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

TL;DR

The work shows that dual conformal symmetry, originally a planar tool in $\mathcal{N}=4$ SYM, yields powerful, unitarity-compatible IBP relations and differential equations for more general theories, including the nonplanar sector. By employing both direct dual-conformal transformations and the embedding formalism, the authors construct compact IBP-generating vectors and DEs that avoid raising propagator powers, facilitating reductions to master integrals and enabling symmetry-based differential equations with $\epsilon$-proportional right-hand sides. They extend the framework to a nonplanar analogue of dual conformal symmetry, derive DEs for nonplanar two-loop integrals, and demonstrate invariance for the full two-loop four-point $\mathcal{N}=4$ amplitude, linking these results to Landau equations and the analytic structure of amplitudes. The findings offer a unifying, symmetry-driven approach to simplifying multi-loop integrals and provide a foundation for systematic nonplanar generalizations and applications to collider phenomenology.

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 $ε$. 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.

Figures (14)

• Figure 1: The double box integrals. Differences of the dual points give momenta flowing in the diagram. The $x_i$ and $y_i$ are dual coordinates the double box. The dual diagram is given by the dashed (blue) diagram.
• Figure 2: The one-loop triangle with outgoing external momenta $p_1, \, p_2, \,-p_1-p_2$ and dual points $x_1,x_2,x_3$. All internal propagators are massless, and the single massive external leg has mass $(p_1+p_2)^2=s$, shown as a thick (red) line. The dashed (blue) lines indicate the dual diagram.
• Figure 3: The one-loop triangle with outgoing external momenta $p_1, \, p_2, \, -p_1-p_2$. All internal propagators are massless, and the massive external legs, shown as thick lines, have masses $p_2^2 =t$ and $(-p_1-p_2)^2=s$. The dashed (blue) line indicates the dual diagram.
• Figure 4: The one-loop triangle that appears in the decay of a Higgs boson to a $b \bar{b}$ quark pair. The outgoing external momenta are $p_1, \, p_2, \, -p_1-p_2$. The Higgs leg, shown as a thick (red) line on the rightmost part of the figure, has squared mass $(-p_1-p_2)^2 = m_H^2$. The bottom-quark lines, appearing in both external legs and internal propagators, are shown as thick (blue) lines with squared mass $m_b^2$.
• Figure 5: The triangle-box diagram.