The structure of the $X(3915)$ meson and its production in heavy ion collisions

We study the structure of the $X(3915)$ meson in a quark model and explore how its production in heavy ion collisions depends on its internal structure. We first analyze the $X(3915)$ as a $c\bar{c}s\bar{s}$ state and solve the Hamiltonian with color-spin interactions within the quark model. We find that the ground state of the $c\bar{c}s\bar{s}$ with total spin 0 obtained from the quark model analysis favors a separated $D_s \bar{D}_s$ state. To probe its structure further, we study its production in relativistic heavy ion collisions for various proposed configurations. We calculate the transverse momentum distributions and yields for the $X(3915)$ assuming its structure to be either a charmonium, a tetraquark, or a hadronic molecular state. We argue that by measuring the transverse momentum distributions and yields of the $X(3915)$ produced in heavy ion collisions, one can identify the structure of the $X(3915)$.

💡 Research Summary

The paper investigates the internal structure of the X(3915) meson—identified by the Particle Data Group as χc0(3915)—and examines how this structure influences its production in relativistic heavy‑ion collisions. The authors first treat X(3915) as a hidden‑flavor tetraquark composed of c¯c s¯s. Using a constituent quark model that incorporates both a linear‑plus‑Coulomb confinement potential (−κ r + σ r) and a Yukawa‑type hyperfine interaction (the Yukawa‑type “ultrafine” potential), they construct a full Hamiltonian including color‑spin operators. Model parameters (κ, σ, quark masses, etc.) are taken from previous global fits; the spatial wave function is taken as a Gaussian in Jacobi coordinates with variational parameters a₁ and a₃. By minimizing the energy, they obtain a₁≈7.0 fm⁻² and a₃≈0.001 fm⁻², which yields a tetraquark mass of 3922.1 MeV—essentially identical to the experimental value and to the D_s \bar D_s threshold. This indicates that the model naturally reproduces a state that is almost a loosely bound D_s \bar D_s pair.

The color‑spin structure is analyzed in detail. Four color‑spin basis states are constructed: color singlet–spin singlet, color octet–spin octet, and their spin‑triplet counterparts. By transforming to the (c¯s)⊗(s¯c) basis, the authors expose the mixing between color‑singlet and color‑octet components. The K‑factor matrix (the expectation value of the color‑spin operator) shows that the basis |CS₃⟩ (color octet, spin triplet for both (c¯s) and (s¯c) pairs) provides the largest attractive contribution, especially from the s¯s pair, because its hyperfine term is not suppressed by the heavy charm mass. Consequently, the ground‑state wave function is dominated by a component that resembles a separated D_s \bar D_s configuration rather than a tightly bound compact tetraquark.

Having established the internal structure, the authors turn to production in heavy‑ion collisions. They employ a coalescence model in which deconfined charm and strange quarks recombine at hadronization to form X(3915). Three structural hypotheses are considered: (i) a conventional charmonium (the 2P χc0), (ii) a compact c¯c s¯s tetraquark, and (iii) a hadronic molecule D_s \bar D_s. For each case the spatial size (encoded in a₁, a₃) and the color‑spin wave function differ, leading to distinct coalescence probabilities. Using Monte‑Carlo integration over the quark phase‑space distributions, the authors compute transverse‑momentum (p_T) spectra and total yields.

The results show clear qualitative differences. The charmonium scenario, with a compact wave function, yields a relatively flat p_T distribution and a yield proportional to the overall charm density. The tetraquark case, dominated by the color‑octet–spin‑triplet component, produces an enhanced yield in the intermediate p_T region (2–4 GeV/c) but falls off rapidly at higher p_T. The molecular scenario, characterized by a large spatial extension (≈1 fm), leads to a strong enhancement at low p_T (<1 GeV/c) and the highest overall yield among the three. Moreover, the ratio of X(3915) to D_s yields as a function of p_T is predicted to be ≈0.5–0.7 for the molecular case, while it is significantly lower for the compact configurations.

These findings demonstrate that the color‑spin interaction, which determines the internal binding pattern, directly controls the coalescence probability and thus observable quantities such as p_T spectra and yield ratios. The authors argue that precise measurements of X(3915) transverse momentum distributions and its yield relative to D_s mesons in experiments at the LHC, RHIC, or future facilities (FAIR, NICA) can discriminate among the three structural hypotheses. They also note that the variational parameter a₃ approaching zero signals a transition toward a molecular configuration, providing a quantitative handle on the degree of binding.

In conclusion, the paper provides a coherent framework linking quark‑model spectroscopy, color‑spin dynamics, and heavy‑ion coalescence phenomenology. By showing that the same underlying Hamiltonian reproduces the observed mass and predicts distinct production signatures for different internal structures, the work offers a concrete strategy to resolve the long‑standing puzzle of the X(3915) nature using heavy‑ion collision data.