Note on the Hopf-algebra-based formula of Yang-Mills-Scalar amplitudes

TL;DR

This paper introduces a recursive Hopf-algebra-based formula for Yang-Mills-Scalar amplitudes involving massive scalars, simplifying calculations and establishing equivalence with previous approaches through explicit examples.

Contribution

It proposes a new recursive HAB formula for YMS amplitudes with massive scalars, connecting it to existing massless scalar results and verifying its consistency.

Findings

The recursive formula simplifies YMS amplitude calculations.

The HAB formula reduces to known massless scalar results in the massless limit.

Explicit calculations confirm the equivalence of different recursive approaches.

Abstract

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 which 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 soft behavior approach. In the massless limit, the HAB formula turns into expressions for YMS amplitudes with massless scalars which was earlier shown to satisfy an alternative recursive expansion formula. In this note, we show the equivalence of these two distinct approaches through explicit calculation on amplitudes with one and two gluons.