Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector

TL;DR

This paper explores how dual conformal symmetry, originally studied in planar supersymmetric theories, influences Feynman integrals in nonplanar and nonsupersymmetric theories like QCD, leading to new relations and differential equations.

Contribution

It demonstrates that dual conformal transformations generate IBP relations and differential equations for nonplanar integrals, extending symmetry concepts beyond planar supersymmetric contexts.

Findings

Dual conformal symmetry preserves unitarity cut conditions.

Generated differential equations often have RHS proportional to epsilon.

Identified a nonplanar analog of dual conformal symmetry.

Abstract

We show that dual conformal symmetry, mainly studied in planar $\mathcal N = 4$ super-Yang-Mills theory, has interesting consequences for Feynman integrals in nonsupersymmetric theories such as QCD, including the nonplanar sector. A simple observation is that dual conformal transformations preserve unitarity cut conditions for any planar integrals, including those without dual conformal symmetry. Such transformations generate differential equations without raised propagator powers, often with the right hand side of the system proportional to the dimensional regularization parameter $\epsilon$. A nontrivial subgroup of dual conformal transformations, which leaves all external momenta invariant, generates integration-by-parts relations without raised propagator powers, reproducing, in a simpler form, previous results from computational algebraic geometry for several examples with up to two loops and five legs. By opening up the two-loop three- and four-point nonplanar diagrams into planar ones, we find a nonplanar analog of dual conformal symmetry. As for the planar case this is used to generate integration-by-parts relations and differential equations. This implies that the symmetry is tied to the analytic properties of the nonplanar sector of the two-loop four-point amplitude of $\mathcal N = 4$ super-Yang-Mills theory.