Associated production of $J/\psi$ plus $Z(W)$ in the improved color evaporation model using the parton Reggeization approach

Associated production of prompt $J/\psi$ mesons and $Z$ or $W$ bosons is very important process for testing perturbative quantum chromodynamics (pQCD), as well as for understanding the mechanism of heavy quarkonium production in the high-energy collisions. The next important issue, the experimental data for spectra and correlation dependencies in the two particles associated production in hard processes at the high energies, motivate us to consider not only the conventional single parton scattering (SPS) scenario of the parton model, but also to take into account the double parton scattering (DPS) mechanism. There are a number of theoretical predictions for cross sections and spectra of such processes, obtained in the collinear parton model (CPM), which are based on the leading order (LO) [1] and the next-to-leading order (NLO) approximations in the strong coupling constant $\alpha_{S}$ [2, 3]. In Refs. [1, 2], the factorization approach of nonrelativistic QCD (NRQCD) [4] was used to describe the prompt $J/\psi$ plus Z-boson production. Oppositely, in Ref. [3], the one loop pQCD calculations were performed as it is used in the color evaporation model (CEM) [5, 6]. In both approaches for $J/\psi$ production mechanism, NRQCD and CEM, the results of calculations in the SPS scenario of the NLO CPM [2, 3] are strongly underestimate the experimental data for $J/\psi$ plus $Z$ production from the ATLAS collaboration at $\sqrt{s}=7\mbox{ and }8$ TeV [7, 8, 9]. These data as well as many others for pair heavy quarkonium production $(J/\psi J/\psi,\Upsilon\Upsilon,\Upsilon J/\psi)$ need inclusion of large DPS contribution in the theoretical framework.

Nowadays, the extractions of parameter $\sigma_{\rm eff}$, based on the DPS pocket formula, have been obtained in different experiments. Values of $\sigma_{\rm eff}=2-25$ mb have been derived [10], though with large errors, with a simple average giving $\sigma_{\rm eff}=15$ mb. It is interesting to study associated $J/\psi$ plus $Z(W)$ beyond the collinear approximation of the parton model, as it was done in [2, 3], i.e. using the high-energy factorization (HEF) or $k_{T}$-factorization approach  [11, 12, 13].

In the present study, we calculate cross section for associated $J/\psi$ plus $Z(W)$ production in the proton-proton collisions in the parton Reggeization approach (PRA) [14, 15, 16], which is a version of the $k_{T}$-factorization approach [11, 12, 13]. The PRA is accurate both in the collinear limit, which drives the transverse-momentum-dependent (TMD) factorization [17] and in the high-energy limit, which is important for Balitsky-Fadin-Kuraev-Lipatov (BFKL) [18, 19, 20, 21] resummation of $\ln(\sqrt{s}/\mu)$-enhanced effects. In the PRA, we have studied successfully a heavy quarkonium production in the proton-(anti) proton collisions at the Tevatron and the LHC using NRQCD approach, see Refs. [22, 23, 24, 25]. The pair production of prompt $J/\psi$ was studied in the PRA using NRQCD approach in the Ref. [26] and using the improved color evaporation model (ICEM) [27] in the Ref. [28]. In our previous study of single and double $J/\psi$ or $\Upsilon$ production, as well as $J/\psi\Upsilon$ pair production, in the PRA using the ICEM  [28, 29], we considered both, the SPS and the DPS production mechanism, and we have found numerical values for parameters of the ICEM $({\cal F}^{\psi},{\cal F}^{\Upsilon}$) at the energy of LHC and the universal DPS parameter $\sigma_{\rm eff}$. These parameters are used during the calculations presented below.

