Tree-level Scattering Amplitudes via Homotopy Transfer

We formalize the computation of tree-level scattering amplitudes in terms of the homotopy transfer of homotopy algebras, illustrating it with scalar $\phi^3$ and Yang-Mills theory. The data of a (gauge) field theory with an action is encoded in a cyclic homotopy Lie or $L_{\infty}$ algebra defined on a chain complex including a space of fields. This $L_{\infty}$ structure can be transported, by means of homotopy transfer, to a smaller space that, in the massless case, consists of harmonic fields. The required homotopy maps are well-defined since we work with the space of finite sums of plane-wave solutions. The resulting $L_{\infty}$ brackets encode the tree-level scattering amplitudes and satisfy generalized Jacobi identities that imply the Ward identities. We further present a method to compute color-ordered scattering amplitudes for Yang-Mills theory, using that its $L_{\infty}$ algebra is the tensor product of the color Lie algebra with a homotopy commutative associative or $C_{\infty}$ algebra. The color-ordered scattering amplitudes are then obtained by homotopy transfer of $C_{\infty}$ algebras.