Soluble theory with massive ghosts

To investigate the unitarity of asymptotically free, higher-derivative theories, like certain models of quantum gravity, I study a prototype in two space-time dimensions. The prototype is a kind of higher-derivative nonlinear $\sigma$ model; it is asymptotically free, exhibits dimensional transmutation, and is soluble in a large-$N$ expansion. The $S$-matrix elements, constructed from the analytic continuation of the Euclidean Green's functions, conserve probability to $\sim O\left(N^{-1\mathrm{}}\right)$, but violate unitarity at $\sim O\left(N^{-2\mathrm{}}\right)$. The model demonstrates that in higher-derivative theories unitarity, or the lack thereof, cannot be decided without explicit control over the infrared limit. Even so, the results suggest that there may exist some (rather special) asymptotically free, higher-derivative theories which are unitary.