Evaluation of conformal integrals

TL;DR

The paper delivers a complete framework to evaluate momentum-space 3-point functions in conformal field theories by expressing all relevant correlators in terms of triple-$K$ integrals and then reducing these to a single master integral $I_{0\{111\}}$. Through a rigorous reduction scheme built on Bessel-function identities and hypergeometric representations, the authors obtain analytic, closed-form results for scalar and tensorial 3-point functions with integer operator dimensions in any dimension, including a full treatment of divergences and regularisation schemes. The master integral is decomposed into divergent, scheme-dependent finite, non-local dilogarithmic, and scale-violating pieces, with the non-local content captured by a dilogarithmic structure encoded in the symmetric function $\\mathcal{L}$. The work also provides explicit tools for scheme changes and connects to tensorial correlators in $d=4$, enabling applications to conformal and holographic calculations and to renormalised correlators in even dimensions. Overall, this establishes a practical, general method for obtaining exact momentum-space 3-point functions in CFTs with broad relevance to AdS/CFT, cosmology, and condensed-matter contexts.

Abstract

We present a comprehensive method for the evaluation of a vast class of integrals representing 3-point functions of conformal field theories in momentum space. The method leads to analytic, closed-form expressions for all scalar and tensorial 3-point functions of operators with integer dimensions in any spacetime dimension. In particular, this encompasses all 3-point functions of the stress tensor, conserved currents and marginal scalar operators.

Figures (1)

• Figure 1: A summary of all triple-$K$ integrals that can be obtained from the master integral $I_{0\{111\}}$ using the reduction scheme. Depending on the values of $n_0$ and $n_1$ the integrals are either finite (represented by squares), linearly divergent (circles), or quadratically divergent (diamonds). Each point represents an infinite series of integrals having the same values of $n_0$ and $n_1$. In particular the master integral $I_{0\{111\}}$ belongs to the series denoted by the red diamond at $(n_0,n_1)=(1,0)$. The linearly divergent integral $I_{2\{111\}}$ belongs to the series denoted by the blue circle at $(0,-1)$, while finite $I_{1\{000\}}$ to the series denoted by the green square at $(-1,-1)$. The three types of integrals are gathered into three separate regions denoted by dashed borders. Since $n_0 \geq n_1$, no integral can appear in the excluded region.