Mixed QCD-EW corrections to $W$-pair production at electron-positron colliders

The discrepancy between the CDF measurement and the Standard Model theoretical prediction for the $W$-boson mass underscores the importance of conducting high-precision studies on the $W$ boson, which is one of the predominant objectives of proposed future $e^+e^-$ colliders. We investigate in detail the production of $W$-boson pairs at $e^+e^-$ colliders, and compute the next-to-next-to-leading order mixed QCD-EW corrections to both the integrated cross section and various kinematic distributions. By employing the method of differential equations, we analytically calculate the two-loop master integrals for the mixed QCD-EW virtual corrections to $e^+e^- \rightarrow W^+W^-$. Utilizing the Magnus transformation, we derive a set of canonical master integrals for each integral family. This canonical basis satisfies a system of differential equations in which the dependence on the dimensional regulator is linearly factorized from the kinematics. We then express all these canonical master integrals as Taylor series in $\epsilon$ up to $\epsilon^4$, with coefficients articulated in terms of Goncharov polylogarithms up to weight four. Upon applying our analytic expressions of these master integrals to the phenomenological analysis of $W$-pair production, we observe that the $\mathcal{O}(\alpha\alpha_s)$ corrections are significantly impactful in the $\alpha(0)$ scheme, particularly in certain phase-space regions. However, these mixed QCD-EW corrections can be heavily suppressed by adopting the $G_{\mu}$ scheme.