Luneburg-lens-like structural Pauli attractive core of the nuclear force at short distances

The nuclear force has been understood to have a repulsive core at short distances, similar to a molecular force, since Jastrow proposed it in 1951 [R. Jastrow, Phys. Rev. 81, 165 (1951)]. The existence of the repulsion was experimentally confirmed from the proton-proton scattering ${}^{1}S_0$ phase shift, which becomes negative beyond 230 MeV. This repulsion is essential for preventing the nucleon-nucleon system from collapsing by attraction. The origin of the repulsion has been considered to be due to the Pauli principle, similar to the repulsion originally revealed in $\alpha -\alpha$ scattering, in many studies including recent lattice QCD calculations. On the other hand, very recently it was shown that an internuclear potential including $\alpha -\alpha$ interactions has a Luneburg-lens-like attraction at short distances rather than repulsion. I show that the nuclear force with an attractive potential at short distances that reproduces the experimental ${}^{1}S_0$ phase shifts well has a Luneburg-lens-like structural Pauli attractive core at short distances and acts as apparent repulsion. The apparent repulsion is caused by the deeply embedded unobservable Pauli forbidden state similar to nucleus-nucleus potentials.

Figure 1

Typical nuclear potentials for the ${}^{1}S_0$ channel, the G3RS potential (dashed line) [12], and the Argonne V18 potential (solid line) [13].

Figure 2

The proton-proton scattering ${}^{1}S_0$ phase shifts calculated with the SPAC potential of Eq. (2) (solid line), the G3RS potential of Eq. (1) (dashed line), and the Reid93 potential [15] (dash-dotted line) are displayed. The experimental data (squares) are from Ref. [45].

Figure 3

(a) The SPAC nuclear force with the Pauli attractive core at the short distances (red solid line) of Eq. (2), the Moscow potential (dashed line) of Ref. [28], and the Luneburg-lens potential (red dotted line) are compared. The horizontal lines in panel (a) indicate the energy of the unobservable Pauli forbidden $0s$ state. The vertical dashed lines are to guide the eye. (b) A magnified SPAC and Moscow potential in panel (a) are compared with the G3RS potential (blue dotted line).

Figure 4

The calculated ${}^{1}S_0$ wave functions of proton-proton scattering at $E_{\mathrm{lab}}=0.1-350$ MeV using (a) the SPAC nuclear force potential with the Pauli attractive core at the short distances and (b) the G3RS potential with the repulsive core at the short distances. The difference of wave functions in panels (a) and (b) are seen in the core region $r<0.5$.

Figure 5

The potentials and the wave functions (in arbitrary units) for the nucleon-nucleon and $\alpha -\alpha$ systems are compared. The SPAC ${}^{1}S_{0}NN$ potential (b) with the SPAC at short distances and the wave function (a) are displayed in comparison with the $\alpha -\alpha$ deep potential with a structural Pauli attraction of Ref. [1] (d) and the Pauli-allowed $s$-wave function (c). The horizontal dashed lines in panels (b) and (d) indicate the energy of the unobservable Pauli forbidden states embedded in the $NN$ and $\alpha -\alpha$ potentials. The wave function of $1s$ in panel (a) and $2s$ in panel (d) have been calculated using the bound-state approximation.