Large-Gap Constraints and CFT-Side Locality Tests

A large higher-spin gap can make low-energy CFT data look hierarchical, but “looks local” is not a theorem until the expansion, Regge bound, spin sector, and finite-gap errors are specified. This page compares CFT-side diagnostics only. It treats approximate bulk locality as a possible later interpretation, never as an input to the test.

Required background. Large-N Crossing, Double-Trace Data, and Contact Ambiguities supplies the order-by-order solution and contact basis. The CFT Regge Limit and Boundedness supplies sheet and growth conditions. Helpful background. Energy Conditions, Causality, and Regge Consistency compares positivity and causal assumptions.

Defining the gap before testing it

Let $\mathcal S_{\rm light}$ be a declared finite set of single-trace primaries that includes the stress tensor and any retained currents or low operators. A typical higher-spin gap is

$$\Delta_{\rm gap} =\inf_{\substack{\mathcal O\notin\mathcal S_{\rm light}\\J_{\mathcal O}>2}} \Delta_{\mathcal O}.$$

The test is meaningful only after answering:

• Is the infimum taken over all symmetry sectors or one OPE?
• Are conserved higher-spin currents absent, or explicitly exempted?
• Is the gap in dimension, twist, or a Mellin mass-like variable?
• Which light scalars and spin-one operators remain below it?
• Is $g\to0$ taken before, after, or jointly with $\Delta_{\rm gap}\to\infty$?

The double-trace towers required by factorization are not included in this single-trace gap. Counting them would make the hypothesis fail by definition.

A gap-suppressed contact hierarchy

After declared light exchanges have been separated, suppose the remaining crossing-symmetric Mellin contribution admits

$$P(s,t;\Delta_{\rm gap}) =\sum_{k=0}^{K} \frac{c_k}{\Delta_{\rm gap}^{\alpha_k}} P_k(s,t)+R_K(s,t),$$

where $P_k$ is a symmetric polynomial of increasing degree, $\alpha_{k+1}>\alpha_k$, and $R_K$ is bounded in a stated domain. This becomes a controlled derivative-like hierarchy only if:

$$|s|,|t|,|u|\le \Lambda^2, \qquad 1\ll\Lambda\ll\Delta_{\rm gap},$$

and the remainder is uniformly smaller than the retained terms there. The powers $\alpha_k$ are not universal: they depend on the definition of the gap, operator normalization, symmetries, and selection rules.

A one-point scaling comparison is not evidence of a hierarchy. At least several gap values and several Mellin kinematic points are needed to distinguish a power law from an accidental cancellation; uncertainty and subleading terms must be included.

Four complementary diagnostics

Polynomial growth in Mellin space

At fixed transverse variable in a declared complex strip, impose

$$|M(s,t)|\le C(t)(1+|s|)^p.$$

This limits the degree of allowed contact polynomials and the number of subtractions in a dispersion relation. The direction in complex $s$, avoided pole sequences, and uniformity in $t$ are part of the hypothesis. A bound verified only on the real axis is not automatically a Regge bound.

Regge behavior on the correct Lorentzian sheet

The position-space Regge limit is an analytic continuation, not the Euclidean limit $z,\bar z\to0$. Its growth is controlled by the leading Regge trajectory under unitarity and boundedness assumptions Costa, Gonçalves, and Penedones 2012, §§2–4. A contact term acceptable in a Euclidean OPE may be excluded by this Lorentzian growth.

Double-trace anomalous dimensions

For low excitation number relative to the gap, a controlled expansion should keep

$$g^2|\gamma_{n,J}|\ll1$$

and organize $\gamma_{n,J}$ according to the same contact hierarchy. Growth with $n$ diagnoses higher polynomial degree. The expansion can fail when $n$ approaches $\Delta_{\rm gap}$, even if it is excellent at fixed $n$. Mixing and low-spin inversion ambiguities must be retained.

Causality and positivity tests

Reflection positivity, averaged-null-energy positivity, and Lorentzian causality can constrain signs or combinations of CFT data. The hypotheses differ. A sign derived for an identical Hermitian correlator cannot be transferred to a nonunitary theory or an arbitrary mixed tensor structure. Causal commutator constraints in appropriate states provide powerful consistency checks Hartman et al. 2016, §§2–4, but translating them into a specific contact coefficient requires the same normalization and Regge assumptions used in the correlator.

No single diagnostic is sufficient in complete generality. Agreement among several independently normalized tests is stronger evidence than any one of them.

Necessary, sufficient, and conditional claims

Observation	Necessary assumptions	Supported CFT statement	Why sufficiency can fail
large single-trace higher-spin gap	stable large-N taxonomy and declared sector	high-spin single-trace operators are parametrically heavy in that sector	says nothing by itself about Mellin growth or correlator uniformity
bounded Mellin growth	valid transform, contour, and complex-direction bound	finite subtraction/contact degree in the tested domain	may hold over a limited domain or without a parametrically large gap
gap-suppressed polynomial coefficients	normalization and multi-point scaling sequence	controlled low-Mellin-variable hierarchy	omitted exchanges or nonuniform limits may enter near the gap
small low-lying $\gamma_{n,J}$	mixing resolved and $n,J$ range declared	perturbative double-trace spectrum in that range	low-spin data do not control all spins or high excitation number
causal/positive sign pattern	unitarity, state, smearing, and tensor sector	stated sign or sum-rule constraint	does not construct a complete correlator

The classic reconstruction arguments show how factorization plus a sparse spectrum constrains crossing Heemskerk et al. 2009, §§3–5. Polynomial boundedness was isolated as an additional effective-field-theory-type condition by Fitzpatrick and Kaplan 2013, §§2–4. Those results motivate the combined test; they do not license dropping its hypotheses.

