Conformal Partial Waves and the Shadow Formalism

The shadow formalism constructs conformally invariant harmonic functions by integrating two three-point functions. The resulting conformal partial wave solves the same Casimir equation as a block but is generally a fixed combination of the physical block and the shadow block. Extracting an OPE contribution therefore requires a monodromy, contour, or asymptotic projection plus a declared normalization.

Required background. Conformal Blocks and Casimir Equations supplies the physical and shadow Casimir solutions. Representations, Intertwiners, and Invariants supplies invariant pairings. Helpful background. Bounded, Compact, and Integral Operators supplies kernel and spectral-measure language.

The scalar shadow transform

A scalar primary $\mathcal O_\Delta$ has a shadow representation of dimension

$$\widetilde\Delta=d-\Delta.$$

In Euclidean signature define, initially where the integral converges,

$$\widetilde{\mathcal O}_{d-\Delta}(x) =\mathcal N_\Delta \int d^dy\, \frac{\mathcal O_\Delta(y)} {(x-y)^{2(d-\Delta)}}.$$

Here $(x-y)^{2a}$ abbreviates $\bigl((x-y)^2\bigr)^a$ on the Euclidean branch. The kernel intertwines the $\Delta$ representation with the $d-\Delta$ representation. Outside the elementary convergence strip it is defined by analytic continuation as a distribution. Contact terms can appear at exceptional dimensions, so the continuation and counterterm prescription are part of the transform.

Applying the transform twice gives

$$\widetilde{\widetilde{\mathcal O}}_\Delta =\mathcal S_\Delta\mathcal O_\Delta.$$

One may choose $\mathcal N_\Delta$ so that $\mathcal S_\Delta=1$, but many sources leave a known scalar factor. Every projector and Plancherel measure must use the same choice. For spin, the kernel also contains inversion tensors or spinor intertwiners and pairs the representation with its reflected dual.

A conformal partial wave

For scalar external operators, a schematic $s$-channel partial wave is

$$\Psi_{\Delta,\ell}^{(s)}(x_i) =\frac1{\mathcal N_{\Delta,\ell}} \int d^dx_0\, \langle\mathcal O_1\mathcal O_2 \mathcal O_{\Delta,\ell}(x_0)\rangle \langle\widetilde{\mathcal O}_{d-\Delta,\ell}(x_0) \mathcal O_3\mathcal O_4\rangle,$$

with all spin indices contracted by the invariant pairing. It is conformally covariant because the integration weights cancel. Acting with the pair Casimir may be moved through the integral, so

$$\mathcal C_{12}\Psi_{\Delta,\ell}^{(s)} =C_{\Delta,\ell}\Psi_{\Delta,\ell}^{(s)}.$$

Euclidean single-valuedness and the equal Casimir eigenvalues imply the decomposition

$$\boxed{ \Psi_{\Delta,\ell}^{(s)} =K_{\Delta,\ell}\,g_{\Delta,\ell}^{(s)} +K_{d-\Delta,\ell}\,g_{d-\Delta,\ell}^{(s)}. }$$

The coefficients $K$ depend on external dimensions, tensor bases, and shadow normalization. A partial wave is therefore not interchangeable with a block. The embedding-space projector construction and its analytic projection are derived in Simmons-Duffin 2014, §§ 2–3.

Projecting the physical block

Near the $s$-channel OPE limit, the two terms behave as

$$g_{\Delta,\ell}^{(s)}\sim u^{\Delta/2}, \qquad g_{d-\Delta,\ell}^{(s)}\sim u^{(d-\Delta)/2}.$$

A monodromy projection retains the component with the declared $u^{\Delta/2}$ behavior. Equivalently, in harmonic analysis one integrates $\Delta$ along a principal-series contour and deforms the contour; poles and residues selected by the physical spectrum yield OPE blocks. This deformation is valid only after arc behavior, pole collisions, and discrete terms are controlled. When $\Delta=d/2$ or the two powers differ by an integer, logarithmic mixing requires a limiting prescription.

On the Euclidean principal series,

$$\Delta=\frac d2+i\nu, \qquad \nu\in\mathbb R,$$

the conformal group admits a Plancherel decomposition with measure $\mu(\Delta,\ell)$. Completeness is a statement about this harmonic basis and its contour. It is not identical to the discrete physical OPE, which is recovered by analytic continuation and residues. Normalizations and spinning pairings are treated systematically in Karateev, Kravchuk, and Simmons-Duffin 2019, §§ 2–4.

