On the scaling critical regularity of the Yang-Mills-Higgs and the Yang-Mills-Dirac system in the Lorenz gauge

TL;DR

This paper establishes the scaling critical regularity for the Yang-Mills-Higgs and Yang-Mills-Dirac systems in the Lorenz gauge by leveraging angular regularity, advancing understanding of their well-posedness at critical thresholds.

Contribution

It introduces a novel approach using angular regularity to achieve scaling critical regularity for both systems, bypassing the need for null structure in the spinor case.

Findings

Achieved scaling critical regularity for YMH system with weighted angular regularity.

Proved scaling critical regularity for YMD system using angular regularity on spinors.

Presented a unified approach to analyze both systems simultaneously.

Abstract

In this paper, we study the local well-posedness of the $(1+3)$-dimensional Yang-Mills-Higgs (YMH) and the Yang-Mills-Dirac (YMD) system in the Lorenz gauge. Since there is some bilinear term in (YMH), which is a lack of null structure, one may obtain the well-posedness at most the energy space. However, we attain the scaling critical regularity of (YMH) by imposing the extra weighted regularity in the angular variables. In (YMD), for the coupled system to persist in time, it is required to impose the angular regularity on the Dirac spinor as much as the Yang-Mills gauge potential and curvature. We can then prove the scaling critical regularity of (YMD) using angular regularity instead of the null structure of the spinor field. In this manner, we present an approach to attack the scaling critical regularity of (YMH) and (YMD) simultaneously. This result is an application of our recent study on the Yang-Mills system in the Lorenz gauge \cite{hong}.