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.
Highlights
There are two non-planar integral families for five particles at two loops, namely the hexa-box integral family a) and the double pentagon integral family b), shown in figure 1
We construct the system of differential equations satisfied by them, and find that it is in canonical form
We fix the boundary constants of the differential equations by exploiting constraints from the absence of unphysical singularities
Summary
To perform the integral reduction [24] for this hexa-box family, we use the program Reduze2 [25], which yields a basis of 73 master integrals. There are 54 planar integrals, 9 are non-planar with up to four external legs (four-point functions with one off-shell leg, which were computed in [26,27,28] in terms of generalized harmonic polylogarithms [29,30,31,32,33]) and 10 that are non-planar with five external legs The latter type of genuine non-planar five-point integrals in the hexa-box integral family are depicted in figure 2. Differential equations for the hexa-box integrals in an alternative basis in terms of pure integrals (containing higher propagator powers) were derived most recently in [34]. We note already here that this basis choice can be done algorithmically [23] by analyzing just the loop integrand
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