⚛️ phenomenology

Scattering lengths of the $J/ψπ$ and $J/ψK$ systems

Using dispersion relations and accounting for chiral symmetry breaking-induced field mixing, this study calculates the attractive $S$ -wave scattering lengths for $J/\psi\pi$ and $J/\psi K$ systems, revealing that their interactions are predominantly governed by soft-gluon exchange rather than coupled-channel mechanisms.

Original authors: Jiang Yan, Xiong-Hui Cao, Meng-Lin Du, Feng-Kun Guo

Published 2026-01-27

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CC BY 4.0

Original authors: Jiang Yan, Xiong-Hui Cao, Meng-Lin Du, Feng-Kun Guo

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 subatomic world as a bustling dance floor. On one side, you have the heavy, slow dancers (like the $J/\psi$ , a "charmonium" particle made of heavy charm quarks). On the other side, you have the light, fast dancers (pions and kaons, which are made of lighter quarks).

This paper asks a simple question: How much do these heavy and light dancers push or pull on each other when they get close? In physics, this "push or pull" is measured by something called a scattering length. If the number is negative, it means they are slightly attracted to each other, like two magnets that are too weak to snap together but still feel a gentle tug.

Here is the story of what the researchers found, explained without the heavy math:

1. The "Ghost" Mix-Up

The scientists started by looking at the rules that govern how these particles interact. They realized that the "heavy dancers" ( $J/\psi$ and a slightly heavier cousin called $\psi'$ ) aren't as distinct as they seem. Because of the way the universe breaks symmetry (a fancy way of saying the rules aren't perfectly balanced), these two particles actually "mix" with each other, like two colors of paint blending together.

To understand the real, physical particles we see in experiments, the researchers had to perform a mathematical "untangling" (diagonalization) to separate the mixed-up versions back into the pure $J/\psi$ and $\psi'$ . They found that even after untangling them, the rules of the game still leave a small "footprint" of that mixing, which affects how the particles interact with the light dancers.

2. The Two Ways They Interact

The paper explores two main ways these heavy and light particles could interact:

The "Glue" Mechanism (Soft-Gluon Exchange): Imagine the heavy dancer is a compact ball of sticky glue. When a light dancer gets close, they don't touch directly; instead, they exchange invisible "glue strands" (gluons) that create a gentle force between them. This is like two people standing near each other feeling a faint static electricity shock.
The "Detour" Mechanism (Coupled-Channel): Imagine the heavy dancer wants to talk to the light dancer, but instead of talking directly, they briefly turn into a completely different pair of dancers (open-charm mesons), have a quick chat, and then turn back. This is a "detour" through a different state of matter.

3. The Results: Pions vs. Kaons

The researchers calculated how strong these interactions are for two types of light dancers: Pions ( $\pi$ ) and Kaons ( $K$ ).

The Pion ( $J/\psi$ + $\pi$ ): The interaction is extremely weak. It's so weak that it's almost like the particles are ignoring each other. This is because of a special rule in physics (chiral symmetry) that makes pions very shy when interacting with heavy particles. The math shows the "scattering length" is tiny (less than -0.0021 femtometers).
The Kaon ( $J/\psi$ + $K$ ): The interaction is stronger, though still weak overall. Why? Because the Kaon is heavier than the Pion (it contains a "strange" quark). This extra weight breaks the "shy" rule slightly, allowing the particles to feel a more noticeable tug. The scattering length is larger (less than -0.028 femtometers).

4. Who Wins the Dance?

The most important discovery in the paper is comparing the two mechanisms mentioned above.

The researchers found that the "Detour" mechanism (turning into other particles) is practically negligible. It's like trying to talk to someone by shouting through a wall; it just doesn't work well here.
The "Glue" mechanism (exchanging gluons) is the dominant force. It is the primary reason the particles interact at all.

The Bottom Line

In simple terms, this paper tells us that when a heavy $J/\psi$ particle meets a light pion or kaon:

They barely interact, but they do feel a very faint, attractive pull.
The pull is slightly stronger with the Kaon than the Pion because the Kaon is heavier.
This interaction happens almost entirely because of the exchange of "glue" (gluons) between them, not because they are taking detours through other particle states.

The authors conclude that this "glue-only" dominance might be a universal rule for how heavy particles interact with light ones, a finding that future experiments and computer simulations (lattice QCD) can test to confirm.

Technical Summary: Scattering Lengths of the $J/\psi\pi$ and $J/\psi K$ Systems

Problem Statement
The low-energy interactions between heavy quarkonia (specifically charmonium states like $J/\psi$ ) and light hadrons (pions $\pi$ and kaons $K$ ) remain a subject of theoretical challenge. While these processes are crucial for understanding nonperturbative Quantum Chromodynamics (QCD) and the phenomenology of $J/\psi$ suppression in relativistic heavy-ion collisions, a quantitative and unified understanding is lacking. A key distinction in these interactions is the absence of common valence quarks between the heavy quarkonium and light hadrons, leading to suppression by the Okubo–Zweig–Iizuka (OZI) rule. Consequently, interactions are mediated primarily by gluonic dynamics rather than quark exchange.

Two primary mechanisms govern these interactions:

Coupled-channel mechanism: Involves coupling to intermediate open-charm meson states (e.g., $D\bar{D}^*$ or $D^*\bar{D}_s$ ) which then rescatter.
Multi-gluon exchange: Viewed as a color van der Waals interaction between color-singlet hadrons, often treated via the QCD multipole expansion.

