1. Zheng-Cheng Gu (PI) Z.C. Gu, arXiv:1308.2488 Sanya Dec, 2013 Majorana Ghosts From topological superconductor to the origin of neutrino mass, three generations and their mass mixing

2. Neutrino Neutrino as a ghost particle: Ghosts hunting: neutrino oscillation Extremely weak interactions with other particles. Almost vanishing rest mass. Billions of neutrinos surrounding us! There exist three generations of (light) neutrinos, identified through their weak doublets: electron, muon and tauon. Weak eigenstates do not coincide with mass eigenstates, oscillation happens during their propagation.

3. Current experimental progress toward ghost hunting Three-generation mixing matrix: (from Michaelmas Term 2009, Prof Mark Thomson) Recent result CP violation (China) (Canada) (Japan)

4. Sterile neutrino and see-saw mechanism A direct Majorana mass term for light neutrino is not allowed in Standard Model(SM) since it breaks electroweak symmetry. Standard mode predicts exactly zero mass for neutrino! GUT scale Why it is so massive and where does the mass come from? How to explain the experimental mass mixing textures from a simple principle at leading order? We assume that mass and mass mixing of light neutrinos are induced through see-saw mechanism. See-saw mechanism and heavy sterile neutrino. M. Gell-Mann, P. Ramond, and R. Slansky(1979), T. Yanagida(1979) S. Wienberg(1979), R. N. Mohapatra and G. Senjanovic(1980) Require neutrino to be a Majorana fermion!

5. An elegant proposal by Ettore Majorana Real solution of Dirac equation(in standard representation) Try to understand the origin of neutrino masses, three generations and compute their mass mixing matrix by investigating the topological nature of a Majorana fermion. We refer a topological Majorana fermion formed by four Majorana zero modes as a Majorana ghost. The main goal of this talk: Neutrinos are promising candidates. But why there are three generations and where do those mystery mixing angles come from?

6. Majorana zero mode: the concept of half degree of freedom Majorana zero mode in the ends Kitaev's Majorana Chain A. Kitaev (2001) Majorana zero mode in vortex core of 2D p+ip TSC N. Read and Green, Phys. Rev. B 61, 10267 (2000). A single Majorana zero mode contains half degree of freedom = half (spinless) fermion

7. Time reversal symmetry of Majorana zero modes Boson has T 2 =1 time reversal symmetry. Fermion=half boson, thus it has T 2 =-1 time reversal symmetry. Majorana zero modes=half fermion, thus it has T 4 =-1 time reversal symmetry. Time reversal symmetry is arguably an unbroken symmetry in a fundamental theory(e.g., it is a local symmetry in quantum gravity), regardless of its spontaneous broken in Standard Model at low energy.

8. A toy model Two copies of Kitaev's Majorana chains withT 2 =-1 time reversal symmetry At first glance, the Majorana spinon is just a Kramers doublet with T 2 =-1. It is incorrect since such a definition will lead to T 2 =1 on a physical site. leads to: time reversal symmetry:

9. T 4 =-1 time reversal symmetry The classification of 1D (symmetry protected) topological orders leads to the T 4 =-1 time reversal symmetry. Local Hilbert space for a many body fermion system: The edge Majorana modes should carry projective representation with T 4 =-1 T 2 =-1for fermion parity odd sector and T 2 =1 for fermion parity even sector. The total symmetry group is extended over fermion parity symmetry group {I,P f }, which is indeed T 4 =1. (X. Chen, Z. C. Gu and X.G. Wen,Phys. Rev. B 84, 235128 (2011)) (X. Chen, Z. C. Gu and X.G. Wen Phys. Rev. B 83, 035107 (2011)) Left end: Right end:

10. Representation theory Majorana ghosts arise on the edge: Projective representation(must be two dimensional):

11. Majorana zero modes in higher dimensions The Majorana zero modes inside the vortex core of a 2D T 2 =-1 symmetry protected topological superconductor (Qi et al, 2008, S Ryu et al, Kitaev, 2009) What happens if we assume that neutrinos are Lorentz spinons formed by four Majorana zero modes? Majorana zero modes inside the Hedgehogs of a 3D T 2 =-1 symmetry protected topological superconductor. (in progress, with Kai Sun et al)

12. A lattice model with Lorentz symmetry at low energy T 4 =-1 A (Lorentz) Majorana fermion per unit cell

13. P 4 =-1 parity symmetry for Majorana zero modes We can define a P 4 =-1 parity symmetry for a pair of Majorana zero modes as well. We only consider the parity action on internal space here, and its spacial action will be included in quantum field theory later. Parity symmetry acts on the spin basis and chiral basis in an expected way.

14. Nontrivial charge conjugation symmetry for Majorana zero modes Since a Majorana particle does not carry charge, its charge conjugation symmetry must be trivial! To our surprise, it turns out that Majorana ghosts formed by Majorana zero modes can have a nontrivial charge conjugation symmetry. Charge conjugation symmetry acts on the spin basis(particle-hole) and chiral basis(particle-antiparticle) in an expected way.

15. CPT super algebra for Majorana ghosts The new definition of CPT symmetries commutes with spin rotation The whole CPT algebra can be generalized into the relativistic theory. The action of CPT symmetries on four Majorana zero modes forms a super algebra

16. Relativistic field theory CPT symmetry for Majorana ghosts the above CPT symmetry satisfies the super algebra.

17. mass term breaks charge conjugation symmetry! How do the Majorana fields transform under such a new definition of CPT symmetry? A comparison of CPT symmetry for Dirac fields

18. The origin of neutrino mass The Fifth force: Z 2 gauge interactions in our universe! The nontrivial charge conjugation symmetry(a Z 2 unitary symmetry) can be promoted to a Z 2 gauge symmetry. The origin of neutrino mass can be explained as the spontaneous breaking of charge conjugation symmetry through Anderson-Higgs mechanism. A poor man’s quantum field theory approach-- a theory without regulation at cut-off, but we hope it can describe the right symmetries, which is enough for our purpose of calculating mass mixing.

19. Why three generations? A simple answer: The number 4 arises because a real Lorentz spinon is made by 4 Majorana zero modes, and the number 2 arises because the local experimental observables are 2 complex fermions: neutrino and anti-neutrino.

20. The origin of three generations Out of four Majorana zero modes, there are three and only three different ways to form complex fermions, corresponding to the three different projective representations with T 4 =-1, (TP) 4 =-1, (TC) 4 =-1 symmetry for neutrino/anti-neutrino Projective representation for d and f fermion

21. Three inequivalent Majorana spinons with Lorentz and (super) CPT symmetry

22. CPT and Lorentz invariant mass term: In Standard Model, we assume three generations of neutrinos have the same representation under CPT. Redefine fields to make them have the same CPT symmetry Mass mixing matrix: For c and f fermion, Lorentz boost is defined as usual while for d fermion, Lorentz boost is defined as:

23. Neutrino mass mixing matrix consistent with experiment excluded by experiment Symmetry: D 4

24. GUT scale Prediction of neutrino masses Predictions: Two possible mass patterns from experiment CP violation leads to small splitting for m 1,m 2

25. Conclusions and future works  Neutrinos as topological Majorana zero modes.  New CPT symmetry for Majorana fields.  The origin of three generations of neutrinos.  The origin of neutrino masses and their mass mixing.  Compute CP violation angle. (to appear, with John Preskill)  Quark CKM matrix. (to appear, with John Preskill)  Realize Standard Model in a lattice model.  Shed new light on quantum gravity.(super cohomology /super tensor category theory)