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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 theMSbar scheme for an arbitrary covariant gauge. In particular we compute theanomalous dimensions of the fields and $\beta$-functions of the four couplingconstants. We note that like Quantum Electrodynamics (QED) in four dimensionsthe matter field anomalous dimension only depends on the gauge parameter at oneloop. As a non-trivial check we verify that the critical exponents derived fromthese renormalization group functions at the Wilson-Fisher fixed point areconsistent with the $\epsilon$ expansion of the respective large $N$ exponentsof the underlying universal theory. Using the Ward-Takahashi identity we deducethe three loop MSbar renormalization group functions for the six dimensionalultraviolet completeness of scalar QED.

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