Conformal $n$-point functions in momentum space

TL;DR

This work provides a general momentum-space representation for scalar $n$-point functions in conformal field theories that solves the conformal Ward identities by introducing a Feynman-integral form with an arbitrary function of $n(n-3)/2$ momentum-space cross-ratios. The construction for $n=4$ reveals a tetrahedral (3-loop) topology with a universal kernel $W_{ ext{alpha}, ext{beta}}$ and a Mellin/spectral structure that connects to position-space Mellin representations and holographic triple-K integrals. The paper analyzes singularities and renormalization, detailing ultralocal and semilocal divergences and their counterterms, and provides examples in free fields and holographic CFTs to illustrate the framework. It also generalizes the approach to arbitrary $n$ and discusses potential momentum-space bootstrap prospects and compact scalar parametrizations to facilitate practical computations and tensor extensions.

Abstract

We present a Feynman integral representation for the general momentum-space scalar $n$-point function in any conformal field theory. This representation solves the conformal Ward identities and features an arbitrary function of $n(n-3)/2$ variables which play the role of momentum-space conformal cross-ratios. It involves $(n-1)(n-2)/2$ integrations over momenta, with the momenta running over the edges of an $(n-1)$-simplex. We provide the details in the simplest non-trivial case (4-point functions), and for this case we identify values of the operator and spacetime dimensions for which singularities arise leading to anomalies and beta functions, and discuss several illustrative examples from perturbative quantum field theory and holography.

Figures (4)

• Figure 1: The 3-loop tetrahedral integral \ref{['intmono']}, where each internal line corresponds to a generalized propagator in \ref{['Den3ab']}.
• Figure 2: Simplifications of the kernel $W_{\alpha,\beta}$: (a) where a propagator in \ref{['Den3ab']} appears with $\gamma_{ij}=d/2+n$ the loop order is reduced by one; (b) with two such propagators we obtain a 1-loop box; (c) for $\gamma_{ij}=-n$, we obtain a 3-point function.
• Figure 3: Three distinct topologies of Feynman diagrams contributing to the connected part of $\langle :\! \phi^4 \!: :\! \phi^4 \!: :\! \phi^4 \!: :\! \phi^4 \!: \rangle$.
• Figure 4: Equivalent electrical circuits where the impedances are related by $z_{ij}=Z_iZ_j/Z_t$. Setting $z_{i4}=s_i$ for $i=1,2,3$ and $(z_{12},z_{23},z_{31})=(z^2/s_3,z^2/s_1,z^2/s_2)$ gives a mapping of Schwinger parameters converting the contact Witten diagram \ref{['impedancerep']} into the form \ref{['kint4']} with $\hat{f}(\hat{u},\hat{v})$ given in \ref{['fhatcontact']}.