Understanding Neutrino Mass, Mixing, and Flavor Symmetries

 
 
New problems appear when there are more than 1 object
Mixing
Mass ratios
type of spectrum
Mass ordering
Relative
CP phases
 
 
Flavor neutrino states:
e
2
3
1
m
1
m
2
m
3
Flavor
states
Mass 
eigenstates
 
Mass eigenstates
e
f
  =  U
PMNS
 
mass
Combinations of mass
states  described by
mixing matrix 
U
PMNS
Flavor states – Weak
interaction states
 
e
2
1
3
mass
|U
e3
|
2
|U
3
|
2
|U
3
|
2
|U
e1
|
2
|U
e2
|
2
f
  =  U
PMNS
 
mass
e
Mass content of
the flavor states
Flavor content of
the mass states

mass
  =  U
PMNS 
+
 
f
 
 
e
 
1
2
3
f
  = U
PMNS  
m
U
e1
    
U
e2
   U
e3
U
1
   
 U
2
    
U
3
U
 
1
    
U
2
     
U
3
 
=
 
1         0        0
  
0       c
23
     - s
23
0      -s
23
       c
23
 I
 = diag (1,  1,  e
i
)
U
PMNS  
= U
23
I

U
13
I

U
12
c
12
      s
12
     0
  
-s
12
     c
12
      0
 0        0        1
c
13
        0       s
13
e
-i
  
 0          1        0
-s
13
e
i
     0      c
13
Not unique parametrization
Convenient for phenomenology, especially for oscillations in matter
Insightful for theory?
I
M
I
M
I
M
 = diag (1, e         , e          )
Dirac phase matrix
i
21
 /2
i
31
 /2
- matrix of Majorana phases
 
e
2
1
3
m
2
31
m
2
21
Normal mass hierarchy 
|U
e3
|
2
|U
3
|
2
|U
3
|
2
|U
e1
|
2
|U
e2
|
2
 
tan
23  
=
 |U
3
|
2   
/ |U
3
|
2
 
sin
13  
=
 |U
e3
|
2
tan
12 
=
  
|U
e2
|
2   
/  |U
e1
|
2
m
2
31 
= m
2
3 
- m
2
1
m
2
21 
= m
2
2 
- m
2
1
Mixing parameters
f
  =  U
PMNS
 
mass
U
PMNS  
= U
23
I

U
13
I

U
12
FLAVOR
Mixing matrix:
1
2
3
e
 
= U
PMNS
MASS
2
 
Octant? With SK atmospheric
lower octant is preferable
NO preference smaller than 2
NO: CP-phase close to

 
 
P. F. Harrison, D. H. Perkins,
W. G. Scott
U
tbm
 = U
23
(
/4) U
12 
 
U
tbm
 =
   2/3       1/3        0
-  1/6       1/3    -   1/2 
-  1/6       1/3        1/2
sin
2
12
 
= 1/3
0.15
0.62
0.78
There is no relation of mixing
with masses  (mass ratios)
0.30 - 0.31
Not accidental
Accidental,  numerology,
 useful for bookkeeping
Accidental symmetry
(still useful)
Lowest order approximation
which corresponds to weakly
broken (flavor)  symmetry
of the Lagrangian
with some other physics
and structures associated
Tri-bimaximal mixing
flavons  other new particles
0.30
 
up           down       charged    neutrinos
quarks    quarks     leptons
10
-1
 
10
-2
 
10
-3
 
10
-4
 
10
-5
1
m
u
 m
t
  =  m
c
2
at m
Z
Koide 
relation
sin
C
 ~   m
d
/m
s
Gatto–Sartori-Tonin relation 
m
2
m
3
~ 0.18

m
21
2

m
32
2
mass ratios
Neutrinos have 
the weakest mass 
hierarchy (if any) 
among fermions 
Related to large
lepton mixing?
m
h  
>   
m
31
2  
> 0.045 eV
 
M
l
  =  M
 
M
l
 = U
lL
m
l
diag 
U
lR
+
 
M
 = U
L
m
diag 
U
L
T
 
U
PMNS
= U
lL
+ 
U
L
for Majorana  
neutrinos
Flavor basis: 
M
l
 = m
l
diag 
U
PMNS
 = U
L
Diagonalization: 
Mixing matrix
Mass spectrum
m
diag  
= (m
1
,  m
2
, m
3
) 
Origin of mixing: 
off-diagonal mass matrices
l 
(1 -

5
) U
PMNS
mass
CC in terms of mass eigenstates: 
m
l
diag  
= (m
e
,  m
,  m
) 
Shape invariance
Relations
between
elements
Mixing
Absolute
values of
matrix
elements
Masses
equalities,
zeros
(in the flavor
basis)
Decoupling is always possible, but the key is that  relations should
be simple to be  consequences of simple symmetries
Opens up a possibility that symmetry is behind TBM
For Cabibbo  mixing:
m
12
 =                     (m
11
 - m
22
)
   sin 
C
1 – 2 sin
2
 
