Soft-core meson-baryon interactions. II. $\pi N$ and $K^{+}N$ scattering

The $\pi N$ potential includes the t-channel exchanges of the scalar mesons σ and $f_0$, vector meson ρ, tensor mesons $f_2$ and $f_2^{\text{'}}$, and the Pomeron as well as the s- and u-channel exchanges of the nucleon N and the resonances Δ, Roper, and $S_{11}$. These baryonic resonances are not generated dynamically. We consider them, at least partially, as genuine three-quark states, and we treat them in the same way as the nucleon. The Roper and $S_{11}$ resonances were needed to find the proper behavior of the phase shifts at higher energies in the corresponding partial waves. The soft-core $\pi N$ model gives an excellent fit to the empirical $\pi N$ S- and P-wave phase shifts up to $T_{\mathrm{lab}}=600$ MeV. Also, the scattering lengths have been reproduced well, and the soft-pion theorems for low-energy $\pi N$ scattering are satisfied. The soft-core model for the $K^{+}N$ interaction is an SU${}_f(3)$ extension of the soft-core $\pi N$ model. The $K^{+}N$ potential includes the t-channel exchanges of the scalar mesons $a_0,\sigma$, and $f_0$, vector mesons $\rho ,\omega$, and ϕ, tensor mesons $a_2,f_2$, and $f_2^{\text{'}}$, and the Pomeron as well as u-channel exchanges of the hyperons $\Lambda ,\Sigma ,\Sigma (1385)$, and $\Lambda (1405)$. The fit to the empirical $K^{+}N$ S-, P-, and D-wave phase shifts up to $T_{\mathrm{lab}}=600$ MeV is reasonable and certainly reflects the present state of the art. Since the various $K^{+}N$ phase shift analyses are also not very consistent, scattering observables are also compared with the soft-core $K^{+}N$ model. A good agreement for the total and differential cross sections as well as the polarizations is found.

Figure 1

Pole potential $V_s$ contains s-channel diagrams; nonpole potential $V_t$ contains t- and u-channel diagrams.

Figure 2

Integral equation for amplitude in the case of a nonpole and pole potential. (a) Integral equation for the total amplitude. (b) Potential in terms of the nonpole and pole potential. (c) Amplitude in terms of the nonpole and pole amplitude Eq. (3.9). (d) Integral equation for the nonpole amplitude Eq. (3.3). (e) Equation for the dressed vertex Eq. (3.10). (f) Equation for the dressed propagator Eqs. (3.12) and (3.11).

Figure 3

Contributions to the $\pi N$ potential from the s-, u- and t-channel Feynman diagrams. The external dashed and solid lines are always the π and N respectively.

Figure 4

Total $\pi N$ $S_{11}$ and $S_{31}$ partial wave potentials $V_L$ as a function of $T_{\mathrm{lab}}$ are given by the solid line. For the $S_{11}$ wave, the resonance pole and total contributions are omitted. The various contributions are (a) long dashed line: ρ, (b) short dashed line: Scalar mesons and Pomeron, (c) dotted line: Nucleon exchange, (d) long dash-dotted line: Δ exchange (e) short dash-dotted line: Tensor mesons, (f) double dashed line: Nucleon or Δ pole, and (g) triple dashed line: Roper pole.

Figure 5

As in Fig. 4, but for the $P_{11}$ and $P_{31}$ partial wave potentials. For the $P_{11}$ wave, the resonance pole and total contributions are omitted.

Figure 6

As in Fig. 4, but for the $P_{13}$ and $P_{33}$ partial wave potentials. For the $P_{33}$ wave, the resonance pole and total contributions are omitted.

Figure 7

Renormalized self-energy ${\Sigma }_{\mathrm{ren}}^{(2)}$ of Eqs. (3.48) and (3.27) for the nucleon and the Δ as a function of $T_{\mathrm{lab}}$. The real part is given by the solid line; the imaginary part, by the dashed line.

