Uncertainty-quantified embedding of the three-body ${}^{22}\mathrm{C}$ model into reaction theory: direct comparison with measured interaction cross sections and the conflicting matter-radius extractions

The neutron-rich nucleus ${}^{22}\mathrm{C}$ is among the heaviest confirmed two-neutron halo nuclei, yet its basic properties are undetermined: the two-neutron separation energy $S_{2n}$ is bounded only to $\lesssim 0.5$ MeV, the ${}^{21}\mathrm{C}$ continuum quantities that fix the core-neutron interaction are poorly measured, and two interaction-cross-section measurements yield matter radii differing by about $2$ fm ($5.4\pm0.9$ fm on a hydrogen target vs $3.44\pm0.08$ fm on a carbon target). A 2026 Bayesian three-body calculation (McGlynn and Hebborn) calibrated the ${}^{20}\mathrm{C}$-$n$ interaction, propagated uncertainties to ${}^{22}\mathrm{C}$ observables, and inferred $S_{2n}\lesssim 0.35$ MeV with an $(s_{1/2})^2$-dominated ground state — but that inference compared theoretical hyperradii to radii derived from cross sections whose quoted uncertainties omit reaction-model systematics. The step that work identifies as required is to fold the uncertainty-quantified three-body predictions into a reaction calculation of the measured cross sections themselves, with fully propagated structure and reaction uncertainties, and establish what the existing data jointly constrain about $S_{2n}$, the matter radius, and the halo configuration — including whether the radius discrepancy is structural or a reaction-analysis artifact.

Background

A two-neutron halo nucleus is a system in which two neutrons orbit a compact inert core in spatially diffuse wavefunctions; when neither the neutron-core pair nor the neutron pair is bound but the three-body system is, the nucleus is called Borromean. The carbon isotope ${}^{22}\mathrm{C}$ (a ${}^{20}\mathrm{C}$ core plus two neutrons) is one of the heaviest confirmed examples, and its properties are exceptionally poorly constrained. The two-neutron separation energy $S_{2n}$, the energy cost to remove the two halo neutrons, is known only as a bound: the AME2020 mass evaluation gives $0.035\pm0.020$ MeV while a direct time-of-flight mass measurement gives $-0.14\pm0.46$ MeV, both consistent with $0\le S_{2n}\lesssim 0.5$ MeV. The relevant continuum of the unbound nucleus ${}^{21}\mathrm{C}$ is also barely characterized: proton-removal data found no low-lying state and only constrained the ${}^{20}\mathrm{C}$+$n$ $s$-wave scattering length (the zero-energy intercept parameter of the $s$-wave phase shift) to $|a_s|<2.8$ fm (Mosby et al. 2013), while the positions of the low-lying $d$-wave resonances rest on sparse invariant-mass spectra. Two measurements of the total interaction cross section of ${}^{22}\mathrm{C}$ — on a hydrogen target (Tanaka et al. 2010) and on a carbon target (Togano et al. 2016, cross section analyzed with a four-body Glauber model, i.e., a high-energy eikonal multiple-scattering description) — lead, when inverted through reaction models, to matter radii of $5.4\pm0.9$ fm and $3.44\pm0.08$ fm respectively, a discrepancy of about $2$ fm. A careful Glauber analysis with full multiple scattering (Nagahisa and Horiuchi 2018) could reproduce the carbon-target result but showed that the two measured cross sections cannot be simultaneously described within that framework, so the discrepancy is not a trivial normalization issue. Effective-field-theory analyses at leading order (Acharya, Ji, and Phillips 2013) had already emphasized the strong model dependence of any radius-based inference. In this situation McGlynn and Hebborn (2026) built the first uncertainty-quantified three-body model of ${}^{22}\mathrm{C}$: within a hyperspherical-harmonics (a three-body coordinate expansion in the hyper-radius and two hyperangles) plus $R$-matrix (a method matching internal bound and scattering wavefunctions to asymptotic channel states) formalism, they Bayesian-calibrated a Woods-Saxon ${}^{20}\mathrm{C}$-$n$ potential on the scarce ${}^{21}\mathrm{C}$ information, propagated the posterior (315 samples) to ${}^{22}\mathrm{C}$ bound-state, scattering, and dipole observables for several assumed values of $S_{2n}$, and found that only calculations with $S_{2n}\lesssim 0.35$ MeV and $s$-wave-dominated ground states are compatible with the smaller, carbon-target-derived radius. Crucially, the authors themselves flag the missing link: their comparison used radii derived from the cross sections by other analyses, whose uncertainties exclude the reaction-model systematics, so the compatibility conclusion — and the resolution of the radius discrepancy — requires the uncertainty-quantified structure predictions to be embedded in a reaction calculation that predicts the measured cross sections directly.

Problem Statement

Embed the Bayesian uncertainty-quantified ${}^{20}\mathrm{C}+n+n$ three-body model of ${}^{22}\mathrm{C}$ (McGlynn and Hebborn, arXiv:2604.06139) into a reaction framework, and by comparing directly with the measured interaction cross sections — not with radii derived from them — determine, with fully propagated uncertainties: (i) the posterior constraints that the existing cross-section data (hydrogen-target and carbon-target measurements) place on the two-neutron separation energy $S_{2n}$, the rms matter radius, and the $s$-wave versus $d$-wave content of the ${}^{22}\mathrm{C}$ ground state, reported both from each measurement separately and from a joint fit of the two; and (ii) whether the two measurements can be jointly reproduced within a consistent reaction treatment — i.e., whether the roughly $2$ fm discrepancy between the derived matter radii reflects ${}^{22}\mathrm{C}$ structure, reaction-analysis systematics, or both. The calculation must propagate the posterior distribution of the core-neutron interaction (the Bayesian calibration on ${}^{21}\mathrm{C}$ information) through the reaction observables, treat the reaction-model uncertainties explicitly, and quantify each source's contribution to the error budget of the predicted cross sections. Completion is defined by (i) and (ii); a quantitative identification of which experimental input ($S_{2n}$, the ${}^{21}\mathrm{C}$ $s$-wave scattering length, or the $d_{3/2}$ resonance) most reduces the remaining uncertainty is a natural but secondary deliverable.

References

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