Constructing the full next-to-leading-order Hamiltonian of the short-range effective theory for single-neutron halo nuclei with a deformed core ($^{11}$Be and $^{17}$C)

Kubushishi and Phillips (arXiv:2606.21547, 2026) established a leading-order (LO) short-range effective theory for deformed s-wave one-neutron halo nuclei, in which the halo neutron is coupled to the rotational degrees of freedom of a deformed core (the Bohr–Mottelson particle-rotor model emerges at LO). Applied to $^{11}$Be and $^{17}$C, LO renormalizes binding energies and reproduces positive-parity spectra and Coulomb-breakup observables fairly well, but the $^{11}$Be $5/2^+$ and $3/2^+$ resonance widths and the $d$-wave asymptotic normalization coefficients (ANCs) remain strongly regulator-dependent, and the rotor expansion parameter $\xi^2 \approx 0.56$–$0.60$ warns of slow convergence. The open problem is to construct the complete next-to-leading-order (NLO) Hamiltonian — the full operator set at the first subleading order of the combined halo ($Q$) and rotor ($\xi$) power counting, unified by $\xi \sim Q$, which the source identifies with the two-derivative $s$-wave contact operators, the $p$-wave final-state interactions in the continuum, and the two-derivative hyperfine operator promoted from N2LO to NLO — and to apply it to $^{11}$Be and $^{17}$C, determining whether the regulator dependence of the $^{11}$Be resonance widths and of the $d$-wave ANCs is removed at NLO and whether convergence holds order by order with quantified truncation errors, thereby establishing (or refuting) this deformed-halo effective field theory (EFT) as a systematic expansion. Intrinsic $d$-wave contact operators (which the source's counting places at N5LO) and the $O(\xi^2)$ non-rigidity corrections to the rigid rotor (which the source defers to a separate future work) lie beyond the NLO completion boundary. As of August 2026 no published work constructs this NLO; the authors state the construction is in progress.

Background

Halo nuclei are weakly bound systems in which one or more valence nucleons extend far beyond a compact nuclear core; the archetypes here are $^{11}$Be (${}^{10}$Be core $+$ neutron, with a $1/2^+$ ground state) and $^{17}$C (${}^{16}$C core $+$ neutron, with $3/2^+$ and $1/2^+$ low-lying s-wave-dominated states). Because the halo radius greatly exceeds both the core size and the range of the nuclear force, such systems are described by Halo effective field theory (Halo EFT), an expansion in the small ratio $Q$ of the halo momentum to the core's internal scale, with neutron–core nuclear dynamics represented by contact interactions (short-range interactions with no spatial extent) plus effective-range parameters (the coefficients of the effective-range expansion, which encode the leading corrections to strictly zero-range scattering); the standard framework is reviewed by Hammer, Ji and Phillips (2017). Key observables are binding energies, resonance widths, asymptotic normalization coefficients (ANCs — the amplitudes of the bound-state wave function's tail, which fix reaction rates at low energy), and Coulomb-breakup cross sections dominated by electric dipole ($B(E1)$) strength.

Standard Halo EFT treats the core as inert and spherical. However, the $^{10}$Be and $^{16}$C cores of $^{11}$Be and $^{17}$C are deformed and exhibit low-lying rotational bands with a $0^+$ ground state and a $2^+_1$ first excited state, so the halo neutron couples to the core's rotation — the physics of the classic particle-rotor model of Bohr and Mottelson. Effective field theories of nuclear rotations have been developed by Papenbrock and Weidenmüller (2020) and by Alnamlah, Coello Pérez and Phillips (2021), which organize corrections to the rigid rotor in an expansion parameter $\xi$: the ratio of the angular velocity associated with the probed rotational scale to that of the first non-rotational core excitation, so that $\xi^2$ is estimated as the ratio of the $2^+_1$ energy to the energy of the first non-rotational state.

