, it seems that quite generally non-perturbative effects [5] dominate in the temperature regime attainable in heavy-ion experiments. Such non-perturbative effects were made explicit by lattice QCD calculations of e.g., effective parton masses [6], hadronic correlators [7], and the gluon condensate [5], below and above the phase transition temperature. So far it is beyond the scope of lattice calculations to compute dynamical quantities. Gluon and quark condensates, however, which describe the non-perturbative imprints of the QCD ground state, have extensively been used in QCD sum rules [8] for studying hadron properties at zero and finite temperature. At zero temperature the quark propagator containing the gluon condensate has been constructed by Lavelle and Schaden [9]. This propagator has been used in a number of investigations [10]. Recently, as an alternative method to lattice and perturbative QCD, it has been suggested in Ref. [11] to include the temperature-dependent gluon condensate into parton propagators at finite temperature. In this way non-perturbative effects can be taken into account within the Greens function technique to compute static [12] as well as dynamic quantities [13]. In the latter work the dilepton production rate, which represents a promising signature for the QGP formation in relativistic heavy-ion collisions, has been calculated using an effective quark propagator containing the gluon condensate. Presently such calculations are, however, not yet self-consistent. One has to introduce in addition an effective quark-photon vertex, which is related to the effective quark propagator by the QED Ward-Takahashi identity. As a first step we intend to calculate the effective quark-photon vertex in the presence of the gluon condensate, both at zero and finite temperature, and to show that it fulfills the Ward identity. In a later step we plan to repeat the dilepton calculation with this vertex, which is, however, a highly non-trivial task in view of the complex form of the result we are going to present. The effective vertex, containing the effects of the gluon condensate in leading order, is given by Fig. 1 and can be written as Γ(P1, P2) = Γ ν 0 + Γ ν G(P1, P2) , (1) where the bare vertex is given by Γ0 = −ieγ ν with the usual QED coupling constant e. 2 For massless bare quarks the one loop correction containing the gluon condensate can be written, using standard Feynman rules, as ΓνG(P1, P2) = − 4 3 ge ∫ dK (2π)4 [γ(P2 / + K/ )γ (P1 / − K/ )γ ] (P1 −K)(P2 +K)2 D̃μρ(K) , (2) where g = 4παs is the strong coupling constant and the four momentum is defined as Q ≡ (q0, ~q) and q = |~q|. The general expression for the non-perturbative gluon propagator containing the gluon condensate is given by [9] D̃μρ(K) = D full μρ (K)−D pert μρ (K) , (3) where the subtraction of the perturbative propagator guarantees that D̃μρ(K) is gauge independent. The effective quark propagator containing the gluon condensate is given in Fig. 2 and can be written as S(P ) = S 0 (P )− ΣG(P ) , (4) where the quark self energy for massless bare quarks reads ΣG(P ) = 4 3 ig ∫ dK (2π)4 D̃μρ(K)γ μ P/−K/ (P −K)2 γ . (5) So far we have discussed general expressions for the vertex and propagator containing the gluon condensate. Next we would like to obtain the zero temperature vertex related to the quark propagator by the Ward-Takahashi identity. At zero temperature, transversality and Lorentz-invariance demand that the gluon propagator has the following form