Gluon polarization tensor and dispersion relation in a weakly magnetized medium

TL;DR

This paper investigates how weak magnetic fields affect gluon behavior in a medium, showing that gluons remain massless and propagate along the light cone, with the lowest order approximation being valid for small fields.

Contribution

It provides a detailed derivation of the gluon polarization tensor in a weak magnetic field and clarifies the spurious nature of certain solutions at lowest order.

Findings

Gluon polarization tensor remains transverse and massless in a weak magnetic field.

Gluons propagate along the light cone despite magnetic field presence.

Spurious solutions indicating a gluon mass are artifacts of the lowest order approximation.

Abstract

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. By comparing with the expression for the gluon polarization tensor valid to all orders in the magnetic field, the existence of a second solution, corresponding to a finite gluon mass, is shown to be spurious and an artifact of the lowest order approximation in the field strength. We also study the strength of the polarization modes for real gluons. We conclude that, provided the spurious solutions are discarded, the lowest order approximation to the gluon polarization and dispersion properties is good as long as the field strength is small compared to the loop fermion mass.