Revisiting Schr\"odinger CFTs: Factorization, Massless Particles, and a Path to the Bootstrap

TL;DR

This paper develops a modern framework for Schr"odinger conformal field theories, introducing harmonic trap geometry, classifying spectra, and establishing foundations for a bootstrap approach, with implications for unitarity bounds and non-renormalization theorems.

Contribution

It introduces harmonic trap geometry and a state-operator correspondence for Schr"odinger CFTs, extending spectral classification and unitarity bounds to include massless and massive states.

Findings

Classified all physical spectra and unitarity bounds in harmonic trap geometry.

Demonstrated that massless states are described by an effective 1d CFT.

Revealed that non-renormalization theorems follow from non-perturbative factorization.

Abstract

We revisit Schr\"odinger CFTs from a modern point of view. We introduce the ''harmonic trap geometry,'' analogous to the cylinder picture in relativistic CFTs, and demonstrate a state-operator correspondence that applies to all operators, including descendant, massless, and ''normal-ordered operators.'' A thermofield double construction plays an extremely important role. We systematically classify all physical spectra in the harmonic trap and their unitarity bounds, extending earlier results to include both massless and massive states of all spins, providing a new analytic treatment of unitarity bounds, and establishing foundations for a bootstrap. In our reformulation, previously known perturbative non-renormalization theorems follow immediately from non-perturbative factorization at fixed points and along RG flows. Massless states are described by an effective 1d CFT, as predicted by DLCQ, and violate the non-renormalization theorems. We include a self-consistent review of Schr\"odinger CFTs in our framework, making the paper accessible to anyone with a field theory background.