@Cacciapaglia:2009ic; @deBlas:2012qp; @Fuchs:2014ola; @Gomez-Bock:2015sia]. In such a case a single-pole approximation is not applicable [@Cacciapaglia:2009ic; @deBlas:2012qp; @Fuchs:2014ola]. Instead, the contributions from the relevant complex poles of the propagators have to be taken into account. The approximation of the propagator matrix in terms of on-shell wave function normalisation factors evaluated at the complex poles and Breit–Wigner propagators allows the implementation of interference contributions using the prescription of Ref. [@Fuchs:2014ola].

We perform in this paper a careful treatment of imaginary parts and analyse the pole structure of matrix-type propagators in detail. While in the no-mixing case every element of an $(n \times n)$ propagator matrix has a single pole, we will demonstrate that in general every single element of an $(n \times n)$ propagator matrix (diagonal and off-diagonal) by itself has $n$ poles. This fact makes the association of the (loop-corrected) mass eigenstates with the propagator poles non-trivial. We will show that in principle all permutations of the $n$ propagator poles with the $n$ mass eigenstates are physically equivalent. For practical purposes, however, it is important to make a choice that leads to numerically stable and well-behaved results.

To be specific, in the following we use the language of the Higgs sector 