Correlators, OPE, and Conformal Blocks

Conformal symmetry reduces correlation functions to a small set of kinematic structures, but the remaining coefficients are meaningful only after operator normalizations, tensor bases, OPE domains, channels, and analytic sheets have been fixed. This chapter turns local operator products into a precise package of conformal data and then shows how one family contributes a block, how convergent channel sums reconstruct correlators, and how associativity plus reflection positivity becomes a bootstrap constraint.

Helpful background. Free-Field OPE Preview separates the general local Wilson expansion from its conformal specialization. The State–Operator Correspondence supplies radial states and cylinder energies. Tempered Distributions and Fourier Calculus supplies the distributional meaning of momentum-space correlators. Multiplets, Invariants, and Selection Rules supplies internal-symmetry sectors.

Enter the correlator problem

Question	Start with	Output
What do covariance and permutations fix for scalar correlators?	Scalar Two- and Three-Point Functions	A normalized two-point metric and scalar three-point coefficients
Which spinning structures are independent?	Spinning Correlators and Tensor Structures	A basis with conservation, parity, chirality, and exchange constraints
What is the independent conformal data?	From the Local OPE to Conformal Data	Spectrum, representations, two-point metric, and three-point tensors
Where may an OPE sum be rearranged?	OPE Convergence, Associativity, and Domain Control	A nested-sphere domain and a qualified remainder estimate
Which variables label four-point kinematics?	Cross Ratios and Four-Point Kinematics	Cross ratios, channel maps, and sheet conventions
How is one family summed?	Conformal Blocks and Casimir Equations	A channel-normalized Casimir eigenfunction with OPE boundary data
What does a shadow integral construct?	Conformal Partial Waves and the Shadow Formalism	A block-plus-shadow harmonic function and its projection
Where does positivity enter crossing?	Crossing Equations and Positivity	A positive scalar sum rule or a positive-semidefinite matrix system
How are several operators and symmetry sectors coupled?	Mixed Correlators and Global-Symmetry Sectors	A closed, basis-covariant crossing system
How are Ward identities solved after Fourier transformation?	Momentum-Space Correlators and Conformal Ward Identities	Form-factor equations with contact, semilocal, and anomaly terms retained

The recommended reading order follows the table from scalar correlators through crossing. The spinning, shadow, mixed, and momentum-space pages can then be read as branches, but each imports the same normalization and domain conventions.

A complete four-point specification

For Euclidean scalar primaries, define

$$u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}.$$

A statement such as “the correlator satisfies crossing” is incomplete until the following data are recorded:

Layer	Required declaration	Why it cannot be inferred later
Spacetime	Dimension, Euclidean or Lorentzian signature, connected configuration space	Reality regions and singular hypersurfaces change
Operators	Scaling dimensions, Spin and internal representations, Hermiticity, statistics	These determine selection rules and positivity
Normalization	Two-point metric and three-point tensor basis	OPE coefficients transform under basis changes
Four-point prefactor	Which powers of $x_{ij}^2$ are removed	The reduced correlator and crossing vector depend on it
OPE channel	Pairing, radial center, and separating sphere	Convergence is channel- and geometry-dependent
Block convention	Casimir normalization and leading OPE asymptotic	A Casimir equation also admits the shadow solution
Analytic data	Branches, continuation path, operator ordering, $i\epsilon$ prescription	Lorentzian orderings live on different boundary values or sheets
Positivity	Reflection-positive inner product and conjugate external ordering	Crossing alone does not make coefficients nonnegative
Distributions	Separated-point part, contact terms, counterterm scheme	Fourier transforms and Ward identities otherwise lose information

For four identical Hermitian scalars $\phi$ of dimension $\Delta_\phi$ in an orthonormal basis, a convenient convention is

$$\langle\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle =\frac{\mathcal G(u,v)} {(x_{12}^2x_{34}^2)^{\Delta_\phi}}.$$

In the $(12)(34)$ channel,

$$\mathcal G(u,v) =1+\sum_{\mathcal O\neq\mathbf1} \lambda_{\phi\phi\mathcal O}^{\,2} g_{\Delta,\ell}(u,v),$$

where the displayed square is nonnegative only under the Hermiticity, orthonormality, reflection-positivity, and identical-pairing assumptions stated above. Bose symmetry selects even spin. Interchanging points $1$ and $3$ gives

$$v^{\Delta_\phi}\mathcal G(u,v) =u^{\Delta_\phi}\mathcal G(v,u).$$

This familiar equation is the end of a chain of justified steps, not its starting assumption. The normalization and positivity convention follows Simmons-Duffin 2017, §§ 5–7; the bootstrap data model and its limitations are reviewed in Poland, Rychkov, and Vichi 2019, §§ III–IV.

Objects that must remain distinct

• A tensor structure is a kinematic invariant fixed by representations and positions.
• An OPE coefficient is dynamical data relative to a two-point metric and structure basis.
• A conformal block sums descendants of one primary in one channel and one normalization.
• A partial wave is a single-valued harmonic-analysis object that generally contains both a block and its shadow.
• A channel sum is meaningful first in its convergence domain and elsewhere by a declared analytic continuation.
• A crossing equation equates channel representations of one correlator; positivity is an additional consequence of a positive inner product and suitable conjugation.

Confusing any adjacent pair changes the conclusion. In particular, single-valuedness of a Euclidean partial wave does not make an individual OPE block single-valued on every Lorentzian sheet, and a positive two-point metric does not make every mixed-correlator coefficient a scalar square.

Handoffs

The general local Wilson expansion remains with Free-Field OPE Preview; this chapter begins when conformal covariance organizes that expansion into families. Conformal Field Theory in One Dimension makes ordering sectors and exact hypergeometric blocks explicit. Numerical Conformal Bootstrap turns crossing vectors into finite approximations and certificates. Analytic and Lorentzian Bootstrap continues the same correlator data to discontinuities, inversion, and Regge limits. Defect, thermal, and supersymmetric chapters reuse the specification table rather than silently changing it.

Review the chapter

Why is a list of dimensions and OPE coefficients insufficient to reproduce a spinning four-point function?

Solution

Spinning correlators also require a tensor-structure basis at each three-point vertex, its normalization and permutation matrices, and a convention for spinning blocks. Degenerate exchanged operators carry matrices of OPE coefficients, while conservation, parity, chirality, and dimension-specific identities can reduce the basis. Analytic channel and branch data are still required.

References

• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI; Open PDF
• Simmons-Duffin, David. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. DOI; Open PDF