Neutrino Masses and Mixing:

1
Neutrino Masses and Mixing

Toward the underlying physics
A. Yu. Smirnov
International Centre for Theoretical Physics,
Trieste, Italy Institute for Nuclear Research,
RAS, Moscow, Russia
Summary of results. Main features.
New symmetry of nature?
Quark-Lepton Symmetry ?
Quark-Lepton Complementarity
Measurements What? Why? Which precision?
2

1. Summary of results
Main features
3
Mass spectrum and mixing
ne
nm
nt

Ue32
?
n2
n3
Dm2sun
n1
mass
mass
Dm2atm
Dm2atm
Ue32
n2
Dm2sun
n1
n3
Inverted mass hierarchy
Normal mass hierarchy
Type of mass spectrum with Hierarchy, Ordering,
Degeneracy absolute mass scale
Type of the mass hierarchy Normal, Inverted
Ue3 ?
CP violating phase
New neutrino states?
A Yu Smirnov
4
Mixing angles
ne
nm
nt

Ue32
Um 32
Ut 32
n3
sin2q13 Ue32
Dm2atm
mass
Ue22
tan2q12 Ue22 / Ue12
n2
n1
Dm2sun
Ue12
tan2q23 Um 32 / Ut 32
Normal mass hierarchy
Dm2atm Dm232 m23 - m22
Dm2sun Dm221 m22 - m21
A Yu Smirnov
5
2-3 masses and mixing

hep-ex/0411038
K2K
SuperKamiokande (atmospheric)
hep-ex/0404034
Dm232 (2.4 0.6/-0.5 ) 10 -3, eV2
Dm232 (1.9 - 3.6 ) 10 -3, eV2
90 C.L.
90 C.L.
sin2 2q23 1.0
6
LMA MSW Solution
KamLAND collaboration hep-ex/0406035

KamLAND versus Solar
KamLAND and Solar
Dm2 (7.9 0.6/-0.5) 10 -5 eV2
Combined fit
1s
tan2 q 0.40 0.10/-0.07
7
Main features

mh (0.04 - 0.3) eV
mh gt Dm232 gt 0.04 eV
The heaviest mass
Hierarchy of mass squared differences
Dm122 / Dm232 0.02 - 0.08
3s
No strong hierarchy of masses
0.09 - 0.05
m2 /m3 gt Dm122 / Dm232 0.19
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
8
Quarks and Leptons

Quarks
Leptons
Mixing
q12 qC 45o
1-2, q12
13o
32o
2-3, q23
2.3o
45o
1-3, q13
0.5o
lt13o
Cabibbo-solar mixing relation - accidental?
Hierarchy of masses
m2 /m3 0.2
Neutrinos
1s
2-3
Charged leptons
m m/mt 0.06
1-2
ms /mb 0.02 - 0.03
Down quarks
1-3
mc /mt 0.005
Up-quarks
0 0.2 0.4 0.6
0.8
sin q
9
Mass Ratios

up down charged
neutrinos quarks quarks leptons
1
mu mt mc2
10-1
Vus Vcb Vub
10-2
10-3
10-4
Mass-mixing
10-5
relation?
at mZ
10
New phase of the field

Very clear program of further phenomenological
and experimental studies precision
measurements
Very unclear and to some extend puzzling
situation from the theoretical point of view
Symmetry? Relations, Regularities,
Ordering? Structure?
Randomness, Anarchy?
11

2. New symmetry
of Nature?
12
Neutrino'' Symmetry?

What can testify ?
Maximal or close to maximal 2-3 mixing
Very small 1 -3 mixing
Quasi-degenerate or partially degenerate spectrum

0.5 - sin2q23 ltlt sin2q23
q13 ltlt q12 x q23
Dm ltlt m
m1 m2 m3
2n data fit
Best fit value sin2 2q23 1.0
m3 m2 ltlt m1 ( inverted hierarchy)
Symmetry which shows up in the neutrino
sector mainly?
- Heidelberg-Moscow - Cosmology? - Large/maximal
mixing lt-gt degenerasy
sin2 2q23 gt 0.91, 90 CL
13
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
14
Deviation from maximal
D_23

