  The Jaynes-Cummings model is a cornerstone of light-matter interactions.
While finite, the model provides an illustrative example of renormalisation in
perturbation theory. We show, however, that exact renormalisation reveals a
rich non-perturbative structure, and that the model provides a physical example
of a theory with a chaotic coupling trajectory and multi-valued beta-function.
We also construct an exact Wilsonian-like renormalisation group flow for the
effective scattering matrix, and show how multi-valued features arise in the
flow. Our results shed light on non-perturbative aspects of renormalisation and
on the structure of the Jaynes-Cummings model itself.
