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

TL;DR

This work derives a concise, symmetry-based expression for conformal 3-point functions in Minkowski momentum space, showing that the Wightman function of three scalars (and extensions to one spinning operator) is given by an Appell $F_4$-type double hypergeometric function with Theta-support factors. The approach combines conformal Ward identities with Lorentzian OPE boundary conditions to produce a closed-form in terms of $F_4$, valid in any $d\ge2$ and for general scaling dimensions, with analytic continuation to all kinematics. It also discusses time-ordered correlators, partial-time ordering, and the relation to the Wightman function, including 2D holomorphic factorization and generalized free field limits. The results pave the way for higher-point functions in momentum space via the momentum-space OPE and have potential applications in inflationary physics, collider bounds, and the study of conserved currents and the stress tensor in Lorentzian CFTs.

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.

Figures (3)

• Figure 1: Two examples of momentum configurations for the Wightman 3-point function. The momenta $p_i$, $p_0$ and $p_f$ add up to zero, and both $p_i$ and $-p_f$ must lie in the light cone indicated in blue for the 3-point function to be non-zero. The intermediate momentum $p_0$ can either be space-like (a) or time-like (b).
• Figure 2: Examples of momentum configurations in the limits (a) $p_f \to 0$ as in eq. \ref{['eq:limit:pf:zero']} and (b) $p_f^2, p_i^2 \to 0_-$ as in eq. \ref{['eq:limit:massless']}. The configuration (a) also shows that the limit $p_f \to 0$ can be reached taking $p_f^2 \to 0_-$ first.
• Figure 3: Examples of momentum configurations for the partially-time-ordered 3-point function \ref{['eq:partialtimeordering']}, which only has support when the momentum $-p_f$ lies in the light-cone shown in blue. In both examples the momenta $p_1$ and $p_2$ are space-like, and one can deform one configuration into the other without any light-cone crossing.