The Chern-Simons Number as a Dynamical Variable

TL;DR

This paper explores the Chern-Simons number as a dynamic variable in electroweak theory, linking gauge field topology with Higgs field geometry to better understand baryon number violation.

Contribution

It introduces a generalized, dynamical perspective of the Chern-Simons number using Hopf invariants and winding numbers, connecting gauge and Higgs field topologies.

Findings

Chern-Simons number should be treated as a quantum dynamical variable.

Construction of Hopf invariant captures geometric information in gauge and Higgs fields.

Discussion on the relation between Hopf and Chern-Simons variables.

Abstract

In the standard electroweak theory that describes nature, the Chern-Simons number associated with the vacua as well as the unstable sphaleron solutions play a crucial role in the baryon number violating processes. We recall why the Chern-Simons number should be generalized from a set of discrete values to a dynamical (quantum) variable. Via the construction of an appropriate Hopf invariant and the winding number, we discuss how the geometric information in the gauge fields is also captured in the Higgs field. We then discuss the choice of the Hopf variable in relation to the Chern-Simons variable.