Non-Abelian gauge couplings at finite temperature and density

With the off-shell one-loop calculation for massless non-Abelian gauge theories in the general covariant gauge we study properties of the gauge coupling renormalized at finite temperature and at finite density, mainly focusing on its dependence on the (finite) baryon-number density ${\rho }_B$ (or on the chemical potential ${\mu }_B$). The strong and severe vertex dependence is shown to come out for the ${\mu }_B$ dependence as well as on the temperature (T) dependence: In terms of the parameter ζ≡${\mu }_B$/T the coupling defined through the three-gluon vertex shows a ln‖ζ‖ behavior, whereas the one defined through the fermion-gluon vertex shows a strong ${\zeta }^2$ behavior. This strong vertex dependence survives even at T=0. Difficulties appearing in the perturbative calculation of physical quantities, indicated by this disaster, are discussed. We also discuss what insight might be gained from the present analysis for the ‘‘(magnetic) screening ’’ of effective charge.