Nonperturbative completion of ultrahigh-energy string/gravitational scattering: the unitary S-matrix for black-hole formation and evaporation at impact parameters near the gravitational radius, and the microscopic origin of entropy suppression

Giddings, Gross and Maharana (Phys. Rev. D 77, 046001 (2008)) analyzed ultrahigh-energy string scattering by treating one string as a test string in the Aichelburg–Sexl shock geometry of the other and sewing tree amplitudes into an eikonal sum, and showed that the gravitational expansion in $R_S/b$ diverges at impact parameters of order the gravitational radius $R_S(E)$, where classical general relativity predicts black-hole formation. They left open the construction of a fully nonperturbative description of such scattering that (i) completes the divergent series at $b\sim R_S(E)$, (ii) yields a detailed unitary S-matrix for the black-hole formation and evaporation regime consistent with quantum mechanics, and (iii) derives microscopically the black-hole thermodynamic estimates $\sigma_T\sim R_S^{D-2}$ and $A(s,t)\sim e^{-S_{BH}}$ for exclusive channels, thereby characterizing the gravitational nonlocality implied by violations of the Froissart and Cerulus–Martin bounds (the high-energy bounds on total cross-sections and fixed-angle amplitudes that follow from locality, unitarity and analyticity in local quantum field theory). The 2024 review by Di Vecchia, Heissenberg, Russo and Veneziano states that whether and how unitarity is preserved in the collapse regime is still an unanswered question. The problem is open: no later work constructs the required nonperturbative S-matrix or refutes its existence.

Background

In quantum gravity, the sharpest formulation of the question “what happens when two particles collide at energies far beyond the Planck scale?” is through the scattering S-matrix, the matrix relating out-states to in-states. Consider two light strings (or particles) colliding at center-of-mass energy $\sqrt{s}\gg M_P$, where $s$ is the Mandelstam invariant (the squared center-of-mass energy) and $M_P$ is the Planck mass. At such trans-Planckian energies the collision is characterized not only by $s$ but also by the impact parameter $b$, the transverse separation of the incoming objects. Classical general relativity introduces a third scale: the gravitational (Schwarzschild) radius $R_S(E)\sim(G_D E)^{1/(D-3)}$ in $D$ spacetime dimensions with Newton constant $G_D$, evaluated at the collision energy $E\sim\sqrt{s}$. For $E\gg M_P$ this radius exceeds the Planck length, and at sufficiently high string scattering energy it also exceeds the string length $\ell_s$ (the string slope parameter $\alpha'\sim\ell_s^2$ sets the string tension), so gravity, not string oscillation, controls the collision.

Since the late 1980s, Amati, Ciafaloni and Veneziano (the “ACV” program) have argued that this regime is the particle-physics window onto black-hole formation; their 2008 paper summarizes the state of the program. The basic tool is the eikonal approximation: at small deflection angle, the exchange of arbitrarily many gravitons between the fast-moving sources exponentiates into a phase $\exp(2i\delta(s,b))$ in impact-parameter space. Corrections to this resummed eikonal organize as a power series in $R_S/b$ (the post-Minkowskian expansion, an expansion in powers of the Newton constant at fixed velocity). The eikonal phase grows as $R_S/b$ becomes large, rescattering and graviton radiation become unsuppressed, and classical collapse criteria — notably the formation of closed trapped surfaces, whose presence signals inevitable black-hole collapse — predict that for $b$ below a critical value of order $R_S$ a finite fraction of the energy collapses into a black hole, which then evaporates via Hawking radiation — the thermal radiation that quantum mechanics makes black holes emit. The eikonal and reduced-action S-matrix constructions of this program and its later extensions are semiclassical effective-action approximations: they are explicitly unitary for impact parameters above a critical value $b_c\sim R_S$, while below $b_c$ the transverse field solutions become complex-valued and unitarity is at best conjectural. They therefore do not by themselves constitute the fully nonperturbative completion the problem requires.

