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

Jiexi Liu, Yi-Jian Du

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.

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

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.
Paper Structure (17 sections, 97 equations)

This paper contains 17 sections, 97 equations.

Table of Contents

  1. Introduction
  2. A review of recursive expansion, HAB formula and soft behaviors
  3. The recursive expansion formula of YMS
  4. Hopf-algebra-based formula
  5. The soft behavior approach to YMS amplitudes
  6. A recursive expansion relation from the HAB formula
  7. Example
  8. General formula for YMS with at least three massive scalars
  9. Boundary case: YMS amplitudes with two massive scalars
  10. Understanding the recursive expansion formula by soft behavior
  11. The relationship between (\ref{['eq:exp-poly']}) and the HAB formulas
  12. The boundary case: amplitudes with two scalars
  13. The base case: amplitudes with one gluon
  14. Amplitudes with two gluons
  15. Conclusions
  16. ...and 2 more sections