The paper has the following structure. In Section 2, the PRA formalism and the ICEM are shortly reviewed. The main formulas of the SPS and DPS scenarios are presented as well as details of the numerical calculations which are based on the Monte-Carlo parton-level event generator KaTie [30]. In Section 3, we compare our predictions and the experimental data for the associated production of prompt $J/\psi$ mesons and $Z(W)$ bosons at the energies $7$ and $8$ TeV . We also present predictions for the energy $\sqrt{s}=13$ TeV, both for the $J/\psi$ plus $Z(W)$ production and for $\Upsilon$ plus $Z(W)$ associated production. Our conclusions are summarized in Section 4.

Now we are in position to compare our theoretical predictions obtained in the ICEM using the PRA within the SPS and the DPS contributions with the data [7, 8, 9] . First of all, we describe phase-space definition of the measured fiducial production cross-section following the geometrical acceptance of the ATLAS detector [7] at $\sqrt{s}=8$ TeV. The Z boson selection via two-lepton decays $Z\to\ell\bar{\ell}\;(\ell=e,\mu)$ includes following criteria: pseudorapidity of each lepton $|\eta^{\ell}|<2.5$, leading lepton has $p_{T}^{\ell_{1}}>25$ GeV and subleading lepton has $p_{T}^{\ell_{2}}>15$ GeV, $|M_{\ell\bar{\ell}}-m_{Z}|<10$ GeV, where $M_{\ell\bar{\ell}}$ is the invariant mass of the lepton pair. Associated with Z boson prompt $J/\psi$ meson satisfies the following conditions: $|y^{\psi}|<2.1$ and $8.5<p_{T}^{\psi}<100$ GeV. Secondly, in case of $J/\psi+W^{\pm}$ production, $W^{\pm}$ is reconstructed in the lepton-neutrino decays $W^{\pm}\to\ell\nu_{\ell}\;(\ell=\mu,\bar{\mu})$. To separate events from $W^{\pm}$ decays criteria are as follows: leptons have $|\eta^{\ell}|<2.4$ and $p^{\ell}_{T}>25$ GeV, to take into account the momentum carried away by the neutrinos, impose a limit on the missing transverse energy $E^{\rm miss}_{T}$, which is defined as a modulus of the vector sum of the transverse momenta of the decay products, in inclusive reactions $E^{\rm miss}_{T}$ is equal to transverse momentum of neutrino $p^{\nu}_{T}$, in measurements $p^{\nu}_{T}>20$ GeV. In addition, the transverse energy of the $W^{\pm}$ boson $M^{W}_{T}$ must be greater than 40 GeV, which is defined as $M^{W}_{T}=\sqrt{2\,p_{T}^{\mu}\,p_{T}^{\nu}\,(1-\cos\Delta\phi_{\mu\nu})}$.

All data for associated $J/\psi$ plus $Z/W$ are presented as ratios of associated production cross section $\sigma(J/\psi+Z/W)$ to inclusive production cross section of the $Z/W^{\pm}$ boson $\sigma(Z/W)$. In our calculations, we define the function

where ${\cal Q}=J/\psi$ or $\Upsilon(1S)$ and $A=Z$ or $W$, inclusive cross sections $\sigma(Z/W+X)$ are from Table. 1, the values of two-muon branchings ${\cal B}(J/\psi\to\mu\bar{\mu})=0.0596$ and ${\cal B}(\Upsilon\to\mu\bar{\mu})=0.0248$ are used.

In the top panel of the Fig. 1, we plot the the ${\cal R}$ function (${\cal Q}=J/\psi,A=Z$), which demonstrates $p_{T}^{\psi}$-dependence for the cross section of $J/\psi$ plus $Z$ boson associated production. The contributions from the SPS and the DPS mechanisms are shown separately. Here and below, the grey boxes around the central lines in the figures indicate upper and lower limits of the cross section obtained due to the variation of the hard scale $\mu$ by the factors $\xi=2$ or $\xi=1/2$ around the central value of the hard scale $\mu=\left(m_{T}^{Z}+m_{T}^{\psi}\right)/2,\ {\rm where}\ m_{T}^{Z}=\sqrt{m_{Z}^{2}+(p_{T}^{Z})^{2}},\ {\rm and}\ m_{T}^{\psi}=\sqrt{m_{\psi}^{2}+(p_{T}^{\psi})^{2}}$. As it is estimated, the SPS contribution dominates only at the very large $p_{T}^{\psi}$. The ratio of the total cross section $\sigma^{\rm SPS}/\sigma^{\rm DPS}\simeq 1/4$ at the $\sigma_{\rm eff}=11$ mb. The hadronization parameter ${\cal F}^{\psi}=0.009$ taken under similar kinematic conditions is used, i.e. in the region of large $p_{T}^{\psi}$ and central rapidity interval. The agreement between our prediction and experimental data looks well excluding only difference at the large $p_{T}^{\psi}$.

