Vector Meson Dominance and gρππ at Finite Temperature from QCD Sum Rules

A Finite Energy QCD sum rule at non-zero temperature is used to determine the q2and the T dependence of the ρππ vertex function in the space-like region. A comparison with an independent QCD determination of the electromagnetic pion form factor Fπ at T 6= 0 indicates that Vector Meson Dominance holds to a very good approximation at finite temperature. At the same time, analytical evidence for deconfinement is obtained from the result that gρππ(q 2, T ) vanishes at the critical temperature Tc, independently of q 2. Also, by extrapolating the ρππ form factor to q2 = 0, it is found that the pion radius increases with increasing T , and it diverges at T = Tc. One of the popular reactions proposed for probing the quark-gluon plasma is dilepton production in high energy heavy ion collisions [1]. An important piece of information required to calculate the dilepton production rates, in the hadronic phase, is the temperature variation of the electromagnetic pion form factor, Fπ(q ). In these calculations it has been usually assumed that Vector Meson Dominance (VMD) remains valid at nonzero temperature and, with a few exceptions [2], that the rho-meson mass and width are temperature independent. It has been argued long ago, though, that all hadronic widths should increase with increasing temperature, and presumably diverge at the critical temperature for deconfinement [2]-[4]. This is expected to hold also for particles which are (hadronically) stable at T = 0, e.g. nucleons and pions. Actual calculations in various frameworks do support such a scenario [5]-[6]. In this sense, the imaginary part of a hadronic Green’s function, i.e. the width, may be viewed as a phenomenological signal for the occurrence of deconfinement. In addition, a QCD sum rule determination of the pion form factor at finite temperature clearly shows that it depends on T in such a way that it vanishes at the critical temperature, while the pion radius diverges there [7]. This determination of Fπ(Q , T ) does not rely on any form of VMD, as it is based on the three-point function associated to the electromagnetic current and two axial-vector currents, thus projecting directly the electromagnetic pion form factor (in the space-like region q = −Q < 0). In this paper we study Finite Energy QCD sum rules (FESR) at T 6= 0 for the three-point function involving the rho-meson interpolating current plus two axial-vector divergences. This allows us to determine the Qand the T dependence of the ρππ coupling, as well as to gauge the validity of VMD at finite temperature. We begin with the determination of gρππ(Q ) at zero temperature (for an earlier analysis using Laplace sum rules see [8]), in order to establish normalizations, as well as to check VMD here. This can be accomplished by using gρππ(Q ) determined from the sum rules, together with VMD, and comparing the resulting pion form factor with the data. At finite temperature, we can also compare it with the direct determination [7], i.e. with a theoretical result not relying on VMD. 2 Since the latter does fit the data very well at T = 0, we can adopt it as the benchmark Fπ(Q , T ) in the absence of experimental data at T 6= 0. Agreement between the two expressions could be taken as evidence in support of VMD at finite T . We consider first the T = 0 correlator Πμ(q) = i 2 ∫ ∫ dx dy ee ·x < 0| T (j π(x) J ρ μ (y) jπ(0) )|0 > = Π1(q ) Pμ +Π2(q ) qμ , (1) where J μ(y) = 1 2 : [ū(y) γμ u(y)− d̄(y) γμd(y)] :, jπ(x) = (mu + md) : d̄(x) iγ5 u(x) :, qμ = (p −p)μ, and Pμ = (p +p)μ. Calculating the imaginary part of the above three-point function in perturbative QCD to leading order in αs and the quark masses gives the result Im Πμ|QCD = 3 4 (mu +md) 2 [(s+ s′ +Q2)2 − 4ss′] [ −QssPμ + ss (s− s)qμ ]