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

Abstract

The existence of three generations of neutrinos (charged leptons/quarks) and their mass mixing are deep mysteries of our universe. It might indicate a profound internal structure of all elementary particles. It is well known that the history of neutrino physics can be traced back to Majorana's elegant work on a real solution of the Dirac equation—known as the Majorana fermion. 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 1D 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 C[over ¯]^{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 (TC[over ¯])^{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 θ_{12}=31.7^{∘},θ_{23}=45^{∘}, and θ_{13}=0^{∘}, which is consistent with an A_{5} flavor symmetry pattern (the golden ratio 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_{0νββ}=m_{1}/sqrt[5].