# Renormalization Group Flows in Non-Relativistic Holographic Effective Field Theories

## TL;DR

This paper introduces a formalism for non-Lorentz invariant quantum field theories with separate space and time cutoffs, and calculates the associated beta-functions in holographic and weakly-coupled Lifshitz theories.

## Contribution

It develops a dual cutoff formalism with independent beta-functions and applies it to holographic Lifshitz and hyperscaling violation theories, extending previous superpotential methods.

## Key findings

- Calculated beta-functions around Lifshitz critical points.
- Extended superpotential formalism for non-relativistic theories.
- Compared holographic and weakly-coupled Lifshitz beta-functions.

## Abstract

We develop a formalism with two different UV cutoff scales, one for space and one for time, appropriate for the richer structure of non-Lorentz invariant quantum field theories. In this formalism there are two different beta-functions for each coupling constant, arising from independent variations of the energy or momentum cutoffs. For holographic non-relativistic theories with rotational invariance, we develop the technique to calculate such beta-functions using a generalization of the superpotential formalism developed in [JHEP ${\bf 1208}$, 164 (2012)]. We then proceed and compute the beta-function around a Lifshitz critical point, as well as for general Lifshitz-invariant theories with hyperscaling violation. Finally, we do a similar computation in a weakly-coupled Lifshitz invariant QFT.