Possible Existence of $^3_ϕ$H, $^4_ϕ$H, $^4_ϕ$He, and $^5_ϕ$He Nuclei

Motivated by recent HAL QCD simulations of the $ϕN$ interaction in the $^4S_{3/2}$ channel and its modification in the $^2S_{1/2}$ channel, we develop a first-principles few-body framework that embeds these potentials into configuration-space Faddeev–Yakubovsky equations. We predict bound $^4_ϕ\mathrm{H}$, $^4_ϕ\mathrm{He}$, and $^5_ϕ\mathrm{He}$ nuclei by performing calculations for $ϕ$-mesic $ϕNNN$ and $ϕNNNN$ systems. Both spin-dependent and spin-independent $ϕN$ interactions are considered, leading to deeply and moderately bound states, respectively. The deeply bound states originate from the strong attraction in the $^2S_{1/2}$ $ϕN$ channel. Coulomb shifts of the binding energies are evaluated. Our findings provide the binding mechanism and demonstrate the importance of short-range $ϕN$ attraction.

💡 Research Summary

This work presents the first systematic few‑body study of light nuclei containing a hidden‑strangeness φ meson, based on the φ–N interaction recently extracted by the HAL QCD collaboration. The authors construct non‑relativistic potentials for the two spin channels of the φ–N system, ⁴S₃/₂ and ²S₁/₂, as a sum of short‑range Gaussian terms and a long‑range two‑pion‑exchange tail. A scaling factor β controls the strength of the ²S₁/₂ channel; HAL QCD analyses suggest β≈6.9 (with statistical and systematic uncertainties) yields a bound φ–N subsystem, whereas β=1 corresponds to spin‑independent interactions with only modest attraction.

For the nucleon–nucleon force the MT I‑III potential is employed, with parameters readjusted when QCD‑motivated nucleon masses are used, so that the deuteron binding energy and np scattering lengths are reproduced. Both physical and QCD‑motivated mass sets are considered throughout the calculations.

The few‑body dynamics are solved exactly using configuration‑space Faddeev–Yakubovsky equations. By exploiting the fact that only one particle (the φ meson) differs from the remaining A‑1 identical nucleons, the number of independent Yakubovsky components is reduced dramatically: 2 equations for A=3, 5 for A=4, and 16 for A=5. The equations are discretized in mass‑scaled Jacobi coordinates, expanded in a partial‑wave basis (only S‑waves are retained), and the radial amplitudes are represented on a Lagrange‑Laguerre mesh. Convergence is achieved to four significant figures with angular momenta limited to ℓ<3.

Results show that the strong short‑range attraction in the ²S₁/₂ channel dominates the binding mechanism. For β≈6.9 the φ–N pair forms a bound state with a few MeV binding energy, and this attraction propagates to the few‑body systems. The φnn and φpp configurations remain unbound because the Pauli principle forces one nucleon into the weakly attractive ⁴S₃/₂ channel. The φnp system, however, supports two bound states with Jπ=0⁻ (ground state) and 1⁻, the former being the most deeply bound because both nucleons couple to the φ in the ²S₁/₂ channel.

The three‑nucleon system φNNN (interpreted as ⁴φH or ⁴φHe) exhibits a single bound state with total isospin T=½ and Jπ=½⁻. The third nucleon is weakly attached (≈4 MeV separation energy) while the first two nucleons are bound to the φ by ≈18–34 MeV. Inclusion of the Coulomb interaction shifts the binding energies of the mirror nuclei by about 0.68 MeV, comparable to the MT I‑III prediction for the ³H–³He difference.

The five‑body system φNNNN (⁵φHe) shows a deeply bound ground state with T=0, Jπ=1⁻ and a nucleon separation energy of ≈23 MeV, indicating that the φ meson is tightly bound inside an α‑particle core. When β is reduced to 1, this state becomes only moderately bound, and the binding pattern of the lighter systems changes accordingly: ³φH becomes nearly unbound, while ⁴φH/⁴φHe retain modest binding.

Tables in the paper list binding energies for both β values and for both mass schemes. For β=6.9 the binding energies (measured from the φ‑separation thresholds) are roughly 18 MeV for φnp, 11–12 MeV for ³φH, 40 MeV for ⁴φH/⁴φHe, and 55–77 MeV for ⁵φHe, depending on whether Coulomb is included. For β=1 the corresponding values shrink dramatically, with ³φH bound by only a few tens of keV and the heavier nuclei bound by 5–6 MeV.

The authors discuss the sensitivity of the results to the short‑range part of the φ–N interaction, emphasizing that the parameter β controls whether the system exhibits deep binding (β≈6.9) or moderate binding (β=1). They also note that the excited “breathing” mode of the α‑particle, which lies just above the ⁴He threshold, may acquire binding when a φ meson is added, but this effect is suppressed for larger β due to increased Coulomb repulsion and compactness of the system.

In conclusion, the study demonstrates that a spin‑dependent φ–N interaction, as suggested by HAL QCD, can generate bound φ‑mesic nuclei with A=3–5. The existence of ³φH, ⁴φH, ⁴φHe, and ⁵φHe is robust against variations in nucleon masses and the inclusion of electromagnetic effects. The work provides a solid theoretical foundation for future experimental searches of φ‑mesic nuclei and highlights the pivotal role of short‑range hidden‑strangeness forces in nuclear binding.