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

This paper investigates double quarkonium production in polarized hadronic collisions using the Color-Singlet Model and TMD factorization to demonstrate that measuring azimuthal modulations in specific kinematic regions at COMPASS and AMBER can provide direct access to quark distributions and enable a further test of the quark Sivers function sign change.

The Gist

Imagine the universe is built from tiny, invisible Lego bricks called quarks. Usually, these bricks are glued together so tightly by a force called "strong interaction" that they never exist alone; they are always stuck in pairs or triplets. When a heavy quark and its anti-quark partner get stuck together, they form a special, short-lived "molecule" called quarkonium (like a J/ψ or an Υ meson).

This paper is a theoretical recipe for predicting what happens when we smash two big "bags" of these bricks (protons or pions) together at high speeds, specifically looking for the rare event where two of these quarkonium molecules are created at the same time.

Here is the breakdown of their work using simple analogies:

1. The Setup: Smashing Bags of Bricks

The authors are studying collisions where two hadrons (particles made of quarks) crash into each other.

• The Goal: They want to see what happens when two heavy quark-antiquark pairs are born from the crash and immediately stick together to form two quarkonium particles.
• The "Clean" Scenario: They focus on a specific, "clean" way this happens. Imagine the quarks are like dancers. Usually, when they crash, they might get tangled with other dancers (gluons) in a messy way. But the authors assume a scenario where the two quark pairs are born perfectly paired up and "colorless" (like wearing matching white outfits) right from the start. This is called the Color-Singlet Model. Because they are so clean, the math is much easier to handle.

2. The Map: Transverse Momentum (The "Sideways" Drift)

In these collisions, the particles don't just fly straight forward; they also drift sideways.

• The Analogy: Imagine two cars driving down a highway. Usually, we only care about how fast they are going forward. But here, the authors are obsessed with how much they are drifting sideways (transverse momentum).
• The Rule: They only look at cases where the sideways drift is very small compared to the total energy of the crash. This allows them to use a special mathematical map called TMD Factorization. Think of this map as a way to separate the "hard crash" (the collision itself) from the "soft drift" (the internal spinning and wobbling of the bricks inside the bags before they even hit).

3. The Spin: The "Sivers" and "Boer-Mulders" Effects

The paper investigates what happens if the "bags" of bricks (the protons) are spinning.

• The Sivers Effect: Imagine the bricks inside a spinning bag don't just spin randomly; they have a preference to drift to the left or right depending on how the bag is spinning. This is the Sivers function. The authors predict that if you smash a spinning proton against a pion, the resulting quarkonium pairs will fly off at specific angles that reveal this hidden drift.
• The Boer-Mulders Effect: This is similar, but it's about how the spin of the quark itself affects its sideways drift.
• The Prediction: The authors calculated that if you measure the angle of the resulting particles, you will see a "wobble" or a specific pattern (like a cosine wave) in the data. This wobble is the fingerprint of these hidden spin-drifts.

4. The Experiments: Where to Look

The authors didn't just do math; they checked if their predictions match real-world experiments.

• COMPASS (CERN): They looked at data from an experiment where a beam of pions hits a target of protons. They found that in this specific setup, the "gluon" (the glue holding quarks together) contribution is tiny. This is great news because it means the data is almost purely showing the behavior of the quarks. Their calculations matched the existing data well.
• LHC Fixed-Target (SMOG/LHCspin): They also looked ahead to future experiments at the Large Hadron Collider (LHC) where they will smash protons into gas targets. Here, the energy is higher. They predict that at these higher energies, the "glue" (gluons) starts to play a bigger role, but the quark signal is still strong enough to be seen.

5. The Big Picture: Testing the Rules of the Universe

Why does this matter?

• The "Sign Change" Test: In physics, there is a rule that says the "Sivers function" (the spin-drift preference) should flip its sign (positive becomes negative) depending on whether you are smashing particles together (like here) or shooting a particle into a target (like in Deep Inelastic Scattering).
• The Claim: The authors argue that measuring double quarkonium production is a perfect, new way to test this rule. Because the math for this process is so similar to a well-known process called the Drell-Yan process (which creates pairs of electrons and positrons), they expect to see the same "sign flip" here. If they see it, it confirms our understanding of how the strong force works.

Summary

In short, this paper provides a detailed map for predicting how two heavy quark "molecules" are created when spinning protons and pions collide. They show that by measuring the angles of these molecules, scientists can peek inside the proton to see how quarks spin and drift sideways. They confirm that current data from CERN supports their theory and predict that future experiments at the LHC will be able to test a fundamental rule about how the universe's strongest force behaves.

Technical Summary

Technical Summary: Quark Polarization and Transverse Momentum Effects on Double Quarkonium Production in Hadronic Collisions

Problem and Motivation
The paper addresses the inclusive production of double quarkonia ($J/\psi$, $\psi(2S)$, and $\Upsilon$ mesons) in polarized hadron-hadron collisions. While single quarkonium production is well-studied, double production offers a unique probe of Transverse Momentum Dependent (TMD) parton distribution functions (TMDs). A critical challenge in applying TMD factorization to hadronic collisions is the potential breakdown of the formalism due to Final State Interactions (FSIs) if the produced heavy quark-antiquark ($Q\bar{Q}$) pairs are not in a purely color-singlet state. The authors focus on the kinematic regime where the transverse momentum of the quarkonium pair ($q_T$) is much smaller than its invariant mass ($M_{QQ}$), a condition required for TMD factorization. Unlike previous studies that focused primarily on the gluon-gluon fusion channel (dominant at LHC collider energies), this work investigates the quark-antiquark annihilation channel ($q\bar{q} \to Q\bar{Q}$), which is expected to be dominant in fixed-target experiments and at lower center-of-mass energies.

