A three-body form factor at sub-leading power in the high-energy limit: planar contributions

We consider two-loop planar contributions to a three-body form factor at the next-to-leading power in the high-energy limit, where the masses of external particles are much smaller than their energies. The calculation is performed by exploiting the differential equations of the expansion coefficients, both for facilitating the linear relations among them, and for deriving their analytic expressions. The result is written in terms of generalized polylogarithms involving a few simple symbol letters. Our method can be readily applied to the calculation of non-planar contributions as well. The result provides crucial information for establishing sub-leading factorization theorems for massive scattering amplitudes in the high-energy limit.