# Are there Goldstone bosons in ${d\leq z+1}$ ?

## TL;DR

This paper investigates the existence of Goldstone bosons in Lifshitz-scaling theories, showing that quantum fluctuations prevent symmetry breaking in field theory for certain dimensions, while holography suggests possible symmetry breaking due to large N effects.

## Contribution

It compares field theory and holographic approaches to symmetry breaking in Lifshitz theories, revealing a discrepancy influenced by large N limits and renormalization subtleties.

## Key findings

- Quantum fluctuations prevent symmetry breaking when d ≤ z+1 in field theory.
- Holography indicates the possibility of symmetry breaking due to large N limit.
- Renormalization of bulk vector modes is crucial for proper holographic description.

## Abstract

We study the viability of spontaneous breaking of continuous symmetries in theories with Lifshitz scaling, according to the number of space-time dimensions $d$ and the dynamical scaling $z$. Then, the answer to the question in the title is no (quantum field theoretically) and yes (holographically). With field theory tools, we show that symmetry breaking is indeed prevented by large quantum fluctuations when $d\leq z+1$, as expected from scaling arguments. With holographic tools, on the other hand, we find nothing that prevents the existence of a vacuum expectation value. This difference is made possible by the large $N$ limit of holography. An important subtlety in this last framework is that in order to get a proper description of a conserved current, renormalization of the temporal mode of the bulk vector requires an alternative quantization. We also comment on the implications of turning on temperature.