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Imprints of nuclear shell structure in exclusive vector meson production

This paper demonstrates that exclusive vector meson production at small $x$ serves as a novel probe of nuclear shell structure, as self-consistent nuclear densities derived from the quark-meson coupling model significantly modify coherent diffractive patterns, particularly for intermediate-mass nuclei like calcium isotopes where $J/\psi$ production offers a clean signal free from saturation effects.

Original authors: Arpita Mondal, Arjun Kumar, Debojit Sarkar

Published 2026-08-25

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CC BY 4.0

Original authors: Arpita Mondal, Arjun Kumar, Debojit Sarkar

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

The atomic nucleus, the dense core at the center of every atom, is not a smooth, uniform ball of matter. It is a complex quantum system where protons and neutrons arrange themselves in specific layers, much like seats in a theater, filling up from the bottom row to the top. This arrangement, known as shell structure, dictates the shape and stability of the nucleus. For decades, physicists have understood that these shells exist, but seeing them directly in action during high-energy collisions has remained elusive. When scientists smash particles together at nearly the speed of light, the resulting data is often a blur of interactions, making it difficult to distinguish the simple, orderly pattern of these shells from the chaotic noise of complex, many-body forces. The upcoming Electron-Ion Collider, a massive machine currently under construction, promises to shine a light on this hidden architecture by firing electrons at heavy nuclei, allowing researchers to map the interior of atoms with unprecedented clarity.

A new study by researchers at the Indian Institute of Technology Bombay and Stony Brook University takes a significant step toward this goal by simulating how these nuclei behave when struck by high-energy photons. The team focused on a specific type of collision where an electron transforms into a photon that briefly becomes a pair of quarks, which then bounce off the nucleus and reassemble into a vector meson, a particle made of a quark and an antiquark. Crucially, the nucleus remains intact after the collision, acting as a mirror that reflects the internal structure of the target. By using a sophisticated computer model that calculates the density of protons and neutrons based on their individual quantum orbits, the researchers predicted exactly how the scattering pattern would look if the shell structure were the dominant factor.

The results reveal that the shell structure leaves a distinct fingerprint on the scattering pattern, specifically in the way the particles scatter at different angles. For lighter nuclei, such as oxygen, the presence of these shells enhances the brightness of secondary peaks in the scattering data, making them stand out more clearly than if the nucleus were a smooth, featureless sphere. For intermediate-sized nuclei, like the calcium isotopes, the effect is even more dramatic. The researchers found that the positions of the dark gaps, or minima, in the scattering pattern shift depending on exactly which shells are filled. When they compared two versions of calcium—one with twenty protons and twenty neutrons, and another with twenty protons and twenty-eight neutrons—they saw the gaps move in opposite directions. This shift is a direct signal of the extra neutrons settling into a new shell, proving that the scattering pattern is sensitive enough to detect the addition of just a few particles in a specific quantum state.

However, the story changes for heavier nuclei, such as lead. In these massive atoms, the influence of the shell structure on the scattering pattern is much weaker and appears only at very specific, sharp angles. The researchers discovered that for heavy targets, the scattering is dominated by the overall size and density of the nucleus rather than the fine details of the shells. This distinction is vital for future experiments. It suggests that for medium-sized nuclei like calcium, scientists can use these scattering patterns as a clean tool to map nuclear shells without interference from other complex forces. Conversely, for heavy nuclei, the same technique can be used to study how the dense matter inside the nucleus behaves under extreme conditions, as the shell effects are small enough to be ignored in the initial analysis.

The study also looked at how different types of particles behave in these collisions. When the researchers simulated the production of a particle called the phi meson, which is larger than the J/psi meson they studied first, the results showed that the scattering pattern became much more sensitive to the non-linear dynamics of the strong force that binds quarks together. In this case, the subtle effects of the nuclear shells are hidden beneath the larger, more dominant effects of the strong force. This means that to see the shell structure in phi meson production, scientists must first account for these powerful interactions. The findings provide a clear roadmap for the Electron-Ion Collider: by choosing the right combination of target nucleus and produced particle, researchers can isolate specific features of nuclear structure. For intermediate-mass nuclei, the J/psi meson offers a pristine view of the shell structure, while for heavy nuclei, the phi meson provides a window into the saturation of the strong force.

Ultimately, this work establishes that the orderly arrangement of protons and neutrons in shells is not just a theoretical concept but a physical reality that shapes how nuclei interact with high-energy light. The simulations show that the differential cross-section, a measure of how likely particles are to scatter at a given angle, carries the imprint of these single-particle orbits. This discovery bridges the gap between the simple picture of independent particles moving in shells and the complex reality of a nucleus where particles constantly interact. By providing a baseline for what the data should look like if only shell structure were at play, the study gives experimentalists a reference point to identify and measure the more subtle, residual correlations that arise when particles interact in groups. As the Electron-Ion Collider comes online, these predictions will serve as a guide, helping physicists decode the intricate geometry of the atomic nucleus and understand how the fundamental building blocks of matter organize themselves into the diverse elements that make up our world.