1). Present bound
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 (?)
15
Deviation of 2-3 mixing from maximal

Dm122 0
Dm122 0
sin2q23 0.45 - 0.47
M. C. Gonzalez-Garcia M. Maltoni, A.S.
16
Ue3 e-id sinq13

U_e3
Last unknown angle in the mixing matrix
Phenomenology Leading effect for SN electron
neutrinos Sub-leading
effects in the solar and atmospheric
neutrinos
A possibility to measure CP violation
Sub-leading effect on the mass matrix
Test of the mechanism of the lepton mixing
enhancement
Test of symmetry
17
U_e3 - expectations

sinq13 sinq12 x sinq23
Naively
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
18
Small U_e3 and symmetry

Zero sinq13 as a consequence of symmetry small
value - result of symmetry violation
A. Joshipura W. Grimus
a b b b d e b e d
Invariant with respect to 2 - 3 permutations,
e.g. Z2
As a consequence, q23 p /4 , q13 0
Zero 1-3 mixing is related to maximal 2-3 mixing!
But the symmetry should be broken (charge lepton
masses)
Value of sinq13 is related to the deviation of
2-3 mixing from maximal
Relation depends on the mechanism of breaking.
It is not possible to introduce symmetry which
leads to q13 0 and is consistent to all
other masses and mixings
V. Berezinsky et al
Planck scale effects q13 (1- 5) 10-4
M. Lindner et al.
Renormalization group effect
19
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
20
Absolute scale of mass

F. Feruglio, A. Strumia, F. Vissani
Sensitivity limit
p
n
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
21
Absolute mass scale
G. L. Fogli et al., hep-ph/0408045

m lt 0.13 eV, 95 C.L.
U. Seljak et al.
m0 gt (0.08 - 0.10) eV
S Si mi
22
P. Minkiwski 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
23
Quasi-degenerate spectrum

Origin?
Different source unrelated to the Dirac mass
matrices of the charged leptons and quarks
Screening of Dirac structure
0 mD 0 m mD 0
MD 0 MD M
Higgs triplet seesaw type II
MD A mD
3x3 matrices
M I
mn mD MD -1 M MD -1 mD A-2 I
24
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?
25
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
26

3. Quark-Lepton
Symmetry?
For mass matrices
I. Dorsner, A. S. Nucl. Phys. B hep-ph/0403305
Mq - Ml ltlt Mq
27
Which terms?
In which terms theory should be formulated?


Observables
Mass matrices
Masses, mass ratios, mixing angles
are fundamental parameters which show certain
symmetry
their properties, symmetries at certain energy
scale
observables are outcome, can have to some
extend, accidental values which do not
reflect the underlying symmetry
Schemes with bi-maximal mixing, or broken
bi-maximal mixing Tri-bimaximal mixing
No apparent regularities -gt anarchy
Quark-Lepton symmetry is strongly broken
Approximate quark-lepton symmetry
28
No special symmetry for neutrinos or leptons
2-3 mixing differs from maximal one

qatm 36 - 380
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
Seesaw mechanism of neutrino mass generation
Large lepton mixing is the artifact of seesaw
29
Singular matrices and Q-L symmetry

Allows to explain difference in mass hierarchies
and mixings of quarks and leptons by small
perturbations of the universal structure
Singular
mass matrices
Det M 0
enhancement
strongly different
of lepton mixing
mass hierarchies
Strong hierarchy of masses of the RH neutrinos
Due to small perturbations of the singular
structure
Seesaw enhancement of lepton mixing
seesaw m 1/M
30
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 gtgt (DYf)ij
f u, d, L, D, M
Specifically
Y0 is unstable det (Y0) 0 (as well as
determinants of sub-matrices) small
perturbations produce significant change of
masses and and mixings
Assume
l4 l3 l2 Y0 l3 l2
l l2 l 1
l 0.2 - 0.3
Other forms are possible
31
Perturbations

