Contact Witten Diagrams

A contact Witten diagram integrates one local bulk vertex against bulk-to-boundary propagators. It is a contribution to a boundary correlator in an AdS perturbation expansion. Its dependence on cross ratios probes the vertex and derivative order, but integration by parts, equation-of-motion terms, and local counterterms prevent a single diagram from defining a unique bulk Lagrangian.

Required background. AdS propagators supplies the normalized external legs.

Helpful background. Four-point cross ratios supplies the conformal kinematics. Predictive EFT power counting supplies the derivative expansion.

The local four-point vertex

For a canonically normalized scalar with interaction

$$S_{\mathrm{int}}=\int_{\mathrm{AdS}}d^{d+1}X\sqrt g\,\frac{\lambda_4}{4!}\phi^4,$$

four source derivatives of the tree-level on-shell action give

$$\langle O(\mathbf x_1)O(\mathbf x_2)O(\mathbf x_3)O(\mathbf x_4)\rangle_{\mathrm{conn}} =-\lambda_4\int d^{d+1}X\sqrt g\prod_{i=1}^4K_\Delta(X;\mathbf x_i),$$

up to the declared Euclidean generating-functional sign. Conformal covariance factors the result into powers of $x_{ij}^2$ times a function of $u=x_{12}^2x_{34}^2/(x_{13}^2x_{24}^2)$ and $v=x_{14}^2x_{23}^2/(x_{13}^2x_{24}^2)$. The remaining integral is a $D$-function Freedman et al. 1999.

Derivative vertices insert contractions of $\nabla K_i$ and yield higher-degree Mellin polynomials. Convergence requires an initial domain in the $\Delta_i$; other values are defined by analytic continuation plus holographic counterterms. At large $N$, the scaling of $\lambda_4$ must match the normalization of $O$ before one quotes an OPE coefficient.

First application: the scalar φ⁴ diagram

For four identical scalars the integral is invariant under all permutations of the external points. After extracting $(x_{12}^2x_{34}^2)^{-\Delta}$, the reduced correlator obeys the corresponding crossing transformations of $u$ and $v$. Its overall coefficient is linear in $\lambda_4C_\Delta^4L^{d+1}$, with additional $L$ powers removed when the bulk field is canonically rescaled.

The OPE limit $u\to0$ contains double-trace families $[OO]_{n,\ell}$. Logarithms at the perturbative order encode their anomalous dimensions; no new single-trace pole is implied by a pure contact vertex. This gives three independent checks: permutation symmetry, dimensional scaling, and the absence of an exchange pole family.

Adversarial control: incomplete integration by parts

Replace a derivative vertex by an integration-by-parts-equivalent expression but discard the boundary term. If the fields have nonvanishing source falloffs or the integral is regulated, that boundary contribution can be a local contact term. The two answers then appear to disagree even though the correctly transformed renormalized action is equivalent. The mismatch is exposed by a changed contact distribution or crossing coefficient.

The evidence ceiling is the coefficient of a perturbative AdS correlator in a stated field basis and scheme. It is not an S-matrix element, an all-orders result, or a unique microscopic interaction. Exchange diagrams add single-particle propagation while retaining contact freedom.

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

• D’Hoker, E., Freedman, D. Z., Mathur, S. D., Matusis, A., and Rastelli, L. (1999), “Graviton Exchange and Complete Four-Point Functions in the AdS/CFT Correspondence,” Nuclear Physics B 562, 353–394. arXiv:hep-th/9903196.
• 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.