Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
Подождите немного. Документ загружается.
1
06 3 Neutrinos beyond the
S
tandard Mode
l
l
epton
fl
avor mixin
g
an
g
les and three
C
P-violatin
g
phases. Thus an e
ff
ective
theory of electroweak interactions at low ener
g
ies totally consists of twenty
-
two flavor parameters which can only be determined from experiments. Why
i
sthenumbero
f
de
g
rees o
ff
reedom so bi
g
in the
fl
avor sector
?
What is th
e
f
undamental ph
y
sics behind these parameters? Such puzzles constitute th
e
flavor problems in modern particle physics.
T
able 3.6
S
ome important discoveries in the developments o
ffl
avor physic
s
Discoveries of le
p
ton flavors,
q
uark flavors and CP violation
1
897 electron
(
Thomson, 1897
)
1
919 proton
(
upanddownquarks
)(
Rutherford, 1919
)
1
932 neutron
(
upanddownquarks
)(
Chadwick, 1932
)
1
933 positron
(
Anderson, 1933
)
1
937 muon
(
Neddermeyer and Anderson, 1937
)
1
947 Kaon
(
strange quark
)(
Rochester and Butler, 1947
)
1
956 electron antineutrino
(
Cowan
et al.
, 1956
)
1
962 muon neutrino
(
Danby
e
tal
.
,
1962
)
196
4
C
P violation in
s
-quark decays
(
Christenso
n
et al.
,
1964
)
1
974 charm quark
(
Auber
t
et a
l
.
,
1974
;
Abrams et a
l
.
,
1974
)
1
975 tau
(
Per
l
et al.
,
1975
)
1
977 bottom quark
(
Herb
et al
., 1977
)
1
995 top quark
(
Ab
e
et al.
,
1995
;A
bachi
e
tal
.
,
1995
)
2
000 tau neutrino
(
Kodama et al.
,
2000
)
200
1
C
P violation in
b
-quark decays
(
Aubert
e
ta
l.
,
2001
;
Abe
e
ta
l.
,
2001
)
Fig. 3.11 is a schematic plot for the mass spectra of six quarks and si
x
l
eptons at the electroweak scale
(
Li and Xing, 2010
)
, where we have used th
e
a
llowed ranges of neutrino masses with a normal hierarchy
(
Gonzalez-Garcia
et al
., 2010
)
and the central values of charged-lepton and quark masses
(
Xin
g
et al.
,
2008
)
. One can see that the span between
m
1
a
n
d
m
t
is at
l
east twe
l
ve
orders of magnitude. Furthermore, the “desert” between the heaviest neutrin
o
ν
3
and the li
g
htest char
g
ed
f
ermio
n
e
−
s
pans at least six orders o
f
ma
g
nitude
.
Why do twelve fermions have such a stran
g
e mass pattern or such hierarch
y
a
nd desert puzzles? The answer to this fundamental question remains open
.
A
s shown in Table 3.1, current neutrino oscillation data onl
y
tell us
m
1
<
m
2
.
It remains unknown whether m
3
i
slar
g
er tha
n
m
2
(
normal hierarchy
)
o
r
s
ma
ll
er t
h
an
m
1
(
inverted hierarchy
)
. The possibility
m
1
≈
m
2
≈
m
3
(
nea
r
d
egeneracy
)
cannot be excluded at present. In contrast, three families of
c
harged fermions have strong mass hierarchies
(
Xin
g
e
tal., 2008
):
3.5 Lepton Flavor Mixing and
C
P Violation 107
Fi
g
.3.11
A
schematic illustration of the flavor “hierarchy” and “desert” problems
in the SM fermion mass spectrum at the electroweak scale
(
Li and Xing, 2010
)
m
e
m
μ
∼
m
u
m
c
∼
m
c
m
t
∼
λ
4
,
m
μ
m
τ
∼
m
d
m
s
∼
m
s
m
b
∼
λ
2
,
(
3.124
)
w
h
ere
λ
≡
s
i
n
θ
C
≈
0
.
2
2wit
h
θ
C
b
eing the Cabibbo angle of quark flavor
m
ixing
(
Cabibbo, 1963
)
. In the standard parametrization of the CKM matrix,
three quark mixin
g
an
g
les exhibit an impressive hierarchy
:
ϑ
1
2
∼
λ
,
ϑ
23
∼
λ
2
,ϑ
13
∼
λ
4
.
(
3.125
)
These two kinds of hierarchies might intrinsically be related to each other
,
because the
fl
avor mixin
g
an
g
les actually measure a mismatch between th
e
m
ass and flavor eigenstates of up- and down-type quarks. For example, th
e
r
elat
i
o
n
s
ϑ
12
≈
m
d
/m
s
,
ϑ
2
3
≈
m
d
/m
b
a
n
d
ϑ
13
≈
m
u
/
m
t
a
r
eco
m
-
patible with Eqs.
(
3.124
)
and
(
3.125
)
. They can be derived from a specific
pattern of up- and down-t
y
pe quark mass matrices with five texture zeros
(
Chkareuli and Froggatt, 1999
)
. On the other hand, it seems quite difficult to
fi
nd a simple way o
f
linkin
g
two lar
g
elepton
fl
avor mixin
g
an
g
le
s
θ
1
2
∼
π
/
6
a
nd θ
23
∼
π
/
4
to small
m
e
/m
μ
an
d
m
μ
/m
τ
.