Methodology
The authors employ a combined framework of TMD factorization and the Color-Singlet (CS) Model, supported by Nonrelativistic QCD (NRQCD) arguments.

• Theoretical Framework: The calculation is performed at the perturbative order $\alpha_s^4$. The CS Model is adopted, assuming the $Q\bar{Q}$ pair is produced directly in a color-singlet state ($^3S_1^{[1]}$). This assumption suppresses color-octet contributions and ensures that only Initial State Interactions (ISIs) occur, preserving TMD factorization (analogous to the Drell-Yan process).
• Formalism: The cross-section is derived using the quark-quark correlator $\Phi^q$, parametrized in terms of leading-twist TMD distributions (unpolarized $f_1$, helicity $g_{1L}$, transversity $h_1$, Sivers $f_{1T}^\perp$, Boer-Mulders $h_1^\perp$, and pretzelosity $h_{1T}^\perp$).
• Calculation: The authors calculate the scattering amplitude for $q\bar{q} \to Q\bar{Q}$ at leading order (LO) in $\alpha_s$. They derive the fully differential cross-section, explicitly identifying the azimuthal modulations arising from the convolution of TMDs. The results are expressed as a sum of structure functions ($F$) multiplied by angular modulations (e.g., $\cos 2(\phi_T - \phi_\perp)$, $\sin(\phi_T - \phi_S)$).
• Phenomenology: The analytical results are applied to specific kinematic regions relevant for the COMPASS/AMBER experiments at CERN and the SMOG/SMOG2 and LHCspin programs at the LHC. The study utilizes recent TMD parameterizations from the MAP Collaboration (MAPTMDPion22, MAP22) and the PV17/PV20 sets. For the poorly known Boer-Mulders function, the authors estimate upper bounds by saturating the positivity constraint.

Key Contributions

1. Analytical Derivation: The paper provides the first analytical expressions for the azimuthal modulations of the double quarkonium cross-section specifically in the quark-antiquark annihilation channel within the TMD formalism. The angular structure is shown to be in strong analogy with the Drell-Yan process, involving convolutions of light quark and antiquark TMDs.
2. Phenomenological Validation: The authors compare their theoretical predictions for the unpolarized cross-section of $\pi^- p \to J/\psi J/\psi X$ with recent COMPASS data. They find good agreement, validating the approach in the kinematic region where the gluon contribution is negligible.
3. Prediction of Asymmetries: The study predicts significant azimuthal asymmetries for double quarkonium production in both unpolarized and polarized configurations.
   • COMPASS/AMBER: Predicts a Sivers asymmetry of 10–15% and Boer-Mulders/Transversity modulations of 5–10% in $\pi^- p$ collisions.
   • LHC Fixed-Target (SMOG/LHCspin): Predicts a smaller, negative Sivers asymmetry (1–2%) for $pp \to J/\psi J/\psi X$ at $\sqrt{s} \approx 70$ GeV, noting that the dominance of the $q\bar{q}$ channel persists up to this energy before gluon fusion becomes dominant at higher energies ($\sqrt{s} \approx 115$ GeV).

Results

• Unpolarized Cross-Section: The calculated unpolarized cross-section for di-$J/\psi$ production in $\pi^- p$ scattering matches COMPASS data within uncertainties. The gluon-induced channel is found to be negligible ($O(10^{-3}$ pb) in this specific kinematic region.
• Azimuthal Modulations:
  • The $\langle \cos 2(\phi_T - \phi_\perp) \rangle$ moment, driven by the Boer-Mulders function, is predicted to be around 5–10% at COMPASS and up to 10–20% at LHC fixed-target energies.
  • The Sivers asymmetry $\langle \sin(\phi_T - \phi_S) \rangle$ is predicted to be large (10–15%) in $\pi^- p$ collisions due to the valence $\bar{u}u$ channel. In $pp$ collisions, the asymmetry is smaller (1–2%) and negative, resulting from the interference of $u$, $d$, and sea-quark Sivers functions.
  • The $\langle \sin(\phi_T + \phi_S - 2\phi_\perp) \rangle$ moment is predicted to be smaller (2–5%) than the Boer-Mulders modulation.
• Energy Dependence: The study confirms that the quark-antiquark channel dominates double quarkonium production at fixed-target energies ($\sqrt{s} \lesssim 70$ GeV), while the gluon-gluon fusion channel becomes dominant at higher LHC energies ($\sqrt{s} \gtrsim 115$ GeV).

Significance and Claims
The authors claim that this work offers a direct pathway to access quark TMDs in a regime where gluon contributions are suppressed or subleading. Specifically:

• Universality Test: The process provides a crucial test of the process dependence of TMDs, particularly the predicted sign change of the quark Sivers function between Semi-Inclusive Deep Inelastic Scattering (SIDIS) and Drell-Yan-like processes (including double quarkonium production).
• Complementarity: The study complements existing analyses that focus solely on the gluon-gluon fusion channel. It demonstrates that quark contributions are non-negligible in fixed-target experiments at the LHC and are the leading mechanism at CERN fixed-target energies.
• Probing Unknowns: By measuring these azimuthal modulations, future experiments (AMBER, LHCspin) can constrain the poorly known Boer-Mulders function ($h_1^\perp$) and the quark Sivers function ($f_{1T}^\perp$) in a kinematic region complementary to SIDIS.
• Theoretical Consistency: The work reinforces the validity of TMD factorization in double quarkonium production under the Color-Singlet assumption, providing a consistent picture across Drell-Yan, SIDIS, and double quarkonium processes.

The paper concludes that a consistent description of azimuthal asymmetries across these different processes is essential for a satisfactory understanding of proton TMDs, their process dependence, and QCD evolution.