C
 
~ 
1/4
For TBM:   m
12
 = - m
13 
,  
 
m
22
 = m
33 
,  
 
m
11
 + m
13
 = m
22
 + m
23
S
4
Relation between matrix elements which
leads to Cabibbo mixing independently of
values of matrix elements
symmetry which produces the relation dihedral D14
C. Hagedorn, 1204.0715
Mixing:  very special  structure which is not related to mass ratios
,
 
e
2
1
1
2
3
3
MASS
2
m
2
23
m
2
32
m
2
21
m
2
21
Inverted mass ordering
Normal mass ordering
MASS
2
FLAVOR
FLAVOR
 
Quark-lepton
symmetry
Unification
Flavor symmetries
 
 
Quasi-
degenerate
 symmetry
Similar to quark spectrum
See-saw
Broken L
e
 – L
 – L
 symmetry
Pseudo-Dirac + 1 Majorana

m
21
2

m
32
2
m
2
m
3

m
21
2

m
32
2

m

m
~                 = 1.6 10
-2
but 1-2  mixing strongly
deviates from maximal
~              = 0.18
m
2
m
3
  ~
rescaling
the weakest
 hierarchy
Fundamental:
principle, symmetry
Accidental: selection of
values of parameters
 
- different patterns
 different possible 
   symmetries
Normal hierarchy
Inverted hierarchy
For two different
Majorana phases
Value of mass 
21
 = 0
21
 = 
Strong dependence
of structure of mass matrix on
the Majorana phases
For
Majorana
neutrinos
 
Establishing NH would favor the  line with
similarity with quark sector,  high scale seesaw, etc.
If so, one would expect  the first quadrant,
23
 < 

In models with residual  (discrete)  flavor symmetries  mixing
does not depend on mass at least in the symmetry limit.
The same mixing pattern can be obtained for NH and IH
The type of hierarchy is fixed by field content
 (scalar  sector),  auxiliary symmetries, etc.
Radiative mechanisms  have different  preferences
those  which are related to charged leptons prefer NH
 broken  (L
e 
– L
 – L
)  symmetry - IH
Seesaw  type I with quark-lepton analogy (symmetry)  leads to NH
 
 
from  symmetry
to anarchy and
randomness
Complicated constructions,
especially if quarks are
included
Not much to add
- String landscape
-  Multiverse?
with
intermediate
possibilities
 
Simple groups  like  S
4 
, A
4
 x Z
2
  can establish equalities of
elements of mass matrix required by TBM
 
Invariance of mass matrices in the flavor basis:
V
i
T
 m
TBM
 V
i
  =  m
TBM
 
1
0     0
0     0     1
0     1     0
U = 
S = 
 -1     2     2
  …    -1     2
  ..     …     -1
charged leptons 
diagonal  due to symmetry 
T = 
 = exp(-2i
/3)
S, T, U –elements of S
4
1    0    0
2
   0
…   …    
T
+
 (m
e
+
 
m
 e
)T  =  m
e
+
 
m
 e
 
1
3
C.S. Lam
V
i
 
  = S, U
i
 
Neutrinos
 

 -permutation
 
Irreducible 
representations:
Order 24,  permutation of 4 elements 
  
Generators:  S, T  
Presentation: 
1 ,  1’
 
,  2,  3,  3’ 
Products  and
invariants
2 x 3 =  2 x 3’ =  3 + 3’ 
1’ x 2 = 2
2 x 2 = 1 + 1’ + 2
1’ x 1’ = 1 
New flavor 
structure
3 x 3’  = 1’ + 2 + 3 + 3’ 
3 x 3 = 3’ x 3’  = 1 + 2 + 3 + 3’ 
 
S
4
 = T
3
 = (ST
2
)
2
 = 1
Symmetry of cube
S
 
 
Mixing appears as a result of different ways of the flavor symmetry
breaking in the  neutrino and charged lepton  (Yukawa) sectors.
G
f
G
l
G
Residual symmetries
of the mass matrices
 M
   M
l
  
A
4
T’
S
4
T
7
T
U, S
Symmetry
transformations
in mass bases
Intrinsic symmetries
of mass matrices
which do not depend
on values of masses
 Z
2
 x Z
2
Z
m
S
 M
 S

T  
= M
 or Z
2
 
Discrete finite groups
Flavons to break  symmetries
E. Ma,
C. S. Lam
....
CP-transformations
can be added
 
S
1
  = diag (1,  -1,  -1) 
S
2
  = diag (- 1,  1, -1) 
S
i
2
 = I
  T = diag(e   ,   e   ,   e    )
i
i
e
i
 = 2
 k
/m   
T
m
 = I
In the mass basis
for  Majorana neutrinos
G
G
l
Z
2
 
x Z
2
The Klein group
Z
m
 

 = 0  
Realized for arbitrary values of neutrino  and charged lepton mass
can not be broken, always exist
m = diag( m
1
, m
2
, m
3
)
m
l
  = diag( m
e
,  m
,  m
 )
can use subgroup
for charged leptons
 
