LSZ-reduction, resonances and non-diagonal propagators: gauge fields

TL;DR

This paper extends the LSZ formalism to gauge theories with mixing, providing a practical method for residue calculation, and illustrates its application through one-loop examples in the Standard Model and Majoron models.

Contribution

It generalizes the LSZ formalism to include mixing of gauge bosons, offering a practical residue calculation method and analyzing the BRST structure of asymptotic states.

Findings

Demonstrates residue factorization at real and complex poles.

Provides a prescription for calculating residues for S-matrix elements.

Applies the formalism to one-loop mixing examples in the Standard Model and Majoron models.

Abstract

We analyze in the Landau gauge mixing of bosonic fields in gauge theories with exact and spontaneously broken symmetries, extending to this case the Lehmann-Symanzik-Zimmermann (LSZ) formalism of the asymptotic fields. Factorization of residues of poles (at real and complex values of the variable $p^2$) is demonstrated and a simple practical prescription for finding the "square-rooted" residues, necessary for calculating $S$-matrix elements, is given. The pseudo-Fock space of asymptotic (in the LSZ sense) states is explicitly constructed and its BRST-cohomological structure is elucidated. Usefulness of these general results, obtained by investigating the relevant set of Slavnov-Taylor identities, is illustrated on the one-loop examples of the $Z^0$-photon mixing in the Standard Model and the $G_Z$-Majoron mixing in the singlet Majoron model.