Regge Limits, Eikonal Scattering, and Causality

In the CFT Regge limit, high center-of-mass energy at fixed impact parameter is encoded by an analytically continued four-point function. In a semiclassical AdS regime it exponentiates into an eikonal phase shift. A negative time delay inside the EFT’s domain signals a causality problem; a sign found only after extrapolating beyond the cutoff instead diagnoses misuse of the truncated vertex.

Required background. Lorentzian inversion and dispersion supplies the analytic assumptions. CFT Regge boundedness supplies the boundary constraint.

Helpful background. AdS cutting rules supplies unitarity. Eikonal Wilson lines, causality and Regge consistency, and high-energy Regge limits supply comparison tools.

Impact parameter and phase shift

Using Regge variables $(\sigma,\rho)$, a holographic correlator has the schematic representation

$$\mathcal G_{\mathrm{Regge}}(\sigma,\rho) \sim\int d\nu\,\beta(\nu) \sigma^{1-j(\nu)}\Omega_{i\nu}(\rho).$$

The leading trajectory $j(\nu)$ and residue $\beta$ determine the AdS impact-parameter amplitude. In the eikonal regime, ladder exchange exponentiates as $\mathcal A\sim e^{2i\delta(S,b)}-1$. Differentiating the phase with respect to energy gives the Shapiro time shift. Einstein graviton exchange produces a positive delay for physical polarizations Cornalba, Costa, and Penedones 2007.

First application: a higher-derivative graviton vertex

Add a curvature-cubed interaction with coefficient $c/\Lambda^4$. Its polarization-dependent correction scales schematically as

$$\delta(S,b)=\delta_{\mathrm{Ein}}(S,b) \left[1+c\,(\Lambda b)^{-4}\mathcal P(\varepsilon_i)+\cdots\right].$$

For some polarization, a finite truncation can make the bracket negative as $b$ approaches $\Lambda^{-1}$. If this occurs at $b\gg\Lambda^{-1}$ with all higher operators suppressed, the EFT is inconsistent. In weakly coupled gravity, avoiding the time advance generally requires new higher-spin states at or below the inferred scale, as shown by Camanho, Edelstein, Maldacena, and Zhiboedov Camanho et al. 2016.

Adversarial control: extrapolate past the cutoff

Evaluate the correction at $b=0.1\Lambda^{-1}$ while retaining only the first higher-derivative term. Every omitted term $(\Lambda b)^{-2k}$ is then large, so a negative truncated phase is not a controlled obstruction. The correct conclusion is EFT breakdown. A genuine causality constraint needs a parametrically separated region with $EL$, spin, curvature, and gap errors all small.

The evidence ceiling is a high-energy consistency constraint within a stated large-$N$, impact-parameter, Regge, and derivative regime. Regge boundedness does not itself prove a large gap or a flat-space limit. Bulk-point singularities probe a different Lorentzian locality diagnostic.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Camanho, X. O., Edelstein, J. D., Maldacena, J., and Zhiboedov, A. (2016), “Causality Constraints on Corrections to the Graviton Three-Point Coupling,” Journal of High Energy Physics 2016(02), 020. arXiv:1407.5597.
• Cornalba, L., Costa, M. S., and Penedones, J. (2007), “Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite $N$ Four-Point Functions,” arXiv:0707.0120.