1
2
3
1
  
2  
3
m
1
      m
2
             m
3
  
 
  S
i
   
 
( U
PMNS
 
S
i
 U
PMNS
+ 
T)
p
  = I
   
D. Hernandez, A.S.
1204.0445
 
If  intrinsic symmetries are residual symmetries of the unique
symmetry group (follow from breaking of unique group)
 bounds on elements 
 of mixing matrix
Two such equations for i = 1,2 fix the mixing matrix completely 
 TBM
For each i the equation gives two relations between mixing parameters
p -integer
Z
2
 
x Z
2
TBM
S
i
  and T are elements of covering group.
By definition product of these elements (taken in the same basis)
also belongs to the  finite discrete group:
In general, the condition allows to fix mixing matrix up to several
possibilities
 
A
4 
Symmetry group of even  permutations 
of 4 elements
Symmetry of tetrahedron
Irreducible representations: 
3
,   
1
,   
1’
,  
1’’
 
E. Ma  
3
x
3
 = 
3
 + 
3
 + 
1
 + 
1
’ + 
1
’’
Products and 
invariants
1
’ x 
1
’’ ~ 
1
S
2
 = 1
Presentation 
of the group:
T
3
 = 1
(ST)
3
 = 1
Generators: S, T
no U = A

G Branco, H P Nilles
 
A
4
T
S
``accidental’’ symmetry 
    due to particular 
    selection of flavon  
    representations and 
    configuration of VEV’s
A

    M
  
TBM-type
   M
l
  
diagonal
In turn, this split originates from 
different flavor assignments of 
the RH components N
c
 and l
c
and different higgs multiplets
 
Charged lepton
Neutrinos
L
l
c
N
c
L
-1
T
1, i, -1
S

M
H
d
H
u
n
k
i
G. Altarelli
D. Meloni 
Flavon sector
T
T
0
S
S
0

0
Driving fields
U(1)
R
0
2
Particular selection
of representations

S 

v
 S  
(1, 1, 1)
 

T 

v
 S
(0, 1, 0)
 
3
1
A
4
1’
1’’
Yukawa sector mass terms
i
i
Z
4
1
1
1
i
i
 at multiplets
 1
 i
-1
-i
-1
1
1
n = 1, 2, …k = 0, 1 
GUT-scale or higher?
Vacuum alignment
symmetry
flavons
 
Yukawa
couplings
VEV’s
Mechanism of 
mass generation
VEV alignment 
-
 different contributions
-
 high order corrections
follow from independent sectors:
All these 
components
should be 
correlated
tune by additional symmetries
Scalar potential
Yukawa sector
``Natural’’ – consequence 
   of symmetry? 
one step constructions do not work
 
No connection to masses
Mass hierarchy from additional symmetries
U(1)  Froggatt- Nielsen mechanism
Discrete symmetries – restricted possibilities  to explain
mass spectrum (degenerate, partially degenerate  spectra)
n
power n is determined by U(1) charges
of the corresponding operators
Missing representations problem
 
Symmetry which left mass matrices invariant for specific mass spectra:
Partially degenerate spectrum
m
1
  = m
2
 ,  m
3
Transformation matrix
S
 = O
2
Relation:
sin
2
2
23
 = +/- sin 
 = cos 
 =       = 1
m
1
m
2
 G
 = SO(2) x Z
2
Majorana phase
1-2 mixing is undefined
Small corrections to mass matrix lead to 1-2 mass splitting
and 1-2 mixing
+/-

/2
maximal
D. Hernandez, A.S.
Certain values of 
 follow from the flavor symmetries
without additional assumptions
Specific transformations which can be responsible
(control) value of the phase
Non-trivial CP phase : from generalized CP transformations
when also flavor is changed
Usually maximal CP violation is predicted
S
n
 D
 
G
f
x H
CP
G
x H
CP
Residual
symmetries
of the mass
matrices
 M
   M
l
A
4
T’
S
4
 G
  = Z
2
 x Z
2
 
no CP violation 
  = 0
 G
 = Z
2
 
 
usually  
  = 0, 
+/-

/2

G
l
x H
l
CP
C. Hagedorn, et al
R. Mohapatra, C C Nishi,
F. Feruglio, C. Hagedorn,
R. Ziegler, E Ma,  G-J.
Ding,  S.F. King, C. Luhn,
A. J. Stuart,  Y-L. Zhou,
(96)
(48)
(27)
 
  
may depend on free parameter and take any value
Y. Farzan, A.S.
 
 
 F. Feruglio
Gui-Jun Ding, S.F. King, Xiang-Gan Liu,  S. Petcov, A V. Titov, M. Tanimoto ,
T. Nomura, H. Okada, T Kobayashi,  O.Popov  Y. Shimizu, P Novichkov,
J. Penedo, T Osaka,  A. Romanino, I. De Medeiros Varzielas, X Wang, S. Zhou ...
Hope was to
Motivated by string theory
Reduce number of parameters, more predictive - ?
Connect masses and mixing - ?
Symmetry related to (orbifold)  compactification of extra dimensions
and primary realized on the moduli fields 
 which describe geometry
of the compactified space.
 
For single modulus field 
 the modular transformation 
 
 
 

  =
a 
  + b
c 
  + d
The 2x2 matrices of integer numbers
a 
  b
c 
  d
with determinant  ad – bc = 1
form the group 
 = SL(2, Z)  [special, linear, integer ...]
Modular group 
N
 is finite subgroup of 
, quotient group of the level N:

N 

Isomorphic to the groups considered previously:
 
3
, 
4
, 
5
  are isomorphic to A
4
, S
4
, A
5
, correspondingly
,.
 
