Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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1
56 4
S
eesaw Mechanisms o
f
Neutrino Masse
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5
P
henomenology o
f
Neutrino Oscillations
So far a number of different neutrino experiments have convincin
g
ly observed
the phenomena of neutrino oscillations, indicating that neutrinos have rest
m
asses and lepton
fl
avors are mixed. This is the
fi
rst compellin
g
experimen
-
tal evidence for new physics beyond the standard model
(
SM
)
of elementary
particle physics. In this chapter we shall first describe the basic properties
o
f
neutrino oscillations in vacuum and then explain the Mikhe
y
ev-
S
mirnov-
Wolfenstein
(
MSW
)
mechanism for neutrino flavor conversions in matter
.
Section 5.2 is devoted to an analysis of quantum coherence in neutrino oscil
-
l
ations. We shall re
f
ormulate neutrino oscillations by means o
f
the lan
g
ua
g
e
of the densit
y
matrix and flavor polarization vector in Section 5.3. Some
ongoing and future accelerator- and reactor-based neutrino oscillation exper-
i
ments will be brie
fly
introduced in
S
ection 5.4.
5
.1 Neutrino Oscillations and Matter E
ff
ect
s
Soon after the discover
y
of electron antineutrinos in 1956, Bruno Pontecorvo
postulated that there might exist the phenomenon of neutrino-antineutrino
oscillations by analo
g
y with the phenomenon o
f
K
0
-
K
0
mixing
(
Pontecorvo,
1
958
)
. Shortly after the discovery of muon neutrinos in 1962, Ziro Maki,
Masami Nakagawa and Shoichi Sakata proposed that two different neutrino
fl
avors could mix with each other and thus the
p
henomenon o
f
ν
e
↔
ν
μ
or
ν
e
↔
ν
μ
t
ransitions might take place
(
Maki et al
.
,
1962
)
. Their ideas poin
t
to the concept of neutrino
(
flavor
)
oscillations: given a beam of neutrino
s
i
nade
fi
nite
fl
avor state at the source, one ma
yfi
nd them in another
fl
avor
s
tate at the detector which is at a certain distance from the source. This kin
d
o
f
quantum phenomenon can naturally occur i
f
three known neutrinos hav
e
n
on-de
g
enerate masses and lepton flavors are mixed. Neutrino oscillations
h
ave been well established in the past twelve years, thanks to a number of
s
o
l
ar, atmosp
h
eric, reactor an
d
acce
l
erator neutrino experiments. Bot
h
t
he
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
1
60 5 Phenomenology o
f
Neutrino
O
scillations
l
on
g
standin
g
solar neutrino puzzle and the observed de
fi
cit o
f
atmospheric
m
uon neutrinos are actuall
y
attributed to neutrino oscillations. In this section
we first outline the basic properties of neutrino oscillations in vacuum and
then describe the phenomenolo
g
yo
f
neutrino oscillations in matter.
5
.1.1 Neutrino
O
scillations in Vacuu
m
Three neutrinos
(
ν
e
,
ν
μ
,ν
τ
)
are defined as the flavor eigenstates, which accord
with three charged leptons
(
e
,
μ
,
τ
)
, in their production processes via the wea
k
c
har
g
ed-current interactions.
S
ince neutrinos are assumed to be massless in
the SM, their flavor and mass eigenstates coincide with each other and thus
l
epton
fl
avors are conserved. But current neutrino oscillation experiment
s
h
ave demonstrated that neutrinos are massive and the SM is incom
p
lete. Be
-
yond the SM we define the mass eigenstates of three neutrinos as
(
ν
1
,ν
2
,
ν
3
),
whose eigenvalues are denoted by
(
m
1
,
m
2
,m
3
)
. In the basis where the mass
ei
g
enstates of char
g
ed leptons are identified with their flavor ei
g
enstates, th
e
phenomenon of neutrino mixing can be described by a
3
×
3
unitary matri
x
V , the so-called Maki-Nakagawa-Sakata
(
MNS
)
matrix
(
Maki
et al
.
,
1962
)
:
⎛
⎝
⎛
⎛
ν
e
ν
μ
ν
τ
⎞
⎠
⎞
⎞
=
⎛
⎝
⎛
⎛
V
e
VV
1
V
e
VV
2
V
e
VV
3
V
μ
VV
1
V
μ
VV
2
V
μ
VV
3
V
τ
VV
1
V
τ
VV
2
V
τ
VV
3
⎞
⎠
⎞
⎞
⎛
⎝
⎛
⎛
ν
1
ν
2
ν
3
⎞
⎠
⎞
⎞
.
