Lorentzian CFT 3-point functions in momentum space

Abstract

In a conformal field theory, two and three-point functions of scalar operators and conserved currents are completely determined, up to constants, by conformal invariance. The expressions for these correlators in Euclidean signature are long known in position space, and were fully worked out in recent years in momentum space. In Lorentzian signature, the position-space correlators simply follow from the Euclidean ones by means of the iǫ prescription. In this paper, we compute the Lorentzian correlators in momentum space and in arbitrary dimensions for three scalar operators by means of a formal Wick rotation. We explain how tensorial three-point correlators can be obtained and, in particular, compute the correlator with two identical scalars and one energy-momentum tensor. As an application, we show that expectation values of the ANEC operator simplify in this approach.

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

“…A comprehensive discussion of this phenomenon has recently appeared in Ref. [24]. We shall see in this paper that the time-ordered product of operators in Minkowski momentum space is simply related to the known Euclidean expression, but also that the Wightman functions are very different objects that do not have a Euclidean counterpart.…”

Convergent momentum-space OPE and bootstrap equations in conformal field theory

“…First steps in resolving generic 3-point functions have been taken in Refs. [4,5], and conformal blocks have been constructed in special cases [8,9,38]. One interesting question in higher dimensions is whether there is a generalization of the region of pointwise convergence.…”

“…We will generally use the letter k to denote the lightcone components of 2D momenta p ± in the following. 4 To evaluate the Fourier transforms, we use the identity…”

“…In this paper we study the operator product expansion (OPE) in general conformal field theories (CFTs) in momentum space, and formulate a bootstrap equation for the CFT data. There have been a number of studies of aspects of CFT in momentum space, including formal developments [1][2][3][4][5], but also following physical motivations, such as the study of anomalies [6][7][8][9], the formulation of Hamiltonian truncation [10][11][12], and early-universe cosmology [13][14][15]. Correlation functions in momentum space exist as the Fourier transform of correlation functions in position space.…”

Conformal n -Point Functions in Momentum Space

“…Nevertheless, through recent efforts, all the necessary prerequisites are now in place. Firstly, the momentum-space 3-point functions of general scalar and tensorial operators are known, including the cases where anomalies and beta functions arise as a result of renormalization [34][35][36][37][38][39][40][41][42][43][44][45][46]. Secondly, momentum-space studies of the 4-point function have yielded special classes of solutions to the conformal Ward identities [15,32,[47][48][49][50][51].…”