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Double quarkonium production in hadronic collisions at fixed-target experiments

This paper presents new analytical expressions and predictions for double quarkonium production in unpolarized and polarized hadronic collisions at fixed-target experiments, utilizing transverse momentum dependent factorization combined with the Color-Singlet Model to describe angular structures, cross sections, and transverse single-spin asymmetries for current and future CERN and LHC facilities.

Original authors: Carlo Flore, Cristian Pisano

Published 2026-07-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Carlo Flore, Cristian Pisano

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. Usually, these bricks are glued together so tightly by a force called the "strong force" 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, heavy "Lego structure" called a quarkonium.

This paper is like a detective story about how these heavy Lego structures are built when two giant "trains" of particles crash into each other in a fixed-target experiment (think of a particle accelerator where one beam hits a stationary target).

Here is the breakdown of what the authors, Carlo Flore and Cristian Pisano, discovered:

1. The Mystery of the "Double" Build

Usually, scientists study how one of these heavy structures is made. But this paper focuses on a rare event where two of them are created at the exact same time in a single crash. It's like smashing two cars together and having two brand-new, identical luxury cars pop out of the wreckage simultaneously.

The authors wanted to understand the "blueprint" (the math) behind how this happens. They looked at two main ways this could occur:

The Glue Method: Two "glue" particles (gluons) smash together to make the pair.
The Brick Method: A quark and an anti-quark (the actual bricks) smash together to make the pair.

2. The "Shadow" Analogy (Transverse Momentum)

To understand the crash, the scientists had to look at the particles not just from the front, but from the side. They used a concept called Transverse Momentum Dependent (TMD) factorization.

Think of it like this: If you throw two balls at each other, you usually just look at how fast they are moving forward. But in this experiment, the scientists are also looking at how much the balls are "wobbling" side-to-side as they fly. This side-to-side wobble holds a secret code about the internal structure of the particles. The authors created new mathematical formulas to decode this "wobble" specifically for the Brick Method (quark-antiquark collision), which is the dominant method at lower-energy experiments.

3. The "Spinning Top" Effect (Spin Asymmetry)

The paper also looked at what happens if the incoming particles are spinning (polarized). Imagine spinning tops crashing into each other. The direction they spin changes how they bounce off.

The authors calculated a specific "twist" in the outcome, called the Sivers asymmetry.

At the COMPASS experiment (using a pion beam): They predicted a huge twist (10–15%). It's like if you spun a top one way, and the resulting crash made the debris fly off mostly to the left. This happens because the specific "flavor" of the particles involved (up and anti-up quarks) aligns perfectly to create this effect.
At the LHC fixed-target experiments (using proton beams): The twist is much smaller (1–2%) and goes the other way (negative). This is because protons are a messy mix of different quarks (up, down, and "sea" quarks) that cancel each other out, making the spin effect much harder to see.

4. The "Compass" vs. The "LHC"

The paper compares two different "playgrounds" for these experiments:

COMPASS/AMBER: These are like a quiet, controlled workshop. Here, the "Brick Method" (quark-antiquark) is the star of the show. The "Glue Method" is so quiet it's almost silent. This makes it the perfect place to study the "wobble" of quarks.
LHC (SMOG2/LHCspin): This is a high-speed, chaotic highway. Here, the "Glue Method" gets louder and starts competing with the "Brick Method." Because the "Brick Method" is less dominant here, the spin effects are smaller, but studying them here helps scientists understand the "glue" particles better.

5. The Bottom Line

The authors successfully wrote down the new rules (mathematical formulas) for how these double-particle crashes happen when quarks collide. They used these rules to predict what experiments at CERN (the European particle physics lab) should see.

Prediction 1: At the COMPASS experiment, they expect to see a strong "spin twist" (Sivers asymmetry) of about 10–15%.
Prediction 2: At the LHC, the twist will be tiny (1–2%), and if they see a bigger twist than that, it would mean the "glue" particles (gluons) are doing something unexpected.

In short, this paper provides a new, clearer map for scientists to navigate the chaotic world of double-particle crashes, helping them figure out exactly how the fundamental building blocks of our universe behave when they collide.

Technical Summary: Double Quarkonium Production in Fixed-Target Experiments

Problem Statement
The production mechanism of quarkonia (bound states of heavy quark-antiquark pairs, $Q\bar{Q}$ ) remains an open issue in Quantum Chromodynamics (QCD). While models such as the Color Evaporation Model, the Color-Singlet Model (CSM), and Non-Relativistic QCD (NRQCD) exist, NRQCD struggles to consistently describe cross-sections and polarization measurements, with long-distance matrix elements retaining large uncertainties. Recent proposals suggest that inclusive double- $J/\psi$ production can probe Transverse Momentum Dependent (TMD) gluon distributions. However, previous analyses focused exclusively on the gluon fusion channel ( $gg \to J/\psi J/\psi$ ). This paper addresses the gap in understanding the quark-antiquark annihilation channel ( $q\bar{q} \to Q\bar{Q}$ ), which is relevant for fixed-target experiments at lower center-of-mass energies, such as COMPASS/AMBER and the LHC fixed-target programs (SMOG2, LHCspin).

