Analytic and Lorentzian Bootstrap

Lorentzian correlators contain causal information that is hidden in a single Euclidean expansion. This chapter uses operator orderings, lightcone limits, and discontinuities to recover CFT data. Analyticity becomes predictive only with the growth, convergence, and subtraction assumptions that justify each contour manipulation.

Enter this chapter

Begin with Lorentzian Correlators and Causal Orderings to connect an operator ordering to a definite analytic continuation.

Helpful background. Lorentzian boundary values and the $i\epsilon$ prescription fix operator orderings; OPE convergence and associativity fix the Euclidean starting domain. The comparison with scattering theory uses causal growth and analytic domains and subtracted dispersion relations.

Topics

1. Lorentzian Correlators and Causal Orderings — Tracks causal boundary values, continuation paths, and sheets.
2. The Lightcone OPE and Large-Spin Expansion — Extracts large-spin information from crossed-channel lightcone singularities.
3. Double-Twist Families and Anomalous Dimensions — Computes corrections to families approaching additive twist at large spin.
4. The Lorentzian Inversion Formula — Recovers spin-dependent OPE data from a double discontinuity.
5. Defect Lorentzian Inversion and Lightcone Expansions — Adapts inversion to defect channels and transverse-spin assumptions.
6. Crossing Kernels and Conformal 6j Symbols — Changes representation channels using normalized conformal partial waves.
7. The CFT Regge Limit and Boundedness — States the high-energy growth bounds that control contour arcs.
8. Dispersion Relations for CFT Correlators — Reconstructs correlators with the required subtractions and ambiguities.
9. Analytic Functionals and Polyakov Blocks — Derives sum rules while checking interchange with the OPE.
10. Tauberian Theorems and Asymptotic Spectral Data — Obtains averaged spectral information from positive transform asymptotics.
11. Multipoint Correlators and Multilightcone Limits — Controls the order of null limits in higher-point expansions.
12. Celestial and Cosmological Correlators: Comparing the Axioms — Identifies which ordinary CFT hypotheses change for celestial or cosmological observables.

How the ideas fit together

Continuing operators around a branch point changes their causal ordering. Suitable combinations of these continuations define a double discontinuity, which supplies the input to Lorentzian inversion. The inversion contour can be deformed only where Regge growth—the behavior in a high-energy limit—makes its distant arc harmless. That condition sets the spin range of the result. Caron-Huot 2017, §§ 2–3.

Dispersion relations use related causal data to reconstruct a whole correlator. They must retain subtraction terms that discontinuities cannot determine. Consequently, extending a large-spin answer to low spin or claiming a unique reconstruction needs a separate argument.

Review the chapter

If two correlators have the same double discontinuity, must they have identical low-spin data?

Solution

No. A contribution invisible to that discontinuity can survive as a subtraction or low-spin ambiguity. The inversion formula constrains the domain in which the arc vanishes; outside it, additional input is needed. Equality of the input does not expand the theorem’s spin range.

Where to go next

Continue to Energy Flow, Light-Ray Operators, and Conformal Colliders to turn Lorentzian stress-tensor data into detector observables.

References

• Caron-Huot, Simon. “Analyticity in Spin in Conformal Theories.” Journal of High Energy Physics 2017, 078 (2017). doi:10.1007/JHEP09(2017)078.