Neutrino Masses and Mixing:

1
Neutrino Masses and Mixing

New Symmetry of Nature?
A. Yu. Smirnov
International Centre for Theoretical Physics,
Trieste, Italy Institute for Nuclear Research,
RAS, Moscow, Russia
Summarizing achievements and highlights
Pattern of lepton mixings New theoretical
puzzle?
Towards the underlying physics
New symmetry of Nature?
Quarks - Lepton symmetry
Flavor symmetry or family symmetry?
Mass spectrum Degenerate or hierarchical?
2
Flavors, Masses, Mixing

Flavor states
Mixing

Flavor states
Mass eigenstate
nm
nt
ne
Eigenstates of the CC weak interactions
m3
n3
?
ns
Sterile neutrinos no weak interactions
mass
m2
n2
Mass eigenstates
m1
n1
n2
n3
n1
m1
m2
m3
Neutrino mass and flavor spectrum
A Yu Smirnov
3
Mixing and Flavor states
n2 sinq ne cosq nm
n1 cosq ne - sinq nm
vacuum mixing angle
coherent mixture of mass eigenstates
ne cosq n1 sinq n2
n1
wave packets
ne
n2
D m 2 2E
Df ---- l
Interference of the parts of wave packets with
the same flavor depends on the phase difference
Df between n1 and n2
Dm2 m22 - m12
4
Reactor experiments CHOOZ, PaloVerde KamLAND

Solar neutrinos
Accelerator experiments
Masses, mixing Dmij2, qi j
Oscillations Conversion in matter
Supernova neutrinos, SN 1987A
Cosmology
Neutrinoless double beta decay
Atmospheric neutrinos
Direct kinematic measurements of neutrino masses
A Yu Smirnov
5
Vacuum Oscillations
Flavors of mass eigenstates do not change
Determined by q
Admixtures of mass eigenstates do not change
no n1 lt-gt n2 transitions
n2
ne
n1
Df Dvphase t
Df 0
Dm2 2E
Dvphase
Dm2 m22 - m12
Due to difference of masses n1 and n2 have
different phase velocities
Oscillation length
ln 2p/Dvphase 4pE/Dm2
oscillations
Amplitude (depth) of oscillations
effects of the phase difference increase which
changes the interference pattern
A sin22q
A. Yu. Smirnov
6
Adiabatic conversion
n1m lt--gt n2m

P sin2 q
n0 gtgt nR
Non-oscillatory transition
n2m n1m
n2 n1
interference suppressed
Resonance
Mixing suppressed
n0 gt nR
Adiabatic conversion oscillations
n2m n1m
n2 n1
n0 lt nR
Small matter corrections
n2m n1m
n2 n1
ne
A. Yu. Smirnov
7
Atmospheric neutrinos
nm - nt vacuum oscillations

ne
p
p
m
nm
nm - ne oscillations in matter
N
e
nm
n
core
cosmic rays
At low energies
r Fm /Fe 2
atmosphere
n
mantle
Parametric effects in nm - ne oscillations
for core crossing trajectories
detector
8
Oscillation parameters
SuperKamiokande
1.9x10-3 lt ?m2 lt 3.0x10-3 eV2 0.90 lt sin22? at
90 C.L.
Preliminary
?m22.4x10-3,sin22?1.00 ?2min37.8/40
d.o.f (sin22?1.02, ?2min37.7/40 d.o.f)
L/E analysis Zenith angle analysis
90 allowed regions
9
Solar Neutrinos
4p 2e- 4He 2ne 26.73 MeV
Adiabatic conversion in matter of the Sun
electron neutrinos are produced
F 6 1010 cm-2 c-1
r (150 0) g/cc
total flux at the Earth
Oscillations in vacuum
n
Oscillations in matter of the Earth
J.N. Bahcall
10
After SNO salt results
P. de Holanda, A.S.

solar data
solar data KamLAND
sin2q13 0.0
KL
Dm2 6.3 10-5 eV2
Dm2 7.1 10-5eV2
tan2q 0.40
tan2q 0.39
11
KamLAND
Kamioka Large Anti-Neutrino Detector
1 kton of LS
Reactor long baseline experiment 150 - 210 km
Liquid scintillation detector
ne p ---gt e n
Epr gt 2.6 MeV
Total rate energy spectrum of events
LMA
precise determination of the oscillation
parameters
10 accuracy
Detection of the Geo-neutrinos
Epr gt 1.3 MeV
12
U_e3
Has important implications for phenomenology
and theory supernova neutrinos can lead
to O(1) effect atmospheric
neutrinos (resonance enhancement,
parametric effects) allows to establish
mass hierarchy LBL experiments
CP-violations requires Ue3 0 key test
of models of large lepton mixing
Ue3 sin q13 e-id ltne n3 gt

