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Implications of nonplanar dual conformal symmetry

Dmitry Chicherin, Johannes M. Henn, Emery Sokatchev

TL;DR

This work investigates directional dual conformal invariance (DDCI) for next-to-planar Feynman integrals and its consequences at the level of integrated quantities, focusing on five-particle scattering. By projecting the dual conformal boost along a lightlike leg, DDCI reduces the independent kinematic variables from four to three and constrains the function space for pentagon functions from a 31-letter to a 10-letter alphabet. The authors verify the DDCI Ward identities for leading poles and analyze subleading terms, showing how IR anomalies are controlled and how the reduced alphabet suffices to capture these structures in explicit two-loop examples. These results suggest a powerful bootstrap program for nonplanar amplitudes and hint at deeper algebraic structures, potentially akin to cluster algebras, that could enable all-order understanding and amplitude decompositions beyond the planar limit.

Abstract

Recently, Bern et al observed that a certain class of next-to-planar Feynman integrals possess a bonus symmetry that is closely related to dual conformal symmetry. It corresponds to a projection of the latter along a certain lightlike direction. Previous studies were performed at the level of the loop integrand, and a Ward identity for the integral was formulated. We investigate the implications of the symmetry at the level of the integrated quantities. In particular, we focus on the phenomenologically important case of five-particle scattering. The symmetry simplifies the four-variable problem to a three-variable one. In the context of the recently proposed space of pentagon functions, the symmetry is much stronger. We find that it drastically reduces the allowed function space, leading to a well-known space of three-variable functions. Furthermore, we show how to use the symmetry in the presence of infrared divergences, where one obtains an anomalous Ward identity. We verify that the Ward identity is satisfied by the leading and subleading poles of several nontrivial five-particle integrals. Finally, we present examples of integrals that possess both ordinary and dual conformal symmetry.

Implications of nonplanar dual conformal symmetry

TL;DR

This work investigates directional dual conformal invariance (DDCI) for next-to-planar Feynman integrals and its consequences at the level of integrated quantities, focusing on five-particle scattering. By projecting the dual conformal boost along a lightlike leg, DDCI reduces the independent kinematic variables from four to three and constrains the function space for pentagon functions from a 31-letter to a 10-letter alphabet. The authors verify the DDCI Ward identities for leading poles and analyze subleading terms, showing how IR anomalies are controlled and how the reduced alphabet suffices to capture these structures in explicit two-loop examples. These results suggest a powerful bootstrap program for nonplanar amplitudes and hint at deeper algebraic structures, potentially akin to cluster algebras, that could enable all-order understanding and amplitude decompositions beyond the planar limit.

Abstract

Recently, Bern et al observed that a certain class of next-to-planar Feynman integrals possess a bonus symmetry that is closely related to dual conformal symmetry. It corresponds to a projection of the latter along a certain lightlike direction. Previous studies were performed at the level of the loop integrand, and a Ward identity for the integral was formulated. We investigate the implications of the symmetry at the level of the integrated quantities. In particular, we focus on the phenomenologically important case of five-particle scattering. The symmetry simplifies the four-variable problem to a three-variable one. In the context of the recently proposed space of pentagon functions, the symmetry is much stronger. We find that it drastically reduces the allowed function space, leading to a well-known space of three-variable functions. Furthermore, we show how to use the symmetry in the presence of infrared divergences, where one obtains an anomalous Ward identity. We verify that the Ward identity is satisfied by the leading and subleading poles of several nontrivial five-particle integrals. Finally, we present examples of integrals that possess both ordinary and dual conformal symmetry.

Paper Structure

This paper contains 20 sections, 85 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Directional dual conformal invariance of next-to-planar Feynman integrals
  3. Implications of DDCI for pentagon functions
  4. Five-particle DDCI variables
  5. Constraints on the pentagon alphabet from DDCI
  6. Checking the DDCI Ward identity for the leading poles: Weight two and three functions
  7. Topology (a)
  8. Topology (i)
  9. Implications of DDCI for the subleading poles
  10. Discussion and outlook
  11. Dual conformal symmetry
  12. Spinor conventions
  13. Conformal inversion and infinitesimal boosts
  14. Spinor-helicity variables
  15. Conformal boosts along a lightlike direction
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Dual space for a planar graph and its nonplanar analog obtained by adding one leg.
  • Figure 2: Integrals of topology $(a)$ with numerators $N_{a_1}$ (left) and $N_{a_2}$ (right). The arrows depict the fermion propagators and fermion external states that form the numerators.
  • Figure 3: Integral of topology (i)
  • Figure 4: Six-point dual space for the hexagon integral (\ref{['IaPole']}).
  • Figure 5: Finite six-dimensional integrals. Dots denote doubled propagators.