Fermion number 1/2 of sphalerons and spectral mirror symmetry
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Regular Article - Theoretical Physics
Fermion number 1/2 of sphalerons and spectral mirror symmetry M. Mehraeen1,2,a , S. S. Gousheh2,b 1 2
Department of Physics, Case Western Reserve University, 10900, Euclid Avenue, Cleveland, OH 44106, USA Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran
Received: 1 February 2020 / Accepted: 8 September 2020 © The Author(s) 2020
Abstract We present a rederivation of the baryon and lepton numbers 21 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 21 of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency 21 of the Dirac sea. The relevance of this analysis to other solutions is also discussed.
1 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]. Their key discovery was the existence of states with halfinteger fermion numbers in theories where all the fundamental fields have integer fermion numbers. As was pointed out in [1–4], in order for a bosonic configuration to have halfinteger fermion numbers, the following two conditions must be simultaneously satisfied: a e-mail:
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1. The existence of a normalizable fermionic zero mode precisely at the bosonic configuration; 2. Mirror symmetry of the entire fermionic spectrum, consisting of the bound and continuum states, at the bosonic configuration, or equivalently, the fermion number conjugation invariance of the system.
Since then, charge fractionalization has been widely studied and has found many applications in different areas, such as particle physics [2–8], condensed matter physics [9–11], polymer physics [12–14], quantum wires [15] and topological insulators [16,17]. One class of solutions that can be found in certain field theories are sphalerons, which are saddle-point solutions in field configuration space [18,19]. An important member of this class of solutions is the ‘S’ sphaleron [20
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