Inclusive production of J/$\psi$ and $\psi'$ mesons at the LHC

We discuss the prompt production of $J/\psi$ mesons in proton-proton collisions at the LHC within a NRQCD $k_{t}$-factorization approach using Kimber-Martin-Ryskin (KMR) unintegrated gluon distributions (UGDF). We include both direct color-singlet production ($gg \to J/\psi$g) as well as a feed-down from $\chi_{c} \to J/\psi \gamma$ and $\psi' \to J/\psi X$. The production of the decaying mesons ($\chi_{c}$ or $\psi'$)is also calculated within NRQCD $k_{t}$-factorization. The corresponding matrix elements for $gg \to J/\psi$, $gg \to \psi'$ and $gg \to \chi_{c}$ include parameters of the nonrelativistic spatial wave functions of quarkonia at $r=0$, which are taken from potential models from the literature. We get the ratio of the corresponding of the cross sections for $\chi_{c}(2)$-to-$\chi_{c}(1)$ much closer to experimental data than obtained in recent analyses. Differential distributions in rapidity of $J/\psi$ and $\psi'$ are calculated and compared to experimental data of the ALICE and LHCb collaborations. We discuss possible onset of gluon saturation effects at forward/backward rapidities. One can describe the experimental data for $J/\psi$ production within model uncertainties with color-singlet component only. Therefore our theoretical results leave only a relatively small room for the color-octet contributions.


Introduction
For a long time there are discrepancies among authors about the production mechanism of J/ψ quarkonia in proton-proton and proton-antiproton collisions. Some authors think that the cross section is dominated by the color-octet contribution. Some authors believe that the color-singlet contribution dominates. The color-octet contribution cannot be calculated from first principle and is rather fitted to the experimental data. Different fits from the literature give different magnitudes of the color-octet contributions. Therefore we concentrate on the color-singlet contribution. In the present paper we wish to calculate the color-singlet contribution as well as possible in the NRQCD k t -factorization and see how much room is left for the more difficult color-octet contribution. In the present approach we concentrate rather on small transverse momenta of J/ψ or ψ ′ relevant for ALICE and LHCb data [1][2][3][4][5]. We expect that color-singlet contributions may dominate in this region of the phase space. Finally ψ ′ quarkonium also has a sizable branching fraction into J/ψX [6]. Fortunately this contribution is much smaller than the direct one as will be discussed in [8]. It was considered recently in an almost identical approach in [9]. 2 Inclusive production of J/ψ and ψ ′ mesons in the NRQCD k t -factorization approach The main color-singlet mechanism for the production of J/ψ and ψ ′ mesons is shown in Fig.1 (left panel). We restrict ourselves to the gluon-gluon fusion mechanism. In the NLO the differential cross section in the k t -factorization can be written as: We calculate the dominant color-single gg → J/ψg contribution taking into account transverse momenta of initial gluons. The corresponding matrix element squared for the gg → J/ψg is The matrix element is taken from [10]. In our calculation we choose the scale of the running coupling constant as: where where here m t is the J/ψ transverse mass. The factorization scale in the calculation was taken as µ 2 F = (m 2 t + p 2 t,g )/2. Similarly we calculate the P-wave χ c meson production. Here the lowest-order subprocess gg → χ c is allowed by positive C-parity of χ c mesons.
In the k t -factorization approach the leading-order cross section for the χ c meson production can be written as: which can also be used to calculate rapidity and transverse momentum distributions of the χ c mesons. In the last equation F g are unintegrated gluon distributions and σ gg→χ c is gg → χ c (off-shell) cross section. The situation is illustrated diagrammatically in Fig.1 (right panel). The ALICE data [4] are shown for comparison.
The matrix element squared for the gg → χ c subprocess is We used the matrix element taken from the Kniehl, Vasin and Saleev paper [7]. For this subprocess the best choice for running coupling constant is: where µ 2 1 = max(q 2 1t , m 2 t ) and µ 2 2 = max(q 2 2t , m 2 t ). Above m t is transverse mass of the χ c meson. The factorization scale for the χ c meson production is fixed as µ 2 F = m 2 t .

Results
In Fig.2 we show differential cross section in rapidity for ψ ′ production at 7 TeV. Our results are compared with ALICE experimental data [4]. In the left panel we present results for Kimber-Martin-Ryskin (KMR) UGDF and in the right panel for mixed Kimber-Martin-Ryskin (KMR) and Kutak-Stasto (KS) UGDFs. Because KMR alone overshoot experimental data for rapidity distribution the best solution is to take the KMR distribution for large x and KS for small x. For ψ ′ meson we have to include only the direct diagram so it's easy to compare our result with experimental data. For J/ψ meson we have to include both diagrams. Below we present results for these two subprocesses. In Fig.3 we show rapidity distribution for direct J/ψ meson production. We present results for three different values of energy: W = 2.76 TeV (left), W = 7 TeV (middle) and W = 13 TeV (right). Our results are compared with ALICE and LHCb experimental data [1][2][3][4][5].

Conclusion
We have calculated the color-singlet contribution in the NRQCD k t -factorization and compared our results with ALICE and LHCb data. Our results in rapidity are almost consistent or even exceed  Figure 4. Rapidity distribution of χ c meson with mixed UGDFs (Kutak-Stasto and KMR). The ALICE and LHCb data points [1][2][3][4][5] for J/ψ are shown for comparison. experimental data. Cross section strongly depends on UGDF and we think the best solution is to use mixed UGDFs (KMR-KS). In our approach only small room is left for color-octet contribution.