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 strongest conclusion 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 compares these assumptions and limitations.

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.