Methodology
The authors investigate the inclusive production of two quarkonium states ( $J/\psi$ , $\psi(2S)$ , $\Upsilon$ ) in inelastic collisions of spin-1/2 hadrons ( $h_1 + h_2 \to Q + Q + X$ ). The analysis relies on the following theoretical framework:

Factorization Scheme: The study employs TMD factorization combined with the Color-Singlet Model (CSM).
Kinematic Regime: The analysis assumes the two final-state quarkonia are almost back-to-back in the transverse plane ( $|\mathbf{q}_T| \ll |\mathbf{K}_\perp|$ ), where $\mathbf{q}_T$ is the sum and $\mathbf{K}_\perp$ is the difference of their transverse momenta.
Partonic Channels: The focus is on the $q\bar{q} \to Q\bar{Q}$ channel. The authors assume the quarkonia are produced in an $S$ -wave state ( $^3S_1^{[1]}$ ).
Validation of CSM: Using HELAC-Onia, the authors verified that Color-Octet (CO) contributions are suppressed by at least $\mathcal{O}(v^3)$ relative to the Color-Singlet channel, justifying the use of CSM. The CO background is estimated to be at most $\sim 10\%$ .
Analytical Derivation: New analytical expressions were derived for the angular structure of the cross section. The fully differential cross section is expressed in terms of structure functions involving azimuthal modulations dependent on the transverse momentum $\mathbf{q}_T$ , the transverse mass $\mathbf{K}_\perp$ , and the hadron spins.
Phenomenological Inputs: Predictions utilize recent TMD parametrizations from the MAP Collaboration (TMDlib), specifically the MAPTMDPion22 set for pions, and PV17/MAP22 sets for protons, all at Next-to-Leading-Logarithmic (NLL) accuracy. The Sivers asymmetry calculations assume the sign change of the Sivers function relative to Semi-Inclusive Deep Inelastic Scattering (SIDIS), consistent with the Drell-Yan process.

Key Contributions

Analytical Formalism: The paper provides the first derivation of the angular structure of the cross section for the $q\bar{q}$ -induced double quarkonium channel. The authors demonstrate that this channel exhibits the same azimuthal modulations and color-flow structure as the Drell-Yan process ( $q\bar{q} \to \gamma^* \to \ell^+\ell^-$ ), making it a complementary reaction for studying quark TMDs.
Unpolarized Cross Sections: New predictions are presented for the unpolarized cross section in both pion-nucleon ( $\pi^- p$ ) and proton-proton ($pp$) collisions.
Spin Asymmetries: The study provides predictions for Transverse Single-Spin Asymmetries (TSSA), specifically the Sivers asymmetry, for both unpolarized and polarized hadronic collisions.
Experimental Predictions: Specific phenomenological predictions are generated for:
- COMPASS/AMBER: $\pi^- p$ collisions at $\sqrt{s} \simeq 18.9$ GeV.
- LHC Fixed-Target (SMOG2/LHCspin): $pp$ collisions at $\sqrt{s} \approx 68.5 - 115$ GeV.

Results

COMPASS/AMBER ( $\pi^- p$ ):
- The gluon-induced channel is negligible ( $\mathcal{O}(10^{-3})$ pb) compared to the $q\bar{q}$ channel.
- The unpolarized cross section predictions using MAP22 and PV17 sets show fair agreement with existing COMPASS data (limited to $\sim 25$ events).
- A sizable Sivers asymmetry of 10–15% is predicted. This is driven by the dominant $\bar{u}\pi u p$ channel, probing the valence region of both the pion and the transversely polarized proton.
LHC Fixed-Target ($pp$):
- The gluon fusion channel becomes significant, contributing 30–40% at $\sqrt{s} \approx 70$ GeV and roughly doubling the $q\bar{q}$ contribution at $\sqrt{s} \approx 115$ GeV.
- The $q\bar{q}$ channel remains dominant only up to $\sqrt{s} \sim 70$ GeV.
- The Sivers asymmetry is predicted to be negative and small (1–2% for di- $J/\psi$ and di- $\psi(2S)$ ), due to the cancellation between $d$ -quark/sea-quark and $u$ -quark contributions in $pp$ collisions.
- For di- $\Upsilon$ , the asymmetry is compatible with zero due to the different kinematic $x$ -range probed.
- A measured asymmetry larger than predicted would signal a non-zero gluon Sivers function.

Significance
The paper establishes double quarkonium production as a viable and complementary probe for TMD physics.

At COMPASS energies, the strong suppression of the gluon channel makes these measurements ideal for isolating and probing quark TMDs (specifically the Sivers function).
At LHC fixed-target energies, the increasing contribution of the gluon channel suggests the process can serve as an important probe for gluon TMDs.
The close correspondence between the $q\bar{q}$ quarkonium channel and the Drell-Yan process validates the use of this reaction to study process dependence in TMDs, offering a kinematic region complementary to SIDIS and standard Drell-Yan measurements.