CHOOZ
atmospheric
CHOOZatmospheric
tan q13
sin22q13
13
Mass spectrum and mixing
ne
nm
nt

Ue32
?
n2
n3
Dm2sun
n1
mass
mass
Dm2atm
Dm2atm
Ue32
n2
Dm2sun
n1
n3
Inverted mass hierarchy (ordering)
Normal mass hierarchy (ordering)
Type of mass spectrum with Hierarchy, Ordering,
Degeneracy absolute mass scale
Type of the mass hierarchy Normal, Inverted
Ue3 ?
A Yu Smirnov
14
Main features

m h (0.04 - 0.4) eV
m h gt Dm232 gt 0.04 eV
Heaviest mass
Hierarchy of mass squared differences
Dm122 / Dm232 0.01 - 0.15
No strong hierarchy of masses
0.22 - 0.08
m2 /m3 gt Dm122 / Dm232 0.18
Bi-large or large-maximal mixing between
neighboring families (1- 2) (2- 3)
1s
2-3
1-2
q12 qC q23 45o
1-3
bi-maximal corrections?
0 0.2 0.4 0.6
0.8
sin q
15
Quarks and Leptons

Quarks
Leptons
Mixing
1-2, q12
13o
32o
q12 qC q23 45o
2-3, q23
2.3o
45o
1-3, q13
0.5o
lt13o
Hierarchy of mass
m2 /m3 gt 0.18
Neutrinos
1s
2-3
Charged leptons
m m/mt 0.06
1-2
ms /mb 0.02 - 0.03
Down quarks
1-3
Up- quarks
mc /mt 0.005
0 0.2 0.4 0.6
0.8
sin q
16
Toward the underlying physics

Neutrino masses and mixing - important message
which we can not understand yet
In which terms, at which level theory
(principles) should be formulated?
Observables
Mass matrices
masses mixing angles are fundamental parameters
which show certain symmetry
their properties, symmetries at certain energy
scale
observables are outcome, can be to some
extend accidental numbers which do not reflect
the underlying symmetry
Schemes with bi-maximal mixing, or broken
bi-maximal mixing Tri-bimaximal mixing
No regularities - anarchy
Smallness of neutrino masses and pattern of
lepton mixing are related?
See-saw
17
Mixing and Mass Matrix
ne nm nt

ne nm nt
Mass matrix for the flavor states nf ( ne, nm
, nt )T, mf is not diagonal
Mixing
nf Un nmass
nmass ( n1, n2, n3 )T
(Majorana)
Diagonalization
mf UnT mdiag Un
mdiag diag ( m1, m2, m3)
In general (symmetry) basis the mass matrix of
the charged leptons is also non-diagonal
Charged
leptons
lS L UlL l mass L
l (e, m, t)T
Lepton mixing --gt the mismatch of rotations of
the neutrinos and the charge leptons
which diagonalize the corresponding mass matrices
Lepton
UP-MNS UlL Un
mixing matrix
18
T. Yanagida M. Gell-Mann, P. Ramond, R.
Slansky S. L. Glashow R.N. Mohapatra, G.
Senjanovic
Seesaw
Smallness of neutrino masses

1 2
NT Y L H NT MR N h.c.
mn
n
N ( nR) c
L (n, l)T
0 mD mDT MR
n N
mD
mD YltHgt
N
If MR gtgt mD
MR
mn - mDT MR-1 mD
Zero charges -gt can have Majorana mass Right
handed components singlets of the SM symmetry
group -gt mass is unprotected by symmetry can
be large -- at the scale of lepton number
violation

Smallness of neutrino mass
Neutrality QEM 0, QC 0
19
New Symmetry of Nature?

What can testify ?
Maximal or close to maximal 2-3 mixing
Very small 1 -3 mixing
Degenerate or partially degenerate spectrum
0.5 - sin2q23 ltlt sin2q23
Dm ltlt m
q13 ltlt q12 x q23
m1 m2 m3
2n data fit
Best fit value sin2 2q23 1.0
m3 m2 ltlt m1 ( inverted hierarchy) ?
sin2 2q23 gt 0.91, 90 CL
20
Degenerate spectrum

Degenerate m0 gt (0.08 - 0.10) eV
Dm/m0 Dmatm2/2m02
Large or maximal mixing
Mass degeneracy
Large scale structure analysis including X-ray
galaxy clusters
Cosmology
m0 0.20 /- 10 eV
S.W Allen, R.W. Schmidt, S. L. Bridle
Heidelberg-Moscow result
mi gt mee
at least for one mass eigenstate
77.7 kg y 4.2s effect
mee (0.29 - 0. 60 ) eV
(3s )
H. Klapdor-Kleingrothaus, et al.
mee(b.f.) 0.44 eV
21
Absolute scale of mass

F. Feruglio, A. Strumia, F. Vissani
Sensitivity limit
p
n
H-M
e
n
x
e
n
Neutrinoless double beta decay
p
mee Sk Uek2 mk eif(k)
Both cosmology and double beta decay have
similar sensitivities
m1
Kinematic searches, cosmology
22
U_e3 - expectations

Naively
sinq13 sinq12 x sinq23
excluded which implies dominant
structures or/and degeneracy in the mass matrix
0.3 - 0.5
solar sector
Atmospheric sector
A bit seriously
sinq13
Mass scales
Dmatm2
Dmsun2
sinq13 Dmsun2/ Dmatm2
0.15 - 0.20
With comparable contribution from the charge
leptons
sin2q13 0.01 - 0.05
If there is no cancellation
23
Relations?

