Analytic result for the nonplanar hexa-box integrals

TL;DR

This work analytically determines all master integrals for one of the two non-planar, massless, five-point two-loop integral families (the hexa-box). By building a 73-term d-log basis and casting the differential equations in canonical form, the authors connect the solution space to the non-planar pentagon alphabet and fix boundary conditions from the absence of unphysical singularities, yielding expressions in terms of iterated integrals in the Euclidean region. They validate the results through comparisons with known subtopologies and independent Mellin-Barnes calculations, achieving high-precision numerical consistency. The study substantiates the conjectured function space for these non-planar integrals and provides a complete analytic resource for a key non-planar two-loop five-point contribution, with future work aimed at Minkowskian regions and the second non-planar family.

Abstract

In this paper, we analytically compute all master integrals for one of the two non-planar integral families for five-particle massless scattering at two loops. We first derive an integral basis of 73 integrals with constant leading singularities. We then construct the system of differential equations satisfied by them, and find that it is in canonical form. The solution space is in agreement with a recent conjecture for the non-planar pentagon alphabet. We fix the boundary constants of the differential equations by exploiting constraints from the absence of unphysical singularities. The solution of the differential equations in the Euclidean region is expressed in terms of iterated integrals. We cross-check the latter against previously known results in the literature, as well as with independent Mellin-Barnes calculations.

Figures (4)

• Figure 1: On the left, with label a), we depict the hexa-box integral family and on the right, with label b), the double pentagon integral family.
• Figure 2: Non-planar integral sectors with genuine five-particle kinematics. The labelling follows that of Bern:2015ple.
• Figure 3: The integration path \ref{['eq: integration path']}, going under the pole at $\tilde{x}=3$, is shown by the thick blue curve. Zeros of the letters \ref{['fullpentagonalphabet']} are marked by red crosses.
• Figure 4: One-loop hexagon integral with chiral numerator $\langle{1|x_{10}x_{02}|3}\rangle$ in region-momenta notations, $|{\lambda_i}\rangle[\tilde{\lambda}_i| = k_i =x_{i-1} - x_i$. The loop-integration $x_0$ is $D$-dimensional, $D=4-2\epsilon$, and the chiral numerator is four-dimensional. Pairs of on-shell momenta are used to represent an off-shell momentum.