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Fermionic pentagons and NMHV hexagon

A. V. Belitsky

TL;DR

The paper develops a nonperturbative framework for NMHV six-point amplitudes in planar $ ext{N}=4$ SYM by computing nonsinglet pentagon transitions on the color flux tube. It derives hole–fermion and gluon–fermion S-matrices and their mirrors, solves bootstrap equations to obtain pentagon form factors, and then translates these into NMHV form factors via a careful OPE treatment. Twist-two contributions from two-particle states are enumerated, including fermion–gluon and hole–antifermion sectors, with explicit measures and form factors. The resulting predictions agree with four-loop NMHV hexagon data, confirming the validity of the pentagon bootstrap approach and revealing the role of ad hoc Zhukowski-based NMHV form factors in matching SU(4) structure. The work advances nonperturbative control over scattering amplitudes in a highly symmetric gauge theory and sets the stage for higher-point generalizations.

Abstract

We analyze the near-collinear limit of the null polygonal hexagon super Wilson loop in the planar N = 4 superYang-Mills theory. We focus on its Grassmann components which are dual to next-to-maximal helicity-violating (NMHV) scattering amplitudes. The kinematics in question is studied within a framework of the operator product expansion that encodes propagation of excitations on the background of the color flux tube stretched between the sides of Wilson loop contour. While their dispersion relation is known to all orders in 't Hooft coupling from previous studies, we find their form factor couplings to the Wilson loop. This is done making use of a particular tessellation of the loop where pentagon transitions play a fundamental role. Being interested in NMHV amplitudes, the corresponding building blocks carry a nontrivial charge under the SU(4) R-symmetry group. Restricting the current consideration to twist-two accuracy, we analyze two-particle contributions with a fermion as one of the constituents in the pair. We demonstrate that these nonsinglet pentagons obey bootstrap equations that possess consistent solutions for any value of the coupling constant. To confirm the correctness of these predictions, we calculate their contribution to the super Wilson loop demonstrating agreement with recent results to four-loop order in 't Hooft coupling.

Fermionic pentagons and NMHV hexagon

TL;DR

The paper develops a nonperturbative framework for NMHV six-point amplitudes in planar SYM by computing nonsinglet pentagon transitions on the color flux tube. It derives hole–fermion and gluon–fermion S-matrices and their mirrors, solves bootstrap equations to obtain pentagon form factors, and then translates these into NMHV form factors via a careful OPE treatment. Twist-two contributions from two-particle states are enumerated, including fermion–gluon and hole–antifermion sectors, with explicit measures and form factors. The resulting predictions agree with four-loop NMHV hexagon data, confirming the validity of the pentagon bootstrap approach and revealing the role of ad hoc Zhukowski-based NMHV form factors in matching SU(4) structure. The work advances nonperturbative control over scattering amplitudes in a highly symmetric gauge theory and sets the stage for higher-point generalizations.

Abstract

We analyze the near-collinear limit of the null polygonal hexagon super Wilson loop in the planar N = 4 superYang-Mills theory. We focus on its Grassmann components which are dual to next-to-maximal helicity-violating (NMHV) scattering amplitudes. The kinematics in question is studied within a framework of the operator product expansion that encodes propagation of excitations on the background of the color flux tube stretched between the sides of Wilson loop contour. While their dispersion relation is known to all orders in 't Hooft coupling from previous studies, we find their form factor couplings to the Wilson loop. This is done making use of a particular tessellation of the loop where pentagon transitions play a fundamental role. Being interested in NMHV amplitudes, the corresponding building blocks carry a nontrivial charge under the SU(4) R-symmetry group. Restricting the current consideration to twist-two accuracy, we analyze two-particle contributions with a fermion as one of the constituents in the pair. We demonstrate that these nonsinglet pentagons obey bootstrap equations that possess consistent solutions for any value of the coupling constant. To confirm the correctness of these predictions, we calculate their contribution to the super Wilson loop demonstrating agreement with recent results to four-loop order in 't Hooft coupling.

Paper Structure

This paper contains 24 sections, 136 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hole--fermion pentagon
  3. Hole--(anti)fermion S-matrix
  4. Anomalous mirror in fermion rapidity
  5. Hole-fermion pentagons and bootstrap
  6. Form factors
  7. Gluon--fermion pentagons
  8. (Anti)gluon--(anti)fermion S-matrix
  9. Anomalous mirror
  10. Gluon--fermion pentagons
  11. Form factors
  12. OPE for NMHV hexagon
  13. Twist-one contribution
  14. Twist-two contributions
  15. Gluon--fermion states
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Pentagon form factor defining the coupling of flux-tube excitations to the Wilson loop contour.
  • Figure 2: The contour for the anomalous mirror transformation of the fermion (left panel) and its graphical representation as the fermion is moved into the crossed channel: it becomes a composite state of antifermion on the small sheet (hence its rapidity is dressed with a check $\check{u}$) and a hole (right panel). See Ref. Basso:2014koa for details.
  • Figure 3: The mirror S-matrix for the fermion and hole as a fusion of the hole-hole and antifermion--hole matrices in the crossed channel.