Renormalizing Two-Neutron Halo Nuclei Without Neutron-Core Interaction

TL;DR

The paper addresses how to renormalize Hongo–Son's Halo EFT for two-neutron halo nuclei in which neutron-core interactions are subleading. It shows that two UV divergences in the trimer self-energy necessitate a second renormalization condition, enabling independent predictions of both the matter and charge radii and revealing a universal radius ratio $\langle r_m^2\rangle/\langle r_c^2\rangle=K_m/K_c$ that depends only on $a$, $B$, and $A$. The authors compare the resulting radii to standard Halo EFT predictions, finding good agreement in the limit of weak neutron-core interaction, and identify a Landau pole that constrains the ultraviolet cutoff, thereby limiting the theory’s range of applicability for certain halo nuclei. They also derive a fully renormalized three-to-three neutron-neutron-core scattering amplitude and discuss its cut structure, linking the Hongo–Son EFT to conventional Halo EFT and providing a concrete procedure to predict radii from a minimal input. Overall, the work clarifies the renormalization structure of this EFT, demonstrates its relation to Halo EFT, and offers practical prescriptions for predicting halo radii and related observables.

Abstract

We consider the Effective Field Theory (EFT) proposed by Hongo and Son to describe two-neutron halo nuclei where the neutron-core interaction is subleading. In this EFT, the ratio of the mean-square matter radius and charge radius is universal in so far that it only depends on the two-neutron separation energy of the nucleus and the neutron-neutron scattering length. By investigating the divergence structure of this theory, we find that one further renormalization condition is required to predict both radii separately. Our renormalization scheme uses one of the mean square radii or the scattering amplitude as input. We use Hongo and Son's theory to calculate the matter radii of the two-neutron halo nuclei $^{11}$Li, $^{14}$Be, $^{17}$B, $^{19}$B, and $^{22}$C and compare to the values obtained with standard Halo EFT. In this comparison we use both the physical value of the neutron-core scattering length and rescaled values. We observe good convergence against Hongo and Son's results for the case of a negligible neutron-core interaction. Similar agreement for the radii is also found in the case of the halo nucleus $^6$He, where the $nc$ interaction is in the p-wave. Our renormalization scheme makes the restriction in the ultraviolet cutoff range from the Landau pole explicit. We calculate the position of the Landau pole for various halo nuclei. In all cases the Landau pole restricts the cutoff to rather low values. Finally, we derive an explicit expression for the three-to-three neutron-neutron-core scattering amplitude and discuss its cut structure.

Figures (7)

• Figure 1: Self-energy diagram of the trimer field.
• Figure 2: Diagrams for the charge form factor of the halo nucleus. All external propagators are amputated. For the detailed kinematics of the right-hand diagram see main text.
• Figure 3: The lines show Hongo and Son's results for the relation between RMS matter and charge radius for the different halo nuclei. Except of $^6$He with the LO $nc$ interaction in a p-wave, all other halos have the $nc$ interaction in the s-wave. This specialty is indicated by a dashed line in the case of $^6$He. The symbols depict results from standard Halo EFT. The color encodes the nucleus using the same color scheme as the lines, while the shape encodes the ratio of the actually used $nc$ scattering length $a_{nc}$ and the physical $nc$ scattering length $a_{nc}^{(0)}$. The right panel shows a zoomed-in version of the left panel. Error bars depicting estimated uncertainties related to numerics and the extraction of the radii are also included. In most cases, these are too small to be visible.
• Figure 4: Landau pole values for ^22C as a function of $\sqrt{\langle r^2_\mathrm{m}\rangle}$ for different values of $B$. The EFT scales for this case are added as horizontal lines. Note that $M^\mathrm{nnc}_\mathrm{lo}$ depends on $B$. We indicate this with two horizontal lines at $B=100keV$ and $B=1keV$ as an upper and lower bound, respectively.
• Figure 5: Diagram for the scattering amplitude in the center-of-mass frame for neutron-neutron-core scattering. The gray blob denotes the dressed trimer propagator and is evaluated at rest.