Kubushishi and Phillips (2026) merged the two expansions into a short-range effective theory for deformed s-wave halos with the combined power counting in $Q$ and $\xi$, unified by taking $\xi \sim Q$. At LO the theory reduces to the particle-rotor model with contact neutron–core interactions and a hyperfine $\vec\sigma\cdot\vec I\,\delta^{(3)}(r)$ coupling of the neutron spin to the rotor angular momentum. For both $^{11}$Be and $^{17}$C, computing the positive-parity spectrum up to the core$(2^+_1)$–neutron threshold for several regulator choices (the regulator is the cutoff function that suppresses the short-distance part of the contact interactions; a properly renormalized calculation must yield predictions independent of it), they showed the LO operator set renormalizes all binding energies, describes the spectra accurately, and yields Coulomb-breakup $B(E1)$ observables in fair agreement with data; adding a single NLO two-derivative operator makes the s-wave ANCs regulator-stable and removes most of the roughly 30 percent regulator variation of the LO dissociation cross section. Three shortcomings remain. First, the widths of the $^{11}$Be $5/2^+$ and $3/2^+$ resonances and the $d$-wave ANCs are strongly regulator-dependent at LO, indicating that higher-order short-range operators are needed to control them; in the source's counting, intrinsic $d$-wave contact interactions enter only at N5LO. Second, only LO in the rotor expansion was kept, while $\xi^2$ is large — approximately 0.56 for the $^{10}$Be core and 0.60 for $^{16}$C — so subleading non-rigidity corrections (deviations of the core from a rigid rotor, e.g. vibration–rotation coupling), which the source defers to a separate future manuscript, are expected to be significant. Third, no order-by-order truncation-error analysis of the full theory exists yet, so it is not established that the expansion is systematic.

Problem Statement

Construct the complete next-to-leading-order (NLO) Hamiltonian of the short-range effective theory for deformed s-wave one-neutron halo nuclei — that is, the full set of operators at the first subleading order, $O(Q)$, of the combined power counting in the halo expansion parameter $Q$ and the rotor expansion parameter $\xi$, unified by $\xi \sim Q$. The operators the source explicitly places at this order are the two-derivative $s$-wave contact operator $C_2(p^2+p'^2)$, the $p$-wave final-state interactions in the continuum, and the two-derivative hyperfine operator $C_2^{hf}(p^2+p'^2)\,\vec I\cdot\vec j$, which the LO promotion of the hyperfine coupling moves from N2LO to NLO. Intrinsic $d$-wave contact operators, which the source's counting assigns to N5LO, and the $O(\xi^2)$ non-rigidity corrections to the leading-order rigid-rotor Hamiltonian, which the source defers to a separate future work, are not part of the NLO completion boundary. Apply the resulting Hamiltonian to $^{11}$Be and $^{17}$C, fitting its low-energy constants (the free coefficients of the contact operators, fixed to experimental data) and computing, for several regulator choices, the positive-parity spectra up to the core$(2^+_1)$–neutron threshold, the $5/2^+$ and $3/2^+$ resonance widths of $^{11}$Be, the $s$-wave and $d$-wave asymptotic normalization coefficients (for $^{17}$C, those of its bound $3/2^+$, $1/2^+$ and $5/2^+$ states, since its $7/2^+$ and $9/2^+$ resonances would require the core $4^+$ state and lie outside the source's model space), and the Coulomb-breakup/$B(E1)$ observables. The application must determine whether the effective theory is a systematic expansion for these two nuclei, specifically: (i) whether, at the complete NLO order, the regulator dependence of the $^{11}$Be $5/2^+$ and $3/2^+$ resonance widths and of the $d$-wave ANCs is removed within the quoted NLO truncation uncertainty, given that the source attributes their renormalization to higher-order short-range operators beyond the single two-derivative contact; and (ii) whether the spectra, $s$-wave ANCs, and breakup observables improve, or remain statistically consistent, order by order, with truncation errors quantified through the combined $Q$–$\xi$ counting. A negative answer counts only after the complete NLO operator set under the source's counting has been enumerated and implemented and a truncation-error analysis performed: such a demonstration that no finite NLO operator set renormalizes these observables at this order, or that the expansion fails to converge order by order, would answer the same question in the negative. The extensions the source lists — applying the EFT to $^{19}$C, extending it to $p$-wave halos such as $^{31}$Ne, and adapting it to proton halos with deformed cores such as $^{22}$Al — are related but logically separate follow-on problems and are not part of this problem's completion boundary.

