Master integrals for mixed QCD-QED corrections to charged-current Drell-Yan production of a massive charged lepton

The two-loop master integrals for the charged-current Drell-Yan process $q\bar{q}^{\prime} \rightarrow \ell \nu$ are computed analytically by using the differential equation method. A suitable choice of master integrals makes it successful to cast the differential equation system into the canonical form. We keep the dependence on charged lepton mass in the building of differential equations and then expand the system against the ratio of small charged lepton mass to large $W$-boson mass. In such a way the final results will contain large logarithms of the form $\log(m_{\ell}^2/m_W^2)$. Finally, all the canonical master integrals are given as Taylor series of $\epsilon = (d - 4)/2$ up to order four, with coefficients expressed in terms of Goncharov polylogarithms up to weight four.