Threshold resummation of the rapidity distribution for Drell-Yan production at $\mathrm{NNLO}+\text{NNLL}$

We consider the production of pairs of lepton through the Drell-Yan process at the LHC and present the most accurate prediction on their rapidity distribution. While the fixed order prediction is already known to next-to-next-to-leading order in perturbative QCD, the resummed contribution coming from threshold region of phase space up to next-to-next-to-leading logarithmic (NNLL) accuracy has been computed in this article. The formalism developed in [1–3] has been used to resum large threshold logarithms in the two dimensional Mellin space to all orders in perturbation theory. We have done a detailed numerical comparison against other approaches that resum certain threshold logarithms in Mellin-Fourier space. Our predictions at NNLL level are close to theirs even though at leading logarithmic and next-to-leading logarithmic level we differ. We have also investigated the impact of these threshold logarithms on the stability of perturbation theory against factorization and renormalization scales. While the dependence on these scales does not get better with resummed results, the convergence of the perturbative series shows a better trend compared to the fixed order predictions. This is evident from the reduction in the K-factor for the resummed case compared to fixed order. We also present the uncertainties on the predictions resulting from parton distribution functions.

Figure 1

Cross sections against ${\mu }_F$ (left), ${\mu }_R$ (middle) and $\mu$ (right) variations at $\mathrm{NNLO}+\mathrm{NNLL}$ for 14 TeV LHC. The bands are obtained by using 7-point scale variation (see text for more details).

Figure 2

Resummed rapidity distribution in Drell-Yan production for the two sets of central scale choices $(M_Z,M_Z)$ and $(M_Z/2,M_Z)$ using MMHT PDFs at 14 TeV LHC. Corresponding bands are obtained using 7-point scale variation around the central scale. The lower panel represents the corresponding K-factors.

Figure 3

Fixed order predictions with the central scale ${\mu }_R={\mu }_F=M_Z$ and resummed prediction with the central scale ${\mu }_R=M_Z/2$, ${\mu }_F=M_Z$ for rapidity $y=0$ and $y=2.4$ using MMHT2014 PDF at each order. The uncertainties are obtained by using 7-point scale variation (see text for more details) around the central scale.

Figure 4

Drell-Yan rapidity distribution for 14 TeV LHC at $q=M_Z$ using MMHT PDFs. The fixed order results are plotted in the left panel and the resummed results in the right panel. Central scale is chosen as ${\mu }_R={\mu }_F=M_Z$ for both and the corresponding bands are obtained using 7-point scale variation (see text for more details) around the central scale. The lower panel represents the corresponding K-factors.

Figure 5

Drell-Yan rapidity distribution for 14 TeV LHC at $y=0$ using MMHT PDFs. The variation of fixed order and resummed results as a function of ${\mu }_R$ are shown separately for $q\overline{q}$ channel and also for all the channels added together.

Figure 6

Same as Fig. 4 but for $q=1\mathrm{TeV}$.

Figure 7

PDF variation at $\mathrm{NNLO}+\mathrm{NNLL}$ using various sets. The y-axis represents the ratio of extremum variation over the central PDF set.

Figure 8

Rapidity distribution at $\mathrm{NNLO}+\mathrm{NNLL}$ for 8 TeV LHC in the invariant mass range $60<q<120\mathrm{GeV}$. The dotted line is the fixed order NNLO contribution, the dashed line represents $\mathrm{NNLO}+\mathrm{NNLL}$ result and the solid line includes EW corrections.

Figure 9

Comparison between the resummed results and the CDF data [80, 81] at $\sqrt{s}=\{1.8\mathrm{TeV},1.96\mathrm{TeV}\}$ in the invariant mass range $66<q<116\mathrm{GeV}$ for two different PDF sets.