Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lorentzian CFT 3-point functions in momentum space

Teresa Bautista, Hadi Godazgar

TL;DR

The paper develops Lorentzian CFT 3-point functions in momentum space by a careful Wick rotation from Euclidean results, clarifying the role of the $i\epsilon$ prescription and causality. It provides two representations for scalar 3-point functions—a momentum-integrated approach and a triple-$K$ (Bessel) form—and analyzes their finiteness, with checks against explicit cases. The work then extends to tensorial correlators by deriving them from scalar correlators via momentum derivatives and applies the framework to compute $\langle O T_{\mu\nu} O\rangle$ and ANEC expectation values in Hofman-Maldacena states, highlighting the natural simplifications in momentum space. Overall, it discusses renormalisation subtleties in Lorentzian signature, contrasts with Euclidean results, and points to future directions for systematic tensor constructions and Lorentzian CFT bootstrap applications.

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 i-epsilon 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.

Lorentzian CFT 3-point functions in momentum space

TL;DR

The paper develops Lorentzian CFT 3-point functions in momentum space by a careful Wick rotation from Euclidean results, clarifying the role of the prescription and causality. It provides two representations for scalar 3-point functions—a momentum-integrated approach and a triple- (Bessel) form—and analyzes their finiteness, with checks against explicit cases. The work then extends to tensorial correlators by deriving them from scalar correlators via momentum derivatives and applies the framework to compute and ANEC expectation values in Hofman-Maldacena states, highlighting the natural simplifications in momentum space. Overall, it discusses renormalisation subtleties in Lorentzian signature, contrasts with Euclidean results, and points to future directions for systematic tensor constructions and Lorentzian CFT bootstrap applications.

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 i-epsilon 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.

Paper Structure

This paper contains 14 sections, 144 equations, 3 figures.

Table of Contents

  1. Introduction
  2. 2-point function
  3. 3-point function of scalars
  4. Momentum-integrated expression
  5. Triple-Bessel expression
  6. Finiteness of the Lorentzian 3-point function
  7. Check
  8. Tensorial correlators
  9. $\langle \mathcal{O} T_{\mu \nu} \mathcal{O} \rangle$ correlator
  10. ANEC expectation values on Hofman-Maldacena states
  11. Discussion
  12. Conventions and formulae
  13. 3-point function of scalars from Fourier transform
  14. 3-point function for $\{d=3,\beta_j=1/2\}$ from the momentum-integrated expression

Figures (3)

  • Figure 1: The two different Wick rotations from Euclidean to Lorentzian time, corresponding to passing either to the right or the left of the branch cut, define the two possible Wightman 2-point functions. The contour on the right corresponds to $\langle \mathcal{O}(x)\mathcal{O}(0)\rangle$ and the contour on the left to $\langle \mathcal{O}(0)\mathcal{O}(x)\rangle$.
  • Figure 2: Closed contour of integration. The contribution from the arc at infinity vanishes.
  • Figure 3: Closed contour of integration for the $p_1^{{E}}$ integral. The contribution from the arc at infinity vanishes, so the integral along the real axis is equal to that on each side of the two branch cuts on the upper-half plane.