Enhanced light signal for the suppression of pile-up events in Mo-based bolometers for the $0\nu\beta\beta$ decay search

Random coincidences of events could be one of the main sources of background in the search for neutrino-less double-beta decay of $^{100}$Mo with macro-bolometers, due to their modest time resolution. Scintillating bolometers as those based on Li$_2$MoO$_4$ crystals and employed in the CROSS and CUPID experiments can eventually exploit the coincident fast signal detected in a light detector to reduce this background. However, the scintillation provides a modest signal-to-noise ratio, making difficult a pile-up pulse-shape recognition and rejection at timescales shorter than a few ms. Neganov-Trofimov-Luke assisted light detectors (NTL-LDs) offer the possibility to effectively increase the signal-to-noise ratio, preserving a fast time-response, and enhance the capability of pile-up rejection via pulse shape analysis. In this article we present: a) an experimental work performed with a Li$_2$MoO$_4$ scintillating bolometer, studied in the framework of the CROSS experiment, and utilizing a NTL-LD; b) a simulation method to reproduce, synthetically, randomly coincident two-neutrino double-beta decay events; c) a new analysis method based on a pulse-shape discrimination algorithm capable of providing high pile-up rejection efficiencies. We finally show how the NTL-LDs offer a balanced solution between performance and complexity to reach background index $\sim$$10^{-4}$ counts/keV/kg/year with 280~g Li$_2$MoO$_4$ ($^{100}$Mo enriched) bolometers at 3034 keV, the Q-value of the double-beta decay, and target the goal of a next generation experiment like CUPID.