Conformal Partial Wave Decomposition

Updated 12 December 2025

Conformal partial wave decomposition is a method for expressing correlation functions in CFT as integrals and sums over irreducible conformal group representations.
It employs the shadow formalism and Mellin-type expansions to reveal OPE data, crossing symmetry, and unitarity through orthogonal basis functions.
The technique bridges AdS/CFT and flat-space amplitudes, utilizing group-theoretic and analytic methods to connect celestial CFT with traditional scattering theory.

Conformal partial wave decomposition is the canonical method for expressing $n$ -point correlation functions in conformal field theory (CFT), and, by extension, celestial amplitudes in four-dimensional Minkowski spacetime, as integrals and sums over basis functions associated with irreducible representations of the conformal group. In the four-point case, this procedure is a Fourier- or Mellin-type expansion onto the eigenfunctions of the quadratic Casimir of $\mathrm{SO}(1,d+1)$ (Euclidean) or $\mathrm{SL}(2,\mathbb{C})$ (Lorentzian signature). The conformal partial waves provide a complete, orthogonal basis, rendering OPE data and dynamical content manifest and facilitating crossing symmetry, unitarity, and analytic control. In the celestial CFT formulation for scattering amplitudes, conformal partial wave decomposition underpins the Mellin transform, contour-deformation techniques, and the explicit structural connection between flat-space amplitudes and CFT data, with crucial roles in both scalar and spinning correlators.

1. Algebraic and Group-Theoretic Foundations

Conformal partial waves $\Psi_{\Delta,\ell}$ are eigenfunctions of the quadratic Casimir operator $C_2$ of the conformal group acting on a pair of points. For four-point functions, the conformal group acts transitively on the cross-ratios $(u,v)$ or in complex variables $(z,\bar z)$ with $u=z\bar z$ , $v=(1-z)(1-\bar z)$ . The defining property is

$C_2\cdot\Psi_{\Delta,\ell}(u,v) = c(\Delta,\ell)\Psi_{\Delta,\ell}(u,v)$

with eigenvalue $c(\Delta,\ell) = \Delta(\Delta-d)+\ell(\ell+d-2)$ . These Casimir eigenfunctions correspond to specific conformal primaries and their descendants exchanged in an OPE channel. In $d=2$ , the decomposition admits an $SL(2,\mathbb{C})$ factorization, with $(\Delta,J)$ labeling scaling dimension and spin (Schomerus, 2021).

The explicit construction utilizes the shadow formalism: a conformal partial wave is written as an integral over the insertion point of a shadow operator, yielding the shadow-kernel representation. In higher dimensions, this yields integral representations over $d$ -dimensional space for the exchanged primary.

2. Partial Wave Expansion and Spectral Representation

A general four-point correlation function or celestial amplitude $A(z_i,\bar z_i)$ admits a decomposition

$A(z_i,\bar z_i) = \sum_{J=0}^\infty \int_{c-i\infty}^{c+i\infty} d\Delta\: C(\Delta, J)\Psi_{\Delta, J}(z, \bar z)$

where $\Psi_{\Delta,J}$ are the SL(2, $\mathbb{C}$ ) or SO(1, $d+1$ ) conformal partial waves, and $C(\Delta, J)$ is the spectral density encoding dynamical OPE coefficients. For celestial CFT, each partial wave is a specific single-valued sum of blocks and shadows, symmetrized as

$\Psi_{\Delta, J} = c_1 F^+_{\Delta, J}(z, \bar z) + c_2 F^-_{\Delta, J}(z, \bar z)$

with $F^+$ and $F^-$ as products of powers of $z$ , $\bar z$ , and Gauss hypergeometric functions, with normalizations rendering $\Psi_{\Delta, J}$ single-valued for the physical region (Nandan et al., 2019, Atanasov et al., 2021).

Completeness and orthogonality are realized via the natural invariant inner product, e.g., for SL(2, $\mathbb{C}$ ): $\langle f, g \rangle = \int d^2z\,|z(1-z)|^{-2} f(z,\bar z)g(z,\bar z)$ with

$\langle \Psi_{\Delta,J}, \Psi_{\Delta', J'} \rangle = 4\pi^2 (1-2h)(2h-1)\,\delta_{J,J'}\,\delta(\nu-\nu')$

where $h = (\Delta+J)/2$ for principal series representations.

3. Extraction of Spectral Density and Discrete/Continuous Structures

The coefficients $C(\Delta, J)$ are projected by inner product: $C(\Delta,J) = \frac{1}{4\pi^2 (1-2h)(2h-1)} \int d^2z\,|z(1-z)|^{-2} \Psi_{\Delta, J}(z, \bar z) \mathcal{A}(z, \bar z)$ For explicit amplitudes, insertion of explicit expressions for $\mathcal{A}$ and $\Psi_{\Delta,J}$ reduces to Mellin–Barnes integrals, the evaluation of which determines the spectral density in terms of residues of $\Gamma$ -functions.

In celestial CFT, the contour deformation in the complex conformal weight plane picks up poles corresponding to exchanged scalar primaries (at $\Delta = 2 + i(\lambda_1+\lambda_2) + 2n$ for scalar exchange), as well as a continuum of contributions associated to so-called light-ray operators of principal continuous series dimension and imaginary spin: $\mathcal{A}(z_i, \bar z_i) = \sum_{n} C_{13,n}C_{24,n} G^{(2)}_{1+n,0}(z, \bar z) + \int_{-\infty}^\infty \frac{d\nu}{2\pi} C_{13}(\nu)C_{24}(\nu) G^{(2)}_{1, -2i\nu}(z, \bar z)$ Here, $G^{(2)}_{\Delta,\ell}$ are 2D conformal blocks, and the discrete and continuous coefficients have precise expressions in terms of OPE data (Atanasov et al., 2021, Nandan et al., 2019).

