Vanishing One-LoopβFunction in GaugedN>4Supergravity

In O$\mathrm{}\left(N\right)\mathrm{}$ extended supergravity theories, the usual arguments for one-loop finiteness (modulo topological terms) cease to apply when the internal symmetry is gauged because of the appearance of a cosmological constant related to the gauge coupling $e$. For $N<~4$, we find that infinite renormalizations are required. Remarkably, the particle content of theories with $N>\mathrm{}4\mathrm{}$ results in a cancellation of these infinities, implying, in particular a vanishing one-loop $\beta \mathrm{}\left(e\right)\mathrm{}$ function.