Correlation functions in the non-relativistic AdS/CFT correspondence

TL;DR

This work analyzes scalar correlation functions in a holographic non-relativistic CFT dual to Schrödinger spacetime with $z=2$ at zero temperature and density. At tree level, scalar $n$-point functions in Sch$_{d+3}$ are obtained by projecting AdS$_{d+3}$ amplitudes in light-cone coordinates onto definite non-relativistic masses, equivalent to a shift $m^2=m_0^2+\beta^2 M^2$ for the bulk-to-boundary propagator. Loop amplitudes, however, do not admit this mapping and must be computed directly in the Schrödinger background with regularization to handle zero modes in the compact $\\partial_\\xi$ direction. The authors explicitly compute the scalar 2- and 3-point functions, showing the holographic 3-point scaling function $\\Psi(y)$ matches the non-relativistic theory of cold atoms at unitarity, providing a nontrivial consistency check for non-relativistic AdS/CFT. Overall, the results support the idea that Schrödinger holography captures universal features of non-relativistic conformal theories in the unitary regime and illuminate the role of mass projections and regularization in connecting AdS and Schrödinger descriptions.

Abstract

We study the correlation functions of scalar operators in the theory defined as the holographic dual of the Schroedinger background with dynamical exponent z=2 at zero temperature and zero chemical potential. We offer a closed expression of the correlation functions at tree level in terms of Fourier transforms of the corresponding n-point functions computed from pure AdS in the lightcone frame. At the loop level this mapping does not hold and one has to use the full Schroedinger background, after proper regularization. We explicitly compute the 3-point function comparing it with the specific 3-point function of the non-relativistic theory of cold atoms at unitarity. We find agreement of both 3-point functions, including the part not fixed by the symmetry, up to an overall normalization constant.

Figures (7)

• Figure 1: Witten diagram that gives the tree level contribution to the 3-point function.
• Figure 2: Witten diagram that gives the tree level contribution to the 4-point function with 3-point contact interactions.
• Figure 3: Witten diagram that gives the tree level contribution to the 6-point function with 3-point contact interactions.
• Figure 4: Witten diagram which gives a 1-loop contribution to the 4-point function
• Figure 5: A Feynman diagram which gives the 3-point function for atoms at unitarity. The lines with a single arrow denote the exact atom propagator $G_{\psi}$, while the line with two arrows represents the fully renormalized diatom propagator $G_{\phi}$.