Covariant color-kinematics duality, Hopf algebras and permutohedra

TL;DR

This paper explores the algebraic and combinatorial structures of BCJ numerators in Yang-Mills-scalar theory, revealing connections to Hopf algebras and permutohedra, and deriving new recursion relations and effective numerators.

Contribution

It uncovers the Hopf algebra and permutohedron structures underlying BCJ numerators and introduces new recursion relations and formulas for effective numerators in heavy-mass limits.

Findings

BCJ numerators exhibit Hopf algebra and permutohedron structures.

New recursion relations for BCJ numerators are derived.

Effective numerators in heavy-mass limit show nontrivial cancellations.

Abstract

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.