Tree-level scattering amplitudes in Yang-Mills theory satisfy a recursion relation due to Berends and Giele which yields e.g., the famous Parke-Taylor formula for maximally helicity violating amplitudes. We show that the origin of this recursion relation becomes clear in the Batalin-Vilkovisky (BV) formalism, which encodes a field theory in an L?-algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that L?-algebra, known as a minimal model. In fact, the quasi-isomorphism contains all the information about the scattering theory. As we explain, the computation of such a minimal model is readily performed in any BV quantizable theory, which, in turn, produces recursion relations for its tree-level scattering amplitudes.
o Branislav, Macrelli Tommaso, Raspollini Lorenzo, Sämann Christian, Wolf Martin (2019) L?-Algebras, the BV Formalism, and Classical Fields, Fortschritte der Physik - Special Issue: Proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M?Theory (Durham University (UK) 12?18 August 2018) 67 (8-9)
We summarise some of our recent works on L??algebras and quasi?groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of L??algebras, we discuss their Maurer?Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin?Vilkovisky formalism. As examples, we explore higher Chern?Simons theory and Yang?Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of L??quasi?isomorphisms, and we propose a twistor space action.