# Testing the parton evolution with the use of two-body final states

## Abstract

We consider the production of \(b{\bar{b}}\) quarks and Drell–Yan lepton pairs under LHC conditions focusing attention on the total transverse momentum of the produced pair and on the azimuthal angle between the momenta of the outgoing particles. Plotting the corresponding distributions in bins of the final-state invariant mass, one can reconstruct the full map of the transverse momentum dependent parton densities in a proton. We give examples of how these distributions can look like at the LHC energies.

Experiments of new generation running at the LHC yield plenty of high precision data. In order to properly interpret these data we need the parton distribution functions to be known with adequately good accuracy. This, in turn, raises the question of a detailed measurement of the parton distributions. In this note we focus attention on two important kinematic observables which enable us to reconstruct the full map of the transverse momentum dependent (TMD), or unintegrated, parton densities. We address the LHC conditions (*pp* collisions at \(\sqrt{s}=7\) TeV), for which we give a number of illustrations.

*q*is for the valence and sea quarks and \(\bar{q}\) stands for the sea anti-quarks. The present calculation of the process (2) is fully identical to that from [2]. We do not consider here higher-order corrections \(q + \bar{q} \rightarrow l^+ + l^- + g\) since they are already taken into account in the \(k_T\)-factorization approach [3, 4, 5, 6, 7] as a part of the evolution of TMD quark densities.

*x*we watch the evolution of parton distributions.

To simulate the \(b{\bar{b}}\) pair production we used the latest JH’2013 parametrization [8] for the TMD gluon densities in a proton. The input parameters of this gluon distribution were fitted to describe the proton structure function \(F_2\). To simulate the production of Drell–Yan lepton pairs we applied complementary TMD valence quark distributions from the same set [8]. The necessary TMD sea quark densities are calculated from the gluon ones in the approximation where the sea quarks occur in the last gluon-to-quark splitting [9].

The results of our calculations are displayed in Figs. 1, 2, 3, 4, 5, 6, and 7. Shown in Figs. 1 and 2 are the spectra of \(b {\bar{b}}\) pair and dilepton transverse momentum \(p_T\) and the azimuthal angle \(\Delta \phi \) plotted for several different intervals of their invariant mass *M*. Here, to make the changes in shape more easily recognizable, we show the normalized differential cross sections. We see that with increasing *M* the maximum in the \(p_T\) spectrum shifts gradually to higher values, and the whole distribution becomes more flat. The \(\Delta \phi \) distribution moves toward \(\Delta \phi \simeq \pi \), which is due to the inequality \(M \gg p_T\). The latter becomes even stronger at high *M* (see Fig. 3). As one can see from Fig. 2, quark distributions follow the same trend as gluon densities.

The observed behavior of the calculated \(p_T\) and \(\Delta \phi \) distributions is related to the different regions of *x* and/or parton transverse momenta probed in the considered *M* bins. In fact, with increasing *M*, the *x* values obtained shifted toward unity, irrespectively of the rapidities of final-state particles, as is demonstrated in Figs. 4 and 5. The latter results show decreasing of the average parton transverse momentum generated in the non-collinear parton evolution. At the highest *M* bin, this average parton transverse momentum becomes small compared to the hard scale (which is order of *M*), so that the collinear kinematics of the partonic subprocesses is reproduced.

Besides the restrictions on the invariant mass, the special kinematical cuts on the final state give us further possibilities to find the region of *x* and/or partonic transverse momenta we desire. It is illustrated in Figs. 6 and 7, where we plot the normalized differential cross sections of the considered subprocesses calculated as functions of *x* and \({\mathbf k}_T^2\) (the longitudinal momentum fraction and transverse momentum of one of the colliding partons) with the additional cuts applied to the rapidity *y* of the final-state quark or lepton pair. As an example, we used \(y < 1\) and \(3< y < 4\). We show that under these cuts one can probe different *x* and/or \({\mathbf k}_T^2\) regions and extract information on the TMD parton distributions at the scale given by *M*. Note that the different \({\mathbf k}_T^2\) regions can be obtained under additional restrictions on the quark or lepton pair transverse momentum \(p_T\) and/or azimuthal angle \(\Delta \phi \).

*x*and/or \({\mathbf k}_T^2\) is reflected in the description of the LHC data. For example, we compared the rapidity distribution of Drell–Yan lepton pair production calculated regarding the kinematical conditions imposed by the LHCb Collaboration [11] using the JH’2013 parton densities (as above) and the ones obtained from the Kimber–Martin–Ryskin (KMR) prescription [12, 13] (see Fig. 8, left panel). The latter is a formalism to construct the TMD parton densities from the known conventional parton distributions. The key assumption is that the \(k_T\) dependence enters at the last evolution step, so that usual DGLAP evolution [14, 15, 16, 17] can be used up to this step. The rapidity distribution of Drell–Yan pair production is sensitive to the

*x*-behavior of TMD partons. One can clearly see that the difference between the TMD parton densities applied (Fig. 8, right panel) leads to a different description of recent LHCb data [11]. Therefore, the LHC experimental data for the processes considered can be used to constrain the TMD parton distributions in a proton.

Additionally, we investigate the dependence of estimates presented above on the parton shower effects using the Monte Carlo event generator cascade [18]. As expected, we observe only a very small contribution of the initial-state parton shower, since in the \(k_T\)-factorization approach it does not influence the transverse momentum of the gluons (because it is determined from the TMD gluon density).

Thus, we conclude that one can map the evolution of parton distributions at the scale *M* from high values of proton longitudinal momentum fraction *x* to low ones by applying different cuts on the final states. This is important for further precise determination of the TMD quark and gluon densities in a proton from the LHC data.

## Notes

### Acknowledgements

This research was supported in part by RFBR Grant 16-32-00176-mol-a and grant of the President of Russian Federation NS-7989.2016.2. We are grateful to DESY Directorate for the support in the framework of Moscow – DESY project on Monte-Carlo implementation for HERA – LHC. We are also grateful to Nikolai Zotov for all his enthusiasm, the many discussions and the very good time we had together. N. Zotov passed away in January 2016; we will miss him.

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