Deviation
of 2-3 mixing
Degeneracy
from maximal
of mass
spectrum
Symmetry which leads to maximal 2-3 mixing e.g.
Permutations, Z2 can give sin q13 0
?
Violation of the symmetry
1-3 mixing
D23 0 sin q13 0
24
Two extreme cases

Quasi-degenerate
spectrum
S Barshay, M. Fukugita, T. Yanagida ...
New symmetry SO(3), A4, Z2 , ...
Hierarchical
Quark -lepton symmetry?
Spectrum
No particular symmetry
Deviation from maximal mixing
Quark-lepton analogy/symmetry
25
Symmetry case

ne nm nt
In the flavor basis
1 0 0 mn m0 0 0 1
dm 0 1 0
dm ltlt m0
Features
Degenerate spectrum m1 m2 - m3
Maximal or near maximal 2-3 mixing
Opposite CP parities of n2 and n3 n2 and n3
form pseudo-Dirac pair
Neutrinoless double beta decay mee m0
Most of the oscillation parameters are not
imprinted in to the dominant structure all
Dm2 the as well as 1-2 and 1-3 mixings are
determined by dm.
dm is due to the radiative corrections?
26
Symmetry
E. Ma, G. Rajasekaran K.S. Babu, J. Valle

A4
Symmetry group of even permutations of 4
elements
Symmetry of tetrahedron (4 faces, 4 vertices)
Platos fire
Irreducible representations 3, 1, 1, 1
(i 1, 2, 3)
H1,2 1
(ni , ei ) 3
(ui , di ) 3
Transformations under A4
u3c, d3c, e3c 1
u1c, d1c, e1c 1
u2c, d2c, e2c 1
Ui, U1c, Di, D1c, Ei, E1c, Nic , ci 3
New heavy quarks leptons and Higgs are
introduced
Explicit asymmetry of charged fermions and
neutrino sector
no nic and Ni
Leads to different mixing of quarks and leptons
27
Model quarks and leptons

Problem to include leptons and quarks in unique
model
Symmetry between leptons and quarks is explicitly
broken
Extra symmetries required (e.g. Z3)
Neutral and charge lepton sectors are different
ei e1c Eic ----- Ei
ni Nic
H1
H2
i 1, 2, 3
lt ci gt
Majorana mass
ME
MN
A4 is broken
Seesaw
Mixing - from the charged leptons in the
symmetry basis
Mixing of quarks UL UL I
1 1 1 UL 1 w w2 1
w2 w
Mixing of leptons ULT UL M 0
Oscillation parameters are determined by dm -
an additional theory independent on A4 required
w exp (-2ip/3)
Masses and mixing angles are unrelated
28
Symmetry approach

Charged leptons
?
Neutrinos
No symmetry
Quarks
Symmetry
Realization requires introduction of
- New leptons and quarks - Extended non-trivial
Higgs sector to break symmetry - Additional
symmetries to suppress unwanted interactions
of new particles
Often - difficult to embed into Grand Unified
schemes
?
or
?
Profound implications
Misleading approach
?
?
29
Do we really see a symmetry?

1). The only serious indication is nearly maximal
2-3 mixing
sin2 2q23 gt 0.91, 90 CL

2). sin2 2q23 is a bad parameter from
theoretical point of view
sin2q23 is better, but for this parameter
sin2q23 (0.35 - 0.65) 90 CL
Relative deviation from maximal mixing (1/2) can
be large
Dsin2q23 /sin2q23 0.7
3). The atmospheric neutrino results may
provide some hint that the mixing is not
maximal
Three neutrino analysis of data is needed
Excess of the e-like events (?)
30
Zenith angle distributions
Up stop ?
Sub-GeV Multi-R ?-like
Sub-GeV e-like
Sub-GeV ?-like
Multi-GeV ?-like PC
Multi-GeV Multi-R ?-like
Up thru ?
Multi-GeV e-like
SuperKamiokande
31
LMA oscillations of atmospheric
neutrinos

