Halo effective field theory (Halo EFT) describes weakly bound halo nuclei through an expansion in the ratio of the halo momentum to the core momentum. The two-body S-wave proton-halo Halo EFT with explicit Coulomb interactions and range corrections was completed by Ryberg, Forssen, Hammer, and Platter, who posed in their conclusions -- but did not answer -- whether a two-proton halo (two protons plus a core, with resonant S-wave proton-core short-range interactions and full Coulomb repulsion) could in principle support an effective three-body bound state, and how its spectrum compares to two-neutron halo nuclei bound by resonant S-wave interactions. This problem asks for the construction of the three-body Halo EFT for the proton-proton-core system (renormalized, with a consistent power counting for Coulomb contributions and the required three-body force) and the determination of its bound-state spectrum. No controlled three-body EFT treatment of this halo topology exists in the published literature.

Background

Near the drip lines, nuclear binding weakens until the last nucleons are only barely attached. Halo nuclei form in this regime: one or more valence nucleons tunnel far outside a tightly bound core, so the spatial extent of the halo greatly exceeds the range of the nuclear force while the valence-core separation energy stays small compared with core excitation energies. When two valence nucleons are present the nucleus is a three-body system (core + nucleon + nucleon); it is called Borromean if the three-body system is bound although no two-body subsystem is bound.

Halo effective field theory (Halo EFT, also called cluster EFT) exploits this scale separation. The core and the valence nucleons are promoted to point-like fields, short-range strong interactions are contact operators organized in powers of the small ratio of the halo momentum to the core momentum (the inverse core radius), and observables are computed with systematic error estimates. Long-range Coulomb interactions between charged clusters must be kept explicit. For a proton plus a charged core this introduces the Coulomb momentum kC=Zcαemμk_C=Z_c\alpha_{\rm em}\mu (with ZcZ_c the core charge, αem\alpha_{\rm em} the fine-structure constant, and μ\mu the proton-core reduced mass); when the Sommerfeld parameter η=kC/p\eta=k_C/p is of order one or larger, Coulomb photon exchange must be resummed inside the proton-core propagator. Ryberg, Forssen, Hammer, and Platter built this two-body S-wave proton-halo EFT including effective-range corrections, computed the charge radius to next-to-leading order and the low-energy radiative-capture cross section (conventionally parameterized by the astrophysical S-factor, which factors out the exponential Coulomb-barrier suppression of the cross section) to fifth order, and confronted it with data for 16^{16}O(p,γ)17(p,\gamma)^{17}F∗^*. They also showed that the existence of proton halos is suppressed relative to neutron halos by the need for two fine tunings in the underlying theory.

For two valence neutrons the corresponding three-body Halo EFT is well developed. If the neutron-core S-wave scattering length is resonant (large compared with the interaction range), the Efimov effect can occur: for three particles with resonant pairwise short-range S-wave interactions, an effective attraction proportional to 1/R21/R^2 in hyperspherical radius RR supports a tower of three-body bound states obeying a geometric spectrum; for finite scattering length only a few universal three-body states survive, and renormalization requires a three-body contact interaction whose coupling runs log-periodically (a limit cycle). Two-neutron halos such as 6^6He and 20^{20}C have been treated in this framework, and the neutron-side program continues to be refined.

The charged analogue is open. In a system of two protons plus a core, the Coulomb repulsion between the valence protons falls off as 1/r1/r, more slowly than the Efimov 1/R21/R^2 attraction, so scale invariance is broken at large distances and only a finite window between the range of the nuclear force and the Coulomb Bohr length can support Efimov-like attraction. Only qualitative estimates of this window exist; a controlled three-body EFT treatment of the halo topology (resonant S-wave proton-core pairs plus charged protons) has not been carried out. Experimentally, 17^{17}Ne (15^{15}O+p+p+p+p) is the only known bound two-proton-halo candidate, and unbound two-proton emitters such as 6^6Be, 12^{12}O, 16^{16}Ne, and 19^{19}Mg realize the same dynamics as resonances, which is why the three-body question also bears on two-proton radioactivity.

Problem Statement

Construct the three-body Halo EFT for a two-proton halo: a core of charge ZcZ_c plus two protons, with resonant S-wave short-range proton-core interactions, the proton-proton pair described by its 1S01S_0 short-range interaction plus Coulomb repulsion, and proton-core Coulomb included -- together with the power counting that organizes the Coulomb contributions at each order and the three-body contact interaction required for renormalization. Within this EFT, determine whether the system can in principle support an effective three-body bound state (and for which ranges of the Coulomb-modified proton-core scattering parameters and the three-body parameter), and characterize the resulting spectrum -- number of bound states, binding energies, and the mean-square matter (halo) radius -- in explicit comparison with the two-neutron halo analogue bound by resonant S-wave interactions. These parts are mandatory and jointly required: a complete resolution must address the construction of the renormalized EFT with its Coulomb power counting and three-body force, the existence determination with its parameter ranges, and the spectrum characterization with the two-neutron comparison, and resolving any subset on its own does not resolve the task. This is the question posed in the conclusions of Ryberg, Forssen, Hammer, and Platter ('Range corrections in Proton Halo Nuclei', Ann. Phys. 2016): 'whether a two-proton halo could in principle support an effective three-body bound state and how the spectrum of this system compares to two-neutron halo nuclei bound by resonant S-wave interactions', which they also note is relevant for studies of two-proton radioactivity.

References

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