Hatsuda Method for Black Hole QNMs

• The Hatsuda method is a high-precision, semi-analytical technique that calculates black hole quasinormal mode frequencies using advanced WKB expansion and Borel summation.
• It employs recursive algebraic techniques and direct matching of WKB-quantized conditions to ensure rapid convergence and robust error control in QNM computations.
• Key parameters such as η, M, Q, and ν critically influence the QNM spectrum, facilitating the transition from classical Schwarzschild to string-inspired black hole regimes.

The Hatsuda method refers to a high-precision, semi-analytical technique for calculating black hole quasinormal mode (QNM) frequencies, developed for broad applicability in gravitational and gauge-theoretical settings. In the context of asymptotically flat hairy electrically charged black holes with a dilaton potential, the Hatsuda method is utilized to extract the QNM spectrum and understand its dependence on black hole parameters, with particular focus on the integration constant $\eta$, the mass $M$, the electric charge $Q$, and the dilaton coupling constant $\nu$ that controls the $U(1)$–dilaton field interaction (Xiong et al., 5 Aug 2025).

1. Quasinormal Mode Calculation Framework

The computation of QNMs for these black hole backgrounds begins with the covariant Klein–Gordon equation for a probe massless scalar field: $$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}\Phi\right) = 0.$$ Upon separation of variables using

$$\Phi(t,x,\theta,\varphi)=\sum_{l,m}\frac{R(x)}{\sqrt{\Psi(x)}}Y_{l,m}(\theta,\varphi)e^{-i\omega t},$$

and choosing appropriate coordinates, the problem reduces to a one-dimensional Schrödinger-type equation in the tortoise coordinate $x_*$: $$\frac{d^2 R(x_*)}{dx_*^2} + \left[\omega^2-V(x)\right]R(x_*)=0,$$ where $dx_*/dx = \eta/f(x)$, and $V(x)$ includes derivatives of the metric functions and conformal factor $\Psi(x)$. The QNM boundary conditions require purely ingoing waves at the event horizon and outgoing waves at spatial infinity.

2. Implementation of the Hatsuda Method

The Hatsuda method employs a very high order WKB expansion for the QNM frequencies, together with Borel summation to resum divergent series contributions. Central features include:

• Calculation of WKB corrections to very high order using recursive algebraic techniques (notably, the BenderWu package).
• Systematic Borel summation to accelerate convergence and regularize the asymptotic WKB series.
• Direct matching of the WKB-quantized eigenvalue condition to impose the QNM boundary conditions.

In the eikonal (large-$l$) limit, the frequency spectrum takes the form: $$\omega \simeq \left(l+\frac{1}{2}\right)\sqrt{f(x_{ph})} - i\left(n+\frac{1}{2}\right)\sqrt{-\frac{f(x_{ph})f''(x_{ph})}{2\eta^2}},$$ where $x_{ph}$ is the photon sphere location and $f(x)$ the metric function, with details determined by the specific black hole configuration.

High-precision QNM frequencies, including both real and imaginary parts, are obtained by resumming higher-order corrections—this enables robust comparison with other semi-analytical and numerical approaches, such as advanced WKB methods with Padé summation.

3. Role of $\eta$, $M$, $Q$, and $\nu$ in Black Hole Dynamics

The metric of this class of hairy black holes incorporates the integration constant $\eta$, entering the definition of mass $M$, electric charge $Q$, and the non-minimal coupling constant $\nu$: $$M = \frac{q^2}{4\eta(\nu-1)} - \frac{\alpha + 3\eta^2}{6\eta^3},\qquad Q = -\frac{q}{4\eta},$$ where $q$ and $\alpha$ are integration and dilaton potential constants, respectively.

