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 iϵ 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.
Highlights
Despite the recent interest in Lorentzian CFTs, an expression of the correlation functions in momentum space beyond 2 points has hitherto been lacking
It is well known that conformal symmetry places restrictions on the form of correlators in a conformal field theory, fixing the 2-point functions, up to a normalisation, and 3-point functions, up to constants that are part of the CFT data
While these correlators can in principle be found by Fourier transforming the Lorentzian correlators in position space, which is already more difficult than the analogous Fourier transform in Euclidean space, we instead show that these correlators can be derived from the Euclidean expressions by a careful Wick rotation
Summary
The 3-point function of scalar operators in position space is given by [23], O1(x1) O2(x2) O3(x3). For small enough N such that βj > 0, or more generically when d is not an even integer, this case is different from the previous, and the expression for the 2-point function to be used is given by (2.21). If we insisted on using the general expression (3.8) for the special βj cases, we would find that the coefficient c({βj}) in (3.9) vanishes due to Γ-functions with negative integer arguments in the denominator, and that the momentum integral diverges. The momentum integral of the final expression could still diverge, but it is always finite (possibly obtained by analytical continuation) because, as we will show in the subsection, the 3-point function is finite without requiring renormalisation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
