Analyticity of the Black Hole S-Matrix

TL;DR

The paper develops a rigorous picture of the Schwarzschild black hole S-matrix analyticity for classical wave scattering in four dimensions. It identifies quasinormal-mode poles and a tail-induced branch cut in the lower half-plane and shows, by causality, the retarded Green's function is analytic in the upper-half plane, while the elastic amplitude acquires a Stokes-induced upper-half-plane branch cut. Extended analyticity is established under complex-x analyticity of the potential, with the black hole singularity behind the horizon controlling the maximal domain. It also analyzes infrared regulator effects, showing polynomial boundedness requires a sharp IR cutoff bound, and tests the framework with the Pöschl–Teller potential and Schwarzschild Regge–Wheeler problem, then discusses implications for Love numbers, EFT, and potential bootstrap program extensions.

Abstract

We establish the analytic structure of the S-matrix in the complex-frequency plane for classical wave scattering on a Schwarzschild background in four space-time dimensions. Our argument relies on the analytic continuation of the gravitational potential, with the singularity behind the horizon playing a crucial role. We find that in the lower half-plane the partial-wave amplitudes are analytic except for the quasinormal-mode poles and the branch cut associated with late-time tails. As a direct consequence of causality, the retarded Green's function and absorption amplitude are analytic in the upper-half plane. We show, however, that Stokes phenomena can obstruct this analyticity domain from carrying over to the elastic amplitude, which instead develops a branch-cut in the upper-half plane. We also determine the effect of infrared (IR) regulators on the analytic structure, showing that polynomial boundedness requires a sharp lower bound on the IR cutoff in terms of the Schwarzschild radius.

Figures (12)

• Figure 1: Analytic structure of Left:$G_R(\omega,x,x')$ and Right:$R(\omega)$ of the black hole. There is a branch cut along the negative imaginary axis for $G_R$ and along the entire imaginary axis for $R$, as well as quasinormal mode poles in the LHP (crosses, locations are approximate).
• Figure 2: $V(x)$ is analytic and decays sufficiently fast in the shaded blue region.
• Figure 3: Integration contour for the integral in Eq. \ref{['eq:complexxint1']}, with $x=e^{i\theta}y$.
• Figure 4: Integration contour $\gamma(x)$ for the iterated integral in Eq. \ref{['eq:chi1angles']} in the complex-$x'$ plane with $x=e^{i\theta_-}y\,\Theta(-y)+e^{i\theta_+}y\,\Theta(y)$. The contour shown has $y>0$ and chooses $y_j$ to be the first integral where the integration variable is positive. $y_n,...,y_{j+1}$ are negative and rotated by $\theta_-$, while $y_j,...,y_1$ and the fixed variable $y$ are positive and rotated by $\theta_+$.
• Figure 5: Summary of the established analyticity domains derived for a potential $V(x)$ satisfying \ref{['eq:Van']} and \ref{['eq:Vbound']}. $A_\text{in}(\omega)$ is analytic in the dotted region and $A_\text{out}(\omega)$ is analytic in the shaded blue region. Therefore $T(\omega)$ is meromorphic in the dotted region and $R(\omega)$ is meromorphic in the overlap of the dotted and blue shaded regions. The regions are defined by diagonal lines at an angle $\pm\theta_\text{max}$ with respect to the real axis.