other spin states should be straightforward.

The paper is organised as follows. After reviewing aspects of higher-order contributions and mixing effects in a system of three scalar states in Section\[sect:LoopMix\], we derive the pole structure of the propagator matrix and resulting relations between the wave function normalisation factors in Section\[sect:poles\]. An analytical derivation of the multi-Breit–Wigner approximation of the full propagators including an uncertainty estimate is given in Section\[sect:fullBWZ\]. We perform a detailed numerical comparison of the two approaches in Section\[sect:NumBWfull\] before we conclude in Section\[sect:conclusions\].

Loop-level mixing of scalar propagators {#sect:LoopMix}
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In a system of $n$ particles which have the same conserved quantum numbers in general a non-trivial mixing between the different states will occur. While at lowest order the mass matrix given in terms of the interaction eigenstates can be diagonalised in order to obtain the mass eigenstates, at the loop level momentum-dependent self-energy corrections enter. These contributions give rise to a momentum-dependent $(n \times n)$ propagator matrix which is in general non-diagonal. The physical masses that are associated with the loop-corrected propagator matrix need to be determined from the (complex) poles of the propagator matrix.

While the case of $(2\times 2)$ mixing provides the minimal setup 