⚛️ phenomenology

Next-to-leading order analysis of $J/ψ+ γ$ production in photon-photon collisions at CEPC

This paper presents a next-to-leading order NRQCD analysis of $J/\psi+\gamma$ production in photon-photon collisions at the CEPC, demonstrating that the direct-photon channel dominates with negligible resolved contributions and that precise polarization measurements in this clean environment can effectively test LDME universality and resolve longstanding polarization puzzles.

Original authors: Ying-Zhao Jiang, Zhan Sun

Published 2026-02-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ying-Zhao Jiang, Zhan Sun

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 you are a detective trying to solve a mystery about how tiny particles called J/ψ mesons (think of them as heavy, exotic atoms made of a charm quark and an anti-charm quark) are created in the universe.

For a long time, physicists have had a "rulebook" for predicting how these particles behave, called NRQCD. But there's a problem: the rulebook works great for predicting how many J/ψ particles are made, but it fails miserably at predicting how they are oriented (their "polarization"). It's like having a recipe that tells you exactly how many cookies you'll bake, but it can't tell you if the cookies will be round, flat, or twisted. This is known as the "J/ψ polarization puzzle."

This paper proposes a new way to solve the mystery by looking at a specific, very clean experiment that could happen at a future giant particle collider called the CEPC (Circular Electron Positron Collider) in China.

Here is the breakdown of their investigation using simple analogies:

1. The Setting: A "Clean Room" vs. A "Mud Pit"

Most experiments happen in "hadronic collisions" (smashing protons together). Imagine this like trying to find a specific coin in a muddy, chaotic swamp. There is so much background noise and debris that it's hard to see what's really happening.

The authors propose using photon-photon collisions (smashing two beams of light together).

The Analogy: This is like moving your investigation from the muddy swamp to a sterile, white laboratory. Because light particles (photons) don't carry the messy "strong force" debris that protons do, the environment is incredibly clean. You can see the J/ψ particle and its partner (a single photon) very clearly without the background noise interfering.

2. The Process: The "Direct" vs. The "Indirect"

When two photons collide, they can create a J/ψ in a few different ways:

Direct: The photons smash together and instantly create the J/ψ.
Resolved (Indirect): The photons act like little bags of smaller particles (quarks and gluons) that then collide to make the J/ψ.

The authors did the math and found that the Direct method is the winner by a huge margin (over 100 times more likely).

The Metaphor: Imagine trying to get a specific toy. The "Resolved" method is like digging through a giant pile of junk to find the toy inside a box. The "Direct" method is like someone handing you the toy directly. The paper shows that at the CEPC, we only need to worry about the "Direct" hand-off; the "junk pile" method is so rare it doesn't matter.

3. The Calculation: Adding "Next-to-Leading Order"

In physics, you can make a rough guess (Leading Order) or a very precise calculation (Next-to-Leading Order, or NLO).

The Analogy: If you are baking a cake, the "Leading Order" is just mixing flour and sugar. The "NLO" is adding the precise amount of baking powder, salt, and checking the oven temperature.
The authors performed this high-precision "NLO" calculation. They found that adding these corrections halves the predicted number of J/ψ particles compared to the rough guess. This is a massive change, showing that the "rough guess" was way off.

4. The Big Discovery: The "Polarization" Mystery

This is the core of the paper. The "polarization" is essentially the direction the J/ψ particle spins or points.

The Old Rulebook (Color Singlet): Predicts the J/ψ should be spinning sideways (transverse).
The New Rulebook (NRQCD with Color Octets): This rulebook includes some "hidden" quantum states (called Color Octets).

The authors tested four different versions of this rulebook (using different sets of data from previous experiments). Here is what they found:

Two versions agreed with the old rulebook (sideways spin).
One version predicted the J/ψ would be almost not spinning at all (unpolarized).
Another version predicted it would be spinning up and down (longitudinal).

The Metaphor: Imagine you have four different weather forecasters. Three of them say "It will rain," but one says "It will be sunny," and another says "It will be snowing." If you go outside and it's sunny, you know the "rain" forecasters were wrong.
The paper shows that the CEPC experiment is sensitive enough to tell which forecaster is right. Specifically, the result depends almost entirely on one specific number in the rulebook (the 3P[8] matrix element).

5. Why This Matters

The authors conclude that this specific experiment (J/ψ + a photon in a clean light-collision) is the perfect test to fix the rulebook.

Because the environment is so clean, and because the result depends heavily on just one specific variable, we won't get confused by other factors.
If the CEPC measures the spin of the J/ψ, we will finally know which version of the NRQCD rulebook is correct.
This will solve the decades-old "polarization puzzle" and help us understand the fundamental forces of nature better.

Summary

In short, this paper says: "Let's go to the cleanest possible lab (CEPC), smash two beams of light together, and count the J/ψ particles. By doing the math very precisely, we can finally figure out exactly how these particles spin, which will fix our broken rulebook for how the universe works."

1. Problem Statement

The paper addresses two major challenges in high-energy physics regarding heavy quarkonium production:

The $J/\psi$ Polarization Puzzle: Despite the success of Non-Relativistic QCD (NRQCD) factorization in describing production rates, it fails to consistently predict the polarization of $J/\psi$ mesons across different production channels. Experimental data (e.g., from CMS) often contradicts theoretical predictions, and different sets of Long-Distance Matrix Elements (LDMEs) that fit production rates yield conflicting polarization results.
The Need for a Clean Probe: Hadronic production channels (like $pp$ collisions) suffer from large theoretical uncertainties due to complex backgrounds and poorly constrained parton distribution functions (PDFs) of gluons within protons. There is a need for a "cleaner" environment to isolate specific NRQCD mechanisms and test the universality of LDMEs.

