Momentum space approach to crossing symmetric CFT correlators

TL;DR

<3-5 sentence high-level summary> The paper develops a momentum-space, crossing-symmetric basis for conformal four-point functions, completing the construction for scalar intermediates with general-spin exchanges by leveraging analytic three-point functions and Polyakov’s factorization. It shows that Polyakov blocks in momentum space are realized by Witten exchange diagrams for scalar intermediates and extends the construction to spinning exchanges using helicity decompositions and differential operators, yielding explicit formulas for two- and three-point functions and for the spinning Polyakov blocks. A key advantage of this basis is the manifest separation of connected and disconnected contributions, which is particularly useful for large $N$ CFTs and their holographic duals, and provides a natural framework for extending Polyakov-type bootstrap ideas to momentum space and cosmological correlators. This work thus supplies analytic, factorization-based building blocks for momentum-space conformal correlators and opens avenues for applications in holography and cosmology, including inflationary collider physics.

Abstract

We construct a crossing symmetric basis for conformal four-point functions in momentum space by requiring consistent factorization. Just as scattering amplitudes factorize when the intermediate particle is on-shell, non-analytic parts of conformal correlators enjoy a similar factorization in momentum space. Based on this property, Polyakov, in his pioneering 1974 work, introduced a basis for conformal correlators which manifestly satisfies the crossing symmetry. He then initiated the bootstrap program by requiring its consistency with the operator product expansion. This approach is complementary to the ordinary bootstrap program, which is based on the conformal block and requires the crossing symmetry as a consistency condition of the theory. Even though Polyakov's original bootstrap approach has been revisited recently, the crossing symmetric basis has not been constructed explicitly in momentum space. In this paper we complete the construction of the crossing symmetric basis for scalar four-point functions with an intermediate operator with a general spin, by using new analytic expressions for three-point functions involving one tensor. Our new basis manifests the analytic properties of conformal correlators. Also the connected and disconnected correlators are manifestly separated, so that it will be useful for the study of large $N$ CFTs in particular.

Figures (6)

• Figure 1: Three-point functions of two scalars and one tensor can be obtained by acting a differential operator ${\mathcal{A}}_{123_m}$ on the cubic Witten diagram of three scalars, with an additional factor $z^s$ multiplied to the vertex in the integrand.
• Figure 2: The Polyakov block for an intermediate spinning operator can be obtained by acting two differential operators ${\mathcal{A}}_{12n_m}$ and ${\mathcal{A}}_{34n_m}$ on the scalar Witten exchange diagram with an additional factor $z^s$ multiplied to the cubic vertices in the integrand.
• Figure 3: Due to momentum conservation, external momenta of correlation functions form a polygon. The soft limit of an external momentum, say $k_4\to0$ (the left figure), corresponds to the OPE limit in position space, where one operator is far separated from the others (the right figure).
• Figure 4: In the collapsed limit, an internal momentum, say ${\boldsymbol{k}}_{12}$, becomes soft compared to external momenta (the left figure). This limit corresponds to the OPE limit in position space, where operators are separated into two groups far away as depicted in the right figure.
• Figure 5: In the collinear limit, two external momenta are parallel and point the same direction.