One loop Volkov propagator in the Lorentz class of gauges

We calculate the ultraviolet divergences in a weak field expansion of the Volkov propagator. This is done for the full Lorentz class of gauges. The expected gauge invariance of the vacuum mass shift in each sideband is recovered. However, the renormalisation of the background induced mass shift is shown to be gauge dependent. In particular, we show that it vanishes in Landau gauge. We find that only in that gauge does the vacuum renormalisation remove all ultraviolet divergences.