(
5.1
)
D
ifferent parametrizations o
f
V
have been presented in Section 3.5.1.
Now we explain how neutrinos chan
g
etheir
fl
avors when propa
g
atin
g
i
n vacuum. A freely propagating neutrino should be on the mass-shell, or
equiva
l
ent
l
yint
h
e mass ei
g
enstate
ν
k
(
t,
x
)(
for
k
=
1
,
2
,
3)
. Therefore, they
m
ust obe
y
the Dirac equation
i
/
∂
//
−
m
k
ν
k
(
t,
x
)
= 0 and the Klein-Gordon
e
q
uat
i
o
n
∂
2
+
m
2
k
ν
k
(
t
,
x
)
=0
(
in the neglect of the spinor structure o
f
the neutrino field which is essentially irrelevant to neutrino oscillations
)
.For
a
stationary neutrino source, which is the case
f
or all the existin
g
neutrino
oscillation experiments, the ener
g
y spectrum of neutrinos is fixed and on
e
a
ctua
ll
y measures t
h
eir propagation in space rat
h
er t
h
an in time
1
.To
b
e
s
peci
fi
c, let us assume that a neutrino beam with ener
gy
E
i
spropa
g
at
i
n
g
i
n the one-dimensional s
p
ace
x
. In this case the correspondin
g
Klein-Gordon
equation can
b
e rewritten as
−
(
E +i
∂
x
∂
)(
E
−
i
∂
x
∂
)
+
m
2
k
ν
k
(
t
,
x
)
=0.In
v
i
ew o
f
−
i
∂
x
∂
ν
k
(
t
,x
)=
p
k
ν
k
(
t
,
x
)
due t
o
ν
k
(
t
,x
)
∝
e
−
i(
Et
−
p
k
x
)
f
o
rfr
ee
n
eu
-
trinos, we
f
urther obtai
n
1
F
or neutrinos from the conventional sources, their oscillations have been treated
in three di
ff
erent wa
y
s: the evolution o
f
neutrino states in time, in space, or in
space and time. They all lead to the same oscillation probabilities in the relativistic
approx
i
mat
i
on
t
≈
x
,
as given in Eq.
(
5.5
)
. However, the scheme of evolution in
space is exclusivel
y
appropriate to understand the recent proposal o
f
M¨ossbauer
neutrino oscillations (Akhmedo
v
et a
l.
, 2008, 2009).
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
6
1
i
∂
x
∂
ν
k
(
x
)
=
m
2
k
2
E
ν
k
(
x
)
,
(
5.2
)
w
h
ere t
h
e approximation
p
k
=
E
2
−
m
2
k
≈
E
has bee
nm
ade a
n
dthete
rm
s
of
O
(
m
2
k
/
E
2
)
have been omitted. Integrating Eq.
(
5.2
)
, one may figure out
the evolution o
f
the mass ei
g
enstates
:
ν
k
(
x
)
=e
−
i
m
2
k
x
/
(2
E
)
ν
k
(
0
)
.Thenth
e
evolution o
f
the
fl
avor ei
g
enstates reads
ν
α
(
x
)=
3
k
=1
V
αk
VV
e
x
p
*
−
i
m
2
k
2
E
x
)
ν
k
(
0
)
.
(
5.3
)
Note t
h
at
i
t
i
s
m
o
r
eco
n
ve
ni
e
n
ttowo
rk
w
i
t
h
t
h
e
n
eut
rin
owavefu
n
ct
i
o
n
s
or state vectors when calculating the neutrino oscillation probabilities. The
o
n
e
-n
eut
rin
o state
|
ν
k
i
s
g
enerated by the Hermitian-conju
g
ate
fi
eld operato
r
ν
†
k
(
x
)
acting on the vacuum state, and likewise for the flavor states. Hence
the evolution of the state vectors is governed by
|
ν
α
(
x
)
=
3
k
=1
V
∗
αk
VV
e
x
p
*
−
i
m
2
k
2
E
x
)
|
ν
k
(
0
)
.
