Conformal 3-point functions and the Lorentzian OPE in momentum space

TL;DR

This paper derives explicit conformal 3-point functions in Minkowski momentum space, revealing their structure as Appell hypergeometric series and extending to operators with spin, thus advancing the understanding of Lorentzian CFT correlators.

Contribution

It provides a closed-form expression for conformal 3-point functions in momentum space using hypergeometric functions, including cases with tensor operators and different operator orderings.

Findings

3-point functions are expressed as Appell $F_4$ hypergeometric series.

Extended the formula to include tensor operators with arbitrary spin.

Discussed relations between Wightman, time-ordered, and partially-time-ordered correlators.

Abstract

In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell $F_4$ type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.