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

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.

💡 Research Summary

This paper presents a comprehensive theoretical investigation into the inclusive production of pairs of heavy quarkonia (such as J/ψ, ψ(2S), and Υ mesons) in collisions of polarized spin-1/2 hadrons (e.g., protons and pions). The primary goal is to establish this process as a novel and clean probe for accessing the transverse momentum dependent (TMD) parton distribution functions (PDFs) of light quarks and antiquarks inside nucleons.

The analysis is strategically confined to a specific kinematic regime where the total transverse momentum of the quarkonium pair (q_T) is much smaller than its invariant mass (M_QQ). This hierarchy of scales is a fundamental requirement for the applicability of TMD factorization. Furthermore, the authors adopt the Color-Singlet Model (CSM) for the quarkonium formation mechanism. In this model, the heavy quark-antiquark pair is produced directly in a color-singlet state with the quantum numbers of the final observed meson. This choice is crucial because it avoids the complications of color-octet production mechanisms, which involve final-state interactions that could potentially violate TMD factorization. The CSM channel is argued to be dominant in the kinematic region of interest, with color-octet contributions being suppressed by powers of the heavy quark relative velocity.

Focusing specifically on the quark-antiquark annihilation channel (q q̅ → Q Q), the paper calculates the fully differential cross section up to order α_s^4 in the strong coupling constant. The calculation is performed within the TMD factorization framework, where the cross section is expressed as a convolution of TMDs from the two incoming hadrons with a perturbatively calculable hard part. The key result is the derivation of the azimuthal angular structure of the cross section. Various modulations (e.g., cos φ, cos 2φ) arise, each associated with specific convolutions of quark TMDs, such as the unpolarized distribution (f1), the Sivers function (f1T⊥), and the Boer-Mulders function (h1⊥). This structure bears a strong formal analogy to the well-known Drell-Yan process.

The phenomenological core of the work applies this formalism to the proposed reaction π^- p → J/ψ J/ψ X, relevant for fixed-target experiments like COMPASS and AMBER at CERN. The authors demonstrate that in the kinematic reach of these experiments, the contribution from the gluon-gluon fusion channel—which dominates at high-energy colliders like the LHC—becomes negligible. This isolation of the quark-antiquark channel provides a direct window into the quark TMDs of the proton, uncontaminated by poorly known gluon TMDs.

A major highlighted implication is the opportunity to perform a crucial test of TMD theory: the predicted sign change of the quark Sivers function between Semi-Inclusive Deep Inelastic Scattering (SIDIS, with future-pointing gauge links) and the Drell-Yan process (with past-pointing gauge links). The π^- p → J/ψ J/ψ process, involving initial-state interactions and thus past-pointing gauge links, should exhibit a Sivers function with the same sign as in Drell-Yan. A measurement of the corresponding asymmetry in this process would therefore provide a third, independent check of this fundamental property, significantly strengthening our understanding of QCD dynamics and process dependence in TMDs.

Finally, the paper discusses the impact of this methodology on ongoing and future fixed-target experiments at the LHC, such as SMOG and LHCspin, where similar studies could be extended to probe the gluon content of the proton under different kinematic conditions. In summary, this research positions double quarkonium production in fixed-target experiments as a unique and powerful tool for precision studies of the three-dimensional partonic structure of hadrons.