Tree-level Scattering Amplitudes via Homotopy Transfer
arxiv(2023)
摘要
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.
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