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-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.