AdS Scattering, Mellin Methods, and Bulk Locality

How much bulk physics can boundary correlators determine? Mellin amplitudes and Lorentzian limits expose interaction scales and scattering data, but approximate locality requires several compatible tests in one controlled energy range. This chapter explains what those tests establish and which reconstruction ambiguities remain.

Enter this chapter

Begin with Imported Mellin Representations and Bulk-Normalization Conventions to distinguish dynamical poles from universal measure factors.

Helpful background. Mellin-space correlators supplies the representation; exchange Witten diagrams supplies the bulk example being interpreted.

Topics

1. Imported Mellin Representations and Bulk-Normalization Conventions — Separate universal Gamma factors from the dynamical amplitude.
2. Mellin Amplitudes and Penedones-Type Flat-Space Limits — Take the normalized large-radius transform to flat-space scattering.
3. Mellin Contact Polynomials and Exchange Poles — Read derivative interactions and exchanged families from Mellin data.
4. Bulk-EFT Information in Double-Trace Data and Large-Spin Anomalous Dimensions — Interpret multi-trace dimensions as bulk multiparticle information.
5. Bulk Causality from Lorentzian Inversion and Dispersive Sum Rules — Connect Lorentzian spectral data to couplings under stated growth bounds.
6. Regge Limits, Eikonal Scattering, and Causality — Test high-energy phase shifts within their causal and EFT domains.
7. Bulk-Point Singularities and Locality Diagnostics — Use Lorentzian singular configurations as qualified locality diagnostics.
8. AdS Wavepackets and Boundary Extraction of Flat-Space Scattering — Prepare localized scattering events with boundary-created wavepackets.
9. Finite-Gap Corrections and Locality Error Budgets — Estimate errors as energy approaches the heavy-state threshold.
10. Bulk-EFT Checks from OPE, Unitarity, and Causality Data — Check crossing, cuts, and causal behavior in a common regime.
11. CFT Criteria for Approximate Bulk Locality — Combine boundary criteria for a controlled local description.
12. CFT-to-Bulk Reconstruction: Uniqueness and Ambiguities — Identify the interactions and completions that measured data leave ambiguous.

How the ideas fit together

A Mellin contact polynomial describes a finite-derivative bulk interaction, while exchange poles encode propagating conformal families. Flat-space scattering requires an additional limit: the AdS radius grows while appropriately scaled Mellin variables and external normalization are tracked Penedones 2011, §§ 2–3.

The local expansion has a separate cutoff. If an omitted state has dimension $\Delta_{\mathrm{gap}}$ and mass approximately $\Delta_{\mathrm{gap}}/L$, a process of energy $E$ must satisfy $EL\ll\Delta_{\mathrm{gap}}$ to suppress its derivative corrections. Here $L$ is the AdS radius. Large $N$ suppresses loops but does not enforce this energy condition.

Claim-domain comparison

Compare the assumptions and limits of these representative conclusions.

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Representative claim domains and validity boundaries for AdS Scattering, Mellin Methods, and Bulk Locality

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
Mellin pole	Declare Gamma measure, contour, and normalization; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: CFT and Mellin data → exchange poles and contacts → Regge and gap controls → flat-space scaling limit → bounded locality claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “recover OPE data and residues” check is counterevidence to the promoted claim.	With the Gamma measure and contour fixed, the pole or residue fails to reproduce the stated OPE data.	Inferring a unique local particle exchange from a Mellin pole alone.	a pole of the declared correlator
flat-space limit	Declare wavepackets and AdS-radius scaling; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: CFT and Mellin data → exchange poles and contacts → Regge and gap controls → flat-space scaling limit → bounded locality claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “unitarity and normalization check” check is counterevidence to the promoted claim.	The scaled wavepacket amplitude has the wrong normalization or fails the applicable scattering unitarity relation in the declared limit.	Treating a limiting scattering distribution as a finite-radius S-matrix equality.	a limiting scattering distribution
bulk locality	Declare gap, Regge bound, and EFT window; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: CFT and Mellin data → exchange poles and contacts → Regge and gap controls → flat-space scaling limit → bounded locality claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “finite-gap remainder estimate” check is counterevidence to the promoted claim.	Within the stated energy window, the finite-gap remainder exceeds the claimed error or the required Regge bound fails.	Promoting a controlled derivative window to exact locality at finite gap.	approximate locality below the gap

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Review the chapter

At fixed large $N$, increase the scattering energy toward the first omitted single-particle mass. Can loop suppression alone preserve the original local EFT prediction?

Solution

No. Heavy-state exchange generates higher-derivative corrections controlled by energy divided by the heavy mass. Those corrections become unsuppressed near the threshold even when loops remain small. The field content or energy range must change.

Where to go next

Continue to Thermal Phases and AdS Black Holes to study how gravitational saddles encode equilibrium states.

References

• Penedones, J. (2011), “Writing CFT Correlation Functions as AdS Scattering Amplitudes,” Journal of High Energy Physics 2011(03), 025. arXiv:1011.1485.