Majorana neutrino and the vacuum of Bogoliubov quasiparticle

Kazuo Fujikawa

doi:10.1016/j.physletb.2018.04.004

Abstract

The Lagrangian of the seesaw mechanism is C violating but the same Lagrangian when re-written in terms of Majorana neutrinos is manifestly C invariant. To resolve this puzzling feature, a relativistic analogue of Bogoliubov transformation, which preserves CP but explicitly breaks C and P separately, was introduced together with the notions of a Bogoliubov quasiparticle and an analogue of the energy gap in BCS theory. The idea of Majorana neutrino as Bogoliubov quasiparticle was then suggested. In this paper, we study the vacuum structure of the Bogoliubov quasiparticle which becomes heavy by absorbing the C-breaking. By treating an infinitesimally small C violating term as an analogue of the chiral symmetry breaking nucleon mass in the model of Nambu and Jona-Lasinio, we construct an explicit form of the vacuum of the Bogoliubov quasiparticle which defines Majorana neutrinos in seesaw mechanism. The vacuum of the Bogoliubov quasiparticle thus constructed has an analogous condensate structure as the vacuum of the quasiparticle (nucleon) in the Nambu–Jona-Lasinio model.

Highlights

It is known that an effective hermitian Lagrangian, which is analogous to BCS theory,L = ν(x)iγμ∂μν(x) − mν(x)ν(x) −ǫ1 [eiα ν T (x)C ν (x) + e−iαν T (x)]ǫ5[eiβ ν γ5ν e−iβ ν γ5 ν (x)], (1)with real parameters m, ǫ1, ǫ5, α and β describes a wide variety of fundamental problems such as the seesaw mechanism for neutrino masses [1, 2, 3, 4, 5, 6, 7] and the neutron-antineutron oscillations [8, 9, 10, 11, 12]
Since the Majorana neutrinos are defined in terms of N(x), we suggested the idea of Majorana neutrino as Bogoliubov quasiparticle [15]
Note that this divergence in the integral is proportional to ǫ25, which is analogous to the case of the model of Nambu and Jona-Lasinio [17], where the divergence is proportional to the induced chiral symmetry breaking mass m2

Summary

Introduction

It is known that an effective hermitian Lagrangian, which is analogous to BCS theory,. We emphasize that the Lagrangian for the Majorana neutrino (10) with the mass eigenvalues such as (13) are in agreement with the common analyses of seesaw mechanism [1, 2, 3, 4, 5, 6, 7], but the charge conjugation is very different since (νL(x))c = CνL(x)T and (νR(x))c = CνR(x)T , which make the C-breaking in (1) unrecognized, are used there. When one identifies N (x) as a Bogoliubov quasiparticle, it would be interesting to see how the vacuum |0 N of the Bogoliubov quasiparticle, which is precisely defined with ǫ1 = 0 in (1), differs from the naive vacuum of the Dirac neutrino |0 (0) with ǫ1 = ǫ5 = 0 in (1) We discuss this problem by following the analysis of Nambu and Jona-Lasinio [17]. We clarify the difference of our relativistic analogue of Bogoliubov transformation and the conventional Bogoliubov transformation which is intrinsically non-relativistic

Conventional Bogoliubov transformation

Vacuum of Bogoliubov quasiparticle

Discussion and conclusion

A Notational convention