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In this paper, we propose an improved method for directly calculatingdouble-copy-compatible tree numerators in (super-)Yang-Mills andYang-Mills-scalar theories. Our new scheme gets rid of any explicit dependenceon reference orderings, restoring a form of crossing symmetry to thenumerators. This in turn improves the computational efficiency of thealgorithm, allowing us to go well beyond the number of external particlesaccessible with the reference order based methods. Motivated by a parallelstudy of one-loop BCJ numerators from forward limits, we explore thegeneralization to include a pair of fermions. To improve the accessiblity ofthe new algorithm, we provide a Mathematica package that implements thenumerator construction. The structure of the computation also provides for astraightforward introduction of minimally-coupled massive particles potentiallyuseful for future computations in both classical and quantum gravity.

Efficient Calculation of Crossing Symmetric BCJ Tree Numerators