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On-Shell Recurrence Relations for One-Loop QCD Amplitudes

Zvi Bern, Lance J. Dixon, David A. Kosower

TL;DR

This work extends on-shell recursion techniques to non-supersymmetric one-loop QCD amplitudes, focusing on rational terms. It identifies and overcomes loop-specific obstacles—boundary terms and double poles—by employing generalized shifts and soft-factor corrections, enabling systematic recurrences for all-plus and single-negative-helicity n-gluon amplitudes. The authors derive compact all-plus and single-minus recurrences, verify them numerically (up to n=15 for all-plus), and produce explicit results up to seven points, illustrating a practical, purely on-shell method complementary to unitarity. The approach holds promise for broader applications to other theories and higher-loop rational terms, potentially enabling more efficient Standard Model amplitude calculations.

Abstract

We present examples of on-shell recurrence relations for determining rational functions appearing in one-loop QCD amplitudes. In particular, we give relations for one-loop QCD amplitudes with all legs of positive helicity, or with one leg of negative helicity and the rest of positive helicity. Our recursion relations are similar to the tree-level ones described by Britto, Cachazo, Feng and Witten. A number of new features arise for loop amplitudes in non-supersymmetric theories like QCD, including boundary terms and double poles. We show how to eliminate the boundary terms, which would interfere with obtaining useful relations. Using the relations we give compact explicit expressions for the n-gluon amplitudes with one negative-helicity gluon, up through n=7.

On-Shell Recurrence Relations for One-Loop QCD Amplitudes

TL;DR

This work extends on-shell recursion techniques to non-supersymmetric one-loop QCD amplitudes, focusing on rational terms. It identifies and overcomes loop-specific obstacles—boundary terms and double poles—by employing generalized shifts and soft-factor corrections, enabling systematic recurrences for all-plus and single-negative-helicity n-gluon amplitudes. The authors derive compact all-plus and single-minus recurrences, verify them numerically (up to n=15 for all-plus), and produce explicit results up to seven points, illustrating a practical, purely on-shell method complementary to unitarity. The approach holds promise for broader applications to other theories and higher-loop rational terms, potentially enabling more efficient Standard Model amplitude calculations.

Abstract

We present examples of on-shell recurrence relations for determining rational functions appearing in one-loop QCD amplitudes. In particular, we give relations for one-loop QCD amplitudes with all legs of positive helicity, or with one leg of negative helicity and the rest of positive helicity. Our recursion relations are similar to the tree-level ones described by Britto, Cachazo, Feng and Witten. A number of new features arise for loop amplitudes in non-supersymmetric theories like QCD, including boundary terms and double poles. We show how to eliminate the boundary terms, which would interfere with obtaining useful relations. Using the relations we give compact explicit expressions for the n-gluon amplitudes with one negative-helicity gluon, up through n=7.

Paper Structure

This paper contains 6 sections, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Review of the Proof
  4. All-Positive Helicity One-Loop Amplitudes
  5. One-Loop Amplitudes with a Single Negative Helicity
  6. Conclusions and Outlook

Figures (3)

  • Figure 1: Terms in the recursive expression (\ref{['jlnrecurse']}) for $A_n^{(1)}(1^+, 2^+,\ldots,n^+)$. The vertices labeled by a $T$ are trees, and the ones labeled by an $L$ are loops.
  • Figure 2: Terms in the recursive expression for $A_5^{(1)}(1^-, 2^+, 3^+, 4^+,5^+)$.
  • Figure 3: The diagrams describing a recursive formula for $A_n^{(1)}(1^-, 2^+, \ldots, n^+)$. Diagram (a) vanishes, while diagram (d) needs to be summed over $j$. For $j=3$ diagram (d) contains both single-pole and double-pole contributions.