Six dimensional ultraviolet completion of the $CP(N)$ $\sigma$ model at two loops

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.