Tree-level Scattering Amplitudes via Homotopy Transfer

Bonezzi, Roberto; Chiaffrino, Christoph; Diaz-Jaramillo, Felipe; Hohm, Olaf

Tree-level Scattering Amplitudes via Homotopy Transfer

Roberto Bonezzi
(
- Humboldt U., Berlin
)
,
Christoph Chiaffrino
(
- Humboldt U., Berlin
)
,
Felipe Diaz-Jaramillo
(
- Humboldt U., Berlin
)
,
Olaf Hohm
(
- Humboldt U., Berlin
)

Dec 14, 2023

72 pages

e-Print:

2312.09306 [hep-th]

Report number:

HU-EP-23/67-RTG

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Abstract: (arXiv)

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.

Note:

72 pages, v2: introduction improved, references added

gauge field theory: Yang-Mills
algebra: Lie
homotopy
scattering amplitude
tree approximation
color
Jacobi identity
plane wave
cyclic
Ward identity

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