Covariant color-kinematics duality, Hopf algebras and permutohedra

Based on the covariant color-kinematics duality, we investigate combinatorial and algebraic structures underlying their Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes in Yang-Mills-scalar (YMS) theory. The closed-formulae for BCJ numerators of YMS amplitudes and the pure-Yang-Mill...

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Bibliographic Details
Published in:arXiv.org 2022-11
Main Authors: Cao, Qu, Dong, Jin, He, Song, Yao-Qi, Zhang
Format: Article
Language:English
Subjects:
Algebra
Amplitudes
Boundaries
Color
Combinatorial analysis
Field theory
Fields (mathematics)
Gluons
Gravitons
Kinematics
Scalars
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Summary:Based on the covariant color-kinematics duality, we investigate combinatorial and algebraic structures underlying their Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes in Yang-Mills-scalar (YMS) theory. The closed-formulae for BCJ numerators of YMS amplitudes and the pure-Yang-Mills (YM) ones exhibit nice quasi-shuffle Hopf algebra structures, and interestingly they can be viewed as summing over boundaries of all dimensions of a combinatorial permutohedron. In particular, the numerator with two scalars and \(n{-}2\) gluons contains Fubini number ( \(F_{n{-}2}\) ) of terms in one-to-one correspondence with boundaries of a \((n{-}3)\)-dimensional permutohedron, and each of them has its own spurious-pole structures and a gauge-invariant numerator (both depending on reference momenta). From such Hopf algebra or permutohedron structure, we derive new recursion relations for the numerators and intriguing "factorization" on each spurious pole/facet of the permutohedron. Similar results hold for general YMS numerators and the pure-YM ones. Finally, with a special choice of reference momenta, our results imply BCJ numerators in a heavy-mass effective field theory with two massive particles and \(n{-}2\) gluons/gravitons: we observe highly nontrivial cancellations in the heavy-mass limit, leading to new formulae for the effective numerators which resemble those obtained in recent works.
ISSN:2331-8422
DOI:10.48550/arxiv.2211.05404