Massive Scalar Quasinormal Modes and Quasibound States of Charged Black Holes with Logarithmic corrections

Abstract

We study how long-range deformations in a black-hole metric affect the oscillation, trapping, and decay of a neutral, minimally coupled massive scalar field. We choose a logarithmic deformation of Reissner-Nordström (RN) sourced by nonlinear electrodynamics. It provides an exact solution of Einstein gravity and admits a finite completion. The same geometry also arises in a phenomenological perfect-fluid dark-matter model. In this setup, we analyze both quasinormal modes (QNMs), which describe relaxation, and quasibound states (QBSs), which describe trapped modes with small horizon absorption. We first show that a small scalar mass moves the potential barrier outward, raises it, and reduces its curvature. First-order WKB then predicts weaker damping at every fixed finite multipole and overtone. We then compare several approaches, namely sixth-order WKB, global matching, a generalized Leaver method (adapted to the logarithmic geometry) and time-domain evolution. All methods yield consistent QNM estimates within their ranges of validity. Global matching and Leaver also agree closely for QBSs. Further analytic estimates with a reference trapped state show that a local Padé approximation gives a binding-energy error of about $0.04\%$, while leading Gamow tunnelling underestimates damping by about $25\%$. Across the sampled QBSs, increasing the scalar mass or nonlinear coupling strengthens binding but shortens the lifetime through greater horizon absorption. Finally, using an appropriate horizon regularization and a common complex radial contour, we demonstrate orthogonality of an independently computed QNM-QBS pair using a suitable relativistic bilinear framework.