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One-Loop Amplitudes in N=4 Super Yang-Mills and Anomalous Dual Conformal Symmetry

Andreas Brandhuber, Paul Heslop, Gabriele Travaglini

TL;DR

This work shows that anomalous dual conformal symmetry at one loop in N=4 SYM imposes strong, calculable constraints on box-coefficient structures. By deriving the box-function anomalies and their relation to tree-level data, the authors obtain a conformal Ward identity framework that reduces almost all one-loop coefficients to a small set of undetermined inputs, namely the symmetric two-mass-hard combinations and three-mass box data, except for the four-mass boxes that remain unconstrained. They explicitly verify these constraints for MHV and NMHV cases and prove the dual conformal covariance of NMHV amplitudes at arbitrary multiplicity via a nontrivial identity among R_{rst}. The results unify infrared behavior with dual conformal anomalies and provide a symmetry-driven reduction of the one-loop amplitude parameter space, consistent with Wilson loop–amplitude duality and advancing precise predictions for general n-point amplitudes.

Abstract

We discuss what predictions can be made for one-loop superamplitudes in maximally supersymmetric Yang-Mills theory by using anomalous dual conformal symmetry. We show that the anomaly coefficient is a specific combination of two-mass hard and one-mass supercoefficients which appears in the supersymmetric on-shell recursion relations and equals the corresponding tree-level superamplitude. We discuss further novel relations among supercoefficients imposed by the remaining non-anomalous part of the symmetry. In particular, we find that all one-loop supercoefficients, except the four-mass box coefficients, can be expressed as linear combinations of three-mass box coefficients and a particular symmetric combination of two-mass hard coefficients. We check that our equations are explicitly satisfied in the case of one-loop n-point MHV and NMHV amplitudes. As a bonus, we prove the covariance of the NMHV superamplitudes at an arbitrary number of points, extending previous results at n <= 9.

One-Loop Amplitudes in N=4 Super Yang-Mills and Anomalous Dual Conformal Symmetry

TL;DR

This work shows that anomalous dual conformal symmetry at one loop in N=4 SYM imposes strong, calculable constraints on box-coefficient structures. By deriving the box-function anomalies and their relation to tree-level data, the authors obtain a conformal Ward identity framework that reduces almost all one-loop coefficients to a small set of undetermined inputs, namely the symmetric two-mass-hard combinations and three-mass box data, except for the four-mass boxes that remain unconstrained. They explicitly verify these constraints for MHV and NMHV cases and prove the dual conformal covariance of NMHV amplitudes at arbitrary multiplicity via a nontrivial identity among R_{rst}. The results unify infrared behavior with dual conformal anomalies and provide a symmetry-driven reduction of the one-loop amplitude parameter space, consistent with Wilson loop–amplitude duality and advancing precise predictions for general n-point amplitudes.

Abstract

We discuss what predictions can be made for one-loop superamplitudes in maximally supersymmetric Yang-Mills theory by using anomalous dual conformal symmetry. We show that the anomaly coefficient is a specific combination of two-mass hard and one-mass supercoefficients which appears in the supersymmetric on-shell recursion relations and equals the corresponding tree-level superamplitude. We discuss further novel relations among supercoefficients imposed by the remaining non-anomalous part of the symmetry. In particular, we find that all one-loop supercoefficients, except the four-mass box coefficients, can be expressed as linear combinations of three-mass box coefficients and a particular symmetric combination of two-mass hard coefficients. We check that our equations are explicitly satisfied in the case of one-loop n-point MHV and NMHV amplitudes. As a bonus, we prove the covariance of the NMHV superamplitudes at an arbitrary number of points, extending previous results at n <= 9.

Paper Structure

This paper contains 18 sections, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. One-loop amplitudes and anomalous box functions
  3. One-loop superamplitudes
  4. Anomalous one-loop amplitudes
  5. Anomalies of one-loop box functions
  6. Conformal constraints on scattering amplitudes
  7. Generic $n$-point superamplitudes
  8. Examples of applications of the conformal equations
  9. Four- and five-point amplitudes
  10. Six-point amplitudes
  11. Analysis of the conformal equations
  12. Comparison with infrared consistency conditions
  13. Solution to the conformal equations
  14. The anomaly equations for MHV and NMHV superamplitudes
  15. MHV (and anti-MHV) amplitudes
  16. ...and 3 more sections

Figures (3)

  • Figure 1: A generic box function. $K_1, K_2, K_3$ and $K_4$ denote the external momenta, and $x_1, x_2, x_3$ and $x_4$ the corresponding region momenta, with $K_i = x_i - x_{i+1}$, $i=1, \ldots , 4$.
  • Figure 2: The 1m, 2me, 2mh, 3m box functions whose dual conformal anomalies are calculated in \ref{['cieq']}. Massive legs are depicted in bold blue.
  • Figure 3: The two cut diagrams which contribute to the 2me box function coefficient of the NMHV amplitude. Here in the corners $M$ represents a tree-level MHV amplitude, $\bar{M}$ the three-point anti-MHV amplitude and $N$ the tree-level NMHV amplitude.