One might ascribe the largeness
of
θ
1
2
a
n
d
θ
2
3
t
o a very weak hierarchy o
f
three neutrino masses and th
e
s
m
a
lln
ess of
θ
13
to the stron
g
mass hierarchy in the char
g
ed-lepton secto
r
(
Fritzsch and Xing, 2006, 2009
)
. There are of course many possibilities of
m
odel buildin
g
to understand the observed lepton
fl
avor mixin
g
pattern, bu
t
n
one of them has experimentall
y
and theoreticall
y
been
j
ustified.
Among a number of concrete flavor puzzles that are currently facing us
,
the
f
ollowin
g
three are particularly intri
g
uin
g.
•
T
he pole masses of three charged leptons satisfy the equality
(
Koide, 1983
)
m
e
+
m
μ
+
m
τ
√
m
e
+
√
m
μ
+
√
m
τ
2
=
2
3
(
3.126
)
1
08 3 Neutrinos beyond the
S
tandard Mode
l
t
oanamazin
g
ly
g
ood de
g
ree o
f
accuracy — its error bar is only o
f
O
(
10
−
5
)
(
Xing and Zhang, 2006
).
•
T
here are two quark-lepton “complementarity” relations in flavor mixing
:
θ
12
+
ϑ
12
≈
θ
23
+
ϑ
23
≈
π
4
,
(
3.127
)
w
hich are compatible with the present experimental data
(
Xing, 2005
).
•
T
wo unitarity trian
g
les o
f
the
C
KM matrix, de
fi
ned by the ortho
g
onality
condition
s
V
ud
V
V
V
∗
ub
VV
+
V
cd
VV
V
∗
cb
VV
+
V
td
VV
V
∗
tb
VV
=0
an
d
V
tb
VV
V
∗
ub
VV
+
V
ts
VV
V
∗
us
VV
+
V
td
VV
V
∗
ud
VV
=0
,
are almost the ri
g
ht trian
g
les. Namely, the common inner an
g
le o
f
thes
e
t
wo triangles satisfies
(
Xing, 2009a
)
α
≡
arg
−
V
ud
VV
V
∗
ub
V
V
V
td
VV
V
∗
tb
VV
≈
π
2
,
(
3.128
)
indicated by current experimental data on quark mixing and CP violation.
S
uch special numerical relations mi
g
ht just be accidental.
O
ne or two o
f
the
m
m
ight also be possible to result from a certain
(
underlying
)
flavor symmetry.
3
.6 Running Behaviors of Neutrino Mass Parameters
The spirit of seesaw mechanisms
(
to be described in Chapter 4
)
is to attribut
e
the small masses of three known neutrinos to the existence of some heav
y
d
e
g
rees o
ff
reedom, such as th
e
S
U
(
2
)
L
g
au
g
e-sin
g
let
f
ermions, th
e
SU
(
2
)
L
gauge-triplet scalars or th
e
SU
(
2
)
L
gauge-triplet fermions. All of them point
to the unique dimension-5 Weinber
g
operator in an e
ff
ective theory a
f
ter th
e
c
orresponding heavy particles are integrated out
(
Weinberg, 1979
):
L
d
=
5
Λ
=
1
2
κ
α
β
α
L
˜
H
˜
H
T
c
β
L
+h
.
c
.,
(
3.129
)
whe
r
e
Λ
is the cuto
ff
scale
,
L
d
enotes the le
f
t-handed lepton doublet
,
˜
H
≡
i
σ
2
H
∗
w
ith
H
bein
g
the SM Hi
gg
s doublet, and
κ
stands for the effectiv
e
n
eutrino coupling matrix. After spontaneous gauge symmetry breaking
,
˜
H
g
ains its vacuum expectation va
l
u
e
˜
H
=
v
/
√
2w
i
th
v
≈
24
6G
e
V
.
W
ear
e
then left with the effective Ma
j
orana mass matrix
M
ν
MM
=
κv
2
/
2 for three ligh
t
n
eutrinos from Eq.
(
3.129
)
. If the dimension-5 Weinberg operator is obtained
i
n the framework of the minimal supersymmetric standard model
(
MSSM
)
,
one will be left wit
h
M
ν
MM
=
κ
(
v
si
n
β
)
2
/
2,
where tan
β
denotes the ratio of th
e
vacuum expectation values of two MSSM Higgs doublets
.
Eq.
(
3.129
)
or its supersymmetric counterpart can provide a simple bu
t
generic way of generating tiny neutrino masses. There are a number of in
-
terestin
g
possibilities o
f
buildin
g
renormalizable
g
au
g
e models to realize th
e
e
ff
ective Weinber
g
mass operator, either radiatively or at the tree level. Th
e
3
.6 Running Behaviors o
f
Neutrino Mass Parameters 109
l
atter case is
j
ust associate
d
wit
h
t
h
ewe
ll
-
k
nown seesaw mec
h
anisms to
b
e
d
iscussed in Section 4.1. Here we assume tha
t
L
d=
5
/Λ
a
rises from an un-
d
erlying seesaw model, whose lightest heavy particle has a mass of
O
(
Λ
)
.
