Gluon polarization tensor and dispersion relation in a weakly magnetized medium

We study the polarization and dispersion properties of gluons moving within a weakly magnetized background at one-loop order. To this end, we show two alternative derivations of the charged fermion propagator in the weak field expansion and use this expression to compute the lowest order magnetic field correction to the gluon polarization tensor. We explicitly show that, in spite of its cumbersome appearance, the gluon polarization tensor is transverse as required by gauge invariance. We also show that none of the three polarization modes develops a magnetic mass and that gluons propagate along the light cone, non withstanding that Lorentz invariance is lost due to the presence of the magnetic field. We also study the strength of the polarization modes for real gluons. We conclude that the lowest order approximation to the gluon polarization and dispersion properties is good as long as the field strength and gluon momentum are not larger than the loop fermion mass. When the fermion mass is the vacuum one, the applicability of these findings for phenomenological studies is rather limited. However, should temperature be accounted for and the fermion mass become the thermal one, conditions met during the plasma phase of a heavy-ion collision, these findings can be potentially very useful to describe gluon mediated processes in the presence of a magnetic field.