  A natural star product for 4-d $\kappa$-Minkowski space is used to
investigate various classes of $\kappa$-Poincar\'e invariant scalar field
theories with quartic interactions whose commutative limit coincides with the
usual $\phi^4$ theory. $\kappa$-Poincar\'e invariance forces the integral
involved in the actions to be a twisted trace, thus defining a KMS weight for
the noncommutative (C*-)algebra modeling the $\kappa$-Minkowski space. The
associated modular group and Tomita modular operator are characterized. In all
the field theories, the twist generates different planar one-loop contributions
to the 2-point function which are at most UV linearly diverging. Some of these
theories are free of UV/IR mixing. In the others, UV/IR mixing shows up in
non-planar contributions to the 2-point function as a polynomial singularity at
exceptional zero external momenta while staying finite at non-zero external
momenta. These results are discussed together with the possibility for the KMS
weight relative to the quantum space algebra to trigger the appearance of KMS
state on the algebra of observables.
