Variational Method for Photon Emission from Quark-Gluon Plasma

Variational method has been applied to estimate Landau-Pomeranchuk-Migdal (LPM) effects on virtual photon emission from the quark gluon plasma as a function of photon mass. The variational method was well tested for the LPM effects in real photon emission by Arnold et. al., For virtual photons, LPM effects arising from multiple scatterings of quarks in the plasma are determined by the integral equations for the transverse vector function (${\bf \tilde{f}(\tilde{p}_\perp)}$) and the longitudinal function ($\tilde{g}({\bf \tilde{p}_\perp})$). We extended the variational method to solve these transverse and longitudinal equations for a variable set \{$p_0,q_0,Q^2$\}, considering bremsstrahlung and $\bf aws$ processes. We solved these equations, also by the self consistent iterations for comparing with the results of variational method. In order to estimate the variational parameter, we obtained empirical fits for the peak positions of the ${\bf p_\perp\cdot\tilde{f}(\tilde{p}_\perp)}$, $p_\perp\tilde{g}({\bf \tilde{p}_\perp})$ distributions from iteration method. We propose that the optimized variational parameter for virtual photon emission is approximately equal to these empirical peak position values. The detailed study showed that the variational method gives reliable results for LPM effects on virtual photon emission for photon virtuality of the order $Q^2/T^2\le 100$. At low $Q^2$, the peak positions for $\tilde{p}_\perp$ distributions of transverse vector functions for virtual photons nearly coincide with the peak positions of corresponding distributions of real photons. We calculated imaginary part of photon retarded polarization tensor as a function of $Q^2/T^2$ using empirical variational parameters