Graviton vertices and the mapping of anomalous correlators to momentum space for a general conformal field theory

We investigate the mapping of conformal correlators and of their anomalies from configuration to momentum space for general dimensions, focusing on the anomalous correlators TOO, TVV - involving the energy-momentum tensor (T) with a vector (V) or a scalar operator (O) - and the 3-graviton vertex TTT. We compute the TOO, TVV and TTT one-loop vertex functions in dimensional regularization for free field theories involving conformal scalar, fermion and vector fields. Since there are only one or two independent tensor structures solving all the conformal Ward identities for the TOO or TVV vertex functions respectively, and three independent tensor structures for the TTT vertex, and the coefficients of these tensors are known for free fields, it is possible to identify the corresponding tensors in momentum space from the computation of the correlators for free fields. This works in general d dimensions for TOO and TVV correlators, but only in 4 dimensions for TTT, since vector fields are conformal only in d = 4. In this way the general solution of the Ward identities including anomalous ones for these correlators in (Euclidean) position space, found by Osborn and Petkou is mapped to the ordinary diagrammatic one in momentum space. We give simplified expressions of all these correlators in configuration space which are explicitly Fourier integrable and provide a diagrammatic interpretation of all the contact terms arising when two or more of the points coincide. We discuss how the anomalies arise in each approach. We then outline a general algorithm for mapping correlators from position to momentum space, and illustrate its application in the case of the VVV and TOO vertices. The method implements an intermediate regularization - similar to differential regularization - for the identification of the integrands in momentum space, and one extra regulator. The relation between the ordinary Feynman expansion and the logarithmic one generated by this approach are briefly discussed.