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 nontrivial 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.
Highlights
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 nðn − 3Þ=2 variables which play the role of momentum-space conformal cross ratios
We provide the details in the simplest nontrivial 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
Summary
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 nðn − 3Þ=2 variables which play the role of momentum-space conformal cross ratios. Motivation.—The structure of correlation functions in a conformal field theory (CFT) is highly constrained by conformal symmetry It has been known since the work of Polyakov [1,2] that the most general 4-point function of scalar primary operators OΔj, each of dimension Δj, takes the form hOΔ1 ðx1ÞOΔ2 ðx2ÞOΔ3 ðx3ÞOΔ4 ðx4Þi 1⁄4 fðu; vÞ Y x2ijδij ; 1≤i
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