In ot
h
er wor
d
s
,
Λ
c
h
a
r
acte
riz
es t
h
e seesaw sca
l
e
.
Above
Λ
t
h
ere ma
y
exis
t
one or more energy thresholds corresponding to the masses of heavier seesa
w
particles
(
King and Singh, 2000; Antusch
et al
., 2002a; Mei, 2005; Antusch
et
al.
,
2005;
C
hao and Zhan
g
, 2007;
S
chmidt, 2007;
C
hakrabortt
y
et al
., 2009
)
.
Belo
w
Λ
t
he energy dependence of the effective neutrino coupling matri
x
κ
i
s
d
escribed by its renormalization-group equation
(
RGE
)
. The evolution o
f
κ
f
r
om
Λ
down to the electroweak scale is
f
ormall
y
independent o
f
an
y
detail
s
of the relevant seesaw model from whic
h
κ
is derived.
3.6.1
O
ne-loop Renormalization-group Equations
A
t the one-loop level
κ
o
beys the RGE
(
Chankowski and Pluciennik, 1993
;
B
abu
et al
., 1993;
A
ntusch
et al.
,
2001; 2002b
)
16
π
2
d
κ
d
t
=
α
κ
κ
+
C
κ
(
Y
l
YY
Y
†
l
YY
)
κ
+
κ
(
Y
l
YY
Y
†
l
YY
)
T
,
(
3.130
)
where
t
≡
l
n
(
μ/
Λ
)
wit
h
μ
b
eing an arbitrary renormalization scale between
t
h
ee
l
ectrowea
k
sca
l
ean
d
t
h
e seesaw sca
l
e, an
d
Y
l
YY
i
st
h
ec
h
ar
g
e
d
-
l
epton
Yukawa couplin
g
matrix. The R
G
Eo
f
Y
l
Y
Y
a
n
dt
h
ose of
Y
u
Y
Y
(
up-type quarks
)
a
nd
Y
d
Y
Y
(
down-type quarks
)
are given by
16
π
2
d
Y
l
YY
d
t
=
α
l
+
C
l
l
C
(
Y
l
YY
Y
†
l
Y
Y
)
Y
l
YY
,
1
6
π
2
d
Y
u
Y
Y
d
t
=
α
u
+
C
u
u
CC
(
Y
u
YY
Y
†
u
YY
)
+
C
d
u
CC
(
Y
d
YY
Y
†
d
Y
Y
)
Y
u
YY
,
16
π
2
d
Y
d
YY
d
t
=
α
d
+
C
u
d
C
(
Y
u
YY
Y
†
u
YY
)
+
C
d
d
C
(
Y
d
YY
Y
†
d
Y
Y
)
Y
d
Y
Y
.
(
3.131
)
In the framework of the
S
Mwehave
C
κ
=
C
d
u
CC
=
C
u
d
C
=
−
C
l
l
C
=
−
C
u
u
CC
=
−
C
d
d
C
=
−
3
2
,
(
3.132
)
a
n
d
α
κ
=
−
3
g
2
2
+
λ
+
2
T
r
3(
Y
u
YY
Y
†
u
YY
)
+3
(
Y
d
YY
Y
†
d
YY
)
+
(
Y
l
YY
Y
†
l
YY
)
,
α
l
=
−
9
4
g
2
1
−
9
4
g
2
2
+Tr
3(
Y
u
YY
Y
†
u
YY
)
+3
(
Y
d
YY
Y
†
d
YY
)
+
(
Y
l
YY
Y
†
l
YY
)
,
α
u
=
−
17
20
g
2
1
−
9
4
g
2
2
−
8
g
2
3
+T
r
3(
Y
u
Y
Y
Y
†
u
Y
Y
)
+3
(
Y
d
YY
Y
†
d
Y
Y
)
+
(
Y
l
YY
Y
†
l
YY
)
,
α
d
=
−
1
4
g
2
1
−
9
4
g
2
2
−
8
g
2
3
+
T
r
3
(
Y
u
YY
Y
†
u
YY
)
+3
(
Y
d
YY
Y
†
d
Y
Y
)
+
(
Y
l
YY
Y
†
l
YY
)
;
(
3.133
)
a
nd in the framework of the M
SS
Mwehave
1
10 3 Neutrinos beyond the
S
tandard Mode
l
C
κ
=
C
d
u
CC
=
C
u
d
C
=
C
l
l
C
3
=
C
u
u
CC
3
=
C
d
d
C
3
=1
,
(
3.134
)
a
n
d
α
κ
=
−
6
5
g
2
1
−
6
g
2
2
+
6Tr
(
Y
u
YY
Y
†
u
YY
)
,
α
l
=
−
9
5
g
2
1
−
3
g
2
2
+
Tr
3(
Y
d
YY
Y
†
d
YY
)
+
(
Y
l
YY
Y
†
l
YY
)
,
α
u
=
−
13
15
g
2
1
−
3
g
2
2
−
16
3
g
2
3
+
3Tr
(
Y
u
YY
Y
†
u
YY
)
,
α
d
=
−
7
1
5
g
2
1
−
3
g
2
2
−
16
3
g
2
3
+Tr
3
(
Y
d
YY
Y
†
d
YY
)
+
(
Y
l
YY
Y
†
l
YY
)
.
