Critical masses and numerical computation of massive scalar quasinormal modes in Schwarzschild black holes

We present a comprehensive analysis of the quasinormal modes (QNMs) of a massive scalar field in Schwarzschild spacetime using two complementary numerical techniques: the Hill-determinant method and Leaver's continued-fraction method. Our study systematically compares the performance, convergence, and consistency of the two approaches across a wide range of field masses and angular momenta. We identify three critical mass thresholds, ${m}_{\mathrm{lim}}$, ${m}_{\mathrm{max}}$, and ${m}_{zd}$, which govern qualitative changes in the QNM spectrum. In particular, long-lived modes emerge at ${m}_{zd}$, where the imaginary part of the frequency vanishes and the mode becomes essentially nondecaying. This phenomenon is robust across multipoles and may have important implications for the phenomenology of massive fields around black holes. Our results provide a detailed numerical characterization of massive scalar QNMs and highlight the complementary strengths of the Hill-determinant and continued-fraction methods, paving the way for future studies of rotating or charged black holes and quasibound states.