Stringy effects in scrambling

TL;DR

The paper extends holographic scrambling analyses to include stringy and Planckian corrections near black hole horizons. By combining elastic eikonal gravity with tree-level stringy corrections and an assessment of inelastic effects, it shows that elastic string effects smear the chaos region and modestly delay scrambling, while inelastic channels provide only subleading contributions for the correlators considered. The framework links high-energy horizon scattering to boundary out-of-time-order correlators, offering concrete formulas for the string-corrected phase δ(s,b) and highlighting regimes where chaos dynamics are smeared or slowed. It also discusses potential observables and connections to weak-coupling Regge behavior, suggesting avenues to diagnose Planckian physics beyond the simplest chaos probes.

Abstract

In [1] we gave a precise holographic calculation of chaos at the scrambling time scale. We studied the influence of a small perturbation, long in the past, on a two-sided correlation function in the thermofield double state. A similar analysis applies to squared commutators and other out-of-time-order one-sided correlators [2-4]. The essential bulk physics is a high energy scattering problem near the horizon of an AdS black hole. The above papers used Einstein gravity to study this problem; in the present paper we consider stringy and Planckian corrections. Elastic stringy corrections play an important role, effectively weakening and smearing out the development of chaos. We discuss their signature in the boundary field theory, commenting on the extension to weak coupling. Inelastic effects, although important for the evolution of the state, leave a parametrically small imprint on the correlators that we study. We briefly discuss ways to diagnose these small corrections, and we propose another correlator where inelastic effects are order one.

Figures (9)

• Figure 1: Locations on the Penrose diagram of the various operators described in the text.
• Figure 2: The path integral contours that define (\ref{['WVWV2']}), (\ref{['WVWV1']}), and (\ref{['WVWV3']}), respectively. The circle is the periodic imaginary time direction, and the folds represent the real-time evolution to produce the $W(t)$ operators. The contour ordering is the same in each case, so the correlators are related by adding or subtracting imaginary time $\beta/2$ to one or two of the operators.
• Figure 3: The one-particle state $W(t_4)|TFD\rangle$ can be represented on any bulk slice.
• Figure 4: The correlation function (\ref{['correlator-to-compute']}) is an inner product of these two states. As explained in the text, changing the ordering of these operators changes an 'in' state to an 'out' state.
• Figure 5: The 'in' and 'out' states in the completely two-sided case (\ref{['WVWV3']}).