(
5.4
)
Fo
r
a
n
eut
rin
obea
min
t
h
e
fl
avo
r
state
|
ν
α
at the source, the probability o
f
observin
g
it in the stat
e
|
ν
β
ν
at a d
i
sta
n
ce
x
=
L
f
r
o
m
t
h
esou
r
ce
i
s
P
(
ν
α
→
ν
β
ν
)
=
|
ν
β
ν
|
ν
α
(
L
)
|
2
=
3
j
=
1
3
k
=1
V
αj
VV
V
βk
VV
V
∗
αk
VV
V
∗
βj
VV
e
x
p
2
−
i
Δ
m
2
k
j
2
E
L
6
=
3
j
=1
|
V
αj
VV
|
2
|
V
βj
VV
|
2
+
2Re
j<
k
V
αj
VV
V
βk
VV
V
∗
αk
VV
V
∗
βj
VV
exp
2
−
i
2
π
L
λ
kj
6
(
5.5
)
w
i
th
Δ
m
2
k
j
≡
m
2
k
−
m
2
j
bein
g
the neutrino mass-squared di
ff
erences an
d
λ
kj
≡
4
π
E/
Δ
m
2
k
j
b
ein
g
the neutrino oscillation len
g
ths. A comparison be
-
tween
L
a
n
d
λ
ji
is sometimes help
f
ul
f
or a rou
g
h estimation o
f
the oscillatio
n
probabilities. For instance, the neutrinos from some astrophysical sources
travel a very long distance
(
L
∼
M
pc
)
before arriving at the detector o
n
the Earth. In this case the oscillation len
g
th
s
λ
k
j
are much smaller than the
propagation
d
istanc
e
L
e
ven for very high neutrino-beam energies
E
T
e
V
a
nd ver
y
small neutrino mass-squared di
ff
erences
|
Δ
m
2
k
j
|
∼
10
−
5
eV
2
.
H
e
n
ce
the oscillatory terms in Eq.
(
5.5
)
should be averaged over many circles of
osci
ll
ations, so t
h
at t
h
e resu
l
tant pro
b
a
b
i
l
ities are rat
h
er simp
l
e
:
P
(
ν
α
→
ν
β
ν
)
=
3
i=
1
|
V
αi
VV
|
2
|
V
βi
VV
|
2
.
(
5.6
)
We see that the
p
henomenon o
f
neutrino oscillations is due to the coherenc
e
a
mong three neutrino mass eigenstates. Eq.
(
5.6
)
is always applicable to th
e
1
62 5 Phenomenology o
f
Neutrino
O
scillations
c
ase in w
h
ic
h
quantum co
h
erence is comp
l
ete
ly l
ost. More
d
iscussions a
b
ou
t
the issues of
q
uantum coherence and decoherence in neutrino oscillations wil
l
be presented in Section 5.2.
T
he probabilities of neutrino oscillations in Eq.
(
5.5
)
can be rewritten a
s
P
(
ν
α
→
ν
β
ν
)=
3
i=1
V
αi
VV
V
∗
βi
VV
2
+2
3
i
<
j
Re
(
V
αi
VV
V
βj
VV
V
∗
αj
VV
V
∗
βi
VV
)
cos
Δ
m
2
ji
L
2
E
+
2
3
i<j
I
m
(
V
αi
VV
V
βj
V
V
V
∗
αj
VV
V
∗
βi
V
V
)
si
n
Δ
m
2
ji
L
2
E
=
3
i
=1
V
αi
VV
V
∗
βi
VV
2
−
4
3
i
<
j
Re
(
V
αi
VV
V
βj
VV
V
∗
αj
VV
V
∗
βi
VV
)
sin
2
Δ
m
2
ji
L
4
E
+
2
3
i
<
j
I
m
(
V
αi
VV
V
βj
V
V
V
∗
αj
VV
V
∗
βi
V
V
)
si
n
Δ
m
2
ji
L
2
E
=
δ
α
β
−
4
3
i
<
j
R
e
(
V
αi
V
V
V
βj
VV
V
∗
αj
VV
V
∗
βi
VV
)
si
n
2
Δ
m
2
ji
L
4
E
+
8
J
γ
αβγ
s
in
Δ
m
2
21
L
4
E
si
n
Δ
m
2
31
L
4
E
s
i
n
Δ
m
2
32
L
4
E
,
(
5.7
)
where the Jarlsko
g
paramete
r
J
h
as been defined in Eq.
(
3.116
)
. The surviva
l
probabilities of neutrino oscillations
(
i.e.
,
β
=
α
)
turn out to b
e
P
(
ν
α
→
ν
α
)
=
1
−
4
i
<
j
|
V
αj
VV
|
2
|
V
βi
VV
|
2
s
i
n
2
Δ
m
2
ji
L
4
E
.
(
5.8
)
In the case of two-flavor neutrino oscillations, Eq.
(
5.8
)
is simplified to
P
(
ν
α
→
ν
α
)
=1−
si
n
2
2
θ
sin
2
1
.
