# Master integrals of a planar double-box family for top-quark pair production

## TL;DR

This paper analytically computes the master integrals for a planar double-box family relevant to top-quark pair production, using differential equations and multiple polylogarithms, providing explicit formulas for complex kinematic scenarios.

## Contribution

It introduces a method to transform differential equations into d-log form and rationalize square roots, enabling explicit analytic solutions for all master integrals involved.

## Key findings

- Analytic expressions for 33 master integrals obtained
- Master integrals expressed in terms of multiple polylogarithms up to weight four
- Boundary conditions fixed by simple integrals and regularity at special points

## Abstract

We calculate analytically the master integrals of a planar double-box family for top-quark pair production using the method of differential equations. With a proper choice of the bases, the differential equations can be transformed to the $d$-log form. The square roots of the kinematic variables in the differential equations can be rationalized by defining two dimensionless variables. We find that all the boundary conditions can be fully fixed either by simple integrals or regularity conditions at some special kinematic points. The analytic results for thirty-three master integrals at general kinematics are all expressed in terms of multiple polylogarithms up to transcendental weight four.