$J/ψ$ production in proton-proton collisions at Spin Physics Detector energies of the JINR Nuclotron-based Ion Collider fAcility

This paper presents a theoretical investigation of inclusive $J/\psi$ production in proton-proton collisions at SPD/NICA energies using TMD gluon densities, revealing the dominance of color-octet mechanisms and distinct $p_T$ hardening patterns that offer crucial guidance for upcoming experimental measurements.

The Gist

Imagine you are trying to understand how a car engine works, but you can't take it apart. Instead, you have to watch the car drive by at different speeds and listen to the noise it makes to guess what's happening inside.

This paper is doing exactly that, but with the smallest building blocks of the universe: protons (the particles inside atoms) and the gluons (the "glue" that holds them together).

Here is the story of the research, broken down into simple concepts:

1. The Setting: A New Race Track

Scientists are building a new, specialized race track called NICA (in Russia). On this track, they will smash two proton beams together. Unlike the massive, super-fast collisions at the Large Hadron Collider (LHC) in Europe, this track is designed for "moderate" speeds.

Think of the LHC as a Formula 1 car going 200 mph, while this new track is like a car going 60 mph. Why go slower? Because at these moderate speeds, the "glue" inside the proton behaves differently. It's a sweet spot where the rules of the quantum world are a bit messy and mysterious, and scientists want to get a better look.

2. The Goal: Catching a "Ghost" Particle

The scientists are looking for a specific particle called J/ψ (pronounced "J-psi").

• The Analogy: Imagine the proton is a busy city. Inside, there are tiny messengers (gluons) zooming around. Occasionally, two messengers crash into each other and create a brand new, short-lived vehicle (the J/ψ).
• The Problem: We can't see the messengers directly. We only see the new vehicle they created. By studying how this vehicle is created (how fast it goes, where it goes), we can figure out what the messengers were doing before the crash.

3. The Tools: Two Different Maps

To predict what will happen on this new track, the scientists used a computer program (a simulator) called PEGASUS. But to run the simulation, they needed a map of where the gluons are and how fast they are moving.

They tried two different maps (theories):

1. The KL Map (KL'2025): This map suggests the gluons are a bit more spread out and move in a specific way.
2. The LLM Map (LLM'2024): This map suggests the gluons are a bit more "jittery" and have a different pattern of movement.

Think of it like two weather forecasters predicting a storm. One says, "It will rain heavily in the center," and the other says, "It will rain heavily in the center but also spread out to the edges."

4. The Findings: What the Simulation Told Them

The scientists ran the simulation for different collision speeds (9, 18, and 27 GeV) and compared the two maps. Here is what they found:

• The "Glue" is Dominant: They found that the J/ψ particle is mostly created by a specific, complex mechanism involving "color octets" (a fancy quantum term for a specific type of glue interaction). It's like finding out that 95% of the time, the new vehicle is built by a specific type of mechanic, not the usual one.
• Speed Changes the Shape: As the collision speed increased, the J/ψ particles started flying in more directions and at higher speeds. This is expected, like how a faster car crash sends debris flying further.
• The Maps Disagree on "Hardness": This is the most important part.
  • The LLM Map predicted that the particles would fly out with a "harder" kick (higher energy) more often.
  • The KL Map predicted a slightly softer kick.
  • Why it matters: When the real experiment starts at NICA, the scientists will measure the actual speed of the particles. Whichever map matches the real data will be the correct one! This will help us understand the true nature of the "glue" inside the proton.

5. Why This Matters

This paper is essentially a practice run. Before the real machine turns on, the scientists are saying: "Here is what we expect to see if our current theories are right. If the real data looks different, we know we need to rewrite the rules of how gluons behave."

It's like a chef tasting a sauce before serving it to the guests. They are checking the recipe (the theory) to make sure it will taste right when the real meal (the experiment) is served.

Summary in One Sentence

This paper uses computer simulations to predict how a specific particle (J/ψ) will behave when protons collide at moderate speeds, helping scientists choose the best "map" to understand the mysterious glue that holds our universe together.

Technical Summary

1. Problem Statement

The paper addresses the need for a theoretical baseline for inclusive $J/\psi$ (charmonium) production in proton-proton ($pp$) collisions at moderate center-of-mass energies ($\sqrt{s} \le 27$ GeV). This energy regime is targeted by the upcoming Spin Physics Detector (SPD) at the NICA collider (JINR).

• The Gap: While high-energy quarkonium production (multi-TeV scale at LHC) is well-constrained by global analyses combining Color-Singlet (CS) and Color-Octet (CO) mechanisms, the moderate-energy regime is less understood.
• The Physics Challenge: In this intermediate energy range:
  • Gluon momentum fractions ($x$) are larger.
  • The phase space for high-$p_T$ recoils is reduced.
  • The sensitivity to the intrinsic transverse momentum ($k_T$) and polarization of gluons becomes significant.
  • Standard collinear factorization may be insufficient; Transverse-Momentum-Dependent (TMD) formalism is required to probe gluon dynamics near the production threshold.

2. Methodology

The authors employ a $k_T$-factorization framework within the Nonrelativistic QCD (NRQCD) factorization approach to simulate $J/\psi$ production.

