Crossing Kernels and Conformal 6j Symbols

A conformal crossing kernel is the change-of-basis coefficient between partial waves associated with different OPE trees. For a four-point function it is the conformal-group analogue of a Racah coefficient, or $6j$ symbol. This representation-theoretic object should not be confused with a conformal block, an OPE coefficient, or the kernel of a dispersion relation.

Required background. The Lorentzian inversion formula provides a practical projection onto direct-channel data. Partial waves and the shadow formalism provide the single-valued basis and its inner product.

Helpful background. Representations, intertwiners, and invariants explains why a change between two association schemes is an intertwiner rather than new dynamical data.

Evidence cutoff. Research-sensitive statements about kernel constructions, meromorphic continuation, and higher-point recoupling reflect primary sources available through 2026-08-09. Later completeness or spinning-kernel results require a renewed source check.

Partial waves form the recoupling basis

Consider four scalar primaries $\mathcal O_i$ in a Euclidean CFT in $d$ dimensions. A single-valued $s$-channel partial wave is a shadow-symmetric combination

$$\Psi^{(s)}_{\Delta,J} =K_{d-\Delta,J}^{(34)}G^{(s)}_{\Delta,J} +K_{\Delta,J}^{(12)}G^{(s)}_{d-\Delta,J},$$

where the $K$ factors depend on the external dimensions and the chosen two- and three-point normalizations. On the scalar principal series,

$$\Delta=\frac d2+i\nu, \qquad \nu\in\mathbb R, \qquad J=0,1,2,\ldots,$$

these partial waves are delta-normalizable with respect to a conformally invariant inner product. Their completeness takes the schematic Plancherel form

$$F(x_i)=\sum_J\int_{d/2-i\infty}^{d/2+i\infty} \frac{d\Delta}{2\pi i}\, \mu(\Delta,J)c(\Delta,J)\Psi_{\Delta,J}(x_i),$$

possibly augmented by discrete terms when the contour is deformed. The Plancherel measure $\mu$, shadow factors, tensor-structure pairings, and quotient by the conformal-group volume are all part of the normalization. Omitting any one of them changes the numerical kernel Karateev, Kravchuk, and Simmons-Duffin 2019, §2.

The $t$-channel partial wave can be expanded in the $s$-channel basis:

$$\Psi^{(t)}_{\Delta',J'} =\sum_J\int_{\mathcal P}\frac{d\Delta}{2\pi i}\, \mu(\Delta,J) \mathcal K_{st}(\Delta,J;\Delta',J') \Psi^{(s)}_{\Delta,J}.$$

Up to the displayed measure convention, the crossing kernel is the normalized overlap

$$\mathcal K_{st}(\Delta,J;\Delta',J') \propto \bigl(\Psi^{(s)}_{\Delta,J},\Psi^{(t)}_{\Delta',J'}\bigr).$$

Writing each partial wave as a gluing of two conformal three-point functions turns this overlap into four three-point structures integrated over internal positions. The resulting tetrahedral contraction is the conformal $6j$ symbol Liu et al. 2019, §3.

What the kernel computes

The kernel is kinematic: it depends on conformal representations and normalization choices, not on the spectrum or OPE coefficients of a particular CFT. Dynamics enters when a correlator supplies a spectral weight in one channel. If

$$F=\sum_{J'}\int_{\mathcal P}d\Delta'\, \rho_t(\Delta',J')\Psi^{(t)}_{\Delta',J'},$$

then its $s$-channel coefficient is obtained by convolution,

$$\rho_s(\Delta,J) =\sum_{J'}\int_{\mathcal P}d\Delta'\, \rho_t(\Delta',J')\mathcal K_{st}(\Delta,J;\Delta',J'),$$

with the same measures as in the completeness relation. Poles of $\mathcal K_{st}$ in $\Delta$ include the double-twist locations generated by the crossed partial wave. After the contour is moved toward a physical OPE, residues contribute direct-channel block coefficients; shadow partners and coincident-pole derivatives must be separated before reading them as CFT data.

For external scalars, a direct calculation in $d=1,2,4$ reduces the kernel to finite combinations of generalized hypergeometric functions. In general dimension, the Lorentzian inversion formula can project a crossed partial wave and exposes the same pole families above its valid spin threshold. The spacetime inversion derivation makes that threshold and its commutator support explicit Simmons-Duffin, Stanford, and Witten 2018, §§3–4. This is a useful equality of two constructions, not permission to discard the inversion formula’s Regge and low-spin conditions.

