The $L_\infty$-algebra of the S-matrix

TL;DR

This paper reveals that the structure constants of vacuum correlation functions in quantum field theory form an $L_$-algebra, linking algebraic structures to S-matrix calculations and generalizing string field theory concepts.

Contribution

It establishes that QFT correlation functions define an $L_$-algebra and connects the LSZ S-matrix prescription to the minimal model theorem, extending algebraic frameworks to general QFTs.

Findings

Correlation functions are structure constants of an $L_$-algebra.

LSZ prescription corresponds to the minimal model theorem.

Recursion relations for amplitudes at tree-level are derived.

Abstract

We point out that the one-particle-irreducible vacuum correlation functions of a QFT are the structure constants of an $L_\infty$-algebra, whose Jacobi identities hold whenever there are no local gauge anomalies. The LSZ prescription for S-matrix elements is identified as an instance of the ``minimal model theorem'' of $L_\infty$-algebras. This generalises the algebraic structure of closed string field theory to arbitrary QFTs with a mass gap and leads to recursion relations for amplitudes (albeit ones only immediately useful at tree-level, where they reduce to Berends-Giele-style relations as shown in recent work).