Note on the Hopf-Algebra-Based Formula of Yang–Mills-Scalar Amplitudes

In this note, we study the Hopf-algebra-based (HAB) formula of Yang–Mills-Scalar (YMS) amplitudes, which expands a YMS amplitude with massive scalars as a combination of propagator matrices that mix massless scalars corresponding to gluons with the original massive scalars. We propose a recursive formula that conveniently expresses the HAB formula. In this formula, gluons are converted into massless scalars. Thus it expresses a YMS amplitude with massive scalars by amplitudes with fewer gluons, massive scalars and massless scalars. We verify this formula by using the soft behavior of amplitudes. We further show the equivalence between the massless limit of the HAB formula and an earlier proposed recursive expansion formula through explicit calculations on amplitudes with one and two gluons.