4. Explicit Examples: Scalar and Gluon Four-Point Amplitudes

For four-point scalar exchange (in celestial CFT), the Mellin transform of the amplitude and the inversion yields

$C^{\mathrm{scalar}}(\Delta, 0) = \frac{g^2 (m/2)^{\Delta-2}}{2\pi \sin\pi(\Delta-2)}\frac{\Gamma(\Delta-2)\Gamma(\Delta-2-i\lambda_1-i\lambda_2)\Gamma(\Delta-2-i\lambda_3-i\lambda_4)}{ \Gamma(\Delta/2-i\lambda_1-i\lambda_2)\Gamma(\Delta/2-i\lambda_3-i\lambda_4)\Gamma(1-\Delta/2+i\lambda_1+i\lambda_2)\Gamma(1-\Delta/2+i\lambda_3+i\lambda_4)}$

with poles in $\Delta$ at prescribed locations. For four-gluon MHV amplitudes, the result for integer spins $J\ge0$ is

$C^{\mathrm{gluon}}(\Delta,J) = i(1-2h)(2h-1)\frac{\Gamma(h-i\lambda_1)\Gamma(h-i\lambda_2)\Gamma(\bar h - i\lambda_3)\Gamma(\bar h - i\lambda_4)}{\Gamma(h+h_{12})\Gamma(h+h_{34})\Gamma(\bar h+\bar h_{12})\Gamma(\bar h + \bar h_{34})}$

demonstrating that, for external gluons, a full tower of spin-J conformal partial waves contribute (Nandan et al., 2019).

5. Crossing Symmetry and Analytic Structure

Conformal partial wave expansions exhibit explicit crossing relations, reflecting permutation symmetry under exchange of operator insertions. The crossing symmetry imposes functional constraints on $C(\Delta,J)$ , with the explicit pole and residue structure ensuring that the t- and u-channel decompositions reproduce the same partial wave coefficients after appropriate re-labeling of Mellin variables (e.g. exchange $\lambda_2 \leftrightarrow \lambda_3$ ). Analytic continuation and contour deformation techniques generalize between channels.

Single-valuedness of the expansion is achieved via specific combinations of blocks and their shadows, and in celestial correlators, the spectral density $C(\Delta,J)$ must ensure hermiticity (unitarity) and appropriate shadow symmetry, with corresponding consistency checks provided in explicit examples (Nandan et al., 2019, Atanasov et al., 2021).

6. Generalizations and Holographic, Integrable, and Flat-Space Connections

The conformal partial wave framework generalizes to arbitrary $d$ , spinning operators, and nontrivial representation content. The shadow-integral and harmonic analysis approaches allow explicit construction for arbitrary spacetime dimension $d$ , with precise group-theoretic normalization and eigenfunction properties (Schomerus, 2021).

From the holographic AdS/CFT perspective, conformal partial waves admit a gravitational dual realization as open Wilson network operators in Hilbert-Palatini formalism, where the partial waves correspond to gauge-invariant combinations of AdS Wilson lines—implementing the same Casimir and Ward identities, and permitting efficient computation of partial waves, blocks, and their generalizations (Bhatta et al., 2016).

In the flat-space limit, the OPE density $c(\Delta,\ell)$ reduces, under a saddle-point approximation at large $\Delta\sim mR$ , to ordinary partial wave amplitudes of scattering theory: the conformal partial wave expansion thus interpolates between AdS boundary correlators and flat-space S-matrix partial wave expansions, with inversion formulas reducing to their Froissart-Gribov and dispersion-theoretic analogs (Rees et al., 2023).

Integrable structures also underpin the decomposition: the conformal Casimir reduces to Calogero-Sutherland-type Hamiltonians, and the full set of partial waves realizes the complete superintegrable structure underlying CFT correlation functions (Schomerus, 2021).

7. Methods for Spinning Correlators and Momentum Space

In $d=4$ , spinning conformal partial waves for traceless symmetric tensors can be uniformly constructed from a finite number of seed scalar blocks via explicit differential operators in the embedding space, thereby reducing computation for arbitrary correlator configurations to algebraic actions on seed functions (Echeverri et al., 2015). The Mellin-space representation reveals polynomial structures (Mack polynomials) encoding spin and crossing, facilitating inversion and OPE extraction for general external and exchanged spins (Sleight et al., 2018).

In momentum space, the decomposition is formulated directly in terms of spin eigenstates in the center-of-mass, with conformal partial waves built from products of vertex functions and Gegenbauer polynomials in the scattering angle, and explicit Appell $F_4$ hypergeometric solutions for the highest-spin component (Gillioz, 2020).

References:

(Nandan et al., 2019) “Celestial Amplitudes: Conformal Partial Waves and Soft Limits”
(Atanasov et al., 2021) “Conformal Block Expansion in Celestial CFT”
(Schomerus, 2021) “Conformal Hypergeometry and Integrability”
(Bhatta et al., 2016) “Holographic Conformal Partial Waves as Gravitational Open Wilson Networks”
(Rees et al., 2023) “Flat-space Partial Waves From Conformal OPE Densities”
(Echeverri et al., 2015) “Deconstructing Conformal Blocks in 4D CFT”
(Sleight et al., 2018) “Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications”
(Gillioz, 2020) “Conformal partial waves in momentum space”