In SUSY, the chiral superfields transform as
I

 (c
 + d)   
 
(
)

I
k
I
k
I
  is the weight of multiplet
(
)  is the representation of 
 element of the group 
N
Appearance  of the weight factor in transformations  is
new element which leads to new consequences
N
 
f
i

 - holomorphic functions of modulus field 
Transformation properties are similar to those of superfields
f
i


 
f
i

  = (c
 + d)   
 
(
)
ij
 f
j

 
k
f
- another key element of formalism
f
i

 form multiplet of 
N
  whose dimension is determined by level N
and weight k
f
For instance for N = 3 and k
f
  = 2, f
i
 form triplet with  components
 Y
1

= 1 + 12q + 36q
2
  +  ...
 Y
2

= - 6q
1/3
 (1 + 7q +   ...  )
 Y
3

= - 18 q
2/3
 (1 + ...   )
q = e
i2

 
  = 0.0117 + i 0.995
Data fit:
Y
i
 = (1, - 0.74,  - 0.27)
weak hierarchy
 
 - holomorphic functions of modulus field 
- Form multiplets of the group 
N
  and transform as superfields:
Y
i


 Y
i

  = (c
 + d)  
 
(
)
ij
 Y
j

 
k
dimension of the multiplet is determined by the level N and weight k
k  is the weight of multiplet
(
)  is the representation of 
 element of the group 
N
Dependence Y
i

 is determined by transformation properties
they are fixed by symmetry
 
1
x 
2
 x 
3
 x 
Y
 = I
Invariance of terms of the superpotential
 
 
Y

1
2
 
3
 
 k
i
 + k
Y
 = 0
requires
for weights
The latter condition acts to some extent as Froggatt- Nielsen
symmetry it gives additional restrictions, forbids some term
texture zeros
 
  - constant
for product of representations
 
        L         E
c
         N
c
       Y      
A
4
    3      1, 1’, 1’’     3        3       3
k      1        -4         -1         2       3
flavon
and

  
are fixed by fitting on m
3
 /m
2
,  1-2 mixing and 1-3 mixing
Predictions:
Y lowest order
modular form
J Griado and F. Feruglio
1807.01125 [hep-ph]
sin 
2 

 = 0.46
 /
  = 1.434

/
  = 1.7

/
  = 1.2
 
        L         E
c
         N
c
       Y
A
4
    3      1, 1’’, 1’     3        3
k      2          2         0        -2
No flavons
Flavor from single modulus field 
is fixed by fitting 12 mixing and 13 mixing
Predictions:
Gui-Jun Ding, S.F. King,
Xiang-Gan Liu
1907.11714 [hep-ph]
sin 
2 

 = 0.58
 /
  = 1.6

/
  = 0.15

/
  = 1.00
Higher order modular forms
Y
(2)
, Y
(4)
 , Y
(6)  
constructed as
products of 
  
Y
(2)
All Yukawas are modular forms
m
1
  =  0.0946 eV
m
2
  =  0.0950 eV
m
3
  =  0.1071 eV
Normal ordering
m
ee
   = 0.095 eV
Cosmological
bound?
 
Several moduli, flavons,
Generic predictions of masses: weak hierarchy,  often quasi-degenerate,
Majorana CP-phases
Minimal  simpest versions do not reproduce data well.
More parameters needed.
Different representations of the same dimension, fine tuning
of corresponding  couplings
Different ways of construction of finite groups
In the end: #  parameters is comparable to # of observables
Use modular symmetry in wrong way?
 
Comparing with usual approach with flavons
 Y
1 
(<
>) , Y
2 
(<
>) ... Y
n 
(<
>)
For a given N, k,  Y - known
functions of the same VEV

<
1
>, <
2
>, ... , <
n
>
- Flavon VEV’s depend on
parameters of potential and
not controlled by symmetry.
Independent for different
representations
Can do the same as
modular symmetry
constructions
 
Y L N
c
 H
u
Usual Yukawa coupling
Y 
i
 
 L N
c
 H
u
Y -constant
 – flavons (dimensionless)
transformation under G –flavor symmetry group:
 
 
 
i
(g)

L 
 
L
(g) L,
N
c 
 
 
N
 (g)
N
c
Invariance: 
L
(g) 
N
(g)

i
 
i
(g)  contains singlet
Y 
 Y,
Modular symmetry
 
Y(
) L N
c
 H
u
All the  factors including coupling transform as
Linear traditional flavor symmetry
f


  (c
 + d)     
f
(
) f

 
f
 k
f
 = 0
k
f
Invariance: 
f
 
 
f
(g)  contains singlet and
Y 
i
 
 

 
Y(
)
F. Feruglio and A. Romanino, Rev.
Mod. Phys. 93 (2021), 1. 015007
1912.0602
 
H.P Nilles, A.Baur, S. Ramos-
Sanchez, P.K. Vaudrevange, A.
Trautner..
Explicit top-down construction from heterotic
string theory compactified on orbifold T
2
 /Z
3
All symmetries used in bottom up approach:
G
traditional  
,  G
modular  
, G
R
 , CP
appear simultaneously in non-trivially unified fashion
Z
9
R
 discrete R symmetry
Z
2
CP
 CP transformations
Kahler potential  (kinetic terms) is involved and introduce
additional parameters, freedom
 
