Analytic and Lorentzian Bootstrap

Lorentzian bootstrap methods extract CFT data from causal orderings, singular limits, and discontinuities rather than from a Euclidean power series alone. Their strength is also their main danger: an answer can change when one changes the continuation path, sheet, Regge bound, contour at infinity, subtraction prescription, or spin range. This chapter develops the methods as conditional statements whose hypotheses remain visible from the first continuation to the final spectral conclusion.

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.

From Euclidean data to Lorentzian information

For an identical real scalar primary normalized by

$$\langle \phi(x)\phi(0)\rangle=\frac{1}{(x^2)^{\Delta_\phi}},$$

write the Euclidean four-point function as

$$\langle\phi_1\phi_2\phi_3\phi_4\rangle =\frac{\mathcal G(z,\bar z)}{(x_{12}^2x_{34}^2)^{\Delta_\phi}}, \qquad u=z\bar z, \qquad v=(1-z)(1-\bar z).$$

In Euclidean signature, $\bar z=z^*$ and the OPE converges in radial domains. A Lorentzian continuation makes $z$ and $\bar z$ independent and gives branch points physical meaning. Boundary values on different sides of a cut encode different Wightman orderings; their difference gives a commutator, and a double discontinuity removes terms that are single-valued around the crossed-channel branch point. This passage from Euclidean single-valuedness to Lorentzian boundary values is the common starting point for lightcone expansions, inversion, and dispersion relations Caron-Huot 2017, §§2–3.

The methods answer different questions:

Method	Controlled input	Typical output	Essential qualification
Lightcone expansion	A crossed-channel singularity with a declared ordering	Large-spin operator families and asymptotic OPE data	Twist gap, order of limits, and asymptotic remainder
Lorentzian inversion	A double discontinuity and a Regge bound	A meromorphic coefficient function, analytic in spin	Valid spin half-plane, arc terms, endpoints, and low-spin additions
Crossing kernel	Principal-series partial waves and their pairings	Change of OPE channel and pole families	Shadow normalization, contour, discrete residues, and tensor basis
CFT dispersion relation	Discontinuities plus growth control	A reconstructed correlator	Subtractions, crossing completion, and contact terms
Analytic functional	A function space on which sums and integrals may be exchanged	Spectral sum rules or bounds	Swapping, endpoint convergence, completeness, and zero structure
Tauberian theorem	A positive spectral measure and transform asymptotics	Averaged or integrated spectral growth	Positivity, averaging scale, and a stated remainder

The same formula can participate in more than one row. For example, the Lorentzian inversion formula can generate a dispersion kernel after its pole data are resummed, but the reconstruction still needs its own convergence and subtraction argument Carmi and Caron-Huot 2020, §§2.2–4.2.

A solvable reference correlator

Generalized free field theory supplies a useful normalization check. Its reduced correlator is

$$\mathcal G_{\mathrm{GFF}}(u,v) =1+u^{\Delta_\phi}+\left(\frac{u}{v}\right)^{\Delta_\phi}.$$

Crossing requires even-spin double-twist primaries $[\phi\phi]_{n,\ell}$ with

$$\Delta_{n,\ell}=2\Delta_\phi+2n+\ell, \qquad n=0,1,2,\ldots, \qquad \ell=0,2,4,\ldots .$$

This one correlator tests several distinct operations. A continuation around $\bar z=1$ fixes the double-discontinuity phases. The lightcone limit recovers the double-twist accumulation points. Inversion recovers poles and residues. A dispersion relation reconstructs the original function only after its contact or subtraction terms are included. An analytic functional should annihilate the appropriate generalized-free spectrum only when its interchange with the OPE sum is justified. A reproducible calculation should compare these routes with frozen conventions.

How to use the chapter

Begin with Lorentzian Correlators and Causal Orderings whenever the sheet or operator order is not already fixed. The lightcone OPE and double-twist corrections then give the most direct asymptotic route. The Lorentzian inversion formula turns the same causal data into a spin-analytic coefficient function; its defect specialization requires a separate channel and transverse-spin analysis.

Crossing kernels organize changes of representation channel. Regge boundedness determines whether contour arcs vanish, and CFT dispersion relations reconstruct correlators with all necessary subtractions. Analytic functionals and Tauberian theorems turn crossing or singular behavior into sum rules and averaged spectral statements. Higher-point limits require an OPE tree and an explicit order of null limits. The final page compares ordinary CFT assumptions with celestial and cosmological correlators without identifying their observables.

A statement is complete only with its domain

Before using an analytic-bootstrap result, record the following scientific information:

• spacetime dimension, signature, external representations, and state;
• normalization of the correlator, blocks, shadows, and OPE coefficients;
• Lorentzian operator ordering, $i\epsilon$ hierarchy, continuation path, and sheet;
• cut and discontinuity convention, including phases for unequal operators;
• OPE or partial-wave channel and its convergence domain;
• Regge parameter, smearing if used, and the bound controlling the arc;
• inversion contour, principal-series measure, valid spin range, and discrete residues;
• subtraction polynomial, contact ambiguity, and crossing completion;
• whether the result is exact, asymptotic, averaged, perturbative, or conditional.

This information is not ceremonial. Removing positivity invalidates many Tauberian and sign conclusions; weakening the Regge bound can restore an arc; changing the continuation path can exchange a commutator for its negative; and continuing a large-spin expression to low spin can miss precisely the terms invisible to the double discontinuity. The modern analytic-bootstrap literature emphasizes both the reach and these qualifications Poland et al. 2019, §IX, pp. 65–66.

What you should be able to do

After completing the chapter, you should be able to:

1. choose a Lorentzian ordering and trace its sheet and discontinuities from a Euclidean correlator;
2. derive a large-spin or inversion result for a solvable correlator, including its contour and spin domain;
3. identify every arc, subtraction, contact, low-spin, and nonperturbative ambiguity left by the derivation;
4. distinguish pointwise spectral information from averaged Tauberian information; and
5. stop at the precise boundary where a celestial, cosmological, scattering, defect, or holographic interpretation needs additional physical input.

References

• Carmi, Dean, and Simon Caron-Huot. “A Conformal Dispersion Relation: Correlations from Absorption.” Journal of High Energy Physics 2020, 009 (2020). doi:10.1007/JHEP09(2020)009.
• Caron-Huot, Simon. “Analyticity in Spin in Conformal Theories.” Journal of High Energy Physics 2017, 078 (2017). doi:10.1007/JHEP09(2017)078.
• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91, 015002 (2019). doi:10.1103/RevModPhys.91.015002.