Majorana Ghosts
From topological superconductor to
the origin of neutrino mass, three
generations and their mass mixing
Zheng-Cheng Gu (PI)
Z.C. Gu, arXiv:1308.2488
Sanya
Dec, 2013
Neutrino
Neutrino as a ghost particle:

Extremely weak interactions with other particles.

Almost vanishing rest mass.

Billions of neutrinos surrounding us!
Ghosts hunting: neutrino oscillation
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.

Current experimental progress
toward ghost hunting
Three-generation mixing matrix:

Recent result

CP violation
(China)
(from Michaelmas Term 2009, Prof Mark Thomson)
(Japan)
(Canada)
Sterile neutrino and see-saw
mechanism
Standard mode predicts exactly zero mass for neutrino!
A direct Majorana mass term for light neutrino is not allowed in Standard
Model(SM) since it breaks electroweak symmetry.
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)
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.
Require neutrino to be a Majorana fermion!
An elegant proposal by Ettore
Majorana
Real solution of Dirac equation(in standard representation)

Neutrinos are promising candidates.
But why there are three generations and where do
those mystery mixing angles come from?

The main goal of this talk:
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.

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
Time reversal symmetry of Majorana
zero modes
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.


Boson has T2=1 time reversal symmetry.

Fermion=half boson, thus it has T2=-1 time reversal symmetry.
Majorana zero modes=half fermion, thus it has T4=-1 time
reversal symmetry.

A toy model
Two copies of Kitaev's Majorana chains withT2=-1 time
reversal symmetry
time reversal symmetry:
leads to:

At first glance, the Majorana spinon is just a Kramers doublet with T2=-1.

It is incorrect since such a definition will lead to T2=1 on a physical site.
T4=-1 time reversal symmetry
The classification of 1D (symmetry protected) topological
orders leads to the T4=-1 time reversal symmetry.
(X. Chen, Z. C. Gu and X.G. Wen Phys. Rev. B 83, 035107 (2011))

Local Hilbert space for a many body fermion system:
T2=-1for fermion parity odd sector and T2=1 for fermion parity even
sector. The total symmetry group is extended over fermion parity symmetry
group {I,Pf}, which is indeed T4=1.

(X. Chen, Z. C. Gu and X.G. Wen,Phys. Rev. B 84, 235128 (2011))
The edge Majorana modes should carry projective representation with
T4=-1

Left end:
Right end:
Representation theory
Majorana ghosts arise on the edge:
Projective representation(must be two dimensional):
Majorana zero modes in higher
dimensions
The Majorana zero modes inside the vortex core of a 2D T2=-1 symmetry
protected topological superconductor (Qi et al, 2008, S Ryu et al, Kitaev, 2009)

Majorana zero modes inside the Hedgehogs of a 3D T2=-1 symmetry
protected topological superconductor. (in progress, with Kai Sun et al)

What happens if we assume that neutrinos are Lorentz
spinons formed by four Majorana zero modes?
A lattice model with Lorentz symmetry
at low energy
T4=-1
A (Lorentz) Majorana
fermion per unit cell
P4=-1 parity symmetry for Majorana
zero modes
We can define a P4=-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.
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.

CPT super algebra for Majorana ghosts
The action of CPT symmetries on four Majorana zero
modes forms a super algebra
The new definition of CPT symmetries commutes with
spin rotation
The whole CPT algebra can be generalized into the
relativistic theory.
Relativistic field theory
CPT symmetry for Majorana ghosts
the above CPT symmetry satisfies the super algebra.
How do the Majorana fields transform under such a new
definition of CPT symmetry?
mass term breaks charge conjugation
symmetry!
A comparison of CPT symmetry for Dirac fields
The origin of neutrino mass
The nontrivial charge conjugation symmetry(a Z2 unitary symmetry) can be
promoted to a Z2 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.

The Fifth force: Z2 gauge interactions in our universe!
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.

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 T4=-1, (TP)4=-1, (TC)4=-1
symmetry for neutrino/anti-neutrino

Projective representation for d and f fermion
Three inequivalent Majorana spinons
with Lorentz and (super) CPT symmetry
Mass mixing matrix:
CPT and Lorentz invariant mass term:
For c and f fermion, Lorentz boost is defined as usual while for d
fermion, Lorentz boost is defined as:

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

Neutrino mass mixing matrix
consistent with experiment
Symmetry: D4
excluded by experiment
Prediction of neutrino masses
Two possible mass patterns from experiment
Predictions:
GUT scale
CP violation leads to small splitting for m1,m2
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)