NNLO $\mathrm{QCD}\times \mathrm{EW}$ corrections to $Z$ production in the $q\overline{q}$ channel

We present the first results for the $\mathcal{O}(\alpha {\alpha }_s)$ corrections to the total partonic cross section of the process $q\overline{q}\to Z+X$, with the complete set of contributions, that include photonic and massive weak gauge boson effects. The results are relevant for the precise determination of the hadronic $Z$ boson production cross section. Virtual and real corrections are calculated analytically using the reduction to the master integrals and their evaluation through differential equations. Real corrections are dealt with using the reverse-unitarity method. They require the evaluation of a new set of two-loop master integrals, with up to three internal massive lines. In particular, three of them are expressed in terms of elliptic integrals. We verify the absence, at this perturbative order, of initial-state mass singularities proportional to a weak massive virtual correction to the quark-gluon splitting.

Figure 1

Corrections factors ${\mathrm{\Delta }}_{u\overline{u}}^{(1,0)}$ and ${\mathrm{\Delta }}_{u\overline{u},K}^{(1,1)}$ with $K=\gamma ,Z,W$, as a function of the partonic variable $z$. The NLO-QCD correction ${\mathrm{\Delta }}_{u\overline{u}}^{(1,0)}$ (gray dashed) is divided by a factor 10. We show the $\mathcal{O}(\alpha {\alpha }_s)$ total correction (black solid), and the contributions of the different subsets $K=\gamma ,Z,W$ in blue, red, and magenta, respectively.