Reconstruction of $\Lambda^0$ Baryons and Polarization Measurements with the Zero-Degree Calorimeter at the Electron-Ion Collider

The Zero-Degree Calorimeter for the upcoming Electron-Ion Collider employs high granularity to detect neutral particles, specifically neutrons and photons, along the proton-beam direction. We demonstrate this detector's capability in reconstructing and identifying $\Lambda^0$ baryons via their decay into a neutron and a $\pi^0$ (which subsequently decays into two photons). For the first time, we show that with realistic simulations, clustering and other algorithms we should be able to reconstruct this challenging decay mode. Additionally, we show the detector's effectiveness in reconstructing the neutron's direction in the $\Lambda^0$ rest frame, enabling determination of the $\Lambda^0$ polarization. These studies set the foundation for future measurements of the Sullivan process, where elastic and inelastic scattering off virtual kaon targets can be explored. Furthermore, this benchmark paves the way for studying more complex decay channels, such as $\Sigma^0 \rightarrow \gamma \Lambda^0$ and $\Xi^0 \rightarrow \pi^0 \Lambda^0$.