AdS Integrals and D-Functions

Scalar contact integrals in AdS are conventionally organized as $D$-functions. Factoring their position dependence defines barred $\bar D$-functions of cross ratios, and differential recursions generate many derivative and exchange results. A useful basis must retain the normalization prefactor, analytic sheet, and permutation map; otherwise equal-looking symbols can differ by Gamma functions and powers of distances.

Required background. Contact diagrams supplies the defining integral. Exchange diagrams supplies the reductions that generate contact terms.

Helpful background. Special functions and boundary data supplies analytic-continuation discipline. Four-point cross ratios supplies $u,v$.

Definition and reduction

With unit-normalized kernels suppressed for clarity, define

$$D_{\Delta_1\Delta_2\Delta_3\Delta_4}(x_i) =\int_{\mathrm{AdS}}d^{d+1}X\sqrt g\prod_{i=1}^4 \left(\frac{z}{z^2+\lvert\mathbf x-\mathbf x_i\rvert^2}\right)^{\Delta_i}.$$

Conformal covariance extracts fixed powers of $x_{ij}^2$ and leaves $\bar D_{\Delta_1\Delta_2\Delta_3\Delta_4}(u,v)$. Schwinger parameters combine denominators, the AdS coordinate integration becomes Gaussian, and the remaining simplex integral gives analytic continuation in the $\Delta_i$. Raising a $\Delta_i$ or differentiating in $u,v$ produces recursion relations D’Hoker and Freedman 2002.

The four external normalizations $C_{\Delta_i}$ are not universally included in $D$. The page’s convention keeps them in the Witten rule and treats $D$ as the geometric integral above.

First application: a four-scalar contact integral

For equal $\Delta$, the $\phi^4$ diagram is proportional to $C_\Delta^4D_{\Delta\Delta\Delta\Delta}$. Permuting the external points maps $(u,v)$ by the usual crossing transformations and reshuffles the extracted distance powers. This checks the barred-function identities without assuming that $\bar D$ itself is invariant.

In the $12$ OPE limit $u\to0$, the expansion contains powers $u^{\Delta+n}$ and logarithms $u^{\Delta+n}\log u$. At first perturbative order the logarithmic coefficient gives the anomalous dimension of $[OO]_{n,\ell}$. Direct parameter integration and recursion from a seed function must agree on that coefficient, providing a normalization-independent short-distance check Freedman et al. 1999.

Adversarial control: mixed D-function conventions

Take a tabulated barred function that absorbed a factor $\Gamma(\Sigma-d/2)/\prod_i\Gamma(\Delta_i)$ and combine it with kernels that assume the factor is absent. Crossing ratios remain correct, so a plot can look plausible, but the known three- or four-point coefficient and the OPE logarithm disagree. A seed contact diagram exposes the error immediately.

The evidence ceiling is an analytically continued basis for specified AdS integrals. It does not choose Lorentzian sheets automatically and does not remove ultraviolet counterterms at loops. AdS cutting rules use spectral rather than $D$-function organization for discontinuities.

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., and Freedman, D. Z. (2002), “Supersymmetric Gauge Theories and the AdS/CFT Correspondence,” in Strings, Branes and Extra Dimensions: TASI 2001. arXiv:hep-th/0201253.
• 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.