Kaon-nucleon scattering states and potentials in the Skyrme model

We study the (anti)kaon nucleon interaction in the Skyrme model. The kaon field is introduced as a fluctuation around the rotating Skyrmion for the nucleon. As an extension of our previous work, we study scattering states and examine phase shifts in various kaon-nucleon channels. Then we study the interaction, where we find that it consists of central and spin-orbit components for isospin channels, $I=0$, 1, with an energy dependence and a nonlocality. The interaction is then fitted to a Shrödinger equivalent local potential for s and p waves.

Figure 1

Phase shifts for the (anti)kaon-nucleon scattering state with $J^P=1/2^{-}$ for the parameter sets A (left) and B (right).

Figure 2

The phase shifts for the $\overline{K}N(J^P=1/2^{-},I=0)$ channel with the three parameter sets A, B, and C.

Figure 3

Phase shifts for the (anti)kaon-nucleon scattering state with $J^P=3/2^{+}$ for the parameter sets A (left) and B (right).

Figure 4

Comparisons between the numerically obtained and the Gaussian-fitted potentials for $u_0^c(N,r)$ (left) and $v_0^c(N,r)$ (right) for the s-wave (upper panels) and p-wave (lower panels) components.

Figure 5

Comparisons between the numerically obtained and the Gaussian-fitted potentials for $u_0^c(WZ,r)$ (left) and $v_0^c(WZ,r)$ (right).

Figure 6

Comparisons between the numerically obtained and the Gaussian-fitted potentials for $u_{\tau }^c(N,r)$ (left) and $v_{\tau }^c(N,r)$ (right).

Figure 7

Comparisons between the numerically obtained and the Gaussian-fitted potentials for $u_0^{LS}(N,r)$ (left) and $v_0^{LS}(N,r)$ (right).

Figure 8

Comparisons between the numerically obtained and the Gaussian-fitted potentials for $u_0^{LS}(WZ,r)$ (left) and $v_0^{LS}(WZ,r)$ (right).

Figure 9

Comparisons between the numerically obtained and the Gaussian-fitted potentials for $u_{\tau }^{LS}(N,r)$ (left) and $v_{\tau }^{LS}(N,r)$ (right).

Figure 10

Comparisons between the numerically obtained and the Gaussian-fitted potentials for $u_l(r)$ (left) and $v_l(r)$ (right).

Figure 11

Nonlocal components of $u_0^c(N,r)$ (left) and $v_0^c(N,r)$ (right) for the s and p waves (solid and dashed lines, respectively).

Figure 12

Total potentials for the $\overline{K}N(J^P=1/2^{-},I=0)$ channel obtained from the bound state (left) and those for the $\overline{K}N(J^P=3/2^{+},I=0)$ channel from a scattering state (right).

Figure 13

The phase shifts for the $\overline{K}N(J^P=1/2^{-},I=0)$ channel obtained from the numerical and fitted potentials with the set A.

Figure 14

The profile functions $F(r)$ for the three parameter sets A, B, and C.

Figure 15

Total potentials for the $\overline{K}N(J^P=1/2^{-},I=0)$ channel from the bound state (left) and the $\overline{K}N(J^P=3/2^{+},I=0)$ channel from a scattering state (right) for the parameter set B, which are derived from the potentials for the set A by the scaling rules Eqs. (32)—(34). The scattering energies are $-32.2$ and 27 MeV for s and p waves, respectively.

Figure 16

Potentials for the $\overline{K}N(J^P=1/2^{-},I=0)$ channel from the bound state (left) and the $\overline{K}N(J^P=3/2^{+},I=0)$ channel from a scattering state (right) for the parameter set D, which are derived from the potentials for the set A by the scaling rules Eqs. (32)—(34). The scattering energies are $-21.0$ and 27 MeV for s and p waves, respectively.

Figure 17

Comparisons between the numerically obtained and the Gaussian-fitted potentials for the local components of $u_0^c(N,\text{local},r)$ (left) and $v_0^c(N,\text{local},r)$ (right).

Figure 18

Comparisons between the numerically obtained and the Gaussian-fitted coefficients for $x_1(r)$ (left) and $y_1(r)$ (right).

Figure 19

Comparisons between the numerically obtained and the Gaussian-fitted coefficients for $x_2(r)$ (left) and $y_2(r)$ (right).

Figure 20

Total potentials for the $\overline{K}N(J^P=1/2^{-},I=0)$ channel from the bound state (left) and the $\overline{K}N(J^P=3/2^{+},I=0)$ channel from a scattering state (right) for the parameter set C, which are derived from the potentials for the set A by the scaling rules Eqs. (32)—(34). The scattering energies are $-81.3$ and 27 MeV for s and p waves, respectively.