Calcium isotopes emerge as a promising place to look for nuclear shell structure in particle collisions, according to a new set of theoretical calculations. The model predicts that coherent J/ψ production would show opposite shifts in a key dip in the spectra of calcium-40 and calcium-48: the second minimum moves toward larger |t| for calcium-40 and toward smaller |t| for calcium-48. The proton number is unchanged between the isotopes, while calcium-48 has eight additional neutrons in the f7/2 shell. The authors identify calcium as a promising intermediate-mass target for probing shell structure.
The result addresses a basic question in nuclear physics: how much of the peaks-and-dips pattern in exclusive vector-meson production can be traced to the way nuclear shells are filled before correlations and other many-body dynamics are included. A diffractive minimum is simply a dip in the calculated cross section, the predicted production rate as momentum transfer changes. The study follows that pattern for coherent J/ψ and ϕ production on four modeled nuclei: oxygen-16, calcium-40, calcium-48 and lead-208.
The nuclear input comes from a self-consistent relativistic-Hartree QMC calculation. In plain terms, the model generates an occupied Dirac orbital for each nucleon and uses those orbitals to produce nuclear densities. Those densities are compared with Woods-Saxon baseline profiles in IPSat and IPNonSat calculations. IPSat represents the saturation treatment used in the comparison, while IPNonSat provides the linearized counterpart.
The isotope pattern is clearest in calcium
Oxygen shows how a shell imprint can change the shape of a spectrum without moving every landmark. In the J/ψ calculation, the first minimum stays essentially unchanged relative to the Woods-Saxon baseline, near 0.10 GeV², while the secondary maximum rises. The ratio of the IPSat plus QMC prediction to the Woods-Saxon baseline reaches about 2 near diffractive minima, corresponding to a roughly two-fold local cross-section enhancement.
The calcium comparison is more distinctive. The second minimum shifts in opposite directions for the two isotopes, and both spectra show enhanced second and higher-order maxima, although the enhancement magnitudes differ. The authors single out calcium as an intermediate-mass candidate for probing shell structure.
The shell imprint does not disappear in the heavier case, but its relative magnitude weakens as nuclear mass increases. For lead-208, the earliest minima nearly coincide between the profiles. QMC mainly changes the magnitudes of the diffractive peaks, while profile differences become appreciable only beyond the second diffractive minimum.
J/ψ and ϕ tell different stories
Meson choice matters. For J/ψ, the small dipole size makes saturation effects negligible in the reported comparison, so the IPSat and IPNonSat Woods-Saxon curves nearly overlap. The authors therefore present coherent J/ψ production, especially for intermediate-mass nuclei such as calcium, as a way to probe shell structure.
For ϕ, the balance is different. Nonlinear saturation effects dominate the comparison. IPNonSat separates from IPSat, with larger cross sections at low |t| and in higher-order peaks, and with the minima shifted. Shell effects remain in the calculation but are sub-leading. The authors therefore frame ϕ as more useful for studying saturation dynamics than for reading nuclear shell structure.
A supplementary comparison with coherent J/ψ data from ultra-peripheral Pb-Pb collisions is reported to show good agreement for the QMC saturation calculation. The supplied analysis does not give a numerical goodness-of-fit measure, so the comparison is a qualitative check rather than a quantified test.
A forecast, not a measurement
This is a model forecast built around four selected nuclei and two mesons, not a direct calcium measurement. Its predictions come from the chosen QMC density, Woods-Saxon baseline, IPSat and IPNonSat frameworks. The analysis reports no statistical uncertainty for the model curves.
The QMC calculation is presented as a baseline for separating shell filling from residual many-body correlations, rather than as a measurement of those correlations. That distinction matters because the reported shifts and enhancements are outputs of the selected model comparison.
The work is a preprint. Its front matter records an arXiv version 2 entry dated 30 Aug 2026.
Paper data and sources
Original title: Imprints of nuclear shell structure in exclusive vector meson production
Authors: Arpita Mondal, Arjun Kumar, Debojit Sarkar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-24
DOI: Not available
Original paper · Full text