Inclusive ${\pi }^{+}d\to p\left(\eta p\right)$ process and the $\eta N$ scattering length

The cross section of the inclusive process ${\pi }^{+}d\to p\left(\eta p\right)$ is calculated as a function of the $\eta p$ invariant mass when the detected proton is moving in the forward direction. The incident pion has a momentum of $p_{\mathrm{lab}}=898.47$ MeV/$c$ for which the $\eta p$ pair are left at rest in the laboratory system which allows one to study the effect of the $\eta p\to \eta p$ final-state interaction in the region of the $N\left(1535\right)S_{11}$ resonance. The sensitivity of the inclusive cross section to different parametrizations of the $\eta N$ final-state interaction is discussed.

Figure 1

Model of the $\pi d\to \eta NN$ process: (a) single-scattering diagram, (b) $N$-exchange diagram, (c) $\pi$-exchange diagram, and (d) $\eta$-exchange diagram.

Figure 2

Differential cross section $d\sigma /d{M}_{\eta p}$ for model B of the $\eta N$ amplitude considering different contributions to the process.

Figure 3

Differential cross section for models A, B, and C as well as considering only the impulse approximation.

Figure 4

The ratio $R$ given by Eq. (1) for models A, B, and C as well as considering only the impulse approximation.