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

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