An effective Lagrangian approach for unstable particles

TL;DR

This work tackles the challenge of gauge-invariantly handling unstable intermediate particles by introducing non-local gauge-invariant effective Lagrangians that resum self-energies while preserving Ward identities. It develops both fermionic and bosonic sectors, deriving dressed propagators and explicit gauge-restoring multi-point vertices, and applies the framework to unstable particles in the Standard Model (W, Z, Higgs, top). The approach yields a practical, minimal alternative to pole-scheme and pinch-technique methods for tree-level calculations with unstable states, with potential extensions to QCD and loop-level corrections. Overall, the method provides a consistent, gauge-invariant pathway to incorporate widths into amplitudes and offers explicit Feynman rules for implementing this in phenomenological analyses and simulations.

Abstract

We propose a novel procedure for handling processes that involve unstable intermediate particles. By using gauge-invariant effective Lagrangians it is possible to perform a gauge-invariant resummation of (arbitrary) self-energy effects. For instance, gauge-invariant tree-level amplitudes can be constructed with the decay widths of the unstable particles properly included in the propagators. In these tree-level amplitudes modified vertices are used, which contain extra gauge-restoring terms prescribed by the effective Lagrangians. We discuss the treatment of the phenomenologically important unstable particles, like the top-quark, the $W$- and $Z$-bosons, and the Higgs-boson, and derive the relevant modified Feynman rules explicitly.