Diagram Normalization and Reproducibility Benchmarks

A Witten-diagram pipeline should reproduce simple analytic fixtures before it is trusted on a complicated correlator. The minimum suite fixes the AdS measure, scalar two-point normalization, cubic contact coefficient, exchange residue, Ward identities, crossing permutations, contour prescription, and numerical precision. Two independent failures should expose any deliberately perturbed convention.

Required background. Dictionary normalization and global data supplies the target coefficients. AdS propagators supplies the first fixture.

Helpful background. Benchmark provenance and reproduction supplies computational records. Basis translation and scheme dependence supplies convention maps.

One declared convention set

Take Euclidean AdS radius $L$, action

$$S=\int d^{d+1}X\sqrt g\left[ \frac12(\nabla\phi)^2+\frac12m^2\phi^2+\frac g{3!}\phi^3\right],$$

standard quantization, and $(-\nabla^2+m^2)G=\delta/\sqrt g$. With $K_\Delta=C_\Delta[z/(z^2+r^2)]^\Delta$, the renormalized two-point coefficient is proportional to $(2\Delta-d)C_\Delta L^{d-1}$. The cubic source derivative gives $-gC_{\Delta_1}C_{\Delta_2}C_{\Delta_3}$ times the known AdS three-kernel integral. These conventions must be stored together; copying only the final coefficient is insufficient Freedman et al. 1999.

For exchange, the residue of the single-trace block must equal the product of the two normalized three-point coefficients. This factorization is independent of contact additions and therefore cross-checks both external normalization and the internal Green function.

First application: three linked fixtures

First, integrate the kernel against a smooth boundary test function and recover its source delta function. Second, evaluate the three-point contact integral and divide by the square roots of the three two-point coefficients to obtain $C_{123}$. Third, calculate the exchange diagram and verify

$$\operatorname*{Res}_{\Delta=\Delta_\chi}\mathcal W_s =C_{12\chi}C_{34\chi}$$

in the chosen conformal-block normalization. A fourth check permutes identical external legs and demands the correct crossing transformation. Numerical quadrature should quote working precision, convergence under domain subdivision, and comparison with the analytic seed.

Adversarial control: perturb measure or normal sign

Change $\sqrt g=L^{d+1}z^{-d-1}$ by one power of $z$, or reverse the outward-normal sign in the renormalized on-shell action. The kernel delta test, two-point coefficient, and exchange residue cannot all remain correct. Tuning $g$ to repair the three-point number leaves at least the two-point and factorization fixtures failing. Requiring multiple independent tests prevents a compensating-error “success.”

The evidence ceiling is reproducibility of a perturbative pipeline in the declared action, quantization, contour, and scheme. It does not validate a bulk EFT beyond its $1/N$, curvature, KK, or derivative cutoff. The next chapter begins with Mellin normalization conventions and carries this discipline into scattering interpretations.

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

• Freedman, D. Z., Mathur, S. D., Matusis, A., and Rastelli, L. (1999), “Correlation Functions in the CFT$_d$/AdS$_{d+1}$ Correspondence,” Nuclear Physics B 546, 96–118. arXiv:hep-th/9804058.
• Witten, E. (1998), “Anti-de Sitter Space and Holography,” Advances in Theoretical and Mathematical Physics 2, 253–291. arXiv:hep-th/9802150.