Fermion number 1/2 of sphalerons and spectral mirror symmetry

M Mehraeen,S S Gousheh

doi:10.1140/epjc/s10052-020-08451-4

Abstract

We present a rederivation of the baryon and lepton numbers frac{1}{2} of the SU(2)_L S sphaleron of the standard electroweak theory based on spectral mirror symmetry. We explore the properties of a fermionic Hamiltonian under discrete transformations along a noncontractible loop of field configurations that passes through the sphaleron and whose endpoints are the vacuum. As is well known, CP transformation is not a symmetry of the system anywhere on the loop, except at the endpoints. By augmenting CP with a chirality transformation, we observe that the Dirac Hamiltonian is odd under the new transformation precisely at the sphaleron, and this ensures the mirror symmetry of the spectrum, including the continua. As a consistency check, we show that the fermionic zero mode presented by Ringwald in the sphaleron background is invariant under the new transformation. The spectral mirror symmetry which we establish here, together with the presence of the zero mode, are the two necessary conditions whence the fermion number frac{1}{2} of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency frac{1}{2} of the Dirac sea. The relevance of this analysis to other solutions is also discussed.

Highlights

In their seminal paper on the subject of charge fractionalization, Jackiw and Rebbi studied the Dirac equation in classical bosonic backgrounds for a number of field theories [1]
Its importance is due to the role that it is believed to play in the early Universe, including the generation of the matter-antimatter asymmetry of the Universe [21,22,23]
The main goal of this paper is to present a rederivation of the half-integer fermion numbers of SU (2)L S sphalerons by adopting an approach that is based on discrete symmetries

Summary

Introduction

In their seminal paper on the subject of charge fractionalization, Jackiw and Rebbi studied the Dirac equation in classical bosonic backgrounds for a number of field theories [1]. In the standard electroweak theory, a single normalizable zero energy solution of the Dirac equation in the sphaleron background was shown to exist for massless fermions in [27] After, this result was extended by Ringwald to the case of massive fermions [28]. Whereas some works have based their arguments on a fermionic zero mode in the limit of vanishing fermion mass [27,33], in this work we have used the Ansatz given by [28], which is an extension to massive fermions within the standard electroweak theory In this case, the Higgs component of the sphaleron plays a nontrivial and essential role, which is beyond an explicit mass term. We assume that in the radial gauge there exists a limiting field

Standard derivation of Fermion number

CP transformation along NCL

The zero mode

Summary and discussion

Dirac Hamiltonian along NCL