(
3.135
)
He
r
e
g
1
,
g
2
a
n
d
g
3
are the
g
au
g
e couplin
g
sandsatis
f
ytheirR
G
E
s
16
π
2
d
g
i
d
t
=
b
i
g
3
i
,
(
3.136
)
where
(
b
1
,b
2
,
b
3
)
=
(
41
/
10
,
−
19
/
6
,
−
7)
in the SM or
(
33
/
5
,
1
,
−
3
)
in the
M
SS
M. In addition,
λ
i
stheHi
gg
s sel
f
-couplin
g
parameter o
f
the
S
Mand
obeys the RGE
(
Antusc
h
et al., 2003
)
16
π
2
d
λ
d
t
=6
λ
2
−
3
λ
3
5
g
2
1
+3
g
2
2
+
3
2
3
5
g
2
1
+
g
2
2
2
+3
g
4
2
+4
λ
Tr
3
(
Y
u
Y
Y
Y
†
u
YY
)
+3
(
Y
d
YY
Y
†
d
YY
)
+
(
Y
l
YY
Y
†
l
YY
)
−
8T
r
3(
Y
u
YY
Y
†
u
YY
)
2
+3
(
Y
d
Y
Y
Y
†
d
YY
)
2
+(
Y
l
Y
Y
Y
†
l
YY
)
2
.
(
3.137
)
The relation between
λ
a
nd the Higgs mass
M
h
M
i
sgivenby
λ
=
M
2
h
M
/
(
2
v
2
),
whe
r
e
v
≈
2
46 GeV is the vacuum expectation value of the Higgs field
.
A
lthou
g
h massive neutrinos are most likely to be Majorana particles, th
e
possibilit
y
that the
y
could be Dirac particles cannot be ruled out at present.
A
combination of the Yukawa interactions of Dirac neutrinos in Eq.
(
3.10
)
a
nd those of charged fermions in Eq.
(
3.47
)
allows one to derive their RGE
s
(
Chen
g
e
tal
.
,
1974; Machacek and Vaughn, 1984
)
. At the one-loop level we
h
ave the
RG
Es o
f
Dirac
f
ermions
:
1
6
π
2
d
Y
ν
YY
d
t
=
α
ν
+
C
ν
ν
(
Y
ν
Y
Y
Y
†
ν
Y
Y
)+
C
l
ν
(
Y
l
YY
Y
†
l
Y
Y
)
Y
ν
Y
Y
,
16
π
2
d
Y
l
YY
d
t
=
α
l
+
C
ν
l
C
(
Y
ν
YY
Y
†
ν
YY
)+
C
l
l
C
(
Y
l
Y
Y
Y
†
l
YY
)
Y
l
YY
,
16
π
2
d
Y
u
YY
d
t
=
α
u
+
C
u
u
CC
(
Y
u
YY
Y
†
u
YY
)
+
C
d
u
CC
(
Y
d
YY
Y
†
d
YY
)
Y
u
YY
,
1
6
π
2
d
Y
d
Y
Y
d
t
=
α
d
+
C
u
d
C
(
Y
u
YY
Y
†
u
YY
)
+
C
d
d
C
(
Y
d
YY
Y
†
d
YY
)
Y
d
YY
.
(
3.138
)
In the framework of the
S
M
3
.6 Running Behaviors o
f
Neutrino Mass Parameters 11
1
C
ν
ν
=
C
l
l
C
=
C
u
u
C
C
=
C
d
d
C
=+
3
2
,
C
l
ν
=
C
ν
l
C
=
C
d
u
CC
=
C
u
d
C
=
−
3
2
,
(
3.139
)
a
n
d
α
ν
=
−
9
20
g
2
1
−
9
4
g
2
2
+
T
SM
TT
,
α
l
=
−
9
4
g
2
1
−
9
4
g
2
2
+
T
SM
TT
,
α
u
=
−
1
7
20
g
2
1
−
9
4
g
2
2
−
8
g
2
3
+
T
SM
TT
,
α
d
=
−
1
4
g
2
1
−
9
4
g
2
2
−
8
g
2
3
+
T
SM
TT
,
(
3.140
)
i
nwhich
T
SM
T
T
=Tr
3
(
Y
u
YY
Y
†
u
YY
)
+3
(
Y
d
YY
Y
†
d
YY
)
+
(
Y
ν
YY
Y
†
ν
YY
)
+
(
Y
l
YY
Y
†
l
YY
)
(
Lindne
r
e
tal.
,
2
005
)
; and in the framework of the MSSM
C
ν
ν
=
C
l
l
C
=
C
u
u
C
C
=
C
d
d
C
=+
3
,
C
l
ν
=
C
ν
l
C
=
C
d
u
CC
=
C
u
d
C
=
+
1
,
(
3.141
)
a
nd
α
ν
=
−
3
5
g
2
1
−
3
g
2
2
+T
r
3
(
Y
u
Y
Y
Y
†
u
Y
Y
)
+
(
Y
ν
Y
Y
Y
†
ν
Y
Y
)
,
α
l
=
−
9
5
g
2
1
−
3
g
2
2
+Tr
(
Y
ν
YY
Y
†
ν
YY
)
+3
(
Y
l
YY
Y
†
l
YY
)
,
α
u
=
−
13
15
g
2
1
−
3
g
2
2
−
16
3
g
2
3
+Tr
3
(
Y
u
YY
Y
†
u
YY
)
+
(
Y
ν
YY
Y
†
ν
YY
)
,
α
d
=
−
7
1
5
g
2
1
−
3
g
2
2
−
16
3
g
2
3
+
T
r
3(
Y
d
YY
Y
†
d
Y
Y
)
+
(
Y
l
YY
Y
†
l
Y
Y
)
.
