Seesaw mechanism and pseudo C-symmetry

Kazuo Fujikawa,Anca Tureanu

doi:10.1140/epjc/s10052-019-7245-y

Kazuo Fujikawa, Anca Tureanu

Open Access

https://doi.org/10.1140/epjc/s10052-019-7245-y

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Abstract

It is shown that the specific “charge conjugation” transformation used to define the Majorana fermions in the conventional seesaw mechanism, namely (nu _{R})^{C}=Coverline{nu _{R}}^{T} for a chiral fermion nu _{R} (and similarly for nu _{L}), is a hidden symmetry associated with CP symmetry, and thus it formally holds independently of the P- and C-violating terms in the CP invariant Lagrangian and it is in principle applicable to charged leptons and quarks as well. This hidden symmetry, however, is not supported by a consistent unitary operator and thus it leads to mathematical (operatorial) ambiguities. When carefully examined, it also fails as a classical transformation law in a Lorentz invariant field theory. To distinguish it from the standard charge conjugation symmetry, we suggest for it the name of pseudo C-symmetry. The pseudo C-symmetry is effective to identify Majorana neutrinos analogously to the classical Majorana condition. The analysis of CP breaking in weak interactions is performed using the conventional CP transformation, which is defined independently of the pseudo C-transformation, in the seesaw model after mass diagonalization. A way to ensure an operatorially consistent formulation of C-conjugation is to formulate the seesaw scheme by invoking a relativistic analogue of the Bogoliubov transformation.

Highlights

Lagrangian are the Majorana fermions that are the exact eigenstates of the charge conjugation by definition
It is shown that the specific “charge conjugation” transformation used to define the Majorana fermions in the conventional seesaw mechanism, namelyC = CνR T for a chiral fermion νR, is a hidden symmetry associated with CP symmetry, and it formally holds independently of the P- and C-violating terms in the CP invariant Lagrangian and it is in principle applicable to charged leptons and quarks as well
To distinguish it from the standard charge conjugation symmetry, we suggest for it the name of pseudo C-symmetry

Summary

Introduction

Lagrangian are the Majorana fermions that are the exact eigenstates of the charge conjugation by definition. We shall use the term charge conjugation in seesaw (and, later on, pseudo C-transformation) for the operation used in defining Majorana neutrinos in the seesaw scheme, and denote it by C. The charge conjugation in seesaw is insensitive to the left-right mass asymmetry in the seesaw Lagrangian (see Eq (12)) This means that, were we able to find a quantum realization of it, that would change a particle of a given helicity to an antiparticle of the opposite helicity, pseudo-C could not be an internal transformation. The purpose of the present paper is to identify the theoretical origin of the pseudo-C conjugation (6) which appears to work regardless of the formal violation of parity (left-right symmetry) and the charge conjugation violation in the CP invariant seesaw model after the diagonalization of the neutrino mass terms. How comes that an ill-defined concept formally leads to correct results?

Seesaw Lagrangian

Pseudo C-symmetry as hidden symmetry associated with CP invariance

Seesaw formulation with Bogoliubov transformation

Discussion and conclusion