Y f Y0 DYf
(DYf)ij ltlt (Y0)ij
f u, d, L, D, M
Perturbation of a given element is proportional
to the element itself
(DY f)ij ( Y0)ij efij
efij lt l
Ansatz for corrections
(1 ef11) l4 (1 ef12) l3
(1 ef13) l2 Yf (1 ef21) l3 (1
ef22) l2 (1 ef23) l (1 ef31)
l2 (1 ef32) l 1
the structure of mass matrix is not changed
Observables (masses and mixings) appear as small
perturbations of the dominant structure given
by Y0 .
Theory is reduced to theory of efij , though
some qualitative features follow from general
form of matrices
32
Quark mixing and masses

Quark and charged lepton mass ratios
Quark mixing
Vus l r(e)
m1 / m2 l2 e
Vcb l e
m2 / m3 l2 r(e)
m1 / m3 l4 e
Vub l2 e
e are combinations of eij in the mass
matrices
Reproduces correct hierarchy of mixings
Vub / Vcb l
r(e) O(1) is the ratio of combinations of
eij
Vcb / Vus e
33

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
34
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
35
An example 1
l 0.26

0 - 0.171 -0.036 el
. . . 0.254 - 0.268 . .
. . . . 0
0 0.233 0 eD .
. . 0.104 0.042 . . .
. . . 0
0 0.0065 0 eM
. . . 0.0010 0.005 . . .
. . . 0
mee 0.0006 eV
m1 0.002 eV
sin2q13 0.008
Generically sinq13 (1 - 3)l 2
Masses of the RH neutrinos
M1 1.3 1010 GeV M2 3.0 1010 GeV M3
8.6 1014 GeV
Strong hierarchy seesaw enhancement of 23-mixing
36
Corrections for quarks
l 0.26

Example 1
0 0.039 - 0.163 ed
. . . - 0.004 - 0.082 . .
. . . . 0
0 0.068 - 0.010 eu
. . . 0.144 0.053 . .
. . . . 0
Example 2
0 - 0.011 - 0.160 ed
. . . - 0.110 - 0.141 . .
. . . . 0
0 0.008 - 0.010 eu
. . . 0.020 - 0.008 . . .
. . . 0
For quarks eij are smaller than 17
37
Origin of mass matrices

Froggatt- Nielsen mechanism, U(1) family symmetry
s q(i) q(j) MF
aijfi fj H
q(i) is the U(1) charge of i-family, aij
O(1), s is the F-N scalar whose VEV violates
U(1) MF is the scale at which F-N operators
are formed
(Yf)ij aij l q(i) q(j)
l ltsgt/ MF
aij a0 (1 e ij)
nearly universal
For the simplest prescription q(1) 2, q(2)
1, q(3) 0, q(s) -1 the required structure
is reproduced
38
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
unit 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
39
H. Minakata, A.S. Phys. Rev. D hep-ph/0405088
Orthogonal approach

4. Quark-Lepton
Complementarity
qsun p/4 - qC
qsun qC p/4
tan2qsun ctg 2qC
S. Petcov, A.S. PL B322, 109 (1994)
Lepton mixing Bi-maximal - CKM?
40
Cabibbo-solar mixing relation

1s
qSun 32.3o /- 2.4o
Solar data KamLAND
qC 12.8o /- 0.15o
Cabibbo angle at mZ
sin2qsun 0.286 /- 0.038
qSun qC 45.1o /- 2.4o
sin2(p/4 - qC ) 0.284 /-0.002
qsun qC p/4
M. Raidal
Quark-Lepton
?
symmetry
Accidental
Grand Unification
equality
Bi-maximal mixing
41
Conditions

1). Lepton mixing Bi-maximal - CKM
Single maximal - Cabibbo
2). Order of rotations
Vleptons Vl Vn R23max R12 CKM
R12max
Different order leads to corrections to QLC
relation
3). Matrix with CP violating phase should not
appear between R12 CKM and R12max
4). Quark-Lepton symmetry which leads to the QLC
relation is realized at high scales --gt
Renormalization group effects should be
small
Renormalization effects are large for SUSY
version and quasi-degenerate mass spectrum
42
Neutrino scenario''
In the symmetry basis

