Unitarity alternatives in the reduced-action model for gravitational collapse

TL;DR

The paper analyzes unitarity in the ACV reduced-action model for transplanckian gravitational scattering, identifying a critical impact parameter $b_c \sim R$ that separates unitary perturbative behavior from a collapse-like regime with elastic suppression. It combines semiclassical analysis, quantum transitions via $S$-matrix eigenstates, and short-distance fluctuation checks to test whether inelastic channels or alternative boundary conditions can restore unitarity. The results show a persistent unitarity deficit for $b<b_c$ in the semiclassical and leading quantum analyses, with inelastic channels providing only partial compensation that depends on the rapidity parameter $y$. The authors argue that UV-sensitive short-distance solutions and string-scale dynamics must be incorporated to recover the lost probability, highlighting the model’s need for a UV completion to address information flow in gravitational collapse. This work clarifies where the reduced-action model remains valid and points to string-theoretic mechanisms as essential for a complete, unitary description of transplanckian scattering.

Abstract

Based on the ACV approach to transplanckian energies, the reduced-action model for the gravitational S-matrix predicts a critical impact parameter b_c ~ R = 2 G sqrt{s} such that S-matrix unitarity is satisfied in the perturbative region b > b_c, while it is exponentially suppressed with respect to s in the region b < b_c that we think corresponds to gravitational collapse. Here we definitely confirm this statement by a detailed analysis of both the critical region b ~ b_c and of further possible contributions due to quantum transitions for b < b_c. We point out, however, that the subcritical unitarity suppression is basically due to the boundary condition which insures that the solutions of the model be ultraviolet-safe. As an alternative, relaxing such condition leads to solutions which carry short-distance singularities presumably regularized by the string. We suggest that through such solutions - depending on the detailed dynamics at the string scale - the lost probability may be recovered.

Figures (6)

• Figure 1: Diagrammatic series of H and multi-H diagrams.
• Figure 2: [Left] Regions of $t$-values spanned by varying $b$ at fixed $y=0.5$. [Right] The inclusive action corresponding to the various solutions.
• Figure 3: [Left] The two solutions $t_2^{(\pm)}$ (red dashed and blue solid) versus $\beta$ for $y=0.2$. They coincide at $b=b_d$. Only $t_2^{(-)}$ for $b < b_c$ is physical (thick line). [Right] Comparison of the previous solutions for $y=0.2$ (blue solid and dashed) with $t_2^{(\pm)}$ in the $y=0$ case (green dotted and dash-dotted), the latter showing a square-root behaviour around the critical point $\beta=0$.
• Figure 4: Numerical evaluation of the unitarity deficit (solid lines) at various values of $y$, and comparison with the quantum v.e.v. squared of the $S$-matrix (dashed lines) illustrating the increasing contribution of the inelastic channels at larger $y$.
• Figure 5: Plot of $\Re \psi_c$ (dashed) and of $\phi_n$ (solid) for $n=0$ (thick) and $n=2$ (thin), the last function being multiplied by a factor of 3. Here $\alpha = 1$ is small in order not to have a huge suppression of the wave functions in the respective classically forbidden regions.