All-order renormalization of propagator matrix for fermionic system with flavor mixing

TL;DR

This paper develops a comprehensive all-order renormalization framework for fermionic propagator matrices with flavor mixing, incorporating complex pole positions and wave-function renormalization consistent with QFT formalism.

Contribution

It provides explicit analytic expressions for renormalization constants in flavor-mixed fermion systems, including unstable cases and WFR bifurcation phenomena.

Findings

Closed-form all-order renormalization expressions

Two-loop expansion of renormalization constants

Generalization to unstable fermions with bifurcation

Abstract

We consider a mixed system of Dirac fermions in a general parity-nonconserving theory and renormalize the propagator matrix to all orders 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 in compliance with the Lehmann-Symanzik-Zimmermann reduction formalism. We present closed analytic all-order expressions and their expansions through two loops for the renormalization constants in terms of the scalar, pseudoscalar, vector, and pseudovector parts of the unrenormalized self-energy matrix, which is computable from the one-particle-irreducible Feynman diagrams of the flavor transitions. We identify residual degrees of freedom in the WFR matrices and propose an additional renormalization condition to exhaust them. We then explain how our results may be generalized to the case of unstable fermions, in which we encounter the phenomenon of WFR bifurcation. In the special case of a solitary unstable fermion, the all-order-renormalized propagator is presented in a particularly compact form.