From Schwinger to Wightman: all conformal 3-point functions in momentum space

All conformal correlation functions of 3 scalar primary operators are constructed in an axiomatic way, relying only on conformal symmetry and causality. The construction makes use of the R-product and its analyticity properties in Minkowski momentum space. The R-product is completely determined by a system of partial differential equations (the Ward identities for conformal transformations) and a boundary condition (permutation symmetry), up to an OPE coefficient. Wightman functions and time-ordered products are then derived from the R-product using operator identities, and the Schwinger function is recovered after a Wick rotation to Euclidean space. This construction does not rely on the Fourier transform of position-space correlation functions.