Fractionalized time reversal, parity and charge conjugation symmetry in topological superconductor: a possible origin of three generations of neutrinos and mass mixing

TL;DR

This paper proposes a topological framework for neutrino generations using Majorana zero modes, explaining their mass mixing and predicting specific neutrino mass ratios and mixing angles consistent with flavor symmetry.

Contribution

It introduces a novel topological model linking Majorana zero modes to neutrino generations and derives neutrino mixing parameters and mass ratios from fractionalized symmetries.

Findings

Neutrino generations correspond to three ways of forming complex fermions from Majorana zero modes.

Predicted neutrino mixing angles are θ12=31.7°, θ23=45°, θ13=0°.

Mass ratios of neutrinos are m1/m3=m2/m3=3/√5, consistent with flavor symmetry.

Abstract

The existence of three generations of neutrinos(charged leptons/quarks) and their mass mixing are deep mysteries of our universe. Recently, Majorana's spirit returns in modern condensed matter physics -- in the context of topological Majorana zero modes in certain classes of topological superconductors(TSCs). In this paper, we attempt to investigate the topological nature of the neutrino by assuming that a relativistic Majorana fermion can be divided into four topological Majorana zero modes at cutoff energy scale, e.g. planck scale. We begin with an exactly solvable $1$D lattice model which realizes a $T^2=-1$ time reversal symmetry protected TSC, and show that a pair of topological Majorana zero modes can realize a $T^4=-1$ time reversal symmetry.Moreover, we find that a pair of topological Majorana zero modes can also realize a $P^4=-1$ parity symmetry and even a nontrivial $\overline C^4=-1$ charge conjugation symmetry. Next, we argue that the origin of three generations of neutrinos(charged leptons and quarks) can be naturally explained as three distinguishable ways of forming a pair of complex fermions(with opposite spin polarizations) out of four topological Majorana zero modes, characterized by the $T^4=-1$, $(TP)^4=-1$ and $(T \overline C)^4=-1$ fractionalized symmetries that each complex fermion carries at cutoff energy scale. Finally, we use a semiclassical approach to compute the neutrino mass mixing matrix at leading order(LO), e.g., in the absence of $CP$ violation correction. We obtain $\theta_{12}=31.7^\circ, \theta_{23}=45^\circ$ and $\theta_{13}=0^\circ$(the golden ratio pattern), which is consistent with an $A_5$ flavor symmetry pattern. We further predict an exact mass ratio of the three mass eigenstates of neutrinos with $m_1/m_3=m_2/m_3=3/\sqrt{5}$ and an effective mass of neutrinoless double beta decay $m_{0\nu\beta\beta}=m_1/\sqrt{5}$.