Gluon polarization tensor in a thermo-magnetic medium

We compute the gluon polarization tensor in a thermo-magnetic environment in the weak and strong magnetic field cases, both at zero and at high temperature. The magnetic field effects are introduced using Schwingeru0027s proper time method. Thermal effects are computed in the Hard Thermal Loop approximation. We find that for the zero temperature case and a non-vanishing quark mass, the coefficients of the tensor structures describing the polarization tensor develop an imaginary part corresponding to the threshold for quark-antiquark pair production. These coefficients are infrared finite and simplify considerably when the quark mass vanishes. In the high temperature case, the quark mass can be safely ignored. Nevertheless, we explicitly maintain this mass finite to separate the unrenormalized vacuum and matter contributions. We discuss how knowledge of this coefficients is useful in particular to study the thermo-magnetic evolution of the strong coupling in the four regimes hereby treated.