Problem Statement
Understanding the geometric structure of atomic nuclei and disentangling single-particle shell structure from genuine many-body correlations is a central challenge at the intersection of nuclear physics and Quantum Chromodynamics (QCD). While heavy-ion collisions constrain nuclear geometry indirectly through final-state correlations, exclusive vector meson production in deep inelastic scattering (DIS) offers a more direct probe of the initial nuclear geometry. However, existing theoretical approaches face limitations: ab initio nucleon configurations, while encoding microscopic structure, do not permit a straightforward interpretation in terms of individual occupied single-particle orbitals, complicating the direct connection between diffractive observables and nuclear shell structure. Furthermore, the computational cost of ab initio methods limits their systematic application across the nuclear chart, particularly for heavy nuclei. Consequently, it remains unclear to what extent the differential $t$ -spectrum in exclusive vector-meson production is governed by the shell structure encoded in the nuclear one-body density versus residual many-body correlations.

Methodology
The authors employ the Quark-Meson Coupling (QMC) model, a QCD-motivated mean-field framework, to compute nuclear densities self-consistently from occupied single-particle orbitals. In this model, non-overlapping MIT bags are bound by self-consistent scalar and vector mean fields that couple to confined light quarks. The nuclear density is constructed from the squared Dirac wave functions of occupied orbitals ( $G_\alpha$ and $F_\alpha$ ) within a relativistic Hartree formulation, accounting for the scalar polarizability of the nucleon and the in-medium modification of quark structure.

These self-consistent QMC densities are calculated for four nuclei: $^{16}\text{O}$ , $^{40}\text{Ca}$ , $^{48}\text{Ca}$ , and $^{208}\text{Pb}$ . These densities are then projected onto the transverse plane to determine the nuclear thickness functions $T_A(b)$ . The authors utilize the color-dipole framework to calculate coherent exclusive $J/\psi$ and $\phi$ photoproduction in electron-nucleus ( $eA$ ) collisions. The scattering amplitude is computed using the IPSat model (incorporating saturation effects) and the linearized IPNonSat model (to quantify non-linear effects). The results are compared against a baseline using the standard two-parameter Woods-Saxon (WS) parametrization, which assumes a structureless, monotonic density profile.

Key Contributions and Results
The paper reports that within the saturation framework, exclusive vector meson production at small $x$ is sensitive to the shell structure of the target nucleus. The key findings are:

Shell Imprints on Density and Thickness: The QMC-derived densities exhibit modulations driven by the specific occupied orbitals (e.g., $s$ -orbitals fixing central density, $d$ -states governing surface diffuseness), which are absent in the smooth Woods-Saxon profiles. While the projection to the transverse plane partially averages these modulations, the resulting thickness functions retain visible imprints, particularly for light and intermediate-mass nuclei.
$J/\psi$ Production as a Shell Probe: Due to the small dipole size of the $J/\psi$ $J / ψ$ , saturation effects are negligible, making the IPSat and IPNonSat results nearly identical for a given density. This renders $J/\psi$ $J / ψ$ production a clean probe of nuclear geometry.
- Light Nuclei ( $^{16}\text{O}$ ): The QMC profile enhances the secondary diffractive lobes relative to the WS baseline.
- Intermediate-Mass Nuclei ( $^{40}\text{Ca}$ vs. $^{48}\text{Ca}$ ): The coherent spectrum shows a clear isotopic dependence. The higher-order diffractive minima shift in opposite directions for the two isotopes (toward larger $|t|$ for $^{40}\text{Ca}$ and smaller $|t|$ for $^{48}\text{Ca}$ ). This shift isolates the effect of the eight additional neutrons occupying the $f_{7/2}$ shell in $^{48}\text{Ca}$ , demonstrating direct sensitivity to shell filling.
- Heavy Nuclei ( $^{208}\text{Pb}$ ): The first diffractive minima coincide with the WS baseline. Shell-structure effects manifest primarily as an enhancement of the diffractive lobe magnitudes, with shifts appearing only at larger $|t|$ and higher-order minima.
$\phi$ Production and Saturation: For the larger $\phi$ dipole, saturation effects are significant. The linearized IPNonSat prediction separates significantly from the saturated IPSat result. In this channel, shell-structure effects constitute a sub-leading modification relative to non-linear QCD dynamics. However, the shell structure must be included in the initial state to properly isolate saturation effects in differential observables.
Quantitative Impact: For $^{16}\text{O}$ , the ratio of the QMC-based cross section to the WS baseline peaks near the diffractive minima, reaching values of order $\sim 2$ , underscoring the significant impact of shell structure on the coherent $t$ -spectrum.

Significance
The authors establish that exclusive vector meson production serves as a probe of the mean-field, single-particle structure of nuclei at small $x$ . The work provides a crucial link between fully correlated many-body systems and independent-particle descriptions in nuclear QCD. Specifically:

Coherent $J/\psi$ production on intermediate-mass nuclei (such as calcium isotopes) is identified as a sensitive probe of nuclear shell structure, capable of isolating shell-filling effects at a fixed proton number.
For light nuclei, these results provide a controlled baseline for isolating and quantifying residual many-body correlations (e.g., clustering), which are expected to be negligible in this regime.
For heavy nuclei, the relative insensitivity of the first few diffraction minima to shell modulations allows for a direct probe of saturation dynamics, provided the initial state geometry is correctly modeled.
The findings offer a necessary input for hydrodynamic modeling and complement information inferred from heavy-ion collisions regarding the nuclear initial state.

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