Giddings, Gross and Maharana (Phys. Rev. D 77, 046001 (2008)) re-examined this territory with string-theoretic methods: they treated one string as a test string propagating through the Aichelburg–Sexl geometry of the other — the gravitational shock wave generated by an ultrarelativistic massless source — expanding the shock profile $\Phi$ in the transverse string fluctuations and truncating at quadratic order, and they sewed string tree amplitudes into an eikonal sum. Their conclusions sharpen the open problem. First, in the regime where momentum transfers reach the string scale, string effects are subdominant to higher-loop gravitational corrections, and the leading gravitational corrections form a power series in $R_S/b$ which “becomes divergent at an impact parameter $b\sim R_S$” — precisely where the semiclassical black-hole regime begins. Second, semiclassical black-hole thermodynamics predicts a total cross-section $\sigma_T\sim[R_S(E)]^{D-2}$ (their eq. 5.1) and exclusive fixed-angle amplitudes suppressed like $e^{-S_{BH}}$ (their eq. 5.2), where $S_{BH}$ is the Bekenstein–Hawking entropy — one quarter of the horizon area in Planck units — of a black hole of mass $\sim\sqrt{s}$. Both estimates violate bounds believed to hold in any local quantum field theory: the Froissart–Martin bound $\sigma_T\le c(\ln E)^{D-2}$ (a consequence of unitarity, analyticity and a mass gap) and the Cerulus–Martin lower bound, which limits how rapidly fixed-angle amplitudes can fall with energy. Since a unitary resolution of Hawking's information problem — whether black-hole evaporation is compatible with quantum-mechanical unitarity — appears to require nonlocality on the scale of the black hole, these violations point to an essential, intrinsically gravitational nonlocality whose mechanism, scale dependence, and role in information recovery are not understood. Constructing the nonperturbative framework that produces these amplitudes, and the unitary S-matrix behind them, is the problem stated below. Two further issues acknowledged in the source accompany it: the possible role of backreaction and higher-order $\alpha'$ corrections beyond the quadratic expansion of $\Phi$, and whether string-theoretic causality effects — causality in string theory being incompletely understood — could allow strings to escape from within horizons in ways not captured by the test-string Aichelburg–Sexl analysis.

Problem Statement

Construct — or rigorously exclude — a fully nonperturbative description of ultrahigh-energy (trans-Planckian) string/gravitational scattering, in string theory or another candidate quantum theory of gravity, that remains valid at impact parameters $b\lesssim R_S(E)$, where $R_S(E)$ is the gravitational radius corresponding to the collision energy. The common objective is a controlled S-matrix for the gravitational-collapse regime; the three requested results below are jointly supporting facets of that single objective, not independent deliverables.

1. Completion of the gravitational perturbation series. The eikonal/post-Minkowskian expansion in $R_S/b$ diverges at $b\sim R_S(E)$. The description must resolve this divergence — resum it, demonstrate its summability, or supersede it with a genuinely nonperturbative object — and, relatedly, settle whether backreaction and higher-order $\alpha'$ corrections (beyond the quadratic expansion of the shock profile $\Phi$ in the test-string Aichelburg–Sexl analysis) modify the quantitative conclusions.

2. Unitarity in the black-hole regime. The construction must determine whether the formation and evaporation of the transient black hole at $b\lesssim R_S(E)$ is described by a detailed, unitary S-matrix consistent with quantum mechanics — the S-matrix form of the information problem, with Hawking-like radiation part of the final state — or establish in a controlled way that unitarity fails. (A related sub-question acknowledged in the source: whether string-theoretic causality effects, not captured by the test-string analysis, allow strings to escape from within horizons.)

3. Microscopic origin of the thermodynamic estimates. It must derive from the underlying string/gravitational degrees of freedom the semiclassical estimates $\sigma_T\sim[R_S(E)]^{D-2}$ for the total cross-section and $A(s,t)\sim e^{-S_{BH}}$ for exclusive channels, showing how black-hole entropy and temperature emerge microscopically in a scattering process, and thereby characterize the mechanism, scale dependence, and role in information recovery of the gravitational nonlocality implied by the violations of Froissart- and Cerulus–Martin-type bounds.

References

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