Possibility of generating a 4-neutron resonance with a $T=3/2$ isospin 3-neutron force

We consider the theoretical possibility of generating a narrow resonance in the 4-neutron system as suggested by a recent experimental result. To that end, a phenomenological $T=3/2$ 3-neutron force is introduced, in addition to a realistic $NN$ interaction. We inquire what the strength should be of the $3n$ force to generate such a resonance. The reliability of the 3-neutron force in the $T=3/2$ channel is examined, by analyzing its consistency with the low-lying $T=1$ states of ${}^4\mathrm{H},{}^4\mathrm{He}$, and ${}^4\mathrm{Li}$ and the ${}^3\mathrm{H}+n$ scattering. The ab initio solution of the $4n$ Schrödinger equation is obtained using the complex scaling method with boundary conditions appropriate to the four-body resonances. We find that to generate narrow $4n$ resonant states a remarkably attractive $3N$ force in the $T=3/2$ channel is required.

Figure 1

Experimental binding energies of the He isotopes compared with the predictions based on AV18 and AV18+UIX Hamiltonians. The UIX $3N$ force is purely repulsive in the $T=3/2$ channel and exhibits attraction only in the $T=1/2$ one. Displayed values are taken from Table II of Ref. [20].

Figure 2

Four-nucleon Jacobi coordinates of K-type and H-type configurations.

Figure 3

Dependence of the eigenenergy distribution on the complex scaling angle $\theta$ for the ${}^4\mathrm{n}$ system with $J^{\pi }=0^{+}$. Two different cases are considered: (a) presence of a narrow resonance at $E_{\mathrm{res}}=3.65-0.66i$ MeV for $W_1(T=3/2)=-28$ MeV and (b) presence of a broad resonance at $E_{\mathrm{res}}=5.88-2.85i$ MeV for $W_1(T=3/2)=-21$ MeV.

Figure 4

(a) Tetraneutron resonance trajectory for the $J^{\pi }=0^{+}$ state. The circles correspond to resonance positions for the $\mathrm{AV}8{}^{\prime }$ and the triangles INOY04'(is-m) potential [28]. Parameter $W_1(T=3/2)$ of the additional 3NF was changed from $-37$ to $-16$ MeV in steps of 1 MeV for calculations based on $\mathrm{AV}8{}^{\prime }$ and from $-36$ to $-24$ MeV in steps of 2 MeV for INOY04'(is-m). To guide the eye the resonance region suggested by the measurement [8] is indicated by the arrow at the top. (b) The same contents as in the upper panel figure ($\mathrm{AV}8{}^{\prime }$), but where the resonance energy (closed circles) and width (shadowed area) are represented as a function of the $W_1(T=3/2)$ parameter.

Figure 5

Tetraneutron resonance trajectories for $J^{\pi }=2^{+}$ and $2^{-}$ states for $W_1(T=3/2)$ values from $-38$ to $-26$ MeV and from $-58$ to $-42$ MeV, respectively.

Figure 6

(a) Calculated energies of the lowest $T=1,J^{\pi }=2^{-}$ states in ${}^4\mathrm{H},{}^4\mathrm{He}$, and ${}^4\mathrm{Li}$ with respect to the strength of $T=3/2 3N$ force, $W_1(T=3/2)$. (b) The same but for $T=1,J^{\pi }=1^{-}$ states. The horizontal dashed lines show the ${}^3\mathrm{He}+N$ and ${}^3\mathrm{H}+N$ thresholds. The solid curve below the corresponding threshold indicates the bound state, while the dotted curve above the threshold stands approximately for the resonance obtained by the diagonalization of $H(\theta =0)$ with the $L^2$ basis functions.

Figure 7

The calculated total cross section of ${}^3\mathrm{H}+n$ represented by the thin black solid line using $W_1(T=3/2)=-10$ MeV. The experimental data [52] are illustrated by the red thick solid line.

Figure 8

Trineutron ${}^3\mathrm{n}$ resonance trajectories for $J=3/2^{-},1/2^{-}$, and $1/2^{+}$ states. The circles correspond to resonance positions for $W_1(T=3/2)$ from $-75$ to $-40$ MeV for $J=3/2^{-}$, from $-90$ to $-50$ MeV for $J=1/2^{-}$, and from $-180$ to $-85$ MeV for $J=1/2^{+}$ in steps of 5 MeV.