Differential equations on unitarity cut surfaces

Abstract

We reformulate differential equations (DEs) for Feynman integrals to avoid doubled propagators in intermediate steps. External momentum derivatives are dressed with loop momentum derivatives to form tangent vectors to unitarity cut surfaces, in a way inspired by unitarity-compatible IBP reduction. For the one-loop box, our method directly produces the final DEs without any integration-by-parts reduction. We further illustrate the method by deriving maximal-cut level differential equations for two-loop nonplanar five-point integrals, whose exact expressions are yet unknown. We speed up the computation using finite field techniques and rational function reconstruction.

Figures (4)

• Figure 1: Left: the scalar box integral with massless lines, $\mathcal{I}_{\rm box}$. Center: the $s$-channel scalar triangle integral, $\mathcal{I}_{\rm triangle}^{(s)}$. Right: the $t$-channel scalar triangle integral, $\mathcal{I}_{\rm triangle}^{(t)}$.
• Figure 2: The one-loop pentagon with massless lines.
• Figure 3: The nonplanar pentabox integral with massless propagators and external legs with zero masses.
• Figure 4: The massless double box. The inverse propagators are labeled on the figure. The irreducible numerators are $z_8 = (l_1 + p_4)^2$ and $z_9 = (l_2 + p_1)^2$. The external kinematic invariants are $(p_1+p_2)^2=s$ and $(p_2+p_3)^2=t$.