The analytic continuation in coupling constant (ACCC) method computes resonance energies and widths of unbound quantum systems by fitting the bound-state energies of an artificially strengthened Hamiltonian with a Padé approximant and continuing the fit back to the physical coupling. Oishi and Kimura (Phys. Rev. C 112, 014305 (2025)) applied ACCC to the two-proton (2p) emitters 6Be and 16Ne within core+p+p three-body models, validating the method against their earlier time-dependent calculations and against data, and explicitly left open a linked set of issues: the microscopic origin of the avoided crossing between the first and second 0+ resonances of 16Ne (coupling of s1/2 and d5/2 channels and of non-resonant continuum channels by the proton-proton interaction); the sensitivity of the computed 2p energies and widths, including the 2+–0+1 Thomas-Ehrman gap, to the core-proton and proton-proton potentials and whether their optimization can remove the deviations from experiment; and the method's reliable width range, currently limited to 10−4≲Γ≲110^{-4}\lesssim\Gamma\lesssim 1 MeV, and how to extend it. No resolution of these items is known in the literature to date.

Background

Two-proton (2p2p) emission is a decay mode of proton-rich nuclei beyond the proton drip line (the boundary of bound nuclear existence on the proton-rich side), in which two protons tunnel through the Coulomb barrier either sequentially or as a correlated pair; the lightest examples, 6^{6}Be (α+p+p\alpha+p+p) and 16^{16}Ne (14^{14}O+p+p+p+p), are prototypes of open three-body quantum systems [Pfützner et al., Rev. Mod. Phys. 84, 567 (2012)]. In such systems a resonance is an eigenstate with complex eigenenergy ER−iΓ/2E_{R}-i\Gamma/2, where ERE_{R} is the resonance position above the threshold and Γ\Gamma its decay width.

Because calculations with scattering boundary conditions are expensive, Oishi and Kimura [Phys. Rev. C 112, 014305 (2025)] used the analytic continuation in coupling constant (ACCC) method, which has a long history in few-body resonance physics [e.g. Tanaka, Suzuki and Varga, Phys. Rev. C 56, 562 (1997); Aoyama, Phys. Rev. C 68, 034313 (2003)]. One adds an attractive auxiliary potential λVex\lambda V_{ex} to the three-body Hamiltonian H^3B=h^(r1)+h^(r2)+H^pp\hat{H}_{3B}=\hat{h}(r_{1})+\hat{h}(r_{2})+\hat{H}_{pp}, where h^\hat{h} is a core-proton Woods-Saxon-plus-Coulomb single-particle Hamiltonian (the Woods-Saxon potential is the standard phenomenological nuclear mean field with central and spin-orbit terms) and H^pp\hat{H}_{pp} contains a proton-proton pairing interaction vppv_{pp}, scaled in the study by a factor fppf_{pp}. For λ≥λ0\lambda\ge\lambda_{0} the system is bound; the bound energies E(λ)≃−w2(λ)E(\lambda)\simeq-w^{2}(\lambda) are fitted by a Padé approximant (a ratio of two polynomials) in x=λ−λ0x=\sqrt{\lambda-\lambda_{0}}, and the fit is evaluated at the physical point λ=0\lambda=0, where xx is purely imaginary, to obtain the complex resonance energy. Only bound-state solvers are needed, which makes ACCC a candidate low-cost tool for systematic surveys along the proton drip line. The same authors had earlier treated 6^{6}Be and 16^{16}Ne by an explicit time-dependent method [Phys. Rev. C 96, 044327 (2017); Phys. Rev. C 111, 044319 (2025)].

