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 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 ₋₈₌, m ₌₀ₗ, and mₙ₃, which govern qualitative changes in the QNM spectrum. In particular, long-lived modes emerge at mₙ₃, where the imaginary part of the frequency vanishes and the mode becomes essentially non-decaying. 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 quasi-bound states.
Alves et al. (Mon,) studied this question.