# Nonrelativistic Yang-Mills Theory for a Naturally Light Higgs Boson

## TL;DR

This paper explores nonrelativistic gauge theories with polynomial shift symmetries as a way to naturally keep the Higgs boson light, analyzing divergences and hierarchy stability in a scalar QED model with dynamical exponent z=3.

## Contribution

It demonstrates that nonrelativistic scalar QED with z=3 can maintain a large hierarchy between scalar mass and high-energy scale, with refined divergence structures in theories lacking boost symmetry.

## Key findings

- Power law divergences have a refined structure without boost symmetry.
- Hierarchy between scalar mass and high-energy scale can be preserved across 7 orders of magnitude.
- Gauge coupling of order 0.1 is compatible with the hierarchy.

## Abstract

We continue the study of the nonrelativistic short-distance completions of a naturally light Higgs, focusing on the interplay between the gauge symmetries and the polynomial shift symmetries. We investigate the naturalness of nonrelativistic scalar quantum electrodynamics with a dynamical critical exponent $z=3$ by computing leading power law divergences to the scalar propagator in this theory. We find that power law divergences exhibit a more refined structure in theories that lack boost symmetries. Finally, in this toy model, we show that it is possible to preserve a fairly large hierarchy between the scalar mass and the high energy naturalness scale across 7 orders of magnitude, while accommodating a gauge coupling of order 0.1.