Weaving the covariant three-point vertices efficiently

TL;DR

This paper presents an efficient algorithm for constructing Lorentz covariant three-point vertices in particle decay processes, utilizing the helicity formalism and systematically handling massless cases and identical particles.

Contribution

The paper introduces a novel, systematic algorithm for weaving covariant three-point vertices, bridging helicity and covariant formalisms, and accommodating off-shell photon couplings.

Findings

Algorithm efficiently constructs covariant vertices for various spins.

Explicit treatment of massless particles and identical particle symmetrization.

Demonstrated with examples for specific spin and helicity values.

Abstract

An efficient algorithm is developed for compactly weaving all the Lorentz covariant three-point vertices in relation to the decay of a massive particle $X$ of mass $m_X$ and spin $J$ into two particles $ M_{1,2}$ with equal mass $m$ and spin $s$. The closely-related equivalence between the helicity formalism and the covariant formulation is utilized so as to identify the basic building blocks for constructing the covariant three-point vertex corresponding to each helicity combination explicitly. The massless case with $m=0$ is worked out straightforwardly and the (anti)symmetrization of the three-point vertex required by spin statistics of identical particles is made systematically. It is shown that the off-shell electromagnetic photon coupling to the states $M_1$ and $M_2$ can be accommodated in this framework. The power of the algorithm is demonstrated with a few typical examples with specific $J$ and $s$ values.