Long-lived quasinormal modes and overtones' behavior of holonomy-corrected black holes

Recently massless test scalar field perturbations of the holonomy-corrected black holes [Z. S. Moreira et al., Quasinormal modes of a holonomy corrected Schwarzschild black hole, Phys. Rev. D 107, 104016 (2023).] were studied in order to estimate quantum corrections to the quasinormal spectrum of a black hole. Here we study both the fundamental mode and overtones of scalar, electromagnetic and Dirac fields with the help of the Leaver method and higher-order WKB formula with Padeapproximants. We observe that the overtones depend on the geometry near the event horizon, while the fundamental mode is localized near the peak of the potential barrier, what agrees with previous studies. We showed that unlike a massless field, the massive one possesses arbitrarily long-lived modes. We also obtain the analytical eikonal formula for quasinormal modes and its extension beyond eikonal approximation as a series in powers of 1/e, where e is the multipole number.