Covariant color-kinematics duality, Hopf algebras, and permutohedra

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 formulas for BCJ numerators of YMS amplitudes and the pure-YM ones exhibit nice quasishuffle 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 $n\ensuremath{-}2$ gluons contains Fubini number (${\mathcal{F}}_{n\ensuremath{-}2}$) of terms in one-to-one correspondence with boundaries of a ($n\ensuremath{-}3$)-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 $n\ensuremath{-}2$ gluons/gravitons: we observe highly nontrivial cancellations in the heavy-mass limit, leading to new formulas for the effective numerators that resemble those obtained in recent works.