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

Zvi Bern, Vittorio Del Duca, Lance J. Dixon, David A. Kosower

TL;DR

The authors compute all four independent seven-point NMHV one-loop amplitudes in ${\cal N}=4$ super-Yang-Mills theory using a unitarity-based approach, reducing the result to a basis of scalar box integrals with compact spinor-coefficient expressions. They uncover striking simplicity and a universal twistor-space structure: box coefficients are coplanar, expressible in a small number of terms, and organized by box topology with symmetries enhancing tractability. An all-$n$ NMHV coefficient is derived for a class of three-mass boxes, illustrating how holomorphic anomaly considerations extend to general NMHV amplitudes. The results provide robust checks via collinear and multi-particle factorization limits and offer a valuable benchmark for future QCD calculations and multi-leg loop analyses.

Abstract

We compute the non-MHV one-loop seven-gluon amplitudes in N=4 super-Yang-Mills theory, which contain three negative-helicity gluons and four positive-helicity gluons. There are four independent color-ordered amplitudes, (- - - + + + +), (- - + - + + +), (- - + + -+ +) and (- + - + - + +). The MHV amplitudes containing two negative-helicity and five positive-helicity gluons were computed previously, so all independent one-loop seven-gluon helicity amplitudes are now known for this theory. We present partial information about an infinite sequence of next-to-MHV one-loop helicity amplitudes, with three negative-helicity and n-3 positive-helicity gluons, and the color ordering (- - - + + ... + +); we give a new coefficient of one class of integral functions entering this amplitude. We discuss the twistor-space properties of the box-integral-function coefficients in the amplitudes, which are quite simple and suggestive.

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

TL;DR

The authors compute all four independent seven-point NMHV one-loop amplitudes in super-Yang-Mills theory using a unitarity-based approach, reducing the result to a basis of scalar box integrals with compact spinor-coefficient expressions. They uncover striking simplicity and a universal twistor-space structure: box coefficients are coplanar, expressible in a small number of terms, and organized by box topology with symmetries enhancing tractability. An all- NMHV coefficient is derived for a class of three-mass boxes, illustrating how holomorphic anomaly considerations extend to general NMHV amplitudes. The results provide robust checks via collinear and multi-particle factorization limits and offer a valuable benchmark for future QCD calculations and multi-leg loop analyses.

Abstract

We compute the non-MHV one-loop seven-gluon amplitudes in N=4 super-Yang-Mills theory, which contain three negative-helicity gluons and four positive-helicity gluons. There are four independent color-ordered amplitudes, (- - - + + + +), (- - + - + + +), (- - + + -+ +) and (- + - + - + +). The MHV amplitudes containing two negative-helicity and five positive-helicity gluons were computed previously, so all independent one-loop seven-gluon helicity amplitudes are now known for this theory. We present partial information about an infinite sequence of next-to-MHV one-loop helicity amplitudes, with three negative-helicity and n-3 positive-helicity gluons, and the color ordering (- - - + + ... + +); we give a new coefficient of one class of integral functions entering this amplitude. We discuss the twistor-space properties of the box-integral-function coefficients in the amplitudes, which are quite simple and suggestive.

Paper Structure

This paper contains 19 sections, 100 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Review of Previous Amplitude Results
  3. Calculational Approach
  4. Results
  5. $A_{7;1}^{{\cal N}=4}(1^-,2^-,3^-,4^+,5^+,6^+,7^+)$
  6. $A_{7;1}^{{\cal N}=4}(1^-,2^-,3^+,4^-,5^+,6^+,7^+)$
  7. $A_{7;1}^{{\cal N}=4}(1^-,2^-,3^+,4^+,5^-,6^+,7^+)$
  8. $A_{7;1}^{{\cal N}=4}(1^-,2^+,3^-,4^+,5^-,6^+,7^+)$
  9. Consistency of the Results
  10. Collinear Behavior
  11. Multi-Particle Factorization
  12. Cancellation of spurious singularities
  13. An All-Multiplicity Coefficient
  14. Twistor-Space Properties
  15. Properties of seven-point box coefficients $c_{ijk}$
  16. ...and 4 more sections

Figures (6)

  • Figure 1: A generalized 'triple cut'. The three propagators cut by the dashed lines are required to be 'open'.
  • Figure 2: Examples of box integral functions $B(i,j,k)$ appearing in 7-point amplitudes; the arguments $i,j,k$ are circled: (a) the one-mass box $B(1,2,7)=F^{\rm 1m}(s_{34},s_{45},s_{345})$, (b) the 'easy' two-mass box $B(1,2,5) = F^{{\rm 2m}e}(s_{345},s_{456},s_{45},s_{712})$, (c) the 'hard' two-mass box $B(1,2,4) = F^{{\rm 2m}h}(s_{56},s_{345},s_{712},s_{34})$, and (d) the three-mass box $B(1,3,5) = F^{{\rm 3m}}(s_{671},s_{456},s_{71},s_{23},s_{45})$.
  • Figure 3: A schematic depiction of multi-particle factorization at one loop.
  • Figure 4: The class of three-mass box functions whose coefficient is given in eq. (\ref{['Alln3m']}).
  • Figure 5: Examples of twistor-space configurations for single-term box coefficients in the helicity amplitude $A_{7;1}^{{\cal N}=4}(1^-,2^-,3^+,4^-,5^+,6^+,7^+)$. In every case, all the points lie in a plane. (a) the easy two-mass box coefficient $c_{125}$, (b) the hard two-mass box coefficient $c_{237}$, (c) the three-mass box coefficient $c_{135}$.
  • ...and 1 more figures