In the bottom panel of the Fig. 1, we plot the normalized $\Delta\phi_{\psi Z}$ spectrum. The spectrum is rather flat due to the large DPS contribution, but at the $\Delta\phi_{\psi Z}\simeq\pi$, it has peak in the data which is described well due to the SPS contribution concentrated in this region of the $\Delta\phi_{\psi Z}$.

Our predictions for $J/\psi$ plus $Z$ associated production spectra at the energy $\sqrt{s}=13$ TeV are shown in the Fig. 2. Parameter ${\cal F}^{\psi}$ is the same as at $\sqrt{s}=8$ TeV, $K$ factor from Table 1.

In the top panel of the Figs. 3 and 4, we plot function (28) (${\cal Q}=J/\psi,\;A=W$) in comparing with the experimental data [8, 9]. Taking into consideration errors of the theoretical calculations coming from the variation of the hard scale $\mu$, we conclude that the agreement between data and our theoretical calculations in the ICEM and the PRA, is even more precise than in case of associated $J/\psi$ plus $Z$ boson production. In the Fig. 5, we present the predictions for cross section of $J/\psi$ plus $W$ associated production at $\sqrt{s}=13$ TeV.

Recently, we have performed calculations in the ICEM using PRA for the single and double prompt $\Upsilon=\Upsilon(1S)$ meson production at the LHC [29]. We found good description of the data taking into account the SPS and the DPS production scenarios, with the universal value of DPS parameter $\sigma_{\rm eff}=11$ mb. Predictions of the prompt $\Upsilon$ plus $Z(W)$ associated production spectra as a function of $p_{T}^{\Upsilon}$ and $\Delta\phi_{\Upsilon Z}$ at $\sqrt{s}=8\mbox{ and }13$ TeV are presented in the Figs. 6. Kinematical cuts on $\Upsilon$ mesons are the same as for $J/\psi$ mesons from Table 1. The hadronization parameter ${\cal F}^{\Upsilon}=0.021$ is taken from the fit of single $\Upsilon$ production [29].

We obtain a quite satisfactory description for the prompt $J/\psi$ mesons $p_{T}$ spectra in the associated $J/\psi$ plus $Z(W)$ bosons production in the ICEM using the PRA at the energy of the LHC, as it was measured by the ATLAS Collaboration [7, 8, 9]. The azimuthal angle difference spectra as functions of $\Delta\phi_{\psi Z}$ and $\Delta\phi_{\psi W}$ are well described too. Both mechanisms, SPS and DPS, have been considered. We don’t use any free parameters to obtain our prediction, the relevant ones, $\sigma_{\rm eff}$, ${\cal F}^{\psi}$ and ${\cal F}^{\Upsilon}$, have been fixed early, when we described single and double prompt $J/\psi$ and $\Upsilon$ production at the LHC [28, 29]. We find the dominant role of the DPS production mechanism in the considered here processes of the associated production, such as $\sigma^{\rm SPS}/\sigma^{\rm DPS}=0.2\sim 0.25$. The predictions for the cross sections and spectra at the energies $\sqrt{s}=8,13$ TeV have been done.

We are grateful to A. Van Hameren for advice on the program KaTie and M. Nefedov for the helpful communication.