Annals of Mathematical Sciences and Applications

Volume 1 (2016)

Number 1

The Chern–Simons number as a dynamical variable

Pages: 123 – 147

DOI: https://dx.doi.org/10.4310/AMSA.2016.v1.n1.a3

Authors

S.-H. Henry Tye (Department of Physics, Jockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong; and the Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, New York, U.S.A.)

Sam S. C. Wong (Department of Physics, Jockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong)

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.

Keywords

Baryon number violation, Chern–Simons number, Hopf invariant, sphaleron

2010 Mathematics Subject Classification

Primary 55P99, 81T13. Secondary 55Q25.

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Published 12 April 2016