References

1. Live-Palm Kubushishi and Daniel R. Phillips, "A short-range effective theory for single-neutron halo nuclei with a deformed core", arXiv:2606.21547 (2026), DOI 10.48550/arxiv.2606.21547, https://arxiv.org/abs/2606.21547
2. Live-Palm Kubushishi and Pierre Capel, "Exploring core excitation in halo nuclei using halo effective field theory: an application to the bound states of $^{11}$Be", arXiv:2507.13585 (2025), DOI 10.48550/arxiv.2507.13585, https://arxiv.org/abs/2507.13585
3. L.-P. Kubushishi and P. Capel, "Coulomb and nuclear breakup of ${}^{11}$Be using an effective-potential theory description at N$^2$LO", Physical Review C 111, 054618 (2025), DOI 10.1103/PhysRevC.111.054618, https://doi.org/10.1103/PhysRevC.111.054618 (preprint arXiv:2406.10168, titled "Study of the Coulomb and nuclear breakup of $^{11}$Be using a Halo-EFT description at N$^2$LO", https://arxiv.org/abs/2406.10168)
4. P. Capel, D. R. Phillips, and H.-W. Hammer, "Dissecting reaction calculations using halo effective field theory and ab initio input", Physical Review C 98, 034610 (2018), DOI 10.1103/PhysRevC.98.034610, https://doi.org/10.1103/PhysRevC.98.034610 (preprint arXiv:1806.02712, titled "Dissecting reaction calculations using Halo EFT and ab initio input", https://arxiv.org/abs/1806.02712)
5. Pierre Capel, Daniel R. Phillips, Andrew Andis, Mirko Bagnarol, Behnaz Behzadmoghaddam, Francesca Bonaiti, Rishabh Bubna, Ylenia Capitani, Pierre-Yves Duerinck, Victoria Durant, Niklas Döpper, Aya El Boustani, Roland Farrell, Maurus Geiger, Michael Gennari, Nitzan Goldberg, Jakub Herko, Tanja Kirchner, Live-Palm Kubushishi, Zhen Li, Simone S. Li Muli, Alexander Long, Brady Martin, Kamyar Mohseni, Imane Moumene, Nicola Paracone, Elad Parnes, Beatriz Romeo, Victor Springer, Isak Svensson, Oliver Thim, and Nuwan Yapa, "Effective field theory analysis of the Coulomb breakup of the one-neutron halo nucleus $^{19}$C", The European Physical Journal A 59, 273 (2023), DOI 10.1140/epja/s10050-023-01181-7, https://doi.org/10.1140/epja/s10050-023-01181-7 (preprint arXiv:2301.06444, https://arxiv.org/abs/2301.06444)
6. Wael Elkamhawy and Hans-Werner Hammer, "Halo EFT for $^{31}$Ne in a spherical formalism", Journal of Physics G: Nuclear and Particle Physics 50, 025103 (2023), DOI 10.1088/1361-6471/aca923, https://doi.org/10.1088/1361-6471/aca923 (preprint arXiv:2207.02838, https://arxiv.org/abs/2207.02838)
7. I. K. Alnamlah, E. A. Coello Pérez, and D. R. Phillips, "Effective field theory approach to rotational bands in odd-mass nuclei", Physical Review C 104, 064311 (2021), DOI 10.1103/PhysRevC.104.064311, https://doi.org/10.1103/PhysRevC.104.064311 (preprint arXiv:2011.01083, titled "An effective field theory approach to fermionic rotational bands in nuclei", https://arxiv.org/abs/2011.01083)
8. T. Papenbrock and H. A. Weidenmüller, "Effective field theory for deformed odd-mass nuclei", Physical Review C 102, 044324 (2020), DOI 10.1103/PhysRevC.102.044324, https://doi.org/10.1103/PhysRevC.102.044324 (preprint arXiv:2005.11865, https://arxiv.org/abs/2005.11865)
9. H.-W. Hammer, C. Ji, and D. R. Phillips, "Effective field theory description of halo nuclei", Journal of Physics G: Nuclear and Particle Physics 44, 103002 (2017), DOI 10.1088/1361-6471/aa83db, https://doi.org/10.1088/1361-6471/aa83db
10. Live-Palm Kubushishi and Pierre Capel, "Insensitivity of the Coulomb breakup of halo nuclei to spectroscopic factors", Physics Letters B 879, 140694 (2026), DOI 10.1016/j.physletb.2026.140694, https://doi.org/10.1016/j.physletb.2026.140694 (preprint arXiv:2603.10339, https://arxiv.org/abs/2603.10339)
11. Matthias Göbel, Hans-Werner Hammer, and Daniel R. Phillips, "Finite-range EFT for the $E1$ strength distribution of ${}^{6}$He", arXiv:2606.32037 (2026), DOI 10.48550/arxiv.2606.32037, https://arxiv.org/abs/2606.32037