# $L_\infty$-algebras and the perturbiner expansion

## TL;DR

This paper demonstrates that the minimal model of the $L_{}$-algebra governing a classical field theory encodes the perturbiner expansion, enabling computation of tree-level scattering amplitudes via homotopy Maurer-Cartan action, confirmed in scalar and Yang-Mills theories.

## Contribution

It introduces a novel method to derive perturbiner expansions from the minimal model of the governing $L_{}$-algebra, linking algebraic structures to scattering amplitude calculations.

## Key findings

- Validated method in bi-adjoint scalar theory
- Validated method in Yang-Mills theory
- Provides a new algebraic approach to scattering amplitudes

## Abstract

Certain classical field theories admit a formal multi-particle solution, known as the perturbiner expansion, that serves as a generating function for all the tree-level scattering amplitudes and the Berends-Giele recursion relations they satisfy. In this paper it is argued that the minimal model for the $L_{\infty}$-algebra that governs a classical field theory contains enough information to determine the perturbiner expansion associated to such theory. This gives a prescription for computing the tree-level scattering amplitudes by inserting the perturbiner solution into the homotopy Maurer-Cartan action for the $L_{\infty}$-algebra. We confirm the method in the non-trivial examples of bi-adjoint scalar and Yang-Mills theories.