Figure 8

S- and P-wave $\pi N$ phase shifts δ as a function of $T_{\mathrm{lab}}$. Empirical phases are from SM95 [21], dots are the multienergy phases, and triangles with error bars are the single-energy phases. NSC $\pi N$ model is given by the solid lines; dashed line is the model without tensor mesons.

Figure 9

As in Fig. 8, but with the empirical phases from KH80 [23].

Figure 10

Contributions to the $K^{+}N$ potential from the u and t channel Feynman diagrams. External dashed and solid lines are always the $K^{+}$ and N, respectively.

Figure 11

Total $K^{+}N$ $S_{01},S_{11},P_{01}$, and $P_{11}$ partial wave potentials $V_L$ as a function of $T_{\mathrm{lab}}$ are given by the solid line. The various contributions are (a) long dashed line: Vector mesons (b) short dashed line: Scalar mesons and Pomeron (c) dotted line: Λ and Σ (d) long dash-dotted line: ${\Sigma }^{*}$ and ${\Lambda }^{*}$ (e). short dash-dotted line: Tensor mesons.

Figure 12

As in Fig. 11, but for the $P_{03},P_{13},D_{03}$, and $D_{13}$ partial wave potentials.

Figure 13

$S_{01}$-, $S_{11}$-, $P_{01}$-, and $P_{11}$-wave $K^{+}N$ phase shifts δ as a function of $T_{\mathrm{lab}}$. Empirical phases are from SP92 [44] multienergy phases (dots) and single-energy phases (filled triangles), [45] single-energy phases (open circles), and [46] single-energy phases (open squares). The NSC $K^{+}N$ model is given by the solid line; the dashed line is the model without tensor mesons.

Figure 14

As in Fig. 13, but for the $P_{03},P_{13},D_{03}$, and $D_{13}$-wave $K^{+}N$ phase shifts.

Figure 15

Total elastic $K^{+}N$ cross section σ as a function of $T_{\mathrm{lab}}$ for both isospin channels. Experimental cross sections are from Ref. 48 (full circles) and Ref. 45 (empty circles). The NSC $K^{+}N$ model is given by the solid line; the dashed line is the model without tensor mesons.

Figure 16

$K^{+}p\to K^{+}p$ differential cross section $d\sigma /d\Omega$ as a function of $\cos \theta$, where θ is the c.m. scattering angle. Experimental differential cross sections are from Ref. 49. The NSC $K^{+}N$ model is given by the solid line; the dashed line is the model without tensor mesons.

Figure 17

$K^{+}n\to K^{+}n$ differential cross section $d\sigma /d\Omega$ as a function of $\cos \theta$, where θ is the c.m. scattering angle. Experimental differential cross sections are from Ref. 50. The NSC $K^{+}N$ model is given by the solid line; the dashed line is the model without tensor mesons.

Figure 18

$K^{+}p\to K^{+}p$ and $K^{+}n\to K^{+}n$ polarizations P as a function of $\cos \theta$, where θ is the c.m. scattering angle. The experimental polarizations are from Ref. 51. The NSC $K^{+}N$ model is given by the solid line; the dashed line is the model without tensor mesons.

Figure 19

${\Theta }^{+}$ resonance included in the NSC $K^{+}N$ model. The total elastic $K^{+}N$ cross section σ is given as a function of $T_{\mathrm{lab}}$. The experimental cross sections are from Ref. 48 (full circles) and Ref. 45 (empty circles). The NSC $K^{+}N$ model is given by the solid line. The dotted, short, and long dashed curves correspond to ${\Gamma }_{{\theta }^{+}}=$1, 10, and 25 MeV, respectively.

Figure 20

Matrix element for the total isospin; m is the $z$ component of the pion isospin and n is the $z$ component of the nucleon isospin.

Figure 21

(a) Baryon emission vertex. (b) Baryon absorption vertex.

Figure 22

(a) ρ emission vertex. (b) ρ absorption vertex.