$\mathrm{QCD}\oplus \mathrm{QED}$ NNLO corrections to Drell-Yan production

We compute the $\mathrm{QCD}\times \mathrm{QED}$ [$\mathcal{O}({\alpha }_s\alpha )$] mixed and ${\mathrm{QED}}^2$ [$\mathcal{O}({\alpha }^2)$] corrections to the production of an on-shell $Z$ boson in hadronic collisions. We obtain them by profiting from the calculation of the pure QCD terms after taking the corresponding Abelian limits. Therefore, we extend the available knowledge up to complete next-to-next-to-leading-order precision in $\mathrm{QCD}\oplus \mathrm{QED}$. We present explicit results for the perturbative coefficients and perform the phenomenological analysis at different collider energies with particular emphasis on the mixed corrections. We study the contribution from the different channels and discuss the scale dependence stabilization effect. We find that the contributions compete with the pure QCD NNLO ones under relevant kinematical conditions for the LHC, despite the fact that they are small, typically at the few per mille level.

Figure 1

Some of the diagrams contributing to NNLO QCD corrections to Drell-Yan.

Figure 2

Diagrams that result after applying the Abelianization procedure to the real NNLO QCD corrections in Fig. 1.

Figure 3

$K$-factors for the different distributions as defined in Eq. (6). The (blue) dashed line corresponds to $K_{\mathrm{QED}}^{\mathrm{NLO}}$, the (blue) dotted line to $K_{\mathrm{QED}}^{\mathrm{NNLO}}$, the solid line to the mixed $K_{\mathrm{QCD}\times \mathrm{QED}}^{\mathrm{NNLO}}$ and the (black) dotted line to the pure NNLO QCD corrections $K_{\mathrm{QCD}}^{\mathrm{NNLO}}$.

Figure 4

Ratio $R$ between the exact and the factorization approximation for the mixed $\mathrm{QCD}\times \mathrm{QED}$ contributions. The inset plot shows the ratio of the cross section computed exactly and with the factorization approximation for the mixed term.

Figure 5

Contribution to the mixed $\mathrm{QCD}\times \mathrm{QED}$ $K$-factor from the different channels. Here the label $q$ accounts for both quarks and antiquarks and $qq$ represents the sum of $q\overline{q}$ and $qq$.

Figure 6

Cross sections corresponding to LO [dashes, $i+j=0$ in Eq. (1)], NLO (dots, $i+j=0$,1) and NNLO (solid, $i+j=0$,1,2) at different factorization and renormalization scales with ${\mu }_R={\mu }_F=\mu$. All results are normalized by the corresponding cross section at $\mu =M_Z$.