Cutkosky rules and 1-loop $\kappa$-deformed amplitudes

In this paper we show that the Cutkosky cutting rules are still valid term by term in the expansion in powers of $\kappa$ of the $\kappa$-deformed 1-loop correction to the propagator. We first present a general argument which relates each term in the expansion to a non-deformed amplitude containing additional propagators with mass $M>\kappa$. We then show the same thing more pragmatically, by reducing the singularity structure of the coefficients in the expansion of the $\kappa$-deformed amplitude, to the singularity structure of non-deformed loop amplitudes, by using algebraic and analytic identities. We will explicitly show this up to second order in $1/\kappa$, but the technique can be generalized to higher orders in $1/\kappa$. Both the abstract and the more direct approach easily generalize to different deformed theories. We will then compute the full imaginary part of the $\kappa$-deformed 1-loop correction to the propagator in a specific model, up to second order in the expansion in $1/\kappa$, highlighting the usefulness of the approach for the phenomenology of deformed models. This explicitly confirms previous qualitative arguments concerning the behaviour of the decay width of unstable particles in the considered model.