All-order renormalization of propagator matrix for unstable Dirac fermions

We consider a system of unstable Dirac fermions in a general parity-nonconserving theory with intergeneration mixing and explain how to renormalize its propagator matrix to all orders in perturbation theory. We work in the pole scheme, in which the squares of the renormalized masses are identified with the complex pole positions and the wave-function renormalization (WFR) matrices are adjusted according to the Lehmann-Symanzik-Zimmermann reduction formalism. The unit-residue property is explicitly verified for the renormalized dressed propagator matrix. Closed analytic expressions for the pole-mass counterterms and WFR matrices in terms of the self-energy functions are presented. We identify residual degrees of freedom in the WFR matrices and propose an additional renormalization condition to exploit them. We demonstrate that, in the presence of instability, the WFR matrices of the in and out states bifurcate in the sense that they are no longer related by pseudo-Hermitian conjugation. The well-known one- and two-loop results for stable fermions are recovered. The all-order renormalized propagator of a single unstable fermion takes a particularly compact form. We also briefly discuss Dirac spinors for unstable fermions.