A reproducible finite-gap test

Given Mellin data at several gap values $G_i$, choose a fixed compact domain $D$ away from poles and fit

$$c_k(G)=A_kG^{-\alpha_k} \left(1+b_kG^{-\beta_k}+\cdots\right).$$

Report:

1. the $G_i$, Mellin points, contours, and pole exclusions;
2. the covariance of the fitted coefficients;
3. competing exponents and the fit window;
4. residuals across all of $D$, not only training points;
5. stability under removing the smallest or largest $G_i$;
6. the order of $g$, $G$, and large-variable limits;
7. a withheld configuration that can falsify the inferred hierarchy.

If the extracted exponent changes substantially when one gap value is removed, the result is an unresolved trend, not a scaling law.

Evidence and interpretation boundary

Each row names the transformation that makes the input comparable to the CFT test.

Input	Canonical source and version	Convention transform	Evidence carried	Allowed CFT conclusion	Optional later interpretation	Does not prove
factorized correlator family	Large-N CFT Data and Vector Models; cited setup frozen in this edition	unit two-point functions and explicit $g$	asymptotic correlator coefficients	order-by-order crossing problem	weakly coupled multiparticle language	a bulk dictionary
higher-spin gap	spectrum file or analytic premise; version and sector required	dimension/twist definition and exception list	exact assumption or bounded spectrum	gap statement in the declared OPE sector	finite low-energy field content	completeness outside that sector
Mellin polynomial hierarchy	Mellin-Space CFT Correlators; displayed convention	Gamma measure, contour, symmetric basis, gap units	analytic or fitted coefficients with errors	finite-domain gap suppression	derivative expansion	local bulk vertices
Regge bound	The CFT Regge Limit and Boundedness; sheet and spin domain required	continuation path and growth normalization	conditional analytic bound	degree/subtraction restriction	high-energy bulk causality	an S-matrix bound without a flat limit
double-trace response	Large-N Crossing and Contact Ambiguities; order in $g$ required	mixing basis and OPE weights	anomalous-dimension averages	perturbative spectral response	two-particle energy shifts	individual eigenvalues when mixing is unresolved
combined diagnostic	this page; evidence checked through 9 August 2026	common $g$, gap, Mellin, and limit conventions	conditional multi-test agreement	stated CFT-side locality diagnostic	Volume 15 may assess approximate locality	that the conditions are universally sufficient

Current evidence boundary

The research-sensitive claims and links on this page were checked through 9 August 2026. The page states structural diagnostics and classic conditional results; it does not freeze a current optimal numerical gap, universal Wilson-coefficient bound, or theorem that the displayed conditions are sufficient for a bulk dual. Updated bounds and unsettled interpretations belong on the analytic and numerical conformal-bootstrap research map.

Common pitfalls

Taking $\Delta_{\rm gap}\to\infty$ at fixed large Mellin variable. The hierarchy is normally controlled only below the gap. State a joint scaling or take the low-variable limit first.

Using necessity as sufficiency. A large gap can be part of an approximate-locality criterion without constructing a dictionary or proving a local completion.

Fitting a power from one ratio. Finite-gap corrections can mimic an exponent. Use multiple values, residuals, and stability tests.

Exercises

Let $c(G)=A/G^4+B/G^6$. Compute the effective exponent $\alpha_{\rm eff}(G)=-d\log|c|/d\log G$ and show why two moderate gap values may not reveal the asymptotic power four.

Solution

For nonzero $c$,

$$\alpha_{\rm eff}(G) =\frac{4A G^{-4}+6B G^{-6}}{A G^{-4}+B G^{-6}} =4+\frac{2B}{AG^2+B}.$$

Only when $|B|\ll|A|G^2$ is the correction small. If $A$ and $B$ have opposite signs, a nearby cancellation can make the effective exponent arbitrarily large or unstable.

References

• Costa, M. S., Gonçalves, V., and Penedones, J. “Conformal Regge Theory.” Journal of High Energy Physics 2012, 091 (2012). arXiv. DOI.
• Fitzpatrick, A. L., and Kaplan, J. “AdS Field Theory from Conformal Field Theory.” Journal of High Energy Physics 2013, 054 (2013). arXiv. DOI.
• Hartman, T., Jain, S., and Kundu, S. “Causality Constraints in Conformal Field Theory.” Journal of High Energy Physics 2016, 099 (2016). arXiv. DOI.
• Heemskerk, I., Penedones, J., Polchinski, J., and Sully, J. “Holography from Conformal Field Theory.” Journal of High Energy Physics 2009, 079 (2009). arXiv. DOI.