Star product and interacting fields on $\kappa$-Minkowski space

In this note we extend the methods developed by Freidel et al. [arXiv:hep-th/0612170] to derive the form of $\phi^4$ interaction term in the case of scalar field theory on $\kappa$-Minkowski space, defined in terms of star product. We present explicit expressions for the $\kappa$-Minkowski star product. Having obtained the the interaction term we use the resulting deformed conservation rules to investigate if they lead to any threshold anomaly, and we find that in the leading order they do not, as expected.