Leptons
Quarks
Seesaw?
Vn R23max R12max
Vu I
Vl VCKM
Vd VCKM
q-l symmetry
Vleptons Vl Vn VCKM R23max R12max
Vquark Vu Vd VCKM
Predictions
sinqsun sin(p/4 - qC) 0.5sin qC ( 2 - 1-
Vcbcos d)
tan2qsun 0.495
more than 1s
large !
sinq13 sinqatm sinqC 0.16
H. Minakata, A.S. R. Mohapatra, P. Frampton
D23 0.5 sin2qC cos2qC Vcbcos d 0.02 /-
0.04
43
Charged Lepton'' scenario

M. Raidal
Leptons
Quarks
Vn VCKM
Vu VCKM
q-l symmetry
Vl R12maxR23max
Vd I
Lopsided
Vleptons R23max R12max VCKM
Vquarks Vu Vd VCKM
Predictions
sinqsun sin (p/4 - qC )
sinq13 - sin qsun Vcb 0.03
H. Minakata, A.S.
D23 cos qsun Vcb cos d lt 0.03
44
Hybrid'' scenario
H. Minakata, A.S.

Leptons
Quarks
Seesaw?
Vn R12max
Vu I
Vl VCKMR23max
Vd VCKM
q-l symmetry
Vleptons Vd Vn R23max VCKM R12max
Vquark Vu Vd VCKM
Predictions
sinqsun sin(p/4 - qC)
sinq13 Vub 0.003
45
Problems
Origins of maximal (bi-maximal) mixing

Propagation of the Cabibbo (CKM-) mixing to
lepton sector
large difference of mass hierarchies me/mm
0.0047, md/ms 0.04 - 0.06, as well as masses
mm and ms at GU scale
Mass matrices are different, the propagation
leads to correction
Dq12 qC md/ms 0.5 - 1.0o
sinqC is the generic parameter of theory of the
fermion masses which appears in various places
(mass ratios, mixing angles)
Cabibbo mixing can be transmitted in more
complicated way
sinqC mm / mt
QLC relation can be accidental
One has very precise pure leptonic relation
qsun qmt p/4
tanqmt mm/mt
46
Future measurements

5. What? Why? Which
precision?
47
Test of the mechanism of the lepton mixing
enhancement
sin2q13

Cabibbo parameter as the fundamental parameter
of theory of fermion masses
sin2qC
0.05
Natural scale
Dm2sun/Dm2atm
0.04
0.03
tan2qC Um3 2
Bimaximal mixing from neutrinos
0.02
Symmetry case
0.01
Double-CHOOZ
JPARC
Test of nm - nt symmetry
Dm2sun/Dm2atm2
sin2qsun Vcb2
0
Bi-maximal mixing from charged leptons
48
D23 0.5 - sin2q23
D23

Present 90 CL bound
0.15
Quark - lepton symmetry
(0.91)
Tests
0.10
D23 sin2q13
(0.96)
Global analysis best fit
Test of nm - nt symmetry
sin2qC
0.05
cosqsun Vcb cos d
Bi-maximal mixing from charged leptons
(0.99)
0.5sin2qC cos2qC Vcb cos d
Bi-maximal mixing from neutrinos
0
- 0.05
d - CP violating phase
(sin2 2q23)
49
D sin2q12 sin2q12 - sin2 (p/4 - qC )
D sin2q12

0.05
sinqsun sinqC ( 2 - 1)
Bi-maximal mixing from neutrinos
0.04
1s upper bound
Large 2-3 mixing from neutrinos
0.03
0.02
Large 2-3 mixing from neutrinos and charged
leptons
0.01
sinqsun sinqC Vcb2/ 2
Bi-maximal mixing from charged leptons
0
- 0.01
- 0.02
50
mass of the heaviest neutrino
mh, eV