 
( U
PMNS
 
S
i
 U
PMNS
+ 
T )
p
  = I
   
D. Hernandez, A.S.
1204.0445
 
Equivalent to
Tr ( U
PMNS
 
S
i
 U
PMNS
 
+ 
T )  = a
   
Tr (W
iU
)  = a
a =  
j

j
  
j
p 
= 1
  
j
- three eigenvalues
  of element W
iU
 
j = 1,2,3
Symmetry group condition
P -integer
Z
2
 
x Z
2
TBM
 
 
|U
i
|
2
  = |U
i
|
2
|U
i
|
2
 =
for column of the mixing matrix:
        1 + a
4 sin
2
 (
k/m)
k, m, p  integers which
determine symmetry group
S
4
 
 =  +/-103
0
 D. Hernandez, A Y S.
1304.7738 [hep-ph]
TBM
G

= Z
2
Two  relations
TBM
1
TBM
2
Schemes  with non-
zero 1-3 mixing can
be obtained
a = Tr(U
PMNS
 
S
i
 U
PMNS
 
+ 
T)
 
 
Nothing should be observed at LHC which is
responsible for neutrino masses
If something is observed against
(excludes) framework
Special value of CP-phases
Light sterile neutrinos
Dark matter
Inflation
Hopeless?
Leptogenesis
Further precise measurements of
the  1-3 and 2-3 mixings (octant)
Proton decay
May or may not help
 
All well established/confirmed
results fit well a framework
with
It is widely
believed that
Connection
exists
Where is
 
Is the (hot) component
of the DM
Mechanism of
generation of
small neutrino
masses is
related to DM
RH neutrinos as DM particles
Neutrino portal connects
DM and neutrinos
The same symmetry is responsible
for smallness of neutrino mass and
stability of the DM
DM particles participate (appear in
loops)  in generation  of neutrino mass
Refraction  on DM produces
the effective neutrino mass
 
R
K*
Lepton
non-universality
R
D*
Systematic way to search for connection is to use effective FT
gauge inv. operators
constructed out of SM
fields , with 
L =2
Closure
L
L
Opening
L
L
D5 operator
m
H
H
New interactions
with new (often
exotic)  particles
Contribute to
R
K*
R
D*
(g -2)
Model building without
model builder
UV completion
 
We can not exclude that neutrino oscillations are
explained the refractive mass squared
Refraction in classical coherent field: energy independent mass:
Problem with cosmology?  Late formation of the field?
Nature of neutrino mass can be related the nature of Dark
matter and the Cosmological evolution
Refraction in cold gas: energy dependent mass at low
energies: avoid the cosmological bound on sum of neutrino
masses from structure formation
 
Flavor neutrinos:
e
Flavor is characteristic of interactions
 
e
are neutrinos associated with or
correspond  to certain charged leptons
The correspondence is established
by the charged current weak interactions
Flavor states -- Weak interaction states
In the presence of vacuum mixing (masses) - ambiguity :
Two ways to determine flavor states
at the Lagrangian
(weak current) level
Phenomenologically
Theoretically
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Neutrino physics involves studying mass, mixing, and flavor symmetries, which present new challenges when dealing with multiple neutrinos. The discussion includes neutrino states, mixing matrices, flavor content, vacuum mixing, standard parametrization, mixing angles, global fits, and the Tri-bimaximal mixing model by Harrison, Perkins, and Scott.


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  1. Mass, mixing and flavor symmetries

  2. Three neutrinos and three problems New problems appear when there are more than 1 object three neutrinos Mixing Relative CP phases Mass ordering Mass ratios type of spectrum

  3. Flavor and mixing

  4. Flavors and mixing Flavor neutrino states: Mass eigenstates 1 2 3 e m1 m2 m3 e Mixing Flavor states Weak interaction states Mass eigenstates Flavor states Combinations of mass states described by mixing matrix UPMNS f = UPMNS mass

  5. Mixing: dual role e |U 3|2 |U 3|2 3 |Ue3|2 mass |Ue2|2 2 1 |Ue1|2 e Flavor content of the mass states mass = UPMNS + f Mass content of the flavor states f = UPMNS mass

  6. Vacuum mixing f = UPMNS m Ue1 Ue2 Ue3 U 1 U 2 U 3 U 1 U 2 U 3 1 2 3 e =

  7. Standard parametrization IM UPMNS = U23I U13I U12 I = diag (1, 1, ei ) Dirac phase matrix 1 0 0 0 c23 - s23 c13 0 s13e-i 0 1 0 c12 s12 0 -s12 c12 0 IM 0 -s23 c23 -s13ei 0 c13 0 0 1 i 21 /2 i 31 /2 IM = diag (1, e , e ) - matrix of Majorana phases Not unique parametrization Convenient for phenomenology, especially for oscillations in matter Insightful for theory?

