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

Abstract

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.

Figures (6)

• Figure 1: Plot of the lapse function $b(r)$. Here we have used the value $m=1$. The event horizon is at $r_+=2.000$ for $A=0$, $r_+=1.610$ for $A=2.00$, and $r_+=0$ for $A=4.71$.
• Figure 2: Plot of the effective potential as a function of $r$. Here, we have used the value $c=1$, $\bar{m}=0.1$ and $\ell=1$. The event horizon is at $r_+=0$ for $A=0.46$, $r_+=0.273$ for $A=0.60$, and $r_+=0.901$ for $A=0.80$.
• Figure 3: The behavior of $\bar{m}_c$ as a function of $A$ for the overtone number $n=0$ (black curve), $n=1$ (blue curve) and $n=2$ (red curve) with $c=1$.
• Figure 4: The behavior of $-Im(\omega)$ for the fundamental mode ($n=0$) as a function of the scalar field mass $\bar{m}$ for different values of the angular number $\ell=30$ (black curve), $\ell=40$ (red curve), $\ell=50$ (blue curve), with $c=1$, $A=0.5$ (top panel), $A=1$ (central panel), and $A=2$ (bottom panel) using the 6th order WKB method with Padé approximants. Here, the WKB method yields critical masses of $\bar{m}_{c}\approx 0.703$, $0.071$ and $0.009$, respectively, via Eq. (\ref{['mass']}).
• Figure 5: The behavior of $Re(\omega)$ for the fundamental mode ($n=0$) as a function of the scalar field mass $\tilde{m}$ for different values of the angular number $\ell=30$ (black curve), $\ell=40$ (red curve), $\ell=50$ (blue curve), with $c=1$, $A=0.5$ (top panel), $A=1$ (central panel), and $A=2$ (bottom panel) using the 6th order WKB method with Padé approximants.