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Multigluon Helicity Amplitudes Involving a Quark Loop

Gregory Mahlon

TL;DR

The paper advances the computation of one-loop QCD amplitudes with a closed quark loop for multi-gluon final states by leveraging double-off-shell quark currents, Weyl-van der Waerden spinors, color factorization, and recursion. It delivers compact, all-plus helicity results for γg→gg...g, e+e−→gg...g, and gg→gg...g, and extends to a case with a single opposite-helicity gluon, validating these against known string-based and SUSY-related relations. The work also provides explicit color-ordered amplitudes and demonstrates consistency with the n-photon limit when colors are suppressed. Together, these results showcase the efficiency of a spinor-recursive, color-factorized approach for complex, multi-leg one-loop amplitudes and pave the way for full helicity analyses and numerical applications, while noting remaining questions about infrared cancellation and massive-quark effects.

Abstract

We apply the solution to the recursion relation for the double-off-shell quark current to the problem of computing one loop amplitudes with an arbitrary number of gluons. We are able to compute amplitudes for photon-gluon scattering, electron-positron annihilation to gluons, and gluon-gluon scattering via a quark loop in the case of like-helicity gluons. In addition, we present the result for the one-loop gluon-gluon scattering amplitude when one of the gluons has opposite helicity from the others.

Multigluon Helicity Amplitudes Involving a Quark Loop

TL;DR

The paper advances the computation of one-loop QCD amplitudes with a closed quark loop for multi-gluon final states by leveraging double-off-shell quark currents, Weyl-van der Waerden spinors, color factorization, and recursion. It delivers compact, all-plus helicity results for γg→gg...g, e+e−→gg...g, and gg→gg...g, and extends to a case with a single opposite-helicity gluon, validating these against known string-based and SUSY-related relations. The work also provides explicit color-ordered amplitudes and demonstrates consistency with the n-photon limit when colors are suppressed. Together, these results showcase the efficiency of a spinor-recursive, color-factorized approach for complex, multi-leg one-loop amplitudes and pave the way for full helicity analyses and numerical applications, while noting remaining questions about infrared cancellation and massive-quark effects.

Abstract

We apply the solution to the recursion relation for the double-off-shell quark current to the problem of computing one loop amplitudes with an arbitrary number of gluons. We are able to compute amplitudes for photon-gluon scattering, electron-positron annihilation to gluons, and gluon-gluon scattering via a quark loop in the case of like-helicity gluons. In addition, we present the result for the one-loop gluon-gluon scattering amplitude when one of the gluons has opposite helicity from the others.

Paper Structure

This paper contains 7 sections, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The double-off-shell quark currents
  3. Construction of Loop Amplitudes
  4. The process $\gamma g \rightarrow gg\cdots g$
  5. The process $e^{+} e^{-} \rightarrow gg\cdots g$
  6. The process $gg \rightarrow gg\cdots g$
  7. Conclusions

Figures (1)

  • Figure 1: The basic diagram for $\gamma g \rightarrow gg\ldots g$. The blob represents the double-off-shell quark current, consisting of the sum of all tree graphs containing exactly $n{-}1$ external gluons.