Topological Mass Generation in Four Dimensions

TL;DR

By reformulating the massless Schwinger model in terms of topological densities $P_2$ and the Chern-Simons current $C_2^{\alpha}$, with $\partial_{\alpha}C_2^{\alpha} = P_2$, the authors connect mass generation to topological and anomalous dynamics, recovering the 2D mass $m^2 = e^2/π$. They extend this topological framework to four dimensions by introducing $P_4$ and $C_4^{\alpha}$ and implementing Stückelberg fields to address potential non-conservation of the axial current, yielding $\partial^2 P_4 + N Λ^2 P_4 = 0$ and hence $m^2 = N Λ^2$. The work discusses the limitations of the 4D construction, notably the need for dynamical anomaly generation and proposes a phenomenological coupling to an $\eta'$ field to resemble QCD-like $U(1)$ physics, treating the setup as an effective theory with higher-dimensional operators. Overall, the paper demonstrates how topological structures can drive mass generation in higher dimensions while highlighting the necessity of explicit dynamics and metric considerations for a fully consistent theory.

Abstract

Schwinger's mechanism for mass generation relies on topological structures of a 2-dimensional gauge theory. In the same manner, corresponding 4-dimensional topological entities give rise to topological mass generation in four dimensions.