Scattering amplitude recursion relations in Batalin-Vilkovisky–quantizable theories

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 8 -algebra. The recursion relation is obtained in the transition to a smallest representative in the quasi-isomorphism class of that L 8 -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.

L∞-algebras and the perturbiner expansion

“…, x n ∈ L. With this definition, it can be shown that the minimal model theorem extends to cyclic L ∞ -algebras. For further details on this, see again the Appendix A of [10].…”

“…This theory should be thought of as a broad generalisation of Chern-Simons theory. In what follows, we shall only sketch the details and refer to [9] (see also [10]). Let L be an L ∞ -algebra with higher order brackets l n .…”

“…The Yang-Mills L ∞ -algebra has been considered in a few papers, e.g., [38][39][40][41]. Let us summarise very briefly its definition following the exposition of [10]. The notation is as in §2.4, except that that we work in dimension d = 4.…”

“…Thus, Ω 0 (R 1,3 , g) is situated in degrees 0 and 3 and Ω 1 (R 1,3 , g) is situated in degrees 1 and 3. The non-vanishing higher order brackets are [10]…”

“…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.…”