0.25
Lower limit of the KATRIN sensitivity
0.20
0.15
Cosmological upper bound
0.10
Lower limit of the quasi-degenerate spectrum
Non-degenerate spectrum
0.05
Dm2atm1/2
Hierarchical spectrum
0
51
Conclusions

One of the main unexpected results is the
pattern of lepton mixing which differs
strongly from the quark mixing
Essentially we have no understanding of physics
behind neutrino masses and mixing considering
whole spectrum of possibilities from randomness
(anarchy) to ordering and symmetry
Key questions on the way to underlying physics
- Is there a new symmetry of nature
behind the pattern of neutrino masses
and micxing? - Is the quark-lepton
symmetry still relevant? - Is the
quark-lepton complementarity real?
To answer these questions one needs to measure
- Absolute mass scale and establish type of
mass hierarchy, - Deviation of 23 mixing
from maximal, - 1 - 3 mixing, -
Deviation from QLC relation
52
Conversion of neutrinos from SN1987A
For normal mass hierarchy
Conversion in the star
F(ne) F0(ne) p DF0
p is the permutation factor
p
Earth matter effect
DF0 F0(nm) - F0(ne)
p depends on distance traveled by neutrinos
inside the earth to a given detector
4363 km Kamioka d 8535
km IMB 10449 km Baksan
C.Lunardini, A.S.
The earth matter effect can partially explain
the difference of Kamiokande and IMB spectra
of events
Normal hierarchy is preferable
H. Minakata, H. Nunokawa, J Bahcall, D Spergel,
A.S.
53
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
54
Speaking about priorities

Final goal
Accomplish determination of the lepton mixing
matrix s13 , d
Modest but still difficult
Reconstruct all parameters of the mass spectrum
and mixing
Is not possible practically
Reconstruct neutrino mass matrix
Hopefully we can understand the underlying
physics even without complete reconstruction
Understand origin of the neutrino mass and
mixing - symmetries, - scales, -
dynamics
or
Physics will not be uncovered even if everything
is known
55
Phenomenology and experiment

Accomplish reconstruction of the neutrino mass
spectrum and lepton mixing, precision
measurements of parameters
Search for new physics
New neutrino states sterile neutrinos
Deviation of 2-3 mixing from maximal
Absolute mass scale, m1, type of the
mass spectrum (hierarchical, degenerate)
New interactions
1 - 3 mixing
Effects of violation of the fundamental
symmetries - CPT, - Lorentz invariance -
Equivalence principle
Type of mass hierarchy, ordering
Majorana phases

Dirac CP-violating phase, d
Nature of neutrinos
56
Leptonic Unitarity Triangle

Ue1 Ue2 U e3 Um1 Um2
Um3 Ut1 Ut2 Ut3
0.78 - 0.88 0.47 - 0. 62 0.0 -
0.23 0.18 - 0.55 0.40 - 0.73 0.57 -
0.82 0.19 - 0.55 0.41 - 0.75 0.55 -
0.82
UPMNS

M.C. Gonzalez-Garcia ,
Global fit of the oscillation data 3s
Ue3 0.16
Ue2 Um2
Ue3Um3
S 2J
nearly best fit values of other angles
Ue1 Um1
Y. Farsan, A.S.
Can we reconstruct the triangle? Can we use it to
determine the CP-violating phase?
J s12 c12 s13 c132 s23 c23 sind
Problem coherence (we deal with coherent states
and not mass eigenstates of
neutrinos)
A Yu Smirnov
57
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.
58
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
59
U_e3 - expectations

