Bulk-EFT Information in Double-Trace Data and Large-Spin Anomalous Dimensions

Double-trace dimensions and OPE coefficients record the energies and overlaps of two-particle states in AdS. At large spin, a light crossed-channel exchange produces a controlled inverse-power tail whose exponent is its twist. This tail diagnoses a long-range bulk force, but its coefficient can be extracted only after operator mixing and all exchanges of the same leading twist are included.

Required background. Exchange Witten diagrams supplies the bulk origin. Double-twist anomalous dimensions supplies the imported CFT result.

Helpful background. The lightcone OPE supplies the expansion. Large-N contact ambiguities supplies short-range freedom.

Binding energy from crossed-channel twist

For identical scalars $O$, the leading two-particle dimensions are

$$\Delta_{[OO]_{n,\ell}}=2\Delta_O+2n+\ell+\gamma_{n,\ell}.$$

Exchange of a crossed-channel primary with twist $\tau_\chi=\Delta_\chi-J_\chi$ gives, at fixed $n$ and large conformal spin $\mathcal J$,

$$\gamma_{n,\ell}^{(\chi)} \sim-\frac{A_{n,\chi}}{\mathcal J^{\tau_\chi}} \left(1+O(\mathcal J^{-2})\right).$$

The coefficient contains the squared OPE coupling and known kinematics. For identical neutral scalars and a positive-norm even-spin exchange, the leading sign is often negative, matching attractive binding; charges and tensor structures can alter that interpretation. The exponent is more robust than a blanket sign claim Fitzpatrick et al. 2013, Komargodski and Zhiboedov 2013.

First application: one light scalar exchange

Insert a scalar $\chi$ of dimension $\Delta_\chi$ in the crossed channel. Expanding its block near the light cone and matching powers of $z$ and $1-\bar z$ yields $\gamma_{0,\ell}\propto-C_{OO\chi}^2\mathcal J^{-\Delta_\chi}$. In AdS, large $\ell$ separates the two particles by a large impact parameter, so the power-law tail is the binding energy from the light field.

Contact interactions contribute only at bounded spin at a fixed derivative order or fall faster in the large-spin expansion. Thus the leading tail separates long-range exchange from local contact freedom. Degenerate $[OO]$ operators require diagonalization; a single correlator measures a projected average rather than an eigenvalue.

Adversarial control: ignore degeneracy or a competing exchange

Let two double traces share $(n,\ell)$ and let a stress tensor have the same leading twist as another crossed-channel operator. Fitting one power law to one correlator assigns the combined coefficient to $\chi$. A rotation in the degenerate subspace changes that coefficient, and including the omitted exchange changes its sign or magnitude. Only the mixing eigenvalues and sum of all leading-twist contributions are invariant.

The evidence ceiling is long-distance bulk-force information in a stated large-spin, large-$N$ regime. It does not reconstruct short-distance vertices or guarantee a higher-spin gap. Lorentzian inversion turns the same discontinuity into a systematic spectral integral.

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

• Fitzpatrick, A. L., Kaplan, J., Poland, D., and Simmons-Duffin, D. (2013), “The Analytic Bootstrap and AdS Superhorizon Locality,” Journal of High Energy Physics 2013(12), 004. arXiv:1212.3616.
• Komargodski, Z., and Zhiboedov, A. (2013), “Convexity and Liberation at Large Spin,” Journal of High Energy Physics 2013(11), 140. arXiv:1212.4103.