  We consider K-mouflage models which are K-essence theories coupled to matter.
We analyse their quantum properties and in particular the quantum corrections
to the classical Lagrangian. We setup the renormalisation programme for these
models and show that K-mouflage theories involve a recursive construction
whereby each set of counter-terms introduces new divergent quantum
contributions which in turn must be subtracted by new counter-terms. This tower
of counter-terms can be constructed by recursion and allows one to calculate
the finite renormalised action of the model. In particular, the classical
action is not renormalised and the finite corrections to the renormalised
action contain only higher derivative operators. We establish an operational
criterion for classicality, where the corrections to the classical action are
negligible, and show that this is satisfied in cosmological and astrophysical
situations for (healthy) K-mouflage models which pass the solar system tests.
We also find that these models are quantum stable around astrophysical and
cosmological backgrounds. We then consider the possible embedding of the
K-mouflage models in an Ultra-Violet completion. We find that the healthy
models which pass the solar system tests all violate the positivity constraint
which would follow from the unitarity of the putative UV completion, implying
that these healthy K-mouflage theories have no UV completion. We then analyse
their behaviour at high energy and we find that the classicality criterion is
satisfied in the vicinity of a high energy collision implying that the
classical K-mouflage theory can be applied in this context. Moreover, the
classical description becomes more accurate as the energy increases, in a way
compatible with the classicalisation concept.

Cosmology and Nongalactic Astrophysics