Conformal partial waves in momentum space

TL;DR

The authors develop a momentum-space conformal partial wave expansion for four scalar primaries in Minkowski space, valid for all $d\geq 3$, with a closed-form expression $G_{\Delta,\ell}(p_1,p_2,p_3)$ that factors into Gegenbauer angular dependence and vertex functions $V_{\Delta,\ell,m}^{[ab]}$ tied to three-point data. The key technical advance is a complete spin-eigenstate basis constructed from momentum-space polarization tensors, enabling a Hilbert-space completeness approach that yields a clean OPE-like decomposition in momentum space. Vertex functions are fixed by conformal Ward identities, with the highest-spin sector solved in terms of Appell $F_4$ functions and lower-spin functions obtained recursively; the ordering of external operators determines the precise linear combination of basis solutions. The framework is validated on the scalar box integral in $d=4$, where a conformal partial wave expansion reproduces known perturbative results, and the authors discuss potential applications to bootstrap-like dispersion relations, forward-limit positivity, and cosmological bootstrap contexts, highlighting the broad relevance of momentum-space CFT techniques.

Abstract

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension $d \geq 3$ (including non-integer $d$). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in $d = 4$ dimensions.

Figures (3)

• Figure 1: Diagrammatic representation of the conformal partial wave \ref{['eq:conformalpartialwave']}, in which each line corresponds to a local primary operator of the CFT carrying a certain momentum. Note that the vertex functions $V_{\Delta,\ell,m}^{[ab]}$ only depend on the scaling dimensions, on the spin (the total spin $\ell$ and its projection $m$ onto a reference direction), and on the momenta of the lines attached to it.
• Figure 2: A possible configuration of momenta in the center-of-mass frame (energy along the vertical axis). The scattering angle $\theta$ corresponds to the angle between the planes spanned by $(p_1, p_2)$ and $(p_3, p_4)$, or equivalently between the vectors $q_{12}$ and $q_{34}$. In this particular example $p_1$ is time-like (backward directed), while $p_2$, $p_3$ and $p_4$ are all space-like, as seen by their position relative to the light cone (dotted lines); the covariant definition \ref{['eq:costheta']} is valid independently of the sign of the $p_i^2$.
• Figure 3: The scalar box diagram corresponding to the 4-point function \ref{['eq:example:4ptfct']} in the theory of a free complex scalar field. The solid lines are propagators of the field $\Phi$, with arrows indicating the charge flow. The external, dashed lines correspond to the composite operators $\Phi^2$ and $\overline{\Phi}^2$. The dual graph in terms of the variables $x_i - x_{i+1} = p_i$ is shown in red, with dotted lines.