The $L_{\infty}$ structure of gauge theories with matter

TL;DR

This paper develops an algebraic framework using $L_{}$-algebras within the Batalin-Vilkovisky formalism to compute tree-level scattering amplitudes in gauge theories with matter, including novel generating functions and explicit amplitude formulas.

Contribution

It introduces a new algebraic approach to gauge theories with matter, deriving recursive perturbiner expansions and generating functions for scattering amplitudes.

Findings

Derived $L_{}$-algebra structures for various gauge theories.

Provided closed-form expressions for fermion and scalar lines with gluons.

Developed a generating function for all tree-level amplitudes using Maurer-Cartan actions.

Abstract

In this work we present an algebraic approach to the dynamics and perturbation theory at tree-level for gauge theories coupled to matter. The field theories we will consider are: Chern-Simons-Matter, Quantum Chromodynamics, and scalar Quantum Chromodynamics. Starting with the construction of the master action in the classical Batalin-Vilkovisky formalism, we will extract the $L_{\infty}$-algebra that allow us to recursively calculate the perturbiner expansion from its minimal model. The Maurer-Cartan action obtained in this procedure will then motivate a generating function for all the tree-level scattering amplitudes. There are two interesting outcomes of this construction: a generator for fully-flavoured amplitudes via a localisation on Dyck words; and closed expressions for fermion and scalar lines attached to $n$-gluons with arbitrary polarisations.