Kinematic Hopf algebra and BCJ numerators at finite α'

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 α' corrections, known as DF²+YM theory, when coupled to two heavy particles which can be removed through a simple factorization limit. The full α' 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.