• Event Generator: Simulations are performed using the PEGASUS event generator, which implements TMD gluon densities.
• TMD Parameterizations: Two recent gluon density sets are compared:
  1. KL′2025: Based on the KMR (Kimber-Martin-Ryskin) scheme.
  2. LLM′2024: Based on the CCFM (Ciafaloni-Catani-Fiorani-Marchesini) evolution scheme.
• NRQCD Formalism: The cross-section is calculated as a sum over intermediate $c\bar{c}$ states ($n$):
  $$d\sigma(pp \to J/\psi+X) = \sum_n d\hat{\sigma}(pp \to c\bar{c}[n]+X) \langle O^{J/\psi}[n] \rangle$$
  • Channels: Both Color-Singlet ($^3S_1^{(1)}$) and Color-Octet ($^1S_0^{(8)}, ^3S_1^{(8)}, ^3P_J^{(8)}$) mechanisms are included.
  • Assumptions: Long-Distance Matrix Elements (LDMEs) are set to unity to isolate the kinematic dependence of Short-Distance Coefficients (SDCs) and the impact of gluon TMDs.
• Simulation Parameters:
  • Energies: $\sqrt{s} = 9, 18, 27$ GeV.
  • Scales: Renormalization ($\mu_R$) and factorization ($\mu_F$) scales are set to $m_T = \sqrt{M_{J/\psi}^2 + p_T^2}$.
  • Uncertainty: Theoretical uncertainty is estimated by varying $\mu_R$ by a factor of 2 ($\mu_R/2$ to $2\mu_R$). Factorization scale variation is not treated as an independent uncertainty in the CCFM approach.
  • Statistics: 12 independent runs with $4 \times 10^5$ events each.

3. Key Contributions

• First Detailed Assessment: This is the first study to provide a quantitative prediction for inclusive $J/\psi$ production at SPD/NICA energies using modern TMD gluon densities.
• Model Comparison: A systematic comparison between KMR-based (KL′2025) and CCFM-based (LLM′2024) TMD evolutions in the near-threshold regime.
• Mechanism Quantification: Explicit quantification of the relative contributions of CS vs. CO channels in the moderate-energy domain.
• Sensitivity Analysis: Demonstration of how $J/\psi$ observables (rapidity, $p_T$ spectra) are sensitive to the underlying gluon $k_T$ broadening and evolution patterns.

4. Key Results

A. Rapidity Distributions ($d\sigma/dy$)

• Shape: Distributions are symmetric and centrally peaked ($|y_{peak}| < 0.07$).
• Energy Dependence: As $\sqrt{s}$ increases from 9 to 27 GeV, the rapidity width (RMS and FWHM) roughly doubles, reflecting the opening of phase space and access to lower gluon $x$ values.
• Model Differences:
  • LLM′2024 predicts systematically higher cross-sections (normalization) than KL′2025 across all energies.
  • LLM′2024 exhibits a slightly broader shape due to its broader intrinsic $k_T$ width and softer small-$x$ behavior.
  • At $\sqrt{s}=9$ GeV, both models show mild suppression near $y=0$ due to near-threshold kinematic constraints.

B. Transverse Momentum Spectra ($d\sigma/dp_T$)

• Behavior: Spectra show a steep falloff with increasing $p_T$. The accessible $p_T$ range expands from $\le 5$ GeV at 9 GeV to $\approx 12$ GeV at 27 GeV.
• Distinct Patterns:
  • KL′2025: Shows a relatively smaller initial amplitude but a harder tail at high $p_T$.
  • LLM′2024: Shows a higher initial amplitude and a "softer" tail (Gaussian-like falloff) at high $p_T$.
  • Crossover: A crossover region is observed around $2 < p_T < 3$ GeV at higher energies, where KL′2025 eventually dominates the tail.
• Normalized Spectra: Fitting the normalized spectra reveals distinct functional forms: KL′2025 follows a power-law rise ($n \approx 1.22$), while LLM′2024 follows a smoother, faster-falling curve due to its Gaussian-like $k_T$ dependence.

C. Intermediate State Contributions

• Dominance of Color-Octet: Across the entire SPD energy range, Color-Octet (CO) mechanisms dominate the production.
• Hierarchy: The contribution hierarchy is $^3P_2^{(8)} > ^3P_0^{(8)} > ^1S_0^{(8)} > ^3P_1^{(8)} > ^3S_1^{(8)}$.
• Color-Singlet Suppression: The CS ($^3S_1^{(1)}$) contribution remains below 1% even at $\sqrt{s} = 30$ GeV, confirming that CO mechanisms are essential for describing $J/\psi$ production at these energies.

D. Total Cross Section vs. Energy

• The total cross section rises monotonically with $\sqrt{s}$.
• The data fits a shifted power law: $\sigma \propto (\sqrt{s} - s_0)^\beta$.
  • LLM′2024: $\beta \approx 1.76$ (softer rise).
  • KL′2025: $\beta \approx 2.09$ (steeper rise).
• At $\sqrt{s} = 9$ GeV, the cross section lies below the extrapolated power law, suggesting near-threshold suppression effects.

5. Significance

• Theoretical Guidance: The results provide a robust theoretical baseline for the SPD experiment at NICA, helping to interpret future measurements of quarkonium production.
• Gluon Dynamics Probe: The study confirms that $J/\psi$ production at moderate energies is a highly sensitive probe of gluon TMDs and their evolution, particularly the intrinsic $k_T$ broadening.
• Model Discrimination: The distinct spectral shapes (hard vs. soft tails) and normalization differences between KL′2025 and LLM′2024 offer clear observables to discriminate between different TMD evolution schemes (KMR vs. CCFM) in the near-threshold region.
• NRQCD Validation: The overwhelming dominance of Color-Octet channels in this regime reinforces the necessity of including CO mechanisms in NRQCD analyses for moderate-energy collisions, contrasting with high-energy regimes where CS contributions can be more significant.