  We extend the recent one loop analysis of the ultraviolet completion of the
$CP(N)$ nonlinear $\sigma$ model in six dimensions to two loop order in the
MSbar scheme for an arbitrary covariant gauge. In particular we compute the
anomalous dimensions of the fields and $\beta$-functions of the four coupling
constants. We note that like Quantum Electrodynamics (QED) in four dimensions
the matter field anomalous dimension only depends on the gauge parameter at one
loop. As a non-trivial check we verify that the critical exponents derived from
these renormalization group functions at the Wilson-Fisher fixed point are
consistent with the $\epsilon$ expansion of the respective large $N$ exponents
of the underlying universal theory. Using the Ward-Takahashi identity we deduce
the three loop MSbar renormalization group functions for the six dimensional
ultraviolet completeness of scalar QED.