The benchmark on 6^{6}Be reproduces both the time-dependent results and the measured 2p2p energy. 16^{16}Ne is the hard case. Experimentally, the ground state 01+0^{+}_{1} lies 1.466(20)1.466(20) MeV above the 14^{14}O+2p+2p threshold with a width bounded only by Γ≤0.08\Gamma\le 0.08 MeV due to detector resolution [Brown et al., Phys. Rev. Lett. 113, 232501 (2014)], a 2+2^{+} state lies at 3.16(2)3.16(2) MeV [Brown et al., Phys. Rev. C 92, 034329 (2015); Marganiec et al., Eur. Phys. J. A 51, 9 (2015)], and a 02+0^{+}_{2} state is reported near 2.1(2)2.1(2) MeV. The core-proton subsystem 15^{15}F contains low-lying s1/2s_{1/2} and d5/2d_{5/2} single-proton resonances whose experimental parameters, especially the broad s1/2s_{1/2}, remain uncertain [Guo et al., Phys. Rev. C 72, 034312 (2005)]. In the ACCC calculation the 01+0^{+}_{1} and 02+0^{+}_{2} energies and widths exhibit an avoided crossing as functions of the pairing factor fppf_{pp}: at fpp=0f_{pp}=0 the lower 0+0^{+} is s1/2s_{1/2}-dominant and the upper d5/2d_{5/2}-dominant, while for fpp>0.8f_{pp}>0.8 the compositions swap; for 0.6≤fpp≤0.80.6\le f_{pp}\le 0.8 the two ACCC solutions cannot be distinguished at all. The physical 2+2^{+}–01+0^{+}_{1} gap (the Thomas-Ehrman shift, an isospin-symmetry-breaking level displacement in mirror nuclei caused by spatially extended weakly bound proton orbitals of low angular momentum) is reproduced only with an angular-momentum-dependent pairing factor fpp(J=2)=0.7f_{pp}(J=2)=0.7, and the known 22+2^{+}_{2} resonance near 7.577.57 MeV is missed entirely, apparently because its width Γ≳1\Gamma\gtrsim 1 MeV exceeds the Padé fit's reliable range, stated by the authors as 10−4≲Γ≲110^{-4}\lesssim\Gamma\lesssim 1 MeV (the lower limit set by the three-body solver's accuracy, the upper by the Padé approximation itself). Excluding the s1/2s_{1/2} channel altogether shifts the computed ground-state energy to 1.7941.794 MeV, far from data, showing strong channel-content sensitivity. Independent Gamow shell model calculations (a shell model built on a basis of resonant and continuum single-particle states) find a 16^{16}Ne ground state dominated by the s1/2s_{1/2} configuration — in tension with the d5/2d_{5/2}-dominant ACCC ground state at fpp=1f_{pp}=1 — and also fail to reproduce the 2+2^{+} excitation energy [Chen et al., Phys. Lett. B 879, 140601 (2026); Li et al., Phys. Rev. C 111, 014302 (2025)], indicating that the relevant physics is not yet under control in any model.

Problem Statement

The problem is to make the ACCC three-body treatment of two-proton-emitting resonances quantitatively reliable, with the 0+0^{+} doublet of 16^{16}Ne as the decisive benchmark. Following the issues explicitly left open by Oishi and Kimura (Phys. Rev. C 112, 014305 (2025)), a complete answer must jointly deliver:

(1) A quantitative microscopic account of the avoided crossing of the 01+0^{+}_{1} and 02+0^{+}_{2} resonances of 16^{16}Ne as the proton-proton pairing strength fppf_{pp} is varied — in particular, how much of the level mixing is attributable to s1/2s_{1/2}–d5/2d_{5/2} channel coupling and how much to coupling with non-resonant continuum channels via vppv_{pp} — replacing the present 'one possible reason' remark, and clarifying the regime 0.6≤fpp≤0.80.6\le f_{pp}\le 0.8 where the two 0+0^{+} solutions cannot be distinguished.

(2) A determination of whether the core-proton potential VcpV_{cp} and the proton-proton interaction vppv_{pp} can be simultaneously optimized within the three-body model so that the computed 2p2p energies and widths of 16^{16}Ne — including the 2+2^{+}–01+0^{+}_{1} Thomas-Ehrman gap and the keV-scale computed 01+0^{+}_{1} width, for which the experimental information is the upper bound Γ≤0.08\Gamma\le 0.08 MeV — agree with experiment, or a demonstration that they cannot.

(3) An extension, or a rigorous delimitation, of the reliable width range of the ACCC/Padé procedure beyond the present 10−4≲Γ≲110^{-4}\lesssim\Gamma\lesssim 1 MeV window (for example by stabilized or higher-order fits, alternative continuation kernels, or hybridization with the complex scaling or time-dependent methods). This item is resolved either by an extension sufficient at least to treat the broad 22+2^{+}_{2} resonance of 16^{16}Ne near 7.67.6 MeV and the fpp<0.4f_{pp}<0.4 regime of 6^{6}Be, or by a rigorous delimitation showing that the ACCC/Padé procedure cannot reach those two cases.

These deliverables support one objective — establishing whether, and over what range of resonance widths, the low-cost bound-state-only ACCC solver yields quantitatively trustworthy 2p2p resonance parameters — and the problem is complete when all three items are answered jointly: items (1) and (2) for 16^{16}Ne, and item (3) for both the 6^{6}Be and 16^{16}Ne benchmark systems.

References

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