(
3.142
)
The R
G
Es o
f
three
g
au
g
e couplin
gs
g
1
,
g
2
a
n
d
g
3
r
emain to
b
et
h
ose
g
ive
n
i
nEq.
(
3.136
).
T
he above RGEs allow us to evaluate the running behavior o
f
κ
or
Y
ν
YY
,
to
g
ether with those o
f
Y
l
Y
Y
,
Y
u
Y
Y
a
n
d
Y
d
Y
Y
,
f
rom the seesaw scale to the electrowea
k
s
cale or vice versa. We shall examine the evolution of neutrino masses, le
p
to
n
flavor mixing angles and CP-violating phases in the subsequent sections.
3.6.2 Evolution o
f
Majorana Neutrino Mass Parameter
s
Without loss o
fg
enerality, we choose the
fl
avor basis wher
e
Y
l
YY
is dia
g
onal
:
Y
l
Y
Y
=
D
l
≡
D
iag
{
y
e
,
y
μ
,y
τ
}
w
ith
y
α
b
eing the eigenvalues o
f
Y
l
Y
Y
.Inthiscas
e
the e
ff
ective Majorana neutrino couplin
g
matri
x
κ
c
an
b
e
d
ia
g
ona
l
ize
db
y
the MN
S
matri
x
V
;i
.e., V
†
κ
V
∗
=
3
κ
≡
Dia
g
{
κ
1
,κ
2
,
κ
3
}
w
i
th
κ
i
bein
g
th
e
eigenvalues o
f
κ.The
n
1
12 3 Neutrinos beyond the
S
tandard Mode
l
d
κ
d
t
=
d
d
t
V
3
κV
3
T
=
˙
V
3
κV
3
T
+
V
˙
3
κV
3
T
+
V
3
κ
˙
V
T
=
1
1
6
π
2
α
κ
V
3
κV
3
T
+
C
κ
D
2
l
V
3
κV
3
T
+
V
3
κV
3
T
D
2
l
,
(
3.143
)
with the help of Eq.
(
3.130
)
. After a definition of the Hermitian matri
x
S
≡
V
†
D
2
l
V
and the anti-Hermitian matrix
T
≡
V
†
˙
V
,
Eq.
(
3.143
)
leadsusto
˙
3
κ
=
1
1
6
π
2
[
α
κ
3
κ
+
C
κ
(
S
3
κ
+
3
κS
3
∗
)]
−
T
3
κ
+
3
κT
3
∗
.
(
3.144
)
Because
3
κ
i
s by definition dia
g
onal and real, the left- and ri
g
ht-hand sides o
f
E
q.
(
3.144
)
must be diagonal and real. We can therefore arrive a
t
˙
κ
i
=
1
16
π
2
(
α
κ
+2
C
κ
Re
S
i
i
)
κ
i
,
(
3.145
)
together with I
m
T
ii
TT
=R
e
T
ii
TT
=Im
S
ii
=0
(
fo
r
i
=
1
,
2
,
3)
.Sincetheoff-
d
iagonal parts of Eq.
(
3.144
)
are vanishing, we hav
e
T
ij
TT
κ
j
−
κ
i
T
∗
ij
T
T
=
C
κ
16
π
2
S
ij
κ
j
+
κ
i
S
∗
ij
(
3.146
)
w
i
th
i
=
j
.There
f
ore,
Re
T
ij
TT
=
−
C
κ
16
π
2
·
κ
i
+
κ
j
κ
i
−
κ
j
Re
S
ij
,
Im
T
ij
TT
=
−
C
κ
1
6
π
2
·
κ
i
−
κ
j
κ
i
+
κ
j
Im
S
i
j
.
(
3.147
)
Due to
˙
V
=
VT
,Eq.
(
3.147
)
actually governs the evolution of
V
with ener
g
ies.
W
e proceed to defin
e
V
≡
PUP
,
in w
h
ic
h
P
≡
D
iag
{
e
i
φ
e
,
e
i
φ
μ
,
e
i
φ
τ
}
,
P
≡
D
ia
g
{
e
i
ρ
,
e
i
σ
,
1
}
,an
d
U
is the
C
KM-like matrix containin
g
three neutrin
o
m
ixin
g
an
g
les and one CP-violatin
g
phase.
