AdS Unitarity Cuts and Cutting Rules

AdS cutting rules compute Lorentzian discontinuities by replacing internal bulk propagators with sums over normalizable intermediate states. They test boundary unitarity and factorization in the AdS spectral basis. Only after a controlled flat-space limit do they reduce to ordinary Cutkosky rules with momentum-conserving delta functions.

Required background. Exchange diagrams supplies internal propagation. Lorentzian Witten diagrams supplies discontinuities and orderings. S-matrix unitarity supplies the distinct flat-space relation.

Helpful background. Cutkosky cutting rules supplies the comparison. Largest-time normalization supplies contour cancellations.

Spectral factorization in AdS

A scalar bulk propagator admits a split representation

$$G_\Delta(X,Y)=\int_{-\infty}^{\infty}d\nu\, \rho_\Delta(\nu)\,\Omega_\nu(X,Y),$$

where $\Omega_\nu$ is a harmonic function that factorizes into two bulk-to-boundary objects integrated over a common boundary point. A cut replaces the pole prescription in $\rho_\Delta$ by its discontinuity. Gluing the resulting on-shell subdiagrams produces a conformal partial-wave sum with positive intermediate-state weights in a unitary theory.

This is the AdS analogue of putting a line on shell, but global AdS has discrete energies and Poincaré AdS uses a continuous radial spectral parameter. The measure and two-point normalization cannot be replaced by a flat-space $\delta(p^2+m^2)$ Meltzer and Sivaramakrishnan 2020.

First application: a one-loop scalar bubble

Cut both internal lines of a bubble connecting two tree-level four-point subdiagrams. The double discontinuity factorizes schematically as

$$\operatorname{dDisc}\mathcal G_{\mathrm{bubble}} =\sum_{n,\ell}\mu_{n,\ell}\, \mathcal A^{\mathrm{tree}}_{12\to[n,\ell]} \mathcal A^{\mathrm{tree}}_{[n,\ell]\to34},$$

where $[n,\ell]$ labels double-trace intermediate states and $\mu_{n,\ell}$ is the AdS spectral norm. The support matches the double-trace thresholds and determines nonlocal loop data. Local polynomial contact terms have no such cut and remain reconstruction ambiguities. Explicit bubble, triangle, and box examples verify this gluing picture Meltzer, Perlmutter, and Sivaramakrishnan 2019.

Adversarial control: insert a flat-space delta function

Replace $\mu_{n,\ell}$ by a flat $d$-momentum phase-space measure at finite $L$. The resulting state count has the wrong energy spacing and normalization, so it fails the boundary conformal-block decomposition and the OPE limit. Agreement can be recovered only in a specified $L\to\infty$ scaling where AdS levels coalesce.

The evidence ceiling is perturbative unitarity of a stated AdS correlator through its discontinuities. Cuts do not fix local counterterms, prove an all-orders theory, or themselves define a flat-space S-matrix. Loop renormalization supplies the complementary local 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

• Meltzer, D., Perlmutter, E., and Sivaramakrishnan, A. (2020), “Unitarity Methods in AdS/CFT,” Journal of High Energy Physics 2020(03), 061. arXiv:1912.09521.
• Meltzer, D., and Sivaramakrishnan, A. (2020), “CFT Unitarity and the AdS Cutkosky Rules,” Journal of High Energy Physics 2020(11), 073. arXiv:2008.11730.