  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.
