Pulse shape discrimination for $\alpha$ event rejection in BEGe-type high-purity germanium detectors

High-purity germanium detectors are widely used in rare-event searches due to their excellent energy resolution and extremely high intrinsic (radio)purity. In experiments searching for neutrinoless double beta decay in $^{76}$Ge such as LEGEND, pulse shape discrimination is required to suppress multi-site $\gamma$ events. In this work, we investigate whether pulse shape discrimination classifiers trained exclusively on $\gamma$ ray data can be used to identify and reject $\alpha$ events, without the need for dedicated $\alpha$ training. In detectors such as LEGEND, the total number of registered $\alpha$ events over the experiment lifetime is expected to be insufficient to train dedicated classifiers, while still contributing to the background. Two approaches based on machine learning are studied: a multilayer perceptron and a projective likelihood classifier. The p+ surface of a point-contact semi-planar germanium detector was exposed to $^{209}$Po and $^{210}$Po sources deposited on a thin gold foil. Two measurement campaigns were performed, yielding $1.36\times10^{5}$ and $1.87\times10^{6}$ $\alpha$ events, respectively. Both classification methods achieve efficient separation of single-site and multi-site $\gamma$ events while strongly reducing the $\alpha$ component. The multilayer perceptron provides the best overall performance, with a signal-like event survival greater than 80%, a background-like event survival below 20%, and an $\alpha$-rejection factor exceeding $2.71\times10^{4}$. These results demonstrate that robust pulse shape discrimination for high-purity germanium detectors can be achieved using training information derived solely from $\gamma$ events, providing a promising strategy for next-generation neutrinoless double beta decay searches.