  8. Mixing angles e Mixing parameters |U 3|2 |U 3|2 3 tan 12 =|Ue2|2 / |Ue1|2 |Ue3|2 sin 13 = |Ue3|2 MASS2 tan 23 = |U 3|2 / |U 3|2 m231 |Ue2|2 2 1 m221 Mixing matrix: |Ue1|2 FLAVOR f = UPMNS mass Normal mass hierarchy 1 2 3 e = UPMNS m231 = m23 - m21 m221 = m22 - m21 UPMNS = U23I U13I U12

  9. Data. Global fit NO preference smaller than 2 Octant? With SK atmospheric lower octant is preferable NO: CP-phase close to

  10. Global fit

  11. TBM: deviations and implications P. F. Harrison, D. H. Perkins, W. G. Scott Tri-bimaximal mixing 2/3 1/3 0 - 1/6 1/3 - 1/2 - 1/6 1/3 1/2 Utbm = U23( /4) U12 sin2 12 = 1/3 Utbm = Accidental, numerology, useful for bookkeeping Not accidental Lowest order approximation which corresponds to weakly broken (flavor) symmetry of the Lagrangian Accidental symmetry (still useful) There is no relation of mixing with masses (mass ratios) with some other physics and structures associated flavons other new particles

  12. Mass hierarchies mh > m312 > 0.045 eV at mZ up down charged neutrinos quarks quarks leptons 1 m2 m3 m212 m322 10-1 ~ 0.18 10-2 mass ratios Neutrinos have the weakest mass hierarchy (if any) among fermions 10-3 10-4 Related to large lepton mixing? Koide relation 10-5 mu mt = mc2 sin C ~ md/ms Gatto Sartori-Tonin relation

  13. Mixing and mass matrices Origin of mixing: off-diagonal mass matrices Ml = M Mixing matrix Diagonalization: Mass spectrum M = U Lm diag U LT m diag = (m1, m2, m3) Ml = UlLmldiag UlR+ mldiag = (me, m , m ) l (1 - 5) UPMNS mass CC in terms of mass eigenstates: UPMNS= UlL+ U L Flavor basis: Ml = mldiag UPMNS= U L

  14. Decoupling of masses and mixing equalities, zeros Shape invariance Relations between elements Mixing Mass matrix of neutrinos (in the flavor basis) Absolute values of matrix elements Masses Opens up a possibility that symmetry is behind TBM Decoupling is always possible, but the key is that relations should be simple to be consequences of simple symmetries

  15. Relations and symmetries Mixing: very special structure which is not related to mass ratios For TBM: m12 = - m13 , m22 = m33 , m11 + m13 = m22 + m23 S4 , sin C m12 = (m11 - m22) 1 2 sin2 C For Cabibbo mixing: ~ 1/4 Relation between matrix elements which leads to Cabibbo mixing independently of values of matrix elements symmetry which produces the relation dihedral D14 C. Hagedorn, 1204.0715

  16. Mass Ordering e 3 2 m221 1 MASS2 MASS2 m223 m232 2 1 m221 3 FLAVOR FLAVOR Inverted mass ordering Normal mass ordering

  17. Theories of mass ordering Accidental: selection of values of parameters Fundamental: principle, symmetry m2 m3 m m but 1-2 mixing strongly deviates from maximal m212 m322 m212 m322 m2 m3 ~ = 0.18 ~ = 1.6 10-2 ~ Similar to quark spectrum Flavor symmetries Quark-lepton symmetry Unification

  18. Pattern of mass matrices For Majorana neutrinos Inverted hierarchy Normal hierarchy 21 = 0 Value of mass 21 = Strong dependence of structure of mass matrix on the Majorana phases - different patterns different possible symmetries For two different Majorana phases

  19. Predicting mass hierarchy In models with residual (discrete) flavor symmetries mixing does not depend on mass at least in the symmetry limit. The same mixing pattern can be obtained for NH and IH Radiative mechanisms have different preferences those which are related to charged leptons prefer NH broken (Le L L ) symmetry - IH Seesaw type I with quark-lepton analogy (symmetry) leads to NH Establishing NH would favor the line with similarity with quark sector, high scale seesaw, etc. If so, one would expect the first quadrant,

  20. Flavor symmetries

  21. Mixing and symmetries with intermediate possibilities

  22. Realizations, problems Simple groups like S4 , A4 x Z2 can establish equalities of elements of mass matrix required by TBM

  23. Deriving the TBM symmetry C.S. Lam Invariance of mass matrices in the flavor basis: ViT mTBM Vi = mTBM Neutrinos Vi = S, Ui 1 0 0 1 0 1 0 0 0 -1 2 2 -1 2 .. -1 1 3 -permutation S = U = charged leptons diagonal due to symmetry 1 0 0 2 0 T = = exp(-2i /3) T+ (me+ m e)T = me+ m e S, T, U elements of S4

  24. S symmetry 4 Order 24, permutation of 4 elements Symmetry of cube Generators: S, T S4 = T3 = (ST2)2 = 1 Presentation: Irreducible representations: 1 , 1 , 2, 3, 3 3 x 3 = 3 x 3 = 1 + 2 + 3 + 3 Products and invariants New flavor structure 3 x 3 = 1 + 2 + 3 + 3 1 x 1 = 1 2 x 3 = 2 x 3 = 3 + 3 2 x 2 = 1 + 1 + 2 1 x 2 = 2