Excess of the e-like events in sub-GeV
Fe Fe0
- 1 P2 ( r c232 - 1)
screening factor
P2 P(Dm212 , q12) is the 2n transition
probability
In the sub-GeV sample
r Fm0 / Fe0 2
The excess is zero for maximal 23- mixing
Searches of the excess can be used to restrict
deviation of the 2-3 mixing from maximal
Zenith angle dependences of the e-like events
0.Peres, A.S.
32
Deviation of 2-3 mixing from maximal

Dm2 0
Dm2 0
sin2q23 0.45 - 0.47
M. C. Gonzalez-Garcia M. Maltoni, A.S.
33
No-symmetry'' approach

I. Dorsner, A. S
2-3 mixing differs from maximal one
No special symmetry for neutrinos or leptons
Approximate quark-lepton symmetry (analogy,
correspondence)
Family structure, weak interfamily connections
no large mixing in the original mass matrices
All quark and lepton mass matrices have similar
structure with flavor alignment
qij mi/mj
qatm 36 - 380
Seesaw mechanism of neutrino mass generation
Large lepton mixing is the artifact of seesaw
34
Quark-Lepton symmetry

Quarks-Lepton symmetry is realized in terms of
the mass matrices (matrices of the Yukawa
couplings).
For the Dirac mass matrices of all quarks and
leptons
(implies large tan b)
YU YD YnD YL Y0
Yf Y0 DYf
( Y0)ij ltlt (DYf)ij
f U, D, L, nD
Specifically
Y0 is unstable det (Y0) 0 (as well as
determinants of sub-matrices) small
perturbations produce significant change of
masses and and mixings
a11 e4 a12 e3 a13 e2 Y
a21 e3 a22 e2 a23 e a31 e2
a32 e 1
e 0.2 - 0.3
Assume
aij 1 O(e)
Observables (masses and mixings) appear as small
perturbations of the dominant structure given
by Y0 .
35

See-saw
Enhances (by factor 2) the mixing which comes
from diagonalization of the neutrino mass matrix
Changes the relative sign of rotations
which diagonalize mass matrices of the charge
leptons and neutrinos
Leads to smallness of neutrino masses
For the RH neutrino mass matrix we take for
simplicity the same form
MR Y0
36
Flipping the sign of rotation

lepton mixing
quark mixing
VCKM U (up) U(down)
VP-MNS U(l) U(n)
For quarks the up and down rotations cancel each
other leading to small mixing whereas for
leptons they sum up producing large mixing
S. Barshay
n
L
U
D
qPMNS
qCKM
It is the see-saw leads to flip of the sign of
rotation which diagonalizes the neutrino mass
matrix
changes the sign of the 23-elements
leads to m(22) gt m(33)
Enhances 22 and 23 elements
37
Expansion parameter

e is determined by inequality of masses of the
s-quark and the muon
(one would expect mm ms in the case of exact
q-l symmetry )
kf e a22 - a23 2
ms/mb kq e3
mm /mt kL e3
These mass ratios can be reproduces for aij 1
O(e) if
e gt 0.25
38
An example

e 0.26
1.00 0.91 1.01
aij(l) . . . 1.26 0.74
. . . . . . 1.00
1.00 1.26 1.00 aij(D
) . . . 1.09 1.05
. . . . . . 1.00
1.0000 1.0026
1.0000 aij(M) . . . 1.0005
1.0007 . . . . . .
1.0000
For quarks deviations of aij from 1 are
smaller than 15
mee 0.001 - 0.005 eV
m1 0.0017 eV
sin2q13 0.005
Generically sinq13 (1 - 3)e 2
Masses of the RH neutrinos
M1 1.2 1010 GeV M2 5.8 1010 GeV M3
8.8 1014 GeV
Strong hierarchy seesaw enhancement of 23-mixing
39
Implications

No special symmetry for neutrinos of for lepton
sector
Y0 can be reproduced with U(1) family
symmetry and charges q, q 1, q 2 charges, if
unut charge is associated with one degree of e
( a la Froggatt-Nielsen)
Violation of the symmetry appears at the level
e2 - e3 (1 - 3) 10-2
If the flavor symmetry is at GU scale its
violation can come e.g. from the string scale
40
Conclusions
One of the main results is an amazing pattern
of the lepton mixing which differs strongly
from the quark mixing

The key question to the underlying physics is if
there is a new (different from quark) symmetry
behind neutrino mass spectrum and mixing
- maximal (near) maximal 2-3 mixing
- degenerate spectrum - probably,
small 1-3 mixing
Nothing special weakly broken quark-lepton
symmetry family structure, small interfamily
mixing. Large lepton mixing -artifact of the
seesaw mechanism
New symmetry of Nature, which is realized
(manifest itself) in the properties of
neutrinos or lepton sector.
?
Neutrinos can shed some light on the fermion
mass problem in general. Hint flavor alignment,
e 0.25 , unstable mass matrices, U(1)
Priorities