${}^6\mathrm{He}$ nucleus in halo effective field theory

Background: In recent years properties of light rare isotopes have been measured with high accuracy. At the same time, the theoretical description of light nuclei has made enormous progress, and properties of, e.g., the helium isotopes can now be calculated ab initio. These advances make those rare isotopes an ideal testing ground for effective field theories (EFTs) built on cluster degrees of freedom.

Purpose: Systems with widely separated intrinsic scales are well suited to an EFT treatment. The Borromean halo nucleus ${}^6\mathrm{He}$ exhibits such a separation of scales. In this work an EFT in which the degrees of freedom are the valence neutrons ($n$) and an inert ${}^4\mathrm{He}$-core ($\alpha$) is employed. The properties of ${}^6\mathrm{He}$ can then be calculated using the momentum-space Faddeev equations for the $\alpha nn$ bound state to obtain information on ${}^6\mathrm{He}$ at leading order (LO) within the EFT.

Results: The $nn$ virtual state and the ${}^2{P}_{3/2}$ resonance in ${}^5\mathrm{He}$ give the two-body amplitudes which are input to our LO three-body Halo EFT calculation. We find that without a genuine three-body interaction the two-neutron separation energy $S_{2n}$ of ${}^6\mathrm{He}$ is strongly cutoff dependent. We introduce a $nn\alpha$ “three-body” operator which renormalizes the system, adjusting its coefficient to reproduce the $S_{2n}$ of ${}^6\mathrm{He}$. The Faddeev components are then cutoff independent for cutoffs of the order of, and above, the breakdown scale of the Halo EFT.

Conclusions: As in the case of a three-body system where only resonant $s$-wave interactions are present, one three-body input is required for the renormalization of the EFT equations that describe ${}^6\mathrm{He}$ at LO. However, in contrast to the $s$-wave-only case, the running of the LO $nn\alpha$ counterterm does not exhibit discrete scale invariance, due to the presence of the $p$-wave $n\alpha$ interaction.

Figure 1

The Faddeev equations for the ${}^6\mathrm{He}$ bound-state problem. The single dashed (or solid) line denotes the $\alpha$ (or $n$) one-body propagator. The thick black and shaded lines represents the $n\alpha$ and $nn$ two-body propagators. The ellipses labeled “$F_{\alpha }$” and “$F_n$” are the corresponding Faddeev components.

Figure 2

The single-channel Faddeev equation for the ${}^6\mathrm{He}$ bound state. The Faddeev component $F_n$ is solved as integral equation containing a two-loop diagram.

Figure 3

The ${}^6\mathrm{He}$ two-neutron separation energy $B_3$ as a function of the cutoff $\Lambda$. The calculation is based on only two-body interactions. The inner panel compares the numerical result (blue solid line) with a polynomial approximation (red dotted line): $B_3/\mathrm{GeV}=-0.00765+0.366{(\Lambda /\mathrm{GeV})}^3$.

Figure 4

The single-channel Faddeev equation for the ${}^6\mathrm{He}$ bound state with the addition of a $nn\alpha$ counterterm. Both sides of the counterterm are in the spectator-$n$ partition.

Figure 5

The $nn\alpha$-counterterm parameter $H_0$ as a function of the cutoff $\Lambda$. $H_0$ is tuned to reproduce $B_3=0.975$ MeV at different values of $\Lambda$.

Figure 6

The Faddeev components $F_{\alpha }$ and $F_n$ as functions of $q$, calculated with cutoff parameters $\Lambda$ at 200 MeV (black dotted line), 400 MeV (orange dot-dot-dashed line), 800 MeV (green dot-dashed line), 1.6 GeV (blue dashed line), and 3.2 GeV (red solid line). The Faddeev components are normalized to provide $F_{\alpha }\left(0\right)=1$.