The dilaton–$U(1)$ coupling $\nu$ is determined by

$$\nu = 1 + \frac{24 Q^2 \eta^4}{\alpha + 3\eta^2 + 6M\eta^3},$$

and the action’s dilaton coupling parameter is $\gamma = \sqrt{(\nu+1)/(\nu-1)}$. For $Q \to 0$, one recovers $\nu \to 1$, the Schwarzschild limit; for large $\eta$ at finite $Q$, the geometry interpolates toward the low-energy limit of string theory ($\gamma\sim 1$).

Each parameter modifies the effective potential $V(x)$, the tortoise coordinate, and the photon sphere, thereby influencing the QNM spectrum.

4. Precision Comparison: Hatsuda vs. High-Order WKB and Padé

Validation of the Hatsuda method is performed through comparison with those obtained by high-order WKB approximations and Padé summation:

• Both approaches start from WKB quantization about the potential peak, relating mode properties to effective photon sphere dynamics.
• The agreement between both methods is quantitatively strong for small $l$ (low angular momentum) and low overtone $n$ values (e.g., $\omega\approx0.833747-0.490338\,i$ in the Schwarzschild limit).
• In the eikonal regime, the theoretical prediction that $\text{Re}(\omega) \sim l\,\Omega_c$ and $\text{Im}(\omega) \sim -(n+1/2)\lambda$ (where $\Omega_c$ is the angular velocity at the photon sphere and $\lambda$ the Lyapunov exponent) is confirmed.

The Hatsuda method achieves rapid convergence and robust error control even as the parameter $\eta$ interpolates between the Reissner–Nordström-like and the string-theory-like regimes.

5. Eikonal Limit and Geometric-Optics Correspondence

In the eikonal ($l \gg 1$) limit, the procedure recovers the correspondence between QNMs and properties of unstable null geodesics:

• Real part of the frequency: $\text{Re}(\omega) \approx l\,\Omega_c$ (orbital frequency of photon sphere).
• Imaginary part: $\text{Im}(\omega) \approx -(n+1/2)\lambda$ (Lyapunov exponent, the instability timescale for geodesic deviation).

It is rigorously established that for large enough $\nu$ (specifically, when $\nu$ approaches values such that $\gamma \sim 1$), the QNM spectrum approaches that of the low-energy effective string theory black hole solution.

Physical Regime	Key Features / QNM Limiting Behavior
$\nu\to 1$ (Schwarzschild)	Standard Schwarzschild QNMs; potential dictated by $\eta$
$\gamma\sim 1$ (String)	QNM spectrum matches low-energy dilaton/string theory limit

6. Parameter Dependence and Physical Transition

The dependence of shadow radius, Lyapunov exponent $\lambda$, and angular velocity $\Omega_c$ on the coupling constant $\nu$ becomes pronounced only when the electric charge $Q$ is near extremality and $\nu\to 1$. In these regimes, changes in $\nu$ significantly affect not only the black hole shadow but also the QNM damping and oscillation frequencies, marking a transition from typical Reissner–Nordström behavior toward string-inspired hairy solutions.

This suggests that measurement of these observables, combined with theoretical calculation via the Hatsuda method, may provide a diagnostic for distinguishing between classical and string-theoretical black holes in scenarios where the dilaton coupling is large.

7. Summary and Significance

The Hatsuda method, employing high-order WKB expansion and Borel resummation, provides precise, parameter-sensitive predictions for the QNM spectrum of asymptotically flat hairy black holes with a dilaton potential (Xiong et al., 5 Aug 2025). This approach:

• Accurately captures the interplay between black hole mass, charge, non-minimal coupling, and geometric structure;
• Verifies the geometric optics limit correspondence between QNM frequencies and photon sphere characteristics;
• Enables robust cross-validation with independent high-order WKB/Padé summation results;
• Demonstrates smooth interpolation from the Schwarzschild to the string-theoretical regimes as dictated by the parameter $\eta$ and coupling $\nu$.

The method thereby establishes a concrete computational and conceptual bridge between semi-analytic gravity, geometric optics, and string-theoretic black hole physics in the analysis of quasinormal modes.