Recursive construction for expansions of tree Yang–Mills amplitudes from soft theorem

Abstract

In this paper, we introduce a fundamentally different approach, based on a bottom-up methodology, for expanding tree-level Yang–Mills (YM) amplitudes into Yang–Mills-scalar (YMS) amplitudes and bi-adjoint-scalar (BAS) amplitudes. Our method relies solely on the intrinsic soft behavior of external gluons, eliminating the need for external aids such as Feynman rules or CHY rules. The recursive procedure consistently preserves explicit gauge invariance at every step, ultimately resulting in a manifest gauge-invariant outcome when the initial expression is already framed in a gauge-invariant manner. The resulting expansion can be directly analogized to the expansions of gravitational (GR) amplitudes using the double copy structure. When combined with the expansions of Einstein–Yang–Mills amplitudes obtained using the covariant color-kinematic duality method from existing literature, the expansions presented in this note yield gauge-invariant Bern–Carrasco–Johansson (BCJ) numerators.