Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Amplitudes in Pure Yang-Mills and MHV Diagrams

Andreas Brandhuber, Bill Spence, Gabriele Travaglini

TL;DR

This work develops a hybrid perturbation framework for Yang-Mills theory in which MHV diagrams (CSW off-shell continuation) are combined with lightcone Yang–Mills methods. The authors show that the one-loop all-minus gluon amplitude in pure Yang-Mills can be generated entirely from MHV vertices and matches the lightcone-gauge result to all orders in the dimensional regularization parameter ε. They then demonstrate that a holomorphic, non-canonical field redefinition introduces a Jacobian that exactly accounts for the all-plus one-loop amplitude, while single-helicity amplitudes arise from a combination of Jacobian effects and equivalence-theorem violations, yielding a coherent, hybrid perturbation theory. The study highlights an intrinsic link between Jacobians, holomorphic field redefinitions, and the structure of loop amplitudes, suggesting new avenues for efficient amplitude computations in non-supersymmetric gauge theories.

Abstract

We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity in non-supersymmetric Yang-Mills theory using MHV diagrams. We argue that the amplitude with all positive helicity gluons arises from a Jacobian which occurs when one performs a Backlund-type holomorphic change of variables in the lightcone Yang-Mills Lagrangian. This also results in contributions to scattering amplitudes from violations of the equivalence theorem. Furthermore, we discuss how the one-loop amplitudes with a single positive or negative helicity gluon arise in this formalism. Perturbation theory in the new variables leads to a hybrid of MHV diagrams and lightcone Yang-Mills theory.

Amplitudes in Pure Yang-Mills and MHV Diagrams

TL;DR

This work develops a hybrid perturbation framework for Yang-Mills theory in which MHV diagrams (CSW off-shell continuation) are combined with lightcone Yang–Mills methods. The authors show that the one-loop all-minus gluon amplitude in pure Yang-Mills can be generated entirely from MHV vertices and matches the lightcone-gauge result to all orders in the dimensional regularization parameter ε. They then demonstrate that a holomorphic, non-canonical field redefinition introduces a Jacobian that exactly accounts for the all-plus one-loop amplitude, while single-helicity amplitudes arise from a combination of Jacobian effects and equivalence-theorem violations, yielding a coherent, hybrid perturbation theory. The study highlights an intrinsic link between Jacobians, holomorphic field redefinitions, and the structure of loop amplitudes, suggesting new avenues for efficient amplitude computations in non-supersymmetric gauge theories.

Abstract

We show how to calculate the one-loop scattering amplitude with all gluons of negative helicity in non-supersymmetric Yang-Mills theory using MHV diagrams. We argue that the amplitude with all positive helicity gluons arises from a Jacobian which occurs when one performs a Backlund-type holomorphic change of variables in the lightcone Yang-Mills Lagrangian. This also results in contributions to scattering amplitudes from violations of the equivalence theorem. Furthermore, we discuss how the one-loop amplitudes with a single positive or negative helicity gluon arise in this formalism. Perturbation theory in the new variables leads to a hybrid of MHV diagrams and lightcone Yang-Mills theory.

Paper Structure

This paper contains 17 sections, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The all-minus amplitudes
  3. Coupling to a scalar
  4. Three-point vertices
  5. The one-loop three-minus vertex
  6. The one-loop splitting amplitude from the one-loop vertex
  7. The four-point all-minus amplitude
  8. The box diagram
  9. The triangle diagrams
  10. The bubble diagram
  11. A holomorphic field redefinition
  12. The all-plus helicity amplitude from a Jacobian
  13. The single-minus and the single-plus helicity amplitudes
  14. The single-plus amplitude
  15. The single-minus amplitude
  16. ...and 2 more sections

Figures (4)

  • Figure 1: The MHV-diagram for the all-minus three-gluon vertex with one leg off shell (drawn with a double line).
  • Figure 2: The box MHV-diagram contributing to the all-minus four-gluon amplitude. A complex scalar runs in the loop, and one has to sum over the two possible internal helicity assignments. This has the effect of doubling the result of a single internal helicity assignment.
  • Figure 3: Bubble and triangle diagrams contributing to the all-minus four-gluon amplitude. The bubble diagram is found to be zero. We also have to include three more triangle diagrams, obtained from the one in the figure by cyclically rotating the external particles.
  • Figure 4: Kinematics of the bubble and triangle integral functions studied in this appendix.