  25. Residual symmetries approach E. Ma, C. S. Lam .... Mixing appears as a result of different ways of the flavor symmetry breaking in the neutrino and charged lepton (Yukawa) sectors. T S4 A4 Gf T7 S Residual symmetries of the mass matrices Z2 x Z2 or Z2 Gl Ml T G M U, S Zm Intrinsic symmetries of mass matrices which do not depend on values of masses Symmetry transformations in mass bases S M S T = M Discrete finite groups Flavons to break symmetries CP-transformations can be added

  26. Intrinsic symmetries Realized for arbitrary values of neutrino and charged lepton mass can not be broken, always exist In the mass basis for charged leptons for Majorana neutrinos m = diag( m1, m2, m3) ml = diag( me, m , m ) G Gl i e i i T = diag(e , e , e ) = 2 k /m S1 = diag (1, -1, -1) S2 = diag (- 1, 1, -1) Tm = I Zm Si2 = I Z2 x Z2 The Klein group can use subgroup = 0 Intrinsic symmetries as residual symmetries

  27. Invariance m1 m2 m3 1 2 3 1 2 3 Si

  28. Symmetry group condition D. Hernandez, A.S. 1204.0445 If intrinsic symmetries are residual symmetries of the unique symmetry group (follow from breaking of unique group) bounds on elements of mixing matrix Si and T are elements of covering group. By definition product of these elements (taken in the same basis) also belongs to the finite discrete group: ( UPMNS Si UPMNS+ T)p = I p -integer For each i the equation gives two relations between mixing parameters Two such equations for i = 1,2 fix the mixing matrix completely TBM TBM Z2 x Z2 In general, the condition allows to fix mixing matrix up to several possibilities

  29. A4symmetry E. Ma G Branco, H P Nilles Symmetry group of even permutations of 4 elements Symmetry of tetrahedron A4 Generators: S, T Presentation of the group: no U = A S2 = 1 T3 = 1 (ST)3 = 1 Irreducible representations: 3, 1, 1 , 1 Products and invariants 3x3 = 3 + 3 + 1 + 1 + 1 1 x 1 ~ 1

  30. A_4 symmetry breaking A4 ``accidental symmetry due to particular selection of flavon representations and configuration of VEV s T S Ml diagonal M TBM-type In turn, this split originates from different flavor assignments of the RH components Nc and lc and different higgs multiplets

  31. An A4model G. Altarelli D. Meloni symmetry Yukawa sector mass terms A4 Z4 1 i -1 -i Charged lepton Neutrinos 3 L L 1 n k i i Hd Hu T 1 at multiplets i Nc i 1 lc i -1 1, i, -1 1 flavons M S n = 1, 2, k = 0, 1 1 1 T = v S(0, 1, 0) S = v S (1, 1, 1) Flavon sector U(1)R Particular selection of representations Vacuum alignment 0 S S0 T T 0 0 Driving fields GUT-scale or higher? 2 -1 1 1

  32. Generic problem m = F(Y, v) Mechanism of mass generation VEV s Yukawa couplings VEV alignment - different contributions - high order corrections TBM follow from independent sectors: Scalar potential Yukawa sector tune by additional symmetries All these components should be correlated ``Natural consequence of symmetry? one step constructions do not work

  33. More problems No connection to masses Mass hierarchy from additional symmetries U(1) Froggatt- Nielsen mechanism n power n is determined by U(1) charges of the corresponding operators Discrete symmetries restricted possibilities to explain mass spectrum (degenerate, partially degenerate spectra) Missing representations problem

  34. Relating to mass degeneracy Symmetry which left mass matrices invariant for specific mass spectra: Partially degenerate spectrum m1 = m2 , m3 Transformation matrix S = O2 D. Hernandez, A.S. G = SO(2) x Z2 m1 m2 Relation: sin22 23 = +/- sin = cos = = 1 maximal +/- /2 Majorana phase 1-2 mixing is undefined Small corrections to mass matrix lead to 1-2 mass splitting and 1-2 mixing

  35. CP-violation, phase Certain values of follow from the flavor symmetries without additional assumptions Non-trivial CP phase : from generalized CP transformations when also flavor is changed

  36. Flavor and CP symmetry Gfx HCP T A4 S4 Flavons (96) (48) Residual symmetries of the mass matrices G x H CP (27) Glx HlCP Sn D M R. Mohapatra, C C Nishi, F. Feruglio, C. Hagedorn, R. Ziegler, E Ma, G-J. Ding, S.F. King, C. Luhn, A. J. Stuart, Y-L. Zhou, Ml no CP violation = 0 G = Z2 x Z2 G = Z2 usually = 0, +/- /2 C. Hagedorn, et al may depend on free parameter and take any value Y. Farzan, A.S.

  37. Linear, Modular, Eclectic

  38. Modular symmetries as flavor symmetries F. Feruglio Gui-Jun Ding, S.F. King, Xiang-Gan Liu, S. Petcov, A V. Titov, M. Tanimoto , T. Nomura, H. Okada, T Kobayashi, O.Popov Y. Shimizu, P Novichkov, J. Penedo, T Osaka, A. Romanino, I. De Medeiros Varzielas, X Wang, S. Zhou ... Motivated by string theory Symmetry related to (orbifold) compactification of extra dimensions and primary realized on the moduli fields which describe geometry of the compactified space. Hope was to Reduce number of parameters, more predictive - ? Connect masses and mixing - ?

