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All Next-to-Maximally-Helicity-Violating One-Loop Gluon Amplitudes in N=4 Super-Yang-Mills Theory

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

TL;DR

This work computes all next-to-MHV one-loop gluon amplitudes in N=4 super-Yang-Mills theory, i.e., amplitudes with three negative-helicity gluons, for arbitrary numbers of external legs. The authors employ a unitarity-based approach, augmented by generalized cuts and infrared consistency constraints, to derive compact expressions for all relevant box-coefficient functions, with four-mass boxes found to vanish in NMHV. A key result is the explicit, helicity-dependent form of three-mass box coefficients from which all other NMHV box coefficients are built, along with novel infrared-derived tree representations for NMHV amplitudes that resemble CSW constructions. The paper also shows that NMHV one-loop coefficients possess a planar twistor-space structure, providing a striking geometric picture and supporting the connection between perturbative amplitudes and twistor-string ideas. Overall, the results offer highly constrained, elegant expressions with consistency checks across soft, collinear, and multi-particle limits, and suggest broader applicability to QCD and potential string-theoretic interpretations.

Abstract

We compute the next-to-MHV one-loop n-gluon amplitudes in N=4 super-Yang-Mills theory. These amplitudes contain three negative-helicity gluons and an arbitrary number of positive-helicity gluons, and are the first infinite series of amplitudes beyond the simplest, MHV, amplitudes. We also discuss some aspects of their twistor-space structure.

All Next-to-Maximally-Helicity-Violating One-Loop Gluon Amplitudes in N=4 Super-Yang-Mills Theory

TL;DR

This work computes all next-to-MHV one-loop gluon amplitudes in N=4 super-Yang-Mills theory, i.e., amplitudes with three negative-helicity gluons, for arbitrary numbers of external legs. The authors employ a unitarity-based approach, augmented by generalized cuts and infrared consistency constraints, to derive compact expressions for all relevant box-coefficient functions, with four-mass boxes found to vanish in NMHV. A key result is the explicit, helicity-dependent form of three-mass box coefficients from which all other NMHV box coefficients are built, along with novel infrared-derived tree representations for NMHV amplitudes that resemble CSW constructions. The paper also shows that NMHV one-loop coefficients possess a planar twistor-space structure, providing a striking geometric picture and supporting the connection between perturbative amplitudes and twistor-string ideas. Overall, the results offer highly constrained, elegant expressions with consistency checks across soft, collinear, and multi-particle limits, and suggest broader applicability to QCD and potential string-theoretic interpretations.

Abstract

We compute the next-to-MHV one-loop n-gluon amplitudes in N=4 super-Yang-Mills theory. These amplitudes contain three negative-helicity gluons and an arbitrary number of positive-helicity gluons, and are the first infinite series of amplitudes beyond the simplest, MHV, amplitudes. We also discuss some aspects of their twistor-space structure.

Paper Structure

This paper contains 12 sections, 58 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Calculational Approach
  4. Results
  5. Consistency of the Results
  6. New Representations of Tree Amplitudes
  7. Twistor-Space Properties
  8. Overview
  9. Twistor properties of three-mass box coefficients
  10. Conclusions and Outlook
  11. Explicit Demonstration of Planarity of Coefficients
  12. Box Integrals

Figures (9)

  • Figure 1: Examples of box integral functions $B(i,j,k,l)$ appearing in seven-point amplitudes; the arguments $i,j,k,l$ are the uncanceled propagators: (a) the one-mass box $B(3,4,5,6)=F^{\rm 1m}(s_{34},s_{45},s_{345})$, (b) the 'easy' two-mass box $B(3,4,6,7) = F^{{\rm 2m}e}(s_{345},s_{456},s_{45},s_{712})$, (c) the 'hard' two-mass box $B(3,5,6,7) = F^{{\rm 2m}h}(s_{56},s_{345},s_{712},s_{34})$, and (d) the three-mass box $B(2,4,6,7) = F^{{\rm 3m}}(s_{671},s_{456},s_{71},s_{23},s_{45})$.
  • Figure 2: A generalized triple cut. The three propagators cut by the dashed lines are required to be 'open'.
  • Figure 3: The box integral functions labeled by the clusters of masses.
  • Figure 4: Schematic depiction of the easy two-mass box coefficients, expressed as a double sum over three-mass box coefficients.
  • Figure 5: An easy two-mass box where positive-helicity legs $a$ and $b$ are buried inside a cluster.
  • ...and 4 more figures