The L∞-algebra of the S-matrix

Abstract

We point out that the one-particle-irreducible vacuum correlation functions of a QFT are the structure constants of an L ∞-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 ∞-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 [1]).

Symmetry factors of Feynman diagrams and the homological perturbation lemma

“…This perspective is useful for a number of applications, in particular for discussing perturbative quantum field theories, see e.g. [4][5][6][7][8][9]. Any L ∞ -algebra comes with a family of quasi-isomorphic (read: equivalent) L ∞ -algebras known as minimal models.…”

“…I.e. Feynman diagrams which do not become disconnected when removing a single arbitrary edge 7. I.e.…”

Loop Amplitudes and Quantum Homotopy Algebras

“…smallest quasi-isomorphic forms) of its L ∞ -algebra. Recently, it was shown that the quasi-isomorphism between both induces recursion relations for these amplitudes [8] (see also [9] for related discussions of the S-matrix in the L ∞ -language, [10] for the tree-level perturbiner expansion, and [11,12] for an L ∞ -interpretation of tree-level onshell recursion relations). In the context of Yang-Mills (YM) theory, this recursion relation is known as the Berends-Giele recursion relation [13].…”

L∞-algebras and the perturbiner expansion

“…Recently, it has been established by Macrelli, Sämann and Wolf [10] that the L ∞ -structure of a classical field theory may also be used to determine the recursion relations for its treelevel scattering amplitudes (see also [11] and [12]). These authors worked out in detail the concrete case of Yang-Mills theory.…”