Loop Witten Diagrams and Bulk EFT Renormalization

Loop Witten diagrams combine ordinary short-distance divergences with AdS boundary behavior and an infinite tower of intermediate CFT states. Renormalization must use local bulk counterterms, include subdivergences, and keep induced boundary contacts distinct from nonlocal correlator data. The result is an EFT prediction at a fixed order in $1/N$ and $E/\Lambda_{\mathrm{bulk}}$, not an ultraviolet completion.

Required background. Bulk interaction scaling and cutoffs supplies the loop order. Contact diagrams and exchange diagrams supply the tree basis. Loops and EFT renormalization supplies generic locality.

Helpful background. One-loop gravity as EFT supplies gravitational power counting. Counterterms and subdivergences supplies recursive subtraction. Background-field gauge fixing, matter matching, and nonlocal form factors supply gauge and nonlocal checks.

Local subtraction in AdS

For a scalar $\lambda_4\phi^4/4!$ theory, the one-loop four-point bubble contains

$$\mathcal W_{\mathrm{bub}} \sim\lambda_4^2\int dX\sqrt g\,dY\sqrt g\, K_1K_2(X)G(X,Y)^2K_3K_4(Y).$$

The divergence arises as $Y\to X$ and is therefore local. In dimensional, spectral, or position-space regularization its pole is canceled by $\delta\lambda_4\int\sqrt g\,\phi^4$, together with every lower-order subdivergence. Curvature allows additional local operators such as $R\phi^2$ and derivative contacts at the relevant EFT order. A nonlocal subtraction would change separated-point data and violate bulk locality.

After subtraction, scale dependence of local contact terms cancels that of the loop. The remaining logarithms and discontinuities determine anomalous dimensions and OPE coefficients. Holographic unitarity reconstructs much of this nonlocal one-loop data from tree correlators, leaving local counterterm freedom Aharony et al. 2017.

First application: a renormalized scalar bubble

Regulate the coincident propagator in $d+1-2\epsilon$ dimensions and write $\lambda_{4,0}=\mu^{2\epsilon}(\lambda_4+\delta\lambda_4)$. Fix $\delta\lambda_4$ by canceling the $1/\epsilon$ coefficient of the contact integral. Expanding the renormalized correlator in the $12$ OPE gives

$$\mathcal G^{(2)}(u,v)\supset \sum_{n,\ell}a^{(0)}_{n,\ell} \left(\frac12\gamma^{(2)}_{n,\ell}\log u +\frac18(\gamma^{(1)}_{n,\ell})^2\log^2u\right) G_{n,\ell}(u,v).$$

The double logarithm is fixed by lower-order data; the single logarithm yields the new anomalous dimension after mixing is resolved. Its regulator independence is a stronger check than cancellation of the integrated pole alone.

Adversarial control: nonlocal or incomplete subtraction

Subtract a function of $u,v$ chosen to cancel the entire bubble, or omit a one-loop subdivergence inside a higher-loop graph. The first removes a physical cut and cannot arise from a local AdS operator; the second leaves regulator dependence and incorrect $\log^2u$ terms. Crossing and the cut calculation expose both failures.

The evidence ceiling is a renormalized correlator through a declared bulk-loop and derivative order. Unknown higher counterterms scale as powers of $E/\Lambda_{\mathrm{bulk}}$, and string, KK, or finite-gap states can enter before Planckian energies. Loop bases and mixing treats the remaining spectral degeneracy.

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

• Aharony, O., Alday, L. F., Bissi, A., and Perlmutter, E. (2017), “Loops in AdS from Conformal Field Theory,” Journal of High Energy Physics 2017(07), 036. arXiv:1612.03891.
• Fitzpatrick, A. L., and Kaplan, J. (2012), “Unitarity and the Holographic S-Matrix,” Journal of High Energy Physics 2012(10), 032. arXiv:1112.4845.