Noncommutative field theory from angular twist

We consider a noncommutative field theory with space-time $\star$-commutators based on an angular noncommutativity, namely a solvable Lie algebra: the Euclidean in two dimension. The $\star$-product can be derived from a twist operator and it is shown to be invariant under twisted Poincar\'e transformations. In momentum space the noncommutativity manifests itself as a noncommutative $\star$-deformed sum for the momenta, which allows for an equivalent definition of the $\star$-product in terms of twisted convolution of plane waves. As an application, we analyze the $\lambda \phi^4$ field theory at one-loop and discuss its UV/IR behaviour. We also analyze the kinematics of particle decay for two different situations: the first one corresponds to a splitting of space-time where only space is deformed, whereas the second one entails a non-trivial $\star$-multiplication for the time variable, while one of the three spatial coordinates stays commutative.