Statistical Framework for Discovery Sensitivity and Majorana Mass Estimation in \(^{136}\)Xe Neutrinoless Double Beta Decay

Neutrinoless double-beta decay ($0\nu\beta\beta$) is a sensitive probe of lepton-number violation and the Majorana nature of neutrinos. In xenon-based experiments, the expected signal rate inside the region of interest (ROI) is extremely small, requiring sensitivity estimates based on Poisson statistics and a careful treatment of detector resolution, background fluctuations, and systematic uncertainties. In this work, we develop a statistical framework relating energy resolution, ROI width, background index, isotope exposure, and discovery sensitivity for $^{136}$Xe-based $0\nu\beta\beta$ experiments. The formalism combines Poisson likelihood methods with realistic background modeling and includes reconstruction-related and final-state interaction (FSI) systematic effects through an effective ROI broadening approach. Using representative detector parameters for LZ, NEXT-100, KamLAND-Zen, and nEXO, we compare expected background counts, required discovery signal statistics, and half-life sensitivities at matched exposure. The corresponding sensitivities are translated into effective Majorana mass reach within both normal- and inverted-hierarchy neutrino mass ordering. The impact of uncertainties associated with the axial-vector coupling constant $g_A$, nuclear matrix elements, and phase-space factors is also examined. Our results show that background suppression, ROI optimization, and control of detector-related systematics are essential for extending sensitivity toward the normal-ordering regime in future $0\nu\beta\beta$ searches.