Abstract
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 MHV amplitudes. We show that the origin of this recursion relation becomes clear in the BV formalism, which encodes a field theory in an $L_\infty$-algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that $L_\infty$-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 quantisable theory, which, in turn, produces recursion relations for its tree-level scattering amplitudes.
Highlights
AND RESULTSWhile string theory has not yet fulfilled its initial promise of a complete and unified description of nature, it has certainly become a successful way of thinking about quantum field theories
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
Summary
While string theory has not yet fulfilled its initial promise of a complete and unified description of nature, it has certainly become a successful way of thinking about quantum field theories. The minimal model L∘ of the L∞-algebra L can always be constructed recursively, and this construction involves a particular map, taking the role of a contracting homotopy, which can be chosen to include the Feynman propagator of the theory. This explains that the minimal model describes the tree-level scattering amplitudes of the original field theory. Particular examples of L∞-algebras include the trivial L∞-algebra L 1⁄4 ⨁k∈ZLk with L 1⁄4 f0g, ordinary Lie algebras with L 1⁄4 L0 and the only nonvanishing product being μ2, as well as differential graded Lie algebras with general L for which μi 1⁄4 0 for i ≥ 3 Quasi-isomorphisms are, in most cases, the appropriate notion of isomorphisms for L∞-algebras
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