Conformal correlators as simplex integrals in momentum space
Adam Bzowski, Paul McFadden, Kostas Skenderis
Abstract
A bstract We find the general solution of the conformal Ward identities for scalar n -point functions in momentum space and in general dimension. The solution is given in terms of integrals over ( n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n -point functions are built in terms of ( n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n -point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves ( n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n ( n − 3) / 2 integrations.