m - ICTP

Standard Model:

G
u
m
α s
α
w
α
d
m
t
m
b
m
e
m
μ
m
c
m
u
θ
d
θ
θ
s
m
τ
m
δ
w
M
H
M
2
m
1
m
3
m
sun
θ
atm
θ
chooz
θ
l
δ
2
δ
3
δ

⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡ −
•
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
•
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
−
1
0
0
0
0
0
0
0
0
1
0
0
0
0
d
d
d
d
i
u
u
u
u
c
s
s
c
c
s
s
c
e
c
s
s
c
V
φ

tan m / m
θ = tan θ = m / m
d d s
⎛ 0 A 0 ⎞
⎜
A
∗
⎜
⎜
⎝ 0
C
B
∗
⎟
B
⎟
D ⎟
⎠
⎛ 0 A 0 ⎞
⎜ ∗ ⎟
⎜ A C B ⎟
⎜ ∗
0 B D⎟
⎝ ⎠
u u c

θ
θ
θ
ϕ
u
d
≅
≈
≅
≅
2
5
11
90
. 4
. 4
0
. 1
o
o
o
m d / ms
m
m
u
c
φ ≈α = 90
0

⎛ 0 A 0 ⎞
⎜
A
∗
⎜
⎜
⎝ 0
C
B
∗
⎟
B
⎟
D⎟
⎠
m m
tanθ ≅ c −e
iφ
s ≈
m m
t
b
0.
042

V
=
⎛
⎜
⎜
⎜
⎝
V
V
1
V
1e
1
μ
τ
V
V
V
2e
2
2
μ
τ
V
V
V
3e
3
3
μ
τ
⎞
⎟
⎟
⎟
⎠

Neutrino
Neutrino mixing
mixing
V=UxP
V=UxP
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
0
0
0
σ
ρ
i
i
e
e
P
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡ −
•
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
•
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
−
1
0
0
0
0
0
0
0
0
1
0
0
0
0
ν
ν
ν
ν
ϕ
c
s
s
c
c
s
s
c
e
c
s
s
c
U
i
l
l
l
l
sun
θ
θ ν ≈
at
θ
θ ≈
angle
reactor
l
−
≈
θ

30
o
36.
8
≤
o
Δm
Δm
θ
≤
21
2
sun
θ
2
32
at
≤
≈
≈
≤
39.
2
o
53.
2
o
−5
8 ⋅10
eV
−3
2
2. 5 ⋅10
eV
2

θ
θ
θ
sun
atm
chooz
= 34.6
= 45
o
o
= 4.6
o
⎛ 0.821 0.568 0.08⎞
⎜ −iδ −iδ
⎟
V =
⎜
−0.047 −0.40 •e − 0.03 + 0.58• e 0.70
⎟
⎜ −iδ −iδ
− 0.047 + 0.40 •e −0.03 −0.58 •e
0.70⎟
⎝ ⎠

⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡ −
•
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
•
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
=
−
1
0
0
0
0
0
0
0
0
1
0
0
0
0
ν
ν
ν
ν
ϕ
c
s
s
c
c
s
s
c
e
c
s
s
c
U
i
l
l
l
l
2
1
tan
m
m
=
ν
θ 07
.
0
tan ≅
=
μ
θ
m
m e
l

Observation:
θ ν
tan
o
≈ 33 __ θ ≈
θν
=
m
m
==> m
m
1
1
2
/ 2
≈
0.
42
45
o

Neutrino masses fixed, fixed,
since
the mass differences are
given by the experiment,
experiment
and the angles are fixed by
the mass matrix

2
32
2
21
4
2
3
2
21
4
2
2
2
21
4
2
1
2
cos
cos
2
cos
cos
2
cos
sin
m
m
m
m
m
m
m
Δ
+
Δ
=
Δ
=
Δ
=
ν
ν
ν
ν
ν
ν
θ
θ
θ
θ
θ
θ

m
m
m
m
1
2
3
2
≈
≈
≈
/ m3
≈
0.
19

Neutrino masses less hierarchical than
the masses of the charged leptons

m
2
/ M = m
τ
ν

Double Beta Decay

M
=
⎡
⎢
⎢
⎢⎣
0
A
0
∗
A
0
B
∗
0⎤
⎥
B
⎥
D⎥⎦

M
M
=
⎡
⎢
⎢
⎢⎣
0
A
0
⎡ 0
⎢
= A
⎢
⎢⎣
0
∗
∗
B
A
0
B
A
C
∗
∗
0⎤
⎥
B
⎥
D⎥⎦
0⎤
⎥
B
⎥
D⎥⎦

θ
θ
1
θ = θ +
θ 1 2
≈ m / m ≈ 1 4
≈ m / m ≈ 2 6
2 2 3
2
μ τ
= = = = = = = > θ ≅
sin θ ≅ 0 .9 7
4 0
o
o
o

V
=
⎛
⎜
⎜
⎜
⎝
V
V
1
V
1e
1
μ
τ
V
V
V
2e
2
2
μ
τ
V
V
V
3e
3
3
μ
τ
⎞
⎟
⎟
⎟
⎠

V
3 e
≈
1
2
•
m
e
mμ
=
0.
049

Conclude: Conclude
Neutrino masses might be Dirac masses or
Majorana masses. masses
Mixing angles for quarks are fixed by ratios of
quark masses. masses.
Very good agreement with
experiment.
experiment

This works also well for leptons. leptons.
One finds, finds,
using
the observed mass differences:
differences
m(1)/m(2)=0.42, m(2)/m(3)=0.19.
m(1): 0.0041 eV
m(2): 0.0097 eV
m(3): 0.051 eV
Atmospheric angle about 40 degrees
Neutrinoless double beta decay: decay:
difficult
Reactor neutrinos: neutrinos:
V (3e) ~ 0.052