Star–triangle normalization check

The scalar star–triangle integral is a useful independent check. If $a_1+a_2+a_3=d$ and the integral is defined by convergence or analytic continuation,

$$\begin{aligned} &\int d^dx_0 \prod_{i=1}^3\frac1{(x_{i0}^2)^{a_i}}\\ &\quad= \pi^{d/2} \prod_{i=1}^3\frac{\Gamma(d/2-a_i)}{\Gamma(a_i)} \frac1{ (x_{23}^2)^{d/2-a_1} (x_{13}^2)^{d/2-a_2} (x_{12}^2)^{d/2-a_3}}. \end{aligned}$$

Both sides have length dimension $-d$, and the exponent at each external point transforms as required. Applying this identity to a scalar three-point kernel verifies that a shadow transform has dimension $d-\Delta$ and fixes its gamma-function normalization. Poles of the gamma functions signal exceptional cases where the distribution needs subtraction or contains contact terms.

Object and domain taxonomy

The following table distinguishes objects that are often all called “blocks.” It is a substantive use of the chapter’s object/domain taxonomy.

Object	Definition and correlator role	Natural domain	Shadow included?	Positivity or completeness status	Ambiguity or distributional term
Tensor structure	Kinematic invariant multiplying primary three-point data	Separated configurations in a chosen basis	No	Neither positive nor complete by itself	Basis changes; contact structures at coincidence
Conformal block $g_{\Delta,\ell}^{(s)}$	Descendants of one irreducible primary in the $s$ channel	OPE branch with declared asymptotic; radial series for $\lvert\rho_s\rvert<1$	No	Positive weight only for suitable Hermitian reflection-positive pairings	Block normalization and short-module subtraction
Shadow block $g_{d-\Delta,\ell}^{(s)}$	Second Casimir solution with shadow OPE power	Same differential-equation domain on another asymptotic branch	It is the shadow component	Not physical OPE data by itself	Branch and collision with the physical root
Conformal partial wave $\Psi_{\Delta,\ell}$	Shadow integral of two three-point functions	Euclidean separated points; principal-series continuation	Yes, block plus shadow	Harmonic completeness only with the Plancherel contour and measure	Shadow normalization, discrete terms, contact poles
Shadow transform	Intertwining integral $\Delta\to d-\Delta$	Convergence strip or analytically continued distributions	Produces the shadow representation	Invertible only modulo its normalization and exceptional kernels	Local counterterms at singular dimensions
Inversion kernel	Pairing that extracts spectral coefficients from a correlator	Domain and contour of the chosen Euclidean or Lorentzian inversion formula	Convention dependent	Completeness requires the full measure, arcs, and possible discrete terms	Subtractions, low-spin pieces, distributional support
Polyakov-type block	Crossing-symmetric combination engineered to have a specified exchange singularity	Depends on Euclidean or dispersive construction	May combine several channel blocks	Not automatically a positive physical contribution	Contact-polynomial or subtraction ambiguity
Contact or semilocal term	Distribution supported when some insertions coincide	Distribution space, not generic separated points	No	Not part of separated-point block completeness	Counterterm scheme or anomaly fixes the allowed term

Each row reconstructs a different part of a correlator. In the identical-scalar convention of this chapter, only a physical block multiplied by its contracted OPE weight contributes one discrete OPE family. A partial wave must first be projected; an inversion kernel extracts rather than contributes; and a contact term is invisible at generic points.

Common pitfalls

A partial wave is a conformal block. The Euclidean partial wave generally contains both $\Delta$ and $d-\Delta$ solutions. Project the declared OPE behavior.

The shadow integral is an ordinary convergent integral for all dimensions. It often requires analytic continuation as a distribution and can acquire contact terms at exceptional parameters.

Principal-series completeness is the physical OPE. The former is a harmonic contour decomposition. The latter emerges after contour deformation, residues, and any discrete or arc terms are handled.

Exercises

Check the scaling dimension of the scalar shadow transform.

Solution

Under $x,y\to\lambda x,\lambda y$, the measure contributes $\lambda^d$, the kernel contributes $\lambda^{-2(d-\Delta)}$, and $\mathcal O_\Delta(y)$ contributes $\lambda^{-\Delta}$. The result scales as $\lambda^{-(d-\Delta)}$, the dimension of the shadow.

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; Open PDF
• Simmons-Duffin, David. “Projectors, Shadows, and Conformal Blocks.” Journal of High Energy Physics 2014, no. 4 (2014): 146. DOI; Open PDF