While the $J/\psi\pi$ interaction is expected to be extremely weak due to chiral symmetry constraints (vanishing at leading order in the chiral limit), the status of the $J/\psi K$ interaction is less clear. The explicit breaking of $SU(3)_f$ symmetry by the strange quark mass may relax these constraints, potentially enhancing the $J/\psi K$ scattering length. Furthermore, the proximity of open-charm strange thresholds suggests coupled-channel effects might be more significant for kaons than for pions. However, systematic quantitative studies of low-energy $J/\psi K$ scattering have been scarce compared to pion or nucleon channels.

Methodology
The authors investigate these interactions within the framework of dispersion relations and chiral effective field theory.

Chiral Lagrangian and Diagonalization:
The study begins with a chiral effective Lagrangian describing interactions between charmonium states and light pseudoscalar mesons. A critical technical step involves addressing the symmetry-breaking terms ( $c_m$ ) in the Lagrangian. The authors demonstrate that these terms induce mixing between bare charmonium fields ( $\psi_1, \psi_2$ ). They perform a diagonalization procedure to identify the physical $J/\psi$ and $\psi'$ states. Crucially, they correct a previous assertion (Ref. [24]) by showing that this diagonalization does not eliminate the contribution of symmetry-breaking terms to physical contact interactions (specifically the $\psi' \to J/\psi\pi\pi$ vertex); rather, it transforms the coefficients while preserving the contact terms in the physical basis.
Scattering Amplitude Calculation:
Using the diagonalized Lagrangian, the authors derive the tree-level amplitude for the crossed-channel process $J/\psi J/\psi \to P\bar{P}$ ( $P=\pi, K$ ). These amplitudes are expressed in terms of low-energy constants (LECs) related to charmonium chromo-polarizabilities ( $\alpha_{AB}$ ).
- The authors utilize constraints from the Cauchy-Schwarz inequality and phenomenological analyses of the $\psi' \to J/\psi\pi\pi$ decay to constrain the chromo-polarizabilities, adopting a lower bound $\alpha_{11} \gtrsim 1.18 \text{ GeV}^{-3}$ .
- To incorporate final-state interactions (FSI), specifically $\pi\pi$ and $K\bar{K}$ rescattering, they employ the Muskhelishvili–Omnès formalism. This involves solving unitarity conditions using dispersion relations and Omnès matrices derived from coupled-channel scattering data.
Crossing Symmetry and Scattering Lengths:
The $S$ -wave scattering amplitudes for $J/\psi P$ at threshold are obtained from the crossed-channel amplitudes via crossing symmetry (evaluating the $J/\psi J/\psi \to P\bar{P}$ amplitude at $s=0$ ). The scattering lengths are then extracted from these threshold amplitudes.
Coupled-Channel Comparison:
To assess the relative importance of the multi-gluon exchange versus the coupled-channel mechanism, the authors compare their results with scattering lengths derived from a coupled-channel model (based on parameters fitted to BESIII data for $Z_c$ and $Z_{cs}$ states). In this comparison, the direct gluon-exchange potential is excluded, isolating the contribution from intermediate open-charm loops.

Key Contributions and Results

Correction of Theoretical Formalism: The paper clarifies the role of chiral symmetry-breaking terms in the charmonium sector, demonstrating that diagonalizing the mass matrix preserves the contact interactions necessary for physical processes like $\psi' \to J/\psi\pi\pi$ .
Quantitative Scattering Lengths: The authors calculate the $S$ $S$ -wave scattering lengths for $J/\psi\pi$ $J / ψ π$ and $J/\psi K$ $J / ψ K$ driven by soft-gluon exchange:
- $a_{J/\psi\pi} \lesssim -0.0021 \text{ fm}$
- $a_{J/\psi K} \lesssim -0.028 \text{ fm}$
  The negative signs indicate attractive interactions, though too weak to support bound states.
Mechanism Dominance: The results show that the $J/\psi K$ interaction is moderately enhanced relative to the pion channel due to explicit chiral symmetry breaking (the strange quark mass). However, when comparing the soft-gluon exchange results with the coupled-channel mechanism results ( $a^{CC}_{J/\psi\pi} \in [-4.2, 6.1] \times 10^{-6} \text{ fm}$ and $a^{CC}_{J/\psi K} \in [-0.01, -8.9 \times 10^{-6}] \text{ fm}$ ), the multi-gluon exchange mechanism is found to be the dominant contributor to the scattering lengths in the near-threshold region.
Universality: The findings align with previous work on $J/\psi$ -nucleon scattering, suggesting that multi-gluon exchange may universally dominate over coupled-channel mechanisms for light hadron scattering off charmonia.

Significance
The paper provides a rigorous, quantitative assessment of low-energy $J/\psi$ interactions with light mesons, resolving ambiguities in the theoretical treatment of chiral symmetry breaking in the charmonium sector. By disentangling the contributions of multi-gluon exchange and coupled-channel dynamics, the study concludes that the former is the primary driver of these interactions, even in the presence of open-charm thresholds. These predictions offer specific targets for future lattice QCD calculations to test the dominance of the soft-gluon exchange mechanism in heavy-light hadron systems. The authors note that while this conclusion holds for charmonia, its applicability to bottomonia remains an open question due to expected differences in chromo-polarizabilities.