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.
Highlights
Conformal invariance imposes strong constraints on the form of correlation functions in any field theory
E-mail: adam.bzowski@fys.kuleuven.be, p.mcfadden@imperial.ac.uk, k.skenderis@soton.ac.uk 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
For special combinations of the operator and spacetime dimensions, these triple-K integrals contain divergences necessitating their regularisation and renormalisation. (For this same reason one cannot generally obtain the momentum-space correlators via a straightforward Fourier transform.) In [26], we presented a complete classification of the divergences and their renormalisation for purely scalar 3-point functions, and in an upcoming paper [27], we will discuss the corresponding renormalisation procedure for tensorial 3-point functions
Summary
Conformal invariance imposes strong constraints on the form of correlation functions in any field theory. In this paper we present a complete method for systematically computing all the triple-K integrals that arise in the evaluation of scalar and tensorial 3-point functions of operators with integer dimensions. This broad class includes many operators of physical interest such as the stress tensor, conserved currents, and marginal scalar operators. The scheme is recursive in nature and is based on simple identities between Bessel functions Since both scalar and tensorial 1-loop 3-point massless Feynman integrals can be re-expressed as triple-K integrals, our procedure generalises and simplifies other momentum-space recursion schemes such as those presented in [28,29,30]. After the relevant triple-K integrals have been evaluated via our recursive procedure, the full scalar and/or tensorial 3-point functions can be reconstructed as described in [25, 26]
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