A
lthou
g
h
P
does not have an
y
physical meaning, its phases have their own RGEs
(
Casas et a
l
., 2000
)
.I
n
c
ontrast
,
P
s
erves
f
or the Ma
j
orana phase matrix. We
fi
nd
T
≡
P
T
P
†
=
P
V
†
˙
VP
†
=
˙
P
P
†
+
U
†
˙
U
+
U
†
P
†
˙
PU
,
(
3.148
)
f
rom which we can obtain six inde
p
endent constraint e
q
uations:
T
11
TT
=
T
11
TT
=i˙
ρ
+
α
U
∗
α
U
1
˙
U
α
U
1
+i
U
α
U
1
˙
φ
α
,
T
22
TT
=
T
22
TT
=i˙
σ
+
α
U
∗
α
U
2
˙
U
α
U
2
+i
U
α
U
2
˙
φ
α
,
T
33
TT
=
T
33
TT
=
α
U
∗
α
U
3
˙
U
α
U
3
+i
U
α
U
3
˙
φ
α
;
3
.6 Running Behaviors o
f
Neutrino Mass Parameters 113
T
12
T
T
=
T
12
T
T
e
i
(
ρ
−
σ
)
=
α
U
∗
α
U
1
˙
U
α
U
2
+
i
U
α
U
2
˙
φ
α
,
T
13
T
T
=
T
13
T
T
e
i
ρ
=
α
U
∗
α
U
1
˙
U
α
U
3
+
i
U
α
U
3
˙
φ
α
,
T
23
TT
=
T
23
TT
e
i
σ
=
α
U
∗
α
U
2
˙
U
α
U
3
+
i
U
α
U
3
˙
φ
α
,
(
3.149
)
where
α
run
so
v
e
r
e
,
μ
an
d
τ
.
Note tha
t
T
ii
TT
= 0 holds and
T
ij
TT
is
g
iven b
y
E
q.
(
3.147
)
. In view o
f
y
e
y
μ
y
τ
,weta
ke
D
2
l
≈
D
ia
g
{
0
,
0
,
y
2
τ
}
as a
n
excellent a
pp
roximation. Then
S
ij
,
T
ij
TT
a
n
d
T
ij
TT
c
an all be ex
p
ressed in terms
o
f
y
2
τ
and the parameters o
f
U
a
n
d
P
.
After a straightforward calculation
,
we obtain the explicit expressions of Eqs.
(
3.145
)
and
(
3.149
)
as follows
:
˙
κ
i
=
κ
i
16
π
2
α
κ
+
2
C
κ
y
2
τ
|
U
τi
UU
|
2
,
(
3.150
)
a
n
d
α
U
∗
α
U
1
i
˙
U
α
UU
1
−
U
α
U
1
˙
φ
α
=˙
ρ,
α
U
∗
α
U
2
i
˙
U
α
UU
2
−
U
α
U
2
˙
φ
α
=˙
σ,
α
U
∗
α
U
3
i
˙
U
α
UU
3
−
U
α
U
3
˙
φ
α
=
0
,
α
U
∗
α
U
1
˙
U
α
UU
2
+i
U
α
U
2
˙
φ
α
=
−
C
κ
y
2
τ
16
π
2
e
i(
ρ
−
σ
)
ζ
−
1
12
Re
U
∗
τ
UU
1
U
τ
UU
2
e
i
(
σ
−
ρ
)
+i
ζ
12
Im
U
∗
τ
UU
1
U
τ
UU
2
e
i(
σ
−
ρ
)
,
α
U
∗
α
U
1
˙
U
α
UU
3
+
i
U
α
U
3
˙
φ
α
=
−
C
κ
y
2
τ
16
π
2
e
i
ρ
ζ
−
1
13
Re
U
∗
τ
UU
1
U
τ
UU
3
e
−
i
ρ
+i
ζ
13
Im
U
∗
τ
UU
1
U
τ
UU
3
e
−
i
ρ
,
α
U
∗
α
U
2
˙
U
α
UU
3
+
i
U
α
U
3
˙
φ
α
=
−
C
κ
y
2
τ
16
π
2
e
i
σ
ζ
−
1
23
R
e
U
∗
τ
UU
2
U
τ
UU
3
e
−
i
σ
+i
ζ
23
Im
U
∗
τ
UU
2
U
τ
UU
3
e
−
i
σ
,
(
3.151
)
whe
r
e
ζ
ij
ζζ
≡
(
κ
i
−
κ
j
)
/
(
κ
i
+
κ
j
)
with
i
=
j
.O
ne can see that those
y
2
τ
-
a
ssociated terms onl
y
consist of the matrix element
s
U
τi
UU
(
fo
r
i
=1
,
2
,
3
)
.I
f
a
parametrization o
f
U
assu
r
es
U
τi
UU
to
b
eassimp
l
easpossi
bl
e, t
h
e resu
l
tan
t
RG
Es o
f
neutrino mixin
g
an
g
les and
C
P-violatin
g
phases will be very concise
.
We find that the FX parametrization advocated in Eq.
(
3.108
)
with
U
=
⎛
⎝
⎛⎛
s
l
s
ν
c
+
c
l
c
ν
e
−
i
φ
s
l
c
ν
c
−
c
l
s
ν
e
−
i
φ
s
l
s
c
l
s
ν
c
−
s
l
c
ν
e
−
i
φ
c
l
c
ν
c
+
s
l
s
ν
e
−
i
φ
c
l
s
−
s
ν
s
−
c
ν
s
c
⎞
⎠
⎞⎞
(
3.152
)
1
14 3 Neutrinos beyond the
S
tandard Mode
l
a
ccor
d
swit
h
t
h
ea
b
ove o
b
servation, w
h
i
l
et
h
e“stan
d
ar
d”
parametrization i
n
E
q.
