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S-matrix and Quantum Tunneling in Gravitational Collapse

M. Ciafaloni, D. Colferai

TL;DR

This work recasts the high-energy gravitational scattering near classical collapse as a tunneling problem in the rho-field space of the ACV reduced-action, identifying a Coulomb-like barrier and a critical impact parameter $b_c \sim R$ that separates perturbative and collapse-like regimes. By quantizing the transverse dynamics and formulating the elastic S-matrix as a tunneling amplitude $\mathcal{T}(b,\alpha)$ with $\alpha=Gs/\hbar$, the authors derive explicit Coulomb-wave solutions for $b=0$ and integral representations for $b>0$, including absorption from inelastic graviton production. The introduction of an absorption parameter $y$ and complexified impact parameter $b^2(1-i y)$ yields a unitary elastic amplitude that smoothly bridges the perturbative and collapse-like regimes, with Airy-function behavior characterizing the transition near $b_c$. Quantum corrections, of order $1/\alpha$, tend to smooth the semiclassical features and demonstrate that $b_c$ is not a true singularity of the quantum amplitude. The framework lays groundwork for incorporating inelastic channels and energy conservation in a complete transplanckian S-matrix theory.

Abstract

Using the recently introduced ACV reduced-action approach to transplanckian scattering of light particles, we show that the $S$-matrix in the region of classical gravitational collapse is related to a tunneling amplitude in an effective field space. We understand in this way the role of both real and complex field solutions, the choice of the physical ones, the absorption of the elastic channel associated to inelastic multigraviton production and the occurrence of extra absorption below the critical impact parameter. We are also able to compute a class of quantum corrections to the original semiclassical $S$-matrix that we argue to be qualitatively sensible and which, generally speaking, tend to smooth out the semiclassical results.

S-matrix and Quantum Tunneling in Gravitational Collapse

TL;DR

This work recasts the high-energy gravitational scattering near classical collapse as a tunneling problem in the rho-field space of the ACV reduced-action, identifying a Coulomb-like barrier and a critical impact parameter that separates perturbative and collapse-like regimes. By quantizing the transverse dynamics and formulating the elastic S-matrix as a tunneling amplitude with , the authors derive explicit Coulomb-wave solutions for and integral representations for , including absorption from inelastic graviton production. The introduction of an absorption parameter and complexified impact parameter yields a unitary elastic amplitude that smoothly bridges the perturbative and collapse-like regimes, with Airy-function behavior characterizing the transition near . Quantum corrections, of order , tend to smooth the semiclassical features and demonstrate that is not a true singularity of the quantum amplitude. The framework lays groundwork for incorporating inelastic channels and energy conservation in a complete transplanckian S-matrix theory.

Abstract

Using the recently introduced ACV reduced-action approach to transplanckian scattering of light particles, we show that the -matrix in the region of classical gravitational collapse is related to a tunneling amplitude in an effective field space. We understand in this way the role of both real and complex field solutions, the choice of the physical ones, the absorption of the elastic channel associated to inelastic multigraviton production and the occurrence of extra absorption below the critical impact parameter. We are also able to compute a class of quantum corrections to the original semiclassical -matrix that we argue to be qualitatively sensible and which, generally speaking, tend to smooth out the semiclassical results.

Paper Structure

This paper contains 15 sections, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The reduced-action approach at semiclassical level
  3. The quantum level and tunneling amplitude
  4. Quantized elastic $\boldsymbol{S}$-matrix
  5. Inelastic $\boldsymbol{S}$-matrix and absorption
  6. Tunneling amplitude neglecting absorption
  7. Calculation of quantum tunneling amplitude
  8. The tunneling wave function and $\boldsymbol{b=0}$ case
  9. Integral representation of tunneling amplitude at $\boldsymbol{b>0}$
  10. Including absorption at quantum level
  11. Perturbative versus collapse-like regimes
  12. Discussion
  13. Stationary-phase estimates of tunneling amplitude
  14. Quadratic expansion
  15. Cubic expansion

Figures (2)

  • Figure 1: Semiclassical solutions $\rho(\tau)/\tau$ showing: (a,b) supercritical branches (solid-blue) at $b^2=1.2\,b_c^2$, (c) the critical one (dash-dotted green) and subcritical ones at $b^2=0.7\,b_c^2$. In the latter case we contrast (m) the singular real-valued one (dashed-red on the left) with (r,i) the regular complex-valued one (dashed, on the right). The black-dotted contour shows the border between free and Coulomb-like evolution for $b\geq b_c$.
  • Figure 2: a) Tunneling amplitude with elastic absorption, for two values of the absorption parameter $y=0$ (dashed-red) and $y=0.5$ (solid-blue), the latter to be compared with the semiclassical result (dash-dotted green). b) Absorbed amplitudes for different values of $\alpha$ and $y$. (The two solid-blue curves in a) and b) are the same amplitudes, just on different scales).