Bulk-EFT Checks from OPE, Unitarity, and Causality Data

A candidate bulk-EFT correlator must satisfy Euclidean crossing and positivity, Lorentzian cut factorization, acceptable Regge growth, and causal phase-shift tests within one common regime. Contact terms can repair some crossing data but cannot cancel a negative physical cut or an acausal high-energy phase without changing the spectrum or leaving the EFT domain.

Required background. AdS cutting rules supplies loop discontinuities. Regge causality supplies the high-energy check.

Helpful background. Crossing and positivity, causality and analytic growth, and Hilbert-space positivity supply the imported conditions.

A boundary-to-bulk validation matrix

Crossing requires the four-point function to agree under external permutations after tensor and prefactor transformations. Reflection positivity requires nonnegative squared OPE coefficients in unitary identical-operator channels. Exchange residues must factorize into normalized three-point coefficients. At loops, the double discontinuity must equal a positive spectral sum of lower-order data. Regge growth must respect the subtractions used in inversion, and the eikonal phase must not produce a controlled time advance.

These tests constrain different analytic pieces. Cuts determine nonlocal logarithms, while crossing-symmetric contact polynomials have no cut. A successful Euclidean fit therefore does not automatically test unitarity or causality Meltzer and Sivaramakrishnan 2020.

First application: test a tree-plus-loop correlator

Take

$$\mathcal G=\mathcal G_{\mathrm{MFT}} +N^{-2}(\mathcal W_{\mathrm{exch}}+\mathcal W_{\mathrm{ct}}) +N^{-4}\mathcal W_{\mathrm{loop}}.$$

First check that the tree exchange residue equals $C_{12\chi}C_{34\chi}$ and that contact terms restore crossing. Next diagonalize any degenerate double traces and verify their squared OPE coefficients remain nonnegative to the controlled order. Compute the loop double discontinuity from the AdS cut and compare it with the explicit loop. Finally continue to the Regge sheet and verify the assumed growth and phase-shift sign for $E<\Lambda$.

A failure should be localized: a wrong residue indicates normalization, a negative cut indicates nonunitary intermediate data, a crossing polynomial mismatch indicates contact choice, and a time advance inside the error-controlled domain indicates missing states or an inconsistent coefficient.

Adversarial control: fit Euclidean data with an acausal model

Tune contact coefficients so a finite set of Euclidean points is matched, while choosing a loop sign that makes the cut negative or a curvature coupling that gives a Regge time advance. The Euclidean residual can be arbitrarily small. The independent Lorentzian tests reject the model because local polynomials cannot change the cut and low-energy fitting does not remove high-energy polarization dependence.

The evidence ceiling is consistency of the tested correlator through declared $1/N$, derivative, spin, and kinematic orders. Passing does not prove a unique completion or all-order locality. CFT locality criteria combine these checks with spectrum and gap data.

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.
• Meltzer, D., and Sivaramakrishnan, A. (2020), “CFT Unitarity and the AdS Cutkosky Rules,” Journal of High Energy Physics 2020(11), 073. arXiv:2008.11730.