  39. Modular symmetries For single modulus field the modular transformation a + b c + d = The 2x2 matrices of integer numbers a b c d with determinant ad bc = 1 form the group = SL(2, Z) [special, linear, integer ...] Modular group N is finite subgroup of , quotient group of the level N: N = ( ) Isomorphic to the groups considered previously: 3, 4, 5 are isomorphic to A4, S4, A5, correspondingly

  40. Modular group N In SUSY, the chiral superfields transform as ,. kI I (c + d) ( ) I kI is the weight of multiplet ( ) is the representation of element of the group N Appearance of the weight factor in transformations is new element which leads to new consequences

  41. Modular forms - another key element of formalism fi( ) - holomorphic functions of modulus field Transformation properties are similar to those of superfields fi( ) fi( ) = (c + d) ( )ij fj( ) kf fi( ) form multiplet of N whose dimension is determined by level N and weight kf For instance for N = 3 and kf = 2, fi form triplet with components Y1( ) = 1 + 12q + 36q2 + ... Y2( ) = - 6q1/3 (1 + 7q + ... ) Y3( ) = - 18 q2/3 (1 + ... ) i2 q = e = 0.0117 + i 0.995 Yi = (1, - 0.74, - 0.27) Data fit: weak hierarchy

  42. Yukawa couplings are modular forms - holomorphic functions of modulus field - Form multiplets of the group N and transform as superfields: k Yi( ) Yi( ) = (c + d) ( )ij Yj( ) k is the weight of multiplet ( ) is the representation of element of the group N dimension of the multiplet is determined by the level N and weight k they are fixed by symmetry Dependence Yi( ) is determined by transformation properties

  43. Invariant superpotentials Invariance of terms of the superpotential Y 1 2 3 - constant requires for product of representations 1x 2 x 3 x Y = I ki + kY = 0 for weights The latter condition acts to some extent as Froggatt- Nielsen symmetry it gives additional restrictions, forbids some term texture zeros

  44. Models J Griado and F. Feruglio 1807.01125 [hep-ph] flavon L Ec Nc Y A43 1, 1 , 1 3 3 3 k 1 -4 -1 2 3 Y lowest order modular form and are fixed by fitting on m3 /m2, 1-2 mixing and 1-3 mixing Predictions: sin 2 = 0.46 / = 1.434 / = 1.7 / = 1.2

  45. Gui-Jun Ding, S.F. King, Xiang-Gan Liu 1907.11714 [hep-ph] Models No flavons Flavor from single modulus field Higher order modular forms Y(2), Y(4) , Y(6) constructed as products of Y(2) All Yukawas are modular forms L Ec Nc Y A43 1, 1 , 1 3 3 k 2 2 0 -2 is fixed by fitting 12 mixing and 13 mixing Predictions: m1 = 0.0946 eV m2 = 0.0950 eV m3 = 0.1071 eV Normal ordering mee = 0.095 eV sin 2 = 0.58 / = 1.6 / = 0.15 / = 1.00 Cosmological bound?

  46. From Model building to Symmetry building Minimal simpest versions do not reproduce data well. Generic predictions of masses: weak hierarchy, often quasi-degenerate, Majorana CP-phases Several moduli, flavons, More parameters needed. Different representations of the same dimension, fine tuning of corresponding couplings Different ways of construction of finite groups In the end: # parameters is comparable to # of observables Use modular symmetry in wrong way?

  47. Modular vs. usual discrete symmetries Comparing with usual approach with flavons Y1 (< >) , Y2 (< >) ... Yn (< >) < 1>, < 2>, ... , < n> For a given N, k, Y - known functions of the same VEV - Flavon VEV s depend on parameters of potential and not controlled by symmetry. Independent for different representations Can do the same as modular symmetry constructions

  48. Comparison F. Feruglio and A. Romanino, Rev. Mod. Phys. 93 (2021), 1. 015007 1912.0602 Usual Yukawa coupling Y L Nc Hu Y -constant Linear traditional flavor symmetry flavons (dimensionless) transformation under G flavor symmetry group: i(g) L L(g) L, Nc N (g)Nc Invariance: L(g) N(g) i i(g) contains singlet Y i L Nc Hu Y Y, Modular symmetry All the factors including coupling transform as f( ) (c + d) f( ) f( ) Invariance: f f(g) contains singlet and kf Y( ) L Nc Hu f kf = 0 Y i Y( )

  49. Eclectic flavor symmetry H.P Nilles, A.Baur, S. Ramos- Sanchez, P.K. Vaudrevange, A. Trautner.. Explicit top-down construction from heterotic string theory compactified on orbifold T2 /Z3 All symmetries used in bottom up approach: Gtraditional , Gmodular , GR , CP Z2CP CP transformations Z9R discrete R symmetry appear simultaneously in non-trivially unified fashion Kahler potential (kinetic terms) is involved and introduce additional parameters, freedom

  50. Back-up

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