Anomalous Decay Rate and Greybody Factors for Regular Black Holes with Scalar Hair

We study the propagation of massive scalar fields in the background of asymptotically flat regular black holes supported by a phantom scalar field with a scalar charge $A$. This parameter regularizes the geometry by removing the central singularity. Focusing on wave dynamics, we analyze scalar perturbations, quasinormal modes, and greybody factors, emphasizing the role of the regularization parameter on the effective potential and the decay properties of the modes. Using WKB methods beyond the eikonal limit, we show that the presence of scalar hair modifies both the oscillation frequencies and damping rates of quasinormal modes. In particular, we demonstrate the occurrence of an anomalous decay rate for massive scalar perturbations: above a critical field mass, the longest-lived modes correspond to lower angular momentum, in contrast with the massless case. We derive analytical expressions for the critical mass and study its dependence on the scalar charge and overtone number. Furthermore, we apply the Horowitz-Hubeny method to compute the quasinormal frequencies and show that the results obtained from the WKB and Horowitz-Hubeny approaches exhibit excellent agreement in the regime where both methods are valid. In addition, we compute reflection and transmission coefficients and analyze the corresponding greybody factors, clarifying how regularity effects imprint themselves on black-hole scattering properties. Our results show that regular black holes with scalar hair exhibit distinctive dynamical signatures that can be probed through quasinormal ringing and wave propagation.

💡 Research Summary

The paper investigates wave propagation of massive scalar fields in the background of asymptotically flat regular black holes that are supported by a phantom scalar field carrying a scalar charge A. This charge regularizes the geometry by eliminating the central singularity, turning the solution into a genuine regular black hole with a well‑defined event horizon. The authors first present the underlying theory, deriving the metric functions and showing how the parameter A controls the horizon radius and the deviation from the Schwarzschild case.

Next, the Klein‑Gordon equation for a massive scalar field (mass (\bar m)) is separated using a standard ansatz, leading to a Schrödinger‑like radial equation with an effective potential
(V_{\text{eff}}(r)=b(r)\big