A. Joshipura
Being small sinq13 is generated by small
perturbations of the dominant structure of mass
matrix responsible for the atmospheric
mass/mixing
l l l l 1 1 l 1 1
(matrix elements are defined up to
coefficients O(1))
l ltlt 1
If there is no fine tuning of 12 and 13
elements
tan2 qsun Dmsun2 2 tan qatm Dmatm2
sin q13 cos
2qsun
0.2
Akhmedov et al
Radiative generation of sinq13
sin q13 0.01 - 0.1
2 me 3 mm
sin q13
0.02
From charged leptons
Type II seesaw minimal SO(10) GUT
sin q13 O(1) sin qc
60
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
61
Neutrino masses and lepton mixing

tan2q23 2l (1 eD23 ) Aseesaw

(eD23 - eM23)(eD22 - 2eD23 - eD232 ) eM22 -
2eM23 - 2eD23eM23 eD232 l2 e
seesaw term
Aseesaw
Seesaw enhancement
small detMR
M2 / M3 l2 e2
eM22 2eM23 O(e 2 )
e are combinations of eij in the mass matrices
Uu3 l2
sinq12 r(e)
m1/m3 l4 e
m2/m3 l2 e
m3 Dmatm2
r(e) O(1) is the ratio of combinations of
eij
mee sin2q12 Dmsol2 (2 - 3) 10-3 eV
62
An example 2
l 0.26
(mn)23 lt 0, (mn)33 gt 0

0 - 0.093 - 0.006 el
. . . 0.262 - 0.262 . . .
. . . 0
0 0.213 0 eD
. . . - 0.200 0.098 . . .
. . . 0
0 0.130 0 eM .
. . 0.264 0.129 . . .
. . . 0
mee 0.0007 eV
m1 0.004 eV
sin2q13 0.001
Masses of the RH neutrinos
M1 2.5 108 GeV M2 2.2 1011 GeV M3
3.8 1014 GeV
Strong hierarchy seesaw enhancement of 23-mixing
63
Large mixing and Degeneracy

Degeneracy parameter
d23 Dm/m Dm2/2m2
Deviation from maximal mixing
D23 0.5 - sin2q23
Degeneracy vs. deviation.
0 1 1 e
D23 d23
1).
D23 0, d23 2e
1 e e 1
maximal mixing, arbitrary mass split
examples
2).
e 1 1 -e
D23 e/2 , d23 0
arbitrary mixing zero mass split
3).
For m 0.25 eV d23 0.03,
D23 0.1 will exclude the possibility 1).
Smaller d21 is associated to larger D21
Simple relation is absent due to presence of
third neutrino Majorana phases
64
Expansion parameter

l is determined by the weakest mass hierarchy of
leptons (muon and tau) and 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 l e 22 - 2e 23
ms/mb kq l3
mm /mt kL l3
These mass ratios can be reproduces if
l gt 0.25
65
Conclusions

One of the main results is an amazing pattern
of the lepton mixing which differs strongly
from the quark mixing
Still approximate quark-lepton symmetry can be
realized in terms of nearly singular mass
matrices which can lead also to enhancement of
the lepton mixing
Hint flavor alignment, l 0.25, family U(1)
, Froggatt-Nielsen mechanism
?
Cabibbo - solar mixing relation if not
accidental challenging and very suggestive
- Quark-lepton complementarity - particular
quark-lepton symmetry, - GUT
Priorities
66
Singular matrices

Democratic matrices
E. Akhmedov, G. C. Branco, F. R. Joaquim J.
Silva-Marcos, 2000 R. Dermisek, 2003
1 1 1 Y0 1 1 1
1 1 1
H. Fritzsch, M. Fukugita, T. Yanagida
After diagonalization all these singular
matrices are reduced to
0 0 0 Y0 0 0 0
0 0 1
One can work immediately in this basis.
Key point is how perturbations are introduced
This can lead to different predictions and
different implications to
theory
67
Single maximal mixing

Leptons
Quarks
Vu I
Vn R23n R12max
Vd VCKM
Vl VCKMR23l
Vquark Vu Vd VCKM
Vleptons R23l VCKM R23n R12max
R23l R23(qC) ,
R23n R23(2qC)
Realization
0 M12 0 M M12 0
0 0 0 M33
maximal 1-2 mixing, enhanced 2-3 mixing if M12
ltlt M33
mu mn diag ,
sinqsun sin(p/4 - qC) sinqC1 - cos(q23n -
q23CKM)/ 2
Reproduces better the QLC relation (qsun 33o)