(
3.107
)
does not. That is why the RGEs of neutrino mixing angles and
CP-violating phases in the standard parametrization are rather complicated
(
Casas
et al.
,
2000;
A
ntusc
h
et al
.
,
2003; Mei, 2005; Lu
o
et al.
,
2005
).
Here we take the FX form of
U
to derive the RGEs of neutrino mas
s
a
nd mixing parameters
(
Xing, 2006
)
. Combining Eqs.
(
3.150
)
,
(
3.151
)
and
(
3.152
)
, we arrive at
˙
κ
1
=
κ
1
16
π
2
α
κ
+2
C
κ
y
2
τ
s
2
ν
s
2
,
˙
κ
2
=
κ
2
16
π
2
α
κ
+
2
C
κ
y
2
τ
c
2
ν
s
2
,
˙
κ
3
=
κ
3
1
6
π
2
α
κ
+2
C
κ
y
2
τ
c
2
,
(
3.153
)
whe
r
e
α
κ
≈
−
3
g
2
2
+6
y
2
t
+
λ
(
SM
)
or
α
κ
≈
−
1
.
2
g
2
1
−
6
g
2
2
+6
y
2
t
(
MSSM
)
;an
d
˙
θ
l
=
C
κ
y
2
τ
1
6
π
2
c
ν
s
ν
c
ζ
−
1
13
c
ρ
c
(
ρ
−
φ
)
+
ζ
1
3
s
ρ
s
(
ρ
−
φ
)
−
ζ
−
1
23
c
σ
c
(
σ
−
φ
)
+
ζ
23
s
σ
s
(
σ
−
φ
)
,
˙
θ
ν
=
C
κ
y
2
τ
16
π
2
c
ν
s
ν
s
2
ζ
−
1
1
2
c
2
(
σ
−
ρ
)
+
ζ
12
s
2
(
σ
−
ρ
)
+
c
2
ζ
−
1
1
3
c
2
ρ
+
ζ
13
s
2
ρ
−
c
2
ζ
−
1
23
c
2
σ
+
ζ
23
s
2
σ
,
˙
θ
=
C
κ
y
2
τ
16
π
2
c
s
s
2
ν
ζ
−
1
13
c
2
ρ
+
ζ
13
s
2
ρ
+
c
2
ν
ζ
−
1
23
c
2
σ
+
ζ
23
s
2
σ
;
(
3.154
)
as well as
˙
ρ
=
C
κ
y
2
τ
16
π
2
3
ζ
12
c
2
ν
s
2
c
(
σ
−
ρ
)
s
(
σ
−
ρ
)
+
3
ζ
13
s
2
ν
s
2
−
c
2
c
ρ
s
ρ
+
3
ζ
23
c
2
ν
s
2
c
σ
s
σ
,
˙
σ
=
C
κ
y
2
τ
16
π
2
3
ζ
12
s
2
ν
s
2
c
(
σ
−
ρ
)
s
(
σ
−
ρ
)
+
3
ζ
13
s
2
ν
s
2
c
ρ
s
ρ
+
3
ζ
23
c
2
ν
s
2
−
c
2
c
σ
s
σ
,
˙
φ
=
C
κ
y
2
τ
16
π
2
c
2
l
−
s
2
l
c
−
1
l
s
−
1
l
c
ν
s
ν
c
ζ
−
1
13
c
ρ
s
(
ρ
−
φ
)
−
ζ
13
s
ρ
c
(
ρ
−
φ
)
−
ζ
−
1
2
3
c
σ
s
(
σ
−
φ
)
+
ζ
23
s
σ
c
(
σ
−
φ
)
+
3
ζ
12
s
2
c
(
σ
−
ρ
)
s
(
σ
−
ρ
)
+
3
ζ
13
s
2
ν
−
c
2
ν
c
2
c
ρ
s
ρ
+
3
ζ
2
3
c
2
ν
−
s
2
ν
c
2
c
σ
s
σ
,
(
3.155
)
w
h
ere
3
ζ
ij
ζ
ζ
≡
ζ
−
1
ij
ζζ
−
ζ
ij
ζζ
=4
κ
i
κ
j
/
κ
2
i
−
κ
2
j
,
c
a
≡
cos
a
an
d
s
a
≡
s
i
n
a
(
for
a
=
ρ
,
σ
,
σ
−
ρ
,
ρ
−
φ
o
r
σ
−
φ
)
. As a byproduct, the RGEs of three unphysical phase
s
φ
α
are listed below
:
˙
φ
e
=
+
C
κ
y
2
τ
16
π
2
c
l
s
−
1
l
c
ν
s
ν
c
ζ
−
1
1
3
c
ρ
s
(
ρ
−
φ
)
−
ζ
13
s
ρ
c
(
ρ
−
φ
)
−
ζ
−
1
2
3
c
σ
s
(
σ
−
φ
)
+
ζ
23
s
σ
c
(
σ
−
φ
)
+
c
2
3
ζ
13
s
2
ν
c
ρ
s
ρ
+
3
ζ
23
c
2
ν
c
σ
s
σ
,
3
.6 Running Behaviors o
f
Neutrino Mass Parameters 115
˙
φ
μ
=
−
C
κ
y
2
τ
1
6
π
2
c
−
1
l
s
l
c
ν
s
ν
c
ζ
−
1
13
c
ρ
s
(
ρ
−
φ
)
−
ζ
13
s
ρ
c
(
ρ
−
φ
)
−
ζ
−
1
23
c
σ
s
(
σ
−
φ
)
+
ζ
2
3
s
σ
c
(
σ
−
φ
)
−
c
2
3
ζ
1
3
s
2
ν
c
ρ
s
ρ
+
3
ζ
23
c
2
ν
c
σ
s
σ
,
˙
φ
τ
=
−
C
κ
y
2
τ
16
π
2
s
2
3
ζ
13
s
2
ν
c
ρ
s
ρ
+
3
ζ
23
c
2
ν
c
σ
s
σ
.
