NNLO QCD$\times$EW corrections to on-shell $Z$ production

TL;DR

The paper provides the first analytic calculation of the mixed ${\cal O}(\alpha\alpha_s)$ QCD-EW corrections to the inclusive production of an on-shell Z boson at hadron colliders. It employs reverse unitarity, IBP reduction, and differential equations to obtain master integrals, including elliptic-integral cases, and expands two-mass kinematics to manage the mass difference between $m_W$ and $m_Z$. Renormalization is treated in both the $G_\mu$ and $\alpha(0)$ schemes, with PDFs evolved to include QED effects, enabling a consistent QCD×QED factorization. The numerical results show a reduction of theoretical uncertainties and a small negative shift in the cross section relative to NNLO-QCD with purely QCD PDFs, highlighting the practical impact of mixed corrections on precision LHC measurements and PDF luminosity calibrations.

Abstract

We present the first analytical results for the ${\cal O}(αα_s)$ corrections to the total cross section for the inclusive production of an on-shell $Z$ boson at hadron colliders. We include the complete set of contributions, with photonic and massive weak gauge boson effects, which have been computed in analytical form and expressed in terms of polylogarithmic and elliptic integrals. We present numerical results, relevant for the precision studies at the LHC. These corrections increase the accuracy of the predictions and contribute to the reduction of the QCD component of the theoretical uncertainty.