Mixing of fermions and spectral representation of propagator

TL;DR

This paper develops a spectral representation for mixing fermion propagators with parity violation, enabling non-perturbative renormalization and modified spin projectors for systems with fermion mixing.

Contribution

It introduces a spectral approach for mixing fermions with parity violation, providing a non-perturbative renormalization method and modified spin projectors.

Findings

Spectral representation for mixing fermions with parity violation is constructed.

Renormalization matrices are obtained in closed form without perturbation theory.

Modified spin projectors are developed for systems with fermion mixing.

Abstract

We develop the spectral representation of propagator for $n$ mixing fermion fields in the case of $\mathsf{P}$-parity violation. The approach based on the eigenvalue problem for inverse matrix propagator makes possible to build the system of orthogonal projectors and to represent the matrix propagator as a sum of poles with positive and negative energies. The procedure of multiplicative renormalization in terms of spectral representation is investigated and the renormalization matrices are obtained in a closed form without the use of perturbation theory. Since in theory with $\mathsf{P}$-parity violation the standard spin projectors do not commute with the dressed propagator, they should be modified. The developed approach allows us to build the modified (dressed) spin projectors for a single fermion and for a system of fermions.