$\mathcal{O}({\alpha }_s^2\alpha )$ threshold corrections to inclusive $Z$-boson production

Abstract

To match the experimental precision targeted by future collider programs, theoretical predictions for Drell-Yan production must incorporate higher-order mixed Quantum Chromodynamics and electroweak corrections. In this work, we present for the first time the $\mathcal{O}({\alpha }_s^2\alpha )$ corrections to the inclusive cross-section for $Z$-boson production in the threshold limit. By systematically evaluating the missing singlet contributions, we obtain the complete set of $\mathcal{O}({\alpha }_s^2\alpha )$ quark form factors, which we use to construct the soft-virtual cross-section by exploiting renormalization group invariance, the K+G evolution equation, and universal soft-collinear factorization. Our results capture the dominant threshold logarithmic enhancements, improve the theoretical scale uncertainties, significantly reduce electroweak input scheme dependence, and establish a crucial step toward complete $\mathcal{O}({\alpha }_s^2\alpha )$ corrections indispensable for future precision programs.

arXiv comment8 pages, 1 figure;