Abstract
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.
Highlights
The general form of position-space n-point correlators in a conformal field theory (CFT) has been known for half a century [1]
We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension
In [2], we presented a counterpart for this result in momentum space: a representation for the general momentum-space n-point function as a Feynman integral over an (n−1)-simplex, featuring an arbitrary function of the cross ratios constructed from the momenta running between the vertices
Summary
The general form of position-space n-point correlators in a conformal field theory (CFT) has been known for half a century [1]. In [2], we presented a counterpart for this result in momentum space: a representation for the general momentum-space n-point function as a Feynman integral over an (n−1)-simplex, featuring an arbitrary function of the cross ratios constructed from the momenta running between the vertices These cross ratios play an equivalent role to those in position space, and ensure the correlator has the same number of degrees of freedom in either basis. We apply our results to a prototypical class of CFT correlators: n-point contact Witten diagrams This illustrates two themes of a general nature: first, the applicability of results from electrical circuit theory relating resistor networks of different topologies; and second, the power of recursive methods when used in combination with the simplex representation. Appendix A presents technical aspects of our proofs of conformal invariance; appendix B reviews the Symanzik trick for conformal integrals; and appendix C derives a new representation for the position-space holographic D-function in terms of a triple-K integral
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