Power corrections in a transverse-momentum cut for vector-boson production at NNLO: the $qg$-initiated real-virtual contribution

We consider the production of a vector boson ($Z$, $W^\pm$ or $\gamma^*$) at next-to-next-to-leading order in the strong coupling constant $\alpha_{\rm S}$. We impose a transverse-momentum cutoff, $q_{\rm T}^{\rm cut}$, on the vector boson produced in the $qg$-initiated channel. We then compute the power corrections in the cutoff, up to the second power, of the real-virtual interference contribution to the cumulative cross section at order $\alpha_{\rm S}^2$. Other terms with the same kinematics, originating from the subtraction method applied to the double-real contribution, have been also considered. The knowledge of such power corrections is a required ingredient in order to reduce the dependence on the transverse-momentum cutoff of the QCD cross sections at next-to-next-to-leading order, when the $q_{\rm T}$-subtraction method is applied. In addition, the study of the dependence of the cross section on $q_{\rm T}^{\rm cut}$ allows as well for an understanding of its behaviour in the small transverse-momentum limit, giving hints on the structure at all orders in $\alpha_{\rm S}$ and on the identification of universal patterns. Our result are presented in an analytic form, using the process-independent procedure described in a previous paper for the calculation of the all-order power corrections in $q_{\rm T}^{\rm cut}$.