J/ψ Emission Sources Cool Down as Photon Energy Rises in Tiny High-Energy Collisions

When a high-energy photon slams into a proton and knocks out a J/ψ meson—the charmonium particle made of a charm quark and its antiquark—the collision is so small that only two particles emerge. Yet a new analysis shows that even this minimal two-body reaction carries a measurable fingerprint of how hot, or how excited, its emission source is. A team at Shanxi Institute of Energy, publishing in Frontiers in Physics, has now charted that fingerprint across a wide energy range and found a surprise: the excitation degree of the J/ψ emission source decreases as the collision energy increases, the opposite of what happens in the sprawling fireballs of heavy-ion collisions.

The study, led by Qi Wang with Jia-Rong Shi and You-Long Wang, dissects squared momentum transfer spectra of J/ψ mesons produced in photon–proton processes inside electron–proton collisions. The experimental data come from three collaborations that measured this reaction at different facilities: GlueX, which probes the near-threshold region at Jefferson Lab; J/ψ-007, which covers intermediate photon energies; and ZEUS, which recorded exclusive J/ψ photoproduction at HERA. Together these datasets span photon energies near 10 GeV and photon–proton center-of-mass energies from roughly 50 GeV up to 290 GeV.

The squared momentum transfer, one of the Mandelstam variables familiar from two-body reaction kinematics, measures how much momentum the incident photon hands over to the emitted J/ψ—or, equivalently, how much recoil momentum the target proton picks up when it is kicked out of the reaction. Because the process is exclusive, with only a J/ψ and a recoil proton in the final state, this quantity is tightly constrained by energy and momentum conservation. Its distribution, however, is not something that can be written down in closed form; it must be built up statistically from the underlying transverse momentum spectra of the emitted particles.

That is where the Monte Carlo machinery comes in. The researchers first generate large numbers of concrete events whose transverse momenta obey either the Erlang distribution or the Tsallis–Lévy function, while respecting conservation of energy and momentum. The Erlang distribution, rooted in the multi-source thermal model, is the convolution of several exponential distributions—one for each participant parton contributing to the transverse momentum of the particle. The number of contributing partons, denoted n, is small, around two to five, and pre-analyses showed it is highly sensitive: as n grows from 1 to 10, the distribution morphs from an exponential curve into a Gaussian-like profile. Neither smaller nor larger values of n describe the data acceptably, so the team fixed n with zero uncertainty and concentrated on the remaining free parameters.

The Tsallis–Lévy function, an extension of Tsallis statistics into high-energy physics, provides the alternative description. Because the photon–proton process produces so few particles per event, the parameter extracted from it cannot honestly be called a final-state temperature; the system is out of equilibrium and the multiplicity is too limited. Instead the researchers call it θ, the average excitation degree of the emission source at the final state, and emphasize that it is a relative measure rather than an absolute observable. For the initial state, they turn to a color string percolation method: in this reaction, the incident photon and a parton in the target proton form a single color string, so the initial-state temperature is computed with one string, a choice the team showed has negligible influence on the result.

Running the Monte Carlo chain—generating events, computing the squared momentum transfer for each, and building the distribution statistically—the team fitted the experimental spectra from all three collaborations. The model curves track the measured differential cross-sections in most cases, across the GlueX photon energy bins from about 8.2 to 11.4 GeV, the ten J/ψ-007 bins from 9.10 to 10.60 GeV, and the ZEUS center-of-mass energy bins from 50–70 GeV up to 230–290 GeV. From the fits they extracted the average transverse momentum of the J/ψ, the initial-state temperature, and the final-state excitation degree, then plotted how each parameter varies with photon energy and with center-of-mass energy.

The headline result is a decreasing trend. Except for the very first data point, the average transverse momentum, the initial-state temperature, and the final-state excitation degree all roughly decline as the photon energy and the photon–proton center-of-mass energy rise. In heavy-ion collisions at similar energies, these same parameters climb with collision energy, reflecting a medium that grows hotter as more energy is deposited. The authors attribute the difference to interaction time: in the small photon–proton system there are no multiple scatterings, so higher energy means a shorter interaction time and lower energy deposition, and the trend also mirrors the shifting balance between soft and hard collision components.

The anomalously low first point gets its own explanation. It corresponds to a photon energy of about 8.74 GeV, sitting in the near-threshold photoproduction region just above the kinetic threshold of 8.2 GeV for the reaction. There, the center-of-mass kinetic energy of the J/ψ is nearly zero, and the dominant mechanism is no longer the generalized parton distribution process involving Compton gluons that prevails at high energies, but rather long-range gluon interactions and non-perturbative hadronic effects inside the nucleon. The final-state J/ψ acquires almost no additional transverse recoil momentum, suppressing the average transverse momentum and, with it, the extracted excitation parameters.

There is also a statistical subtlety worth noting. Each photon–proton event contains only two particles in the final state, yet the analysis applies the Erlang distribution and the Tsallis–Lévy function to spectra built from enormous numbers of events. The authors argue this is analogous to the grand canonical ensemble in statistical physics: because particles in different events are produced under the same or similar conditions at a given collision energy, they obey certain statistical laws even though they do not interact and the multiplicity per event is very low. On that view, the concept of excitation degree remains meaningful even for two-body reactions, as a reflection of the average kinetic energy of disorganized motion.

The work confirms and extends the team’s previous research on initial- and final-state temperatures from squared momentum transfer spectra, pushing the energy range higher. The authors caution that more explicit conclusions about how the parameters evolve with energy will require experimental datasets with significantly higher statistics. Future analyses, they say, will include additional datasets to systematically compare excitation functions across small and large collision systems—testing whether the cooling trend seen in photon–proton reactions holds as a general signature of how tiny systems differ from the quark–gluon fireballs of heavy-ion physics.

Subject of Research: High-energy nuclear physics

Article Title: Analysis of the excitation degree of J/ψ emission sources from squared momentum transfer spectra in high-energy collisions

Article References: Analysis of the excitation degree of J/ψ emission sources from squared momentum transfer spectra in high-energy collisions. (n.d.). https://doi.org/10.3389/fphy.2026.1957749

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1957749

Keywords: J/psi meson, photon-proton collisions, Erlang distribution, Tsallis-Levy function, Monte Carlo, GlueX, ZEUS, excitation degree, squared momentum transfer, Frontiers in Physics, Analysis, excitation