Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory

Understand

Tree-level color-ordered Yang-Mills (YM) amplitudes can be decomposed in terms of $(n-2)!$ bi-scalar (BS) amplitudes, whose expansion coefficients form a basis of Bern-Carrasco-Johansson (BCJ) numerators.

• By the help of the recursive expansion of Einstein-Yang-Mills (EYM) amplitudes, the BCJ numerators are given by polynomial functions of Lorentz contractions which are conveniently described by graphic rule.
• In this work, we extend the expansion of YM amplitudes to off-shell level.
• We define different types of off-shell extended numerators that can be generated by graphs.