Bulk Phase Shift and Singularity

High-energy scattering of a light particle off a black hole at fixed impact parameter is described by an eikonal phase, which encodes the resulting time delay and angular deflection. This bulk phase shift admits a holographic interpretation as of the thermal momentum-space two-point function of a scalar operator in the CFT in the Regge limit. At small impact parameter, the phase shift acquires an imaginary part signaling inelastic scattering, obscuring the interpretation of time delay and deflection which become complex-valued. However, in a holographic CFT these quantities can also be extracted from the so called bulk-cone singularities of the position space thermal correlator. Extending this analysis to small impact parameters, we find that the position-space correlator develops a singularity precisely when a null geodesic reflects off the black hole singularity and reaches the opposite boundary. This provides a more robust identification of bulk time delay and angular deflection through singularities of position-space correlators.

💡 Research Summary

The paper investigates high‑energy scattering of a light probe off an AdS‑Schwarzschild black hole at fixed impact parameter, focusing on the eikonal phase shift δ that encodes the probe’s time delay Δt and angular deflection Δθ. In the bulk, δ is expressed as δ = p_t Δt − p_θ Δθ and can be written as an integral over the radial coordinate involving the black‑hole metric function f(r). For pure AdS the phase shift reduces to a simple analytic form, while the presence of the black‑hole mass μ introduces a non‑trivial dependence on the turning point r_T determined by b² = f(r_T) r_T².

The authors then map this bulk phase shift to the dual conformal field theory (CFT) by showing that δ appears in the Fourier transform of the finite‑temperature two‑point function G_T(ω,ℓ) on the sphere. In the Regge (eikonal) limit, where ω,ℓ≫1 with the ratio b = ω/ℓ held fixed, the thermal correlator takes the form G_T⁺ = G_T⁰ e^{i δ(b)}. Using a WKB approximation for a massive scalar field in the black‑hole background, they demonstrate that the scalar’s radial phase S(r) satisfies the same equation as the null geodesic, establishing the identification ℓ S(b) = ω T(b) − ℓ Θ(b). Consequently, the time delay T(b) and angular deflection Θ(b) are given by derivatives of the eikonal phase, T = ∂_b S, Θ = S − b ∂_b S.

When the impact parameter falls below a critical value b_c, the turning point disappears, the phase shift acquires a large imaginary part, and the bulk picture suggests inelastic processes such as absorption or capture by the black hole. In this regime Δt and Δθ become complex, making a direct physical interpretation problematic.

To resolve this, the authors study the position‑space thermal Wightman function obtained by Fourier‑inverting the momentum‑space correlator. They find that for b < b_c the correlator develops a singularity precisely when a null geodesic, after reflecting off the black‑hole singularity, emerges on the opposite boundary. This “bulk‑cone singularity” occurs at spacetime points (t,θ) satisfying t = T(b) and θ = Θ(b). The singularity is a consequence of analytic continuation to the second sheet required by Regge kinematics and causality. Notably, the two‑sided Wightman function does not exhibit this singularity because double‑trace operator contributions cancel the stress‑tensor sector’s divergence.

The paper concludes that, in the inelastic regime, bulk time delay and angular deflection are more robustly extracted from the singularity structure of the boundary correlator rather than directly from the complex phase shift. This provides a clearer holographic dictionary linking black‑hole interior dynamics to CFT observables and suggests further investigations into rotating black holes, higher‑spin exchanges, and non‑thermal states. The appendices supply technical details on the WKB derivation, integral evaluations in arbitrary dimensions, Gegenbauer polynomial expansions, and the treatment of negative impact parameters.