Dual conformal symmetry, integration-by-parts reduction, differential equations, and the nonplanar sector

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.

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 of the double-box integral. 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 $b\overline{b}$ 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.

Figure 6

The pentabox integral.

Figure 7

The one-loop box diagram and its dual diagram.

Figure 8

The crossed triangle box, with two massless legs $p_1$ and $p_2$ and one massive leg shown as a thick line. We remove the rightmost part of the diagram enclosed in a (red) ellipse, in order to open up the diagram into a one-loop planar diagram.

Figure 9

This figure is obtained from Fig. 8 by removing the rightmost part enclosed in the (red) ellipse, including the massive leg. The result is a planar diagram, allowing dual coordinates $x_i$ to be introduced. Each of the dashed (blue) lines corresponds to one of the six propagators in the integral.

Figure 10

The two-loop nonplanar crossed box. The part of the diagram enclosed in a red ellipse will later be removed, so that the diagram is broken up into a one-loop planar diagram.

Figure 11

The planar one-loop diagram obtained by removing the bottom left part of Fig. 10. This allows one to introduce dual coordinates $x_i$. Each of the dashed lines corresponds to one of the six propagators in the integral.

Figure 12

Weight diagram for the integrand (6.19) under the conformal boost (6.23).

Figure 13

Weight diagram for the integrand (6.20) under the conformal boost (6.23).

Figure 14

An illustrative multiloop diagram, where a similar analysis as for the nonplanar double box in Fig. 10 identifies a hidden symmetry.