# Abelian Higgs model at four loops, fixed-point collision and deconfined criticality

## TL;DR

This paper performs a four-loop analysis of the abelian Higgs model to determine the critical number of scalar fields where the phase transition changes from second to first order, with implications for deconfined quantum criticality.

## Contribution

It provides the first four-loop calculation of the abelian Higgs model in $4- ext{epsilon}$ expansion and estimates the critical number of fields in three dimensions, revealing a transition from second to first order.

## Key findings

- Critical number of scalar fields $n_c$ is approximately 12.2 in 3D.
- Transition changes from second to first order below $n_c$.
- Strong evidence for a smooth interpolation of $n_c$ between 2 and 4 dimensions.

## Abstract

The abelian Higgs model is the textbook example for the superconducting transition and the Anderson-Higgs mechanism, and has become pivotal in the description of deconfined quantum criticality. We study the abelian Higgs model with $n$ complex scalar fields at unprecedented four-loop order in the $4-\epsilon$ expansion and find that the annihilation of the critical and bicritical points occurs at a critical number of $n_c \approx 182.95\left(1 - 1.752\epsilon + 0.798 \epsilon^2 + 0.362\epsilon^3\right) + \mathcal{O}\left(\epsilon^4\right)\nonumber$. Consequently, below $n_c$, the transition turns from second to first order. Resummation of the series to extract the result in three-dimensions provides strong evidence for a critical $n_c(d=3)$ which is significantly below the leading-order value, but the estimates for $n_c$ are widely spread. Conjecturing the topology of the renormalization group flow between two and four dimensions, we obtain a smooth interpolation function for $n_c(d)$ and find $n_c(3)\approx 12.2\pm 3.9$ as our best estimate in three dimensions. Finally, we discuss Miransky scaling occurring below $n_c$ and comment on implications for weakly first-order behavior of deconfined quantum transitions. We predict an emergent hierarchy of length scales between deconfined quantum transitions corresponding to different $n$.