Kinematic Hopf algebra and BCJ numerators at finite ${\alpha }^{\prime }$

Abstract

In this letter, starting from a kinematic Hopf algebra, we first construct a closed-form formula for all Bern-Carrasco-Johansson (BCJ) numerators in Yang-Mills (YM) theory with infinite orders of ${\alpha }^{\prime }$ corrections, known as $\mathrm{D}{\mathrm{F}}^2+\mathrm{YM}$ theory, when coupled to two heavy particles which can be removed through a simple factorization limit. The full ${\alpha }^{\prime }$ dependence appears simply in massive physical propagator factors, with factorization strongly constraining the construction. The intricate structure induced by the massive poles also naturally leads us to find a novel closed-form and local expression for BCJ numerators in usual pure YM theory, based directly on the kinematic Hopf algebra.

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