Finite-Gap Corrections and Locality Error Budgets

A finite gap to heavy single-trace or higher-spin operators defines a finite bulk cutoff, not exact locality. Integrating out a state of dimension $\Delta_{\mathrm{gap}}\simeq M_{\mathrm{gap}}L$ generates higher-derivative contact terms. Their error grows with energy and spin even while curvature remains weak and $N$ remains large.

Required background. Higher-spin gaps and Einstein obstructions supplies the gap criterion. Bulk interaction scaling supplies loop power counting. Regge causality supplies high-spin growth.

Helpful background. Large-gap locality tests supplies the CFT input. EFT truncation diagnostics supplies error estimation.

From operator gap to derivative expansion

Let $\Lambda=M_{\mathrm{gap}}=\Delta_{\mathrm{gap}}/L$. At energies $E\ll\Lambda$, a four-point Mellin amplitude admits

$$M=M_{\mathrm{light}} +c_0+c_2\frac{s}{\Delta_{\mathrm{gap}}^2} +c_4\frac{s^2+t^2+u_M^2}{\Delta_{\mathrm{gap}}^4}+\cdots,$$

with crossing combinations and coefficients determined by the heavy spectrum. Naturalness of $c_{2k}$ is an assumption; symmetries may suppress terms and large degeneracies may enhance them. Spin introduces another expansion because large $\ell$ probes the Regge trajectory rather than fixed-angle low energy Heemskerk et al. 2009.

First application: first omitted contact correction

Suppose the EFT retains terms through $2k$ derivatives and the first omitted term is natural. For an observable dominated by physical energy $E$,

$$\epsilon_{\mathrm{gap}} \sim\left(\frac{E}{\Lambda}\right)^{2k+2} =\left(\frac{EL}{\Delta_{\mathrm{gap}}}\right)^{2k+2}.$$

The full error budget also includes $N^{-2}$ bulk loops, $\alpha'/L^2$ string corrections when distinct from the measured gap, KK thresholds, numerical/OPE uncertainty, and Regge enhancement at large spin. These terms should be reported separately rather than added as one unexplained percentage.

Adversarial control: approach the gap at weak curvature

Keep $L/\ell_p$ enormous but raise $EL$ from $2$ to $0.8\Delta_{\mathrm{gap}}$. Quantum gravity from small $N$ remains suppressed, yet the derivative error becomes order one. A two-derivative correlator can fail well before Planckian curvature. Raising spin toward the leading higher-spin trajectory causes an analogous nonuniform breakdown.

The evidence ceiling is an observable-specific local EFT through a stated derivative, loop, energy, and spin order. A finite gap never licenses exact locality, and a large scalar gap without a higher-spin gap is insufficient for Einstein gravity. The combined OPE and causality checks test the retained EFT output.

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

• Camanho, X. O., Edelstein, J. D., Maldacena, J., and Zhiboedov, A. (2016), “Causality Constraints on Corrections to the Graviton Three-Point Coupling,” Journal of High Energy Physics 2016(02), 020. arXiv:1407.5597.
• Heemskerk, I., Penedones, J., Polchinski, J., and Sully, J. (2009), “Holography from Conformal Field Theory,” Journal of High Energy Physics 2009(10), 079. arXiv:0907.0151.