Abstract
We present a study of the correlation function of three stress-energy tensors in d dimensions using free field theory realizations, and compare them to the exact solutions of their conformal Ward identities (CWI's) obtained by a general approach in momentum space. The identification of the corresponding form factors is performed within a reconstruction method, based on the identification of the transverse traceless components (Ai) of the same correlator. The solutions of the primary CWI' s are found by exploiting the universality of the Fuchsian indices of the conformal operators and a re-arrangement of the corresponding inhomogenous hypergeometric systems. We confirm the number of constants in the solution of the primary CWI's of previous analysis. In our comparison with perturbation theory, we discuss scalar, fermion and spin 1 exchanges at 1-loop in dimensional regularization. Explicit checks in d=3,4,5 prove the consistency of this correspondence. By matching the 3 constants of the CFT solution with the 3 free field theory sectors available in d=4, the general solutions of the conformal constraints is expressed just in terms of ordinary scalar 2- and 3-point functions (B0,C0). We show how the renormalized d=4TTT vertex separates naturally into the sum of a traceless and an anomaly part, the latter determined by the anomaly functional and generated by the renormalization of the correlator in dimensional regularization. The result confirms the emergence of anomaly poles and effective massless exchanges as a specific signature of conformal anomalies in momentum space, directly connected to the renormalization of the corresponding gravitational vertices, generalizing the behaviour found for the TJJ vertex in previous works.
Highlights
Exact results in four dimensional conformal field theories (CFT’s) have gathered a lot of attention along the years, mostly because the enlarged SO(2, 4) symmetry of such theories has been essential for determining the structure of the correlators, especially for 2- and 3-point functions
While the equivalence between the CFT and the free field theory solutions is obviously expected at some level, the search for an exact match between the two approaches provides a simplification of the results and offers some physical intuition about the origin of the conformal anomaly, once we move to momentum space
We have presented a comparative study of the 3-graviton vertex T T T in CFT’s in momentum space
Summary
Exact results in four dimensional conformal field theories (CFT’s) have gathered a lot of attention along the years, mostly because the enlarged SO(2, 4) symmetry of such theories has been essential for determining the structure of the correlators, especially for 2- and 3-point functions. These are derived by imposing on them the corresponding conformal Ward identities (CWI’s), which in even spacetime dimensions are broken by the conformal anomaly [1]. This works extends the perturbative results contained in [7], where some of the methods used for the transition to momentum space have been extensively discussed and to which we refer for further details
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