All-order renormalization of propagator matrix for unstable Dirac fermions

Abstract

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.

Flavour mixing transport theory and resonant leptogenesis

“…[89]. This is sufficient for our purposes since we do not consider gauge interactions [90,91]. In this section we denote renormalised quantities by a hat.…”

LSZ-reduction, resonances and non-diagonal propagators: Fermions and scalars

“…the literature procedure of calculating coefficients ζ for fermions and scalars. Our analysis is closest in spirit to the one given in [18,19,20]; there are, however, some differences. First, we follow the philosophy of keeping the renormalization scheme as general as possible.…”

“…In particular, we do not impose any concrete renormalization conditions on the two-point functions. Second, we offer a technical improvement in comparison with the analyzes of [8,18,19,20], where the cofactor matrix was used to get the formulae for ζ. By contrast, the coefficients ζ in our approach are expressed directly in terms of properly normalized eigenvectors of certain "mass-squared matrices", so that the case of degenerated eigenvalues is naturally covered by our prescription.…”

Fermion propagator diagonalization and eigenvalue problem

“…One should point out the results of [12,13], where explicit formulas for matrix propagator in all orders of perturbation theory were derived. Similar results were obtained in [14], where models with dynamic generation of fermion masses were discussed.…”