The authors propose investigating the process $\gamma\gamma \to J/\psi + \gamma$ at the Circular Electron Positron Collider (CEPC). This process offers a cleaner environment than hadroproduction because the initial state involves nearly on-shell virtual photons from lepton bremsstrahlung, suppressing strong-interaction backgrounds.

2. Methodology

The study employs a rigorous theoretical framework combining NRQCD factorization with Next-to-Leading Order (NLO) QCD calculations.

Factorization Framework: The differential cross-section is factorized into Short-Distance Coefficients (SDCs), calculable perturbatively in $\alpha_s$ , and universal Long-Distance Matrix Elements (LDMEs), which describe the non-perturbative transition of a $c\bar{c}$ pair into a physical $J/\psi$ .
Process Decomposition: The authors analyze three photon-initiated channels:
1. Direct: $\gamma + \gamma \to c\bar{c} + \gamma$
2. Single-Resolved: $\gamma + g/q \to c\bar{c} + \gamma$
3. Double-Resolved: $g/q + g/q \to c\bar{c} + \gamma$
  Result: The study demonstrates that the direct channel dominates by over two orders of magnitude, allowing the authors to neglect resolved photon contributions and focus exclusively on the direct process.
NLO Calculations:
- Leading Order (LO): Pure QED process $\gamma\gamma \to c\bar{c}[^3S_1^{[1]}] + \gamma$ .
- Next-to-Leading Order (NLO): Includes virtual corrections to the LO process and real emission processes involving gluons:
  - $\gamma\gamma \to c\bar{c}[^3S_1^{[1]}] + \gamma$ (virtual corrections)
  - $\gamma\gamma \to c\bar{c}[^1S_0^{[8]}] + \gamma + g$
  - $\gamma\gamma \to c\bar{c}[^3P_J^{[8]}] + \gamma + g$
- Renormalization: Ultraviolet (UV) divergences are handled via the on-mass-shell (OS) scheme. Infrared (IR) singularities in the $^3P_J^{[8]}$ channel are absorbed into the renormalization of the corresponding NRQCD matrix elements.
Simulation Setup:
- Collider: CEPC with $\sqrt{s} = 240$ GeV.
- Kinematic Cuts: Photon transverse momentum $p_t^\gamma > 1$ GeV and rapidity $|y_\gamma| < 2.65$ .
- LDME Sets: Four representative sets (Set 1–4) from literature [10, 19, 75, 76] are used, all of which successfully describe $J/\psi$ hadroproduction rates but differ in their polarization predictions.
- Tools: Calculations were performed using a custom Mathematica-Fortran package (incorporating FeynArts, FeynCalc, FIRE, Apart) and cross-checked with the FDC package.

3. Key Contributions

First NLO NRQCD Polarization Analysis: This is the first study to present NLO QCD corrections specifically for the polarization of $J/\psi$ in the $\gamma\psi + \gamma$ channel at an $e^+e^-$ collider.
Demonstration of Direct Channel Dominance: The authors quantitatively prove that resolved photon contributions are negligible in this specific process, unlike in inclusive $J/\psi$ production, thereby eliminating uncertainties associated with photon PDFs.
Sensitivity Analysis: The study isolates the specific sensitivity of the polarization parameter $\lambda_\theta$ to the ratio of LDMEs, specifically the $^3P_J^{[8]}$ state relative to the color-singlet $^3S_1^{[1]}$ state.

4. Key Results

Cross Sections:
- The NLO corrections to the Color-Singlet Model (CSM) cross section are significant, reducing the LO prediction by approximately 50%.
- Color-Octet (CO) contributions to the total integrated cross section are small but non-negligible.
- At the CEPC design luminosity ( $300 \text{ fb}^{-1}$ ), the process is expected to yield $\mathcal{O}(10^3)$ observable $J/\psi$ events per year (assuming a 10% dileptonic branching ratio).
Polarization Patterns ( $\lambda_\theta$ ):
- LDME Dependence: The polarization predictions are highly sensitive to the choice of LDME set.
  - Sets 1 & 2: Predict moderate to strong transverse polarization, consistent with CSM predictions.
  - Set 3: Predicts near-unpolarized or weakly longitudinal behavior.
  - Set 4: Predicts near-unpolarized to slightly transverse behavior.
- Mechanism: The divergence arises primarily from the $^3P_J^{[8]}$ color-octet mechanism. The NLO calculation reveals that the $^3P_J^{[8]}$ contribution introduces a negative term proportional to the LO cross section. Since the LO process is transversely polarized, this negative term suppresses the transverse component and enhances the longitudinal component.
- Insensitivity: The polarization is found to be insensitive to the $^1S_0^{[8]}$ and $^3S_1^{[8]}$ LDMEs in this specific channel, making it a "clean" probe for the $^3P_J^{[8]}$ matrix element.

5. Significance

Resolving the Polarization Puzzle: The process $\gamma\gamma \to J/\psi + \gamma$ serves as a critical benchmark. Because the polarization depends almost exclusively on the $^3P_J^{[8]}$ LDME (unlike hadroproduction which involves all three CO states), measuring the polarization at CEPC can directly constrain this specific matrix element.
Testing Universality: By comparing the LDMEs extracted from hadroproduction (which fit rates but fail polarization) with those required to fit the clean $\gamma\gamma$ data, this study can test the universality of NRQCD LDMEs. If the same LDMEs cannot describe both hadroproduction and this clean channel, it would indicate a fundamental flaw in the current NRQCD factorization framework.
Feasibility at CEPC: The study confirms that the CEPC, with its high luminosity and clean environment, is an ideal facility to collect the necessary statistics to distinguish between the different theoretical scenarios and potentially resolve the long-standing $J/\psi$ polarization discrepancy.