(
3.156
)
It is easy to check that the relationshi
p
˙
φ
=˙
ρ
+˙
σ
+
˙
φ
e
+
˙
φ
μ
+
˙
φ
τ
h
olds
.
T
h
at is w
hy
φ
α
should not be i
g
nored in derivin
g
the R
G
Es o
f
those phys
-
i
cal parameters, althou
g
h these three phases can
fi
nally be rotated away b
y
rephasing the charged-lepton fields.
S
ome discussions on the basic
f
eatures o
f
R
G
Es o
f
three neutrino masses
,
three flavor mixin
g
an
g
les and three CP-violatin
g
phases are in order.
(
a
)
The running behaviors of three neutrino masse
s
m
i
(
or equivalently
κ
i
)
a
re essentia
lly
i
d
entica
l
an
dd
etermine
dby
α
κ
,
un
l
ess tan
β
i
s
l
ar
g
eenou
gh
i
n the MSSM to make th
e
y
2
τ
-
associated term com
p
etitive with the
α
κ
t
erm
.
In our phase convention, ˙
κ
i
or ˙
m
i
(
for
i
=
1
,
2
,
3
)
are independent of th
e
C
P-violatin
g
phas
e
φ
.
(
b
)
Among three neutrino mixing angles, only the derivative o
f
θ
ν
c
onta
i
n
s
a
term proportiona
l
t
o
ζ
−
1
12
.N
ote that
ζ
−
1
ij
ζζ
=(
m
i
+
m
j
)
2
/
Δ
m
2
i
j
w
i
th
Δ
m
2
ij
≡
m
2
i
−
m
2
j
h
olds. Current solar and atmospheric neutrino oscillation data yiel
d
Δ
m
2
21
≈
7
.
7
×
10
−
5
e
V
2
an
d
Δ
m
2
32
≈
Δ
m
2
31
≈
2
.
4
×
10
−
3
eV
2
.So
θ
ν
i
s
i
n
genera
l
more sensitive to ra
d
iative corrections t
h
a
n
θ
l
an
d
θ
. The evolution of
θ
ν
c
an be suppressed through the fine-tuning of
(
σ
−
ρ
)
. The smallest neutrino
m
ixing angl
e
θ
l
may get radiative corrections even if its initial value is zero
,
a
nd thus it can be radiatively generated from other neutrino mixing angle
s
a
nd CP-violating phases
(
Antusc
h
et al.
, 2003; Mei an
d
Xin
g
, 2004; Luo
et
a
l
.
,
2005; Luo and Xing, 2006a
)
.
(
c
)
The running behavior of
φ
is quite di
ff
erent
f
rom those o
f
ρ
a
n
d
σ
,
because it includes a
p
eculiar term
p
ro
p
ortional to
s
−
1
l
.
This term, which
do
min
ates
˙
φ
w
h
en
θ
l
is su
ffi
ciently small, becomes diver
g
ent in the limi
t
θ
l
→
0. Indeed
,
φ
i
s not well-defined if
θ
l
is exactly vanishing. But bot
h
θ
l
a
n
d
φ
c
an be radiatively
g
enerated. We may require tha
t
˙
φ
r
e
m
a
in
fi
ni
te
when θ
l
approaches zero, implying that the following necessary condition ca
n
be extracted
f
rom the expression o
f
˙
φ
i
nEq.
(
3.155
)
:
ζ
−
1
13
c
ρ
s
(
ρ
−
φ
)
−
ζ
1
3
s
ρ
c
(
ρ
−
φ
)
−
ζ
−
1
23
c
σ
s
(
σ
−
φ
)
+
ζ
2
3
s
σ
c
(
σ
−
φ
)
=
0
.
(
3.157
)
It turns out that
t
a
n
φ
=
3
ζ
13
si
n
2
ρ
−
3
ζ
23
si
n
2
σ
ζ
−
1
13
+
ζ
1
3
+
3
ζ
1
3
c
os
2
ρ
−
ζ
−
1
23
+
ζ
2
3
+
3
ζ
2
3
cos 2
σ
(
3.158
)
ho
l
ds
.N
ote t
h
at t
h
e
ini
t
i
a
l
va
l
ue of
θ
l
,i
f
it is exactl
y
zero or extremel
y
small,
m
ay immediately drive φ
t
oitsquasi-fixed poin
t
(
Luo and Xing, 2006b
)
.In