Universal Properties of Weakly Bound Two-Neutron Halo Nuclei

We construct an effective field theory of a two-neutron halo nucleus in the limit where the two-neutron separation energy $B$ and the neutron-neutron two-body virtual energy ${\epsilon }_n$ are smaller than any other energy scale in the problem, but the scattering between the core and a single neutron is not fine-tuned, and the Efimov effect does not operate. The theory has one dimensionless coupling which formally runs to a Landau pole in the ultraviolet. We show that many properties of the system are universal in the double fine-tuning limit. The ratio of the mean-square matter radius and charge radius is found to be $⟨r_m^2⟩/⟨r_c^2⟩=Af({\epsilon }_n/B)$, where $A$ is the mass number of the core and $f$ is a function of the ratio ${\epsilon }_n/B$ which we find explicitly. In particular, when $B\gg {\epsilon }_n$, $⟨r_m^2⟩/⟨r_c^2⟩=\frac{2}{3}A$. The shape of the $E1$ dipole strength function also depends only on the ratio ${\epsilon }_n/B$ and is derived in explicit analytic form. We estimate that for the ${}^{22}\mathrm{C}$ nucleus higher-order corrections to our theory are of the order of 20% or less if the two-neutron separation energy is less than 100 keV and the $s$-wave scattering length between a neutron and a ${}^{20}\mathrm{C}$ nucleus is less than 2.8 fm.

Figure 1

The self-energy of the dimer.

Figure 2

The Feynman diagram determining the charge form factor of the halo nucleus. The double line represents the halo nucleus, the single line the core, and the dotted line the neutron dimer, whose propagator is given in Eq. (4).

Figure 3

The Feynman diagram determining the neutron form factor of the halo nucleus.

Figure 4

The effective vertex of the dimer-photon coupling.

Figure 5

The Feynman diagram for the $E1$ dipole strength function.

Figure 6

The $E1$ dipole strength function, plotted as a function of $\omega /B$, for $B=3{\epsilon }_n$, $B={\epsilon }_n$, and $B=\frac{1}{3}{\epsilon }_n$. The functions are so normalized by $N$ that the area under the theoretical curve, extended to $\omega /B=\infty$, is 1.