Algebraic consistency and explicit construction of one-loop Bern-Carrasco-Johansson numerators of Yang-Mills and related theories

We study the algebraic structure of one-loop Bern-Carrasco-Johansson numerators in Yang-Mills and related theories. Starting from the propagator matrix that connects color-ordered integrands to numerators, we identify the consistency conditions that ensure the existence of Jacobi-satisfying numerator solutions and determine the unique construction. The relation between one-loop numerators and forward-limit tree numerators is clarified, together with the additional physical conditions required for a consistent double-copy interpretation. We propose a two-step expansion strategy for obtaining explicit one-loop numerators. The Yang-Mills integrand is first decomposed into scalar-loop Yang-Mills-scalar building blocks, which are then expanded into biadjoint scalar integrands. We derive explicit results for up to three external gluons, showing how the kinematic consistency conditions uniquely determine the coefficients in each case. Similar results for Einstein-Yang-Mills and gravity amplitudes are also presented.