The following diagram shows the four-point change of channel and its continuation to higher-point OPE trees. Follow the representation label through each recoupling: the same local move becomes an edge of a larger tree rather than a new kind of OPE coefficient.

Schematic recoupling map. A conformal $6j$ symbol changes one binary association of four representations into another; compositions relate higher-point OPE trees, subject to the same shadow, measure, and contour conventions.

An equivalent algebraic reading is:

Stage	Basis object	Operation	Required check
Four-point $s$ channel	$\Psi^{(s)}_{\Delta,J}$	Expand a crossed partial wave	Principal-series measure and shadow pairing
Four-point $t$ channel	$\Psi^{(t)}_{\Delta',J'}$	Project onto the $s$ basis	External-order and tensor-structure convention
Channel change	$\mathcal K_{st}$	Integrate the product of four three-point structures	Normalization and discrete residues
Higher-point tree	Products of partial waves	Compose local recouplings	Equality of two recoupling sequences

Recoupling consistency

A legitimate change of basis must be invertible on the chosen harmonic-analysis space. Schematically,

$$\begin{aligned} &\sum_{J'}\int_{\mathcal P}d\Delta'\, \mu(\Delta',J') \mathcal K_{st}(\Delta,J;\Delta',J')\\ &\qquad\times \mathcal K_{ts}(\Delta',J';\widetilde\Delta,\widetilde J) =\frac{\delta_{J\widetilde J}\delta(\nu-\widetilde\nu)} {\mu(\Delta,J)} \end{aligned}$$

in a convention where shadow-equivalent labels have been quotiented once. For five representations, comparing two sequences of elementary recouplings gives the conformal analogue of the pentagon identity. These relations are independent checks on a proposed kernel. They are stronger than matching only the locations of a few double-twist poles.

A mean-field check

Take a crossed scalar partial wave and project it into the direct channel. Its kernel has pole pairs at the dimensions associated with the two external pairings; for identical scalars, one family begins at

$$\Delta=2\Delta_\phi+J+2n, \qquad n=0,1,2,\ldots .$$

The residues reproduce the corresponding mean-field OPE data after the block, shadow, and Plancherel normalizations are combined. This check tests both pole locations and residues. A calculation that finds only the locations has not fixed the $6j$ symbol.

Common pitfalls

Replacing partial waves by blocks. A Euclidean conformal block is not single-valued and does not by itself furnish the principal-series orthogonal basis. The shadow combination and its normalization are essential.

Moving the contour without recording residues. Discrete terms crossed during a contour deformation are part of the channel transform. Dropping them can erase low-lying or shortened representations.

Calling any large-spin coefficient a crossing kernel. Large-spin residues may be extracted from a kernel, but they do not specify its full meromorphic dependence, inverse transform, or recoupling identities.

Ignoring tensor structures. With spinning external operators, the kernel is a matrix between three-point-structure bases. Each shadow transform carries its own pairing matrix and possible parity label.

Exercises

Assume the $t$-channel spectral density consists of one delta-normalized partial wave. Use the channel-change equation to identify the $s$-channel spectral density, then state the two extra steps required to obtain physical block coefficients.

Solution

The direct-channel density is the appropriate column of $\mathcal K_{st}$. To obtain a physical OPE expansion, deform the principal-series contour and retain every crossed pole residue, then separate each partial wave into its physical block and shadow block with the declared $K$ factors. Regge restrictions enter if Lorentzian inversion is used to evaluate the kernel.

References

• Karateev, Denis, Petr Kravchuk, and David Simmons-Duffin. “Harmonic Analysis and Mean Field Theory.” Journal of High Energy Physics 2019, no. 10 (2019): 217. doi:10.1007/JHEP10(2019)217.
• Liu, Junyu, Eric Perlmutter, Vladimir Rosenhaus, and David Simmons-Duffin. “d-Dimensional SYK, AdS Loops, and 6j Symbols.” Journal of High Energy Physics 2019, no. 3 (2019): 052. doi:10.1007/JHEP03(2019)052.
• Simmons-Duffin, David, Douglas Stanford, and Edward Witten. “A Spacetime Derivation of the Lorentzian OPE Inversion Formula.” Journal of High Energy Physics 2018, no. 7 (2018): 085. doi:10.1007/JHEP07(2018)085.