Quark polarization and transverse momentum effects on double quarkonium production in hadronic collisions

TL;DR

This work develops a TMD-factorized description of inclusive double quarkonium production in polarized hadron collisions, focusing on the quark-antiquark annihilation channel within the Color-Singlet NRQCD framework in the regime where $q_T \ll M_{QQ}$. It derives analytic expressions for azimuthal cross-section modulations as convolutions of leading-twist quark TMDs (including $f_1^q$, $g_{1L}^q$, $h_1^{\perp q}$, $f_{1T}^{\perp q}$, etc.) with calculable hard factors $H$ and $\Delta H$, highlighting Drell–Yan–like angular structures. Phenomenological results for COMPASS and fixed-target LHC programs show that quark TMDs can produce measurable modulations, with a particularly large Sivers effect in $\pi^-p$ at COMPASS, while gluon contributions are suppressed at low energies but become relevant at higher fixed-target energies. The study demonstrates the potential of di-quarkonium production to access quark TMDs, test process-dependence and sign-change hypotheses, and complement gluon-TMD studies in Drell–Yan-like processes and future LHC fixed-target programs.

Abstract

We investigate the inclusive production of double quarkonia ($J/ψ$, $ψ(2S)$, $Υ$ mesons) in polarized hadron-hadron collisions, considering the kinematic configuration where the transverse momentum of each pair of bound states is much smaller than its invariant mass. Supported by nonrelativistic QCD arguments, we adopt the Color-Singlet Model for the quarkonium formation mechanism and assume the validity of transverse momentum dependent factorization. In strong analogy with dilepton production in the Drell-Yan processes, the azimuthal modulations of the cross section, calculated to the order $α_s^4$ in the QCD coupling constant, can be expressed as convolutions of transverse momentum dependent distributions of light quarks and antiquarks inside the incoming hadrons. By adopting very recent parameterizations of these distributions, we show that a phenomenological study of these quantities in $π^-p\to J/ψ\,J/ψ\,X $ in the kinematic region covered by the COMPASS and AMBER fixed-target experiments at CERN, where the gluon contribution is found to be negligible, would provide direct access to the quark distributions. In particular, this will offer the possibility of a further sign change test of the quark Sivers function of the proton. The impact of our findings on similar studies about the gluon content of the proton in present and future fixed-target experiments at the LHC, like SMOG and LHCspin, is also demonstrated.

Figures (7)

• Figure 1: Representative cut diagram for the process $h_1(P_1) + h_2(P_2) \to \pazocal{Q}(K_1) + \pazocal{Q}(K_2) + X$ through quark-antiquark annihilation at LO. The upper and lower blue ovals describe the correlators $\Phi^q(x_1, \bm k_{1{ T}}^2)$ and $\overline\Phi^q(x_2, \bm k_{2{ T}}^2)$, respectively.
• Figure 2: Feynman diagrams contributing to the scattering amplitude for the partonic process $q \,\overline q \to \pazocal{Q} \, \pazocal{Q}$ at order $\alpha_s^2$. The other two crossed diagrams, in which the directions of the arrows in the heavy (anti)quark lines are reversed, are not shown explicitly.
• Figure 3: Comparison with COMPASS data COMPASS:2022djq of the unpolarized cross section for di-$J/\psi$ production in $\pi^- p$ scattering, as a function of the transverse momentum of the $J/\psi$-pair, $q_{ T}$, at $\sqrt{s} = 18.9$ GeV.
• Figure 4: Predictions for (a) $\lvert \langle \cos 2(\phi_{ T} - \phi_\perp)\rangle \rvert$ and $\lvert \langle \sin(\phi_{ T} + \phi_S - 2\phi_\perp)\rangle \rvert$, and (b) the Sivers asymmetry $\langle \sin(\phi_{ T} - \phi_S )\rangle = A_{UT}^{\sin(\phi_{ T} - \phi_S )}$ for di-$J/\psi$ production in $\pi^- p$ collisions at $\sqrt{s} = 18.9$ GeV at COMPASS. Computations are performed saturating the bound for $h_1^{\perp q}$ (see Eq. \ref{['eq:BM-pos-bound']}) and adopting the same $q_T$-binning as for the unpolarized cross section COMPASS:2022djq.
• Figure 5: Predictions for the quark induced unpolarized cross-section for (a) di-$J/\psi$, (b) di-$\psi(2S)$ and (c) di-$\Upsilon$ production at fixed-target experiments at LHCb for different values of $\sqrt{s}$ and in different $Y_{\pazocal{Q}\pazocal{Q}}$, $z$ and $M_{\pazocal{Q}\pazocal{Q}}$ intervals. The MAP22 TMDs Bacchetta:2022awv have been employed in the computation.