Majorana ghosts: From topological superconductor to the origin of neutrino masses, three generations and their mass mixing

The existence of three generations of neutrinos and their mass mixing are the deep mysteries of our universe. 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. A cutting-edge step towards understanding the nature of neutrino has been taken by the experimental discovery of neutrino mass mixing during the past decade, which indicates neutrino has a small but non-vanishing mass. A natural way to explain the origin of this small mass is the so-called seesaw mechanism, which requires the neutrino to be a Majorana fermion. Recently, Majorana’s spirit returns in modern condensed matter physics – in the context of Majorana zero modes in certain classes of topological superconductors(TSCs). In this paper, we attempt to investigate the topological nature of the neutrino by establishing a connection between the Majorana fermion and Majorana zero modes – assuming a relativistic Majorana fermion is made up of four Majorana zero modes. We begin with an exactly solvable 1D condensed matter model which realizes a T 2 = −1 time reversal symmetry protected TSC. We show that the pair of Majorana zero modes on each end will realize a T 4 = −1 representation of the time reversal symmetry and carry 1/4 spin. We find that a pair of Majorana zero modes can realize a P 4 = −1 parity symmetry as well and even a nontrivial C = −1 charge conjugation symmetry. The CPT symmetries for a Majorana fermion made up of four Majorana modes form a super algebra. We then generalize the CPT super algebra into quantum field theory and point out that the nontrivial charge conjugation symmetry can be promoted to a Z2 gauge symmetry, whose spontaneously breaking leads to the origin of the (right-handed) neutrino mass. The Z2 gauge symmetry indicates the existence of the fifth force in our universe, which is possible to be detected in future LHC experiment. Finally, we show that the origin of three generations of neutrinos can be naturally explained as three distinguishable ways to form a pair of complex fermions(a particle and an anti-particle) out of four Majorana zero modes, characterized by the T 4 = −1, (TP )4 = −1 and (TC)4 = −1 fractionalized symmetries that particles/anti-particles carry. Together with the Z2 gauge (minimal coupling) principle, we are able to determine the mass mixing matrix with no fitting parameter at leading order(in the absence of the CP violation and charged lepton contribution). We obtain θ12 = 31.7 ◦, θ23 = 45 ◦ and θ13 = 0 ◦(known as the golden ratio pattern), which are intrinsically close to the current experimental results. We further predict an exact mass ratio for the three mass eigenstates with m1/m3 = m2/m3 = 3/ √ 5.