2
7
Δ
m
2
eV
2
·
L
km
·
G
eV
E
,
(
5.9
)
whe
r
e
θ
a
n
d
Δ
m
2
r
epresent t
h
e neutrino mixin
g
an
gl
ean
d
mass-square
d
d
ifference, respectivel
y
. Current experimental data on neutrino oscillations
c
an all be interpreted by means of Eq.
(
5.9
)
as a good approximation, simply
because
|
Δ
m
2
3
2
|
≈
|
Δ
m
2
3
1
|
Δ
m
2
21
a
n
d
|
V
e
VV
3
|
2
1 hold
(
see Section 3.
1
f
or details
)
. For illustration, Table 5.1 lists six known neutrino oscillation
experiments toget
h
er wit
h
t
h
eir surviva
l
pro
b
a
b
i
l
ities an
d
neutrino mixin
g
a
n
g
les in the two-
fl
avor neutrino mixin
g
scheme.
Recall that the one-
p
article stat
e
|
ν
i
is generated by the neutrino fiel
d
operato
r
ν
i
(
x
)
rather than its Hermitian conjugate. Hence one may easily
read off the probabilities of antineutrino oscillations from Eq.
(
5.7
)
through
the re
p
lacement
s
V
→
V
∗
a
nd
J
→−
J
:
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
63
Table 5.1 Survival probabilities and mixing factors of neutrino oscillations for six
known ex
p
eriments in the two-
fl
avor a
pp
roximation
E
xperiment
S
urvival probability Mixin
gf
actors
S
olar
ν
e
→
ν
e
1
−
sin
2
2
θ
12
s
i
n
2
Δ
m
2
2
2
1
L
4
E
s
i
n
2
2
θ
12
=4
|
V
e
VV
1
|
2
|
V
e
VV
2
|
2
K
a
mLAN
D
ν
e
→
ν
e
1
−
s
in
2
2
θ
1
2
s
i
n
2
Δ
m
2
2
1
L
4
E
s
i
n
2
2
θ
1
2
=4
|
V
e
V
V
1
|
2
|
V
e
V
V
2
|
2
A
tmospheric
ν
μ
→
ν
μ
1
−
s
in
2
2
θ
23
s
i
n
2
Δ
m
2
3
2
L
4
E
s
i
n
2
2
θ
2
3
=4
|
V
μ
VV
3
|
2
1
−|
V
μ
VV
3
|
2
K2K
ν
μ
→
ν
μ
1
−
si
n
2
2
θ
23
sin
2
Δ
m
2
32
L
4
E
sin
2
2
θ
2
3
=4
|
V
μ
VV
3
|
2
1
−|
V
μ
VV
3
|
2
M
IN
OS
ν
μ
→
ν
μ
1
−
si
n
2
2
θ
23
sin
2
Δ
m
2
32
L
4
E
sin
2
2
θ
2
3
=4
|
V
μ
VV
3
|
2
1
−|
V
μ
VV
3
|
2
C
HOOZ
ν
e
→
ν
e
1
−
si
n
2
2
θ
1
3
sin
2
Δ
m
2
31
L
4
E
sin
2
2
θ
1
3
=4
|
V
e
VV
3
|
2
1
−|
V
e
VV
3
|
2
P
(
ν
α
→
ν
β
)=
δ
α
β
−
4
3
i<
j
R
e
(
V
αi
VV
V
βj
VV
V
∗
αj
VV
V
∗
βi
VV
)
si
n
2
Δ
m
2
ji
L
4
E
−
8
J
γ
αβ
γ
s
i
n
Δ
m
2
2
1
L
4
E
s
in
Δ
m
2
3
1
L
4
E
s
i
n
Δ
m
2
32
L
4
E
.
(
5.10
)
O
n the other hand, the probabilities o
f
ν
β
ν
→
ν
α
osc
i
llat
i
o
n
sca
n
also be
r
ead
off from Eq.
(
5.7
)
through the interchange of
α
a
n
d
β
:
P
(
ν
β
ν
→
ν
α
)
=
δ
α
β
−
4
3
i
<j
Re
(
V
αi
VV
V
βj
VV
V
∗
αj
VV
V
∗
βi
VV
)
sin
2
Δ
m
2
ji
L
4
E
−
8
J
γ
αβ
γ
s
in
Δ
m
2
2
1
L
4
E
s
i
n
Δ
m
2
3
1
L
4
E
s
in
Δ
m
2
3
2
L
4
E
.
(
5.11
)
A
difference between P
(
ν
α
→
ν
β
)
an
d
P
(
ν
α
→
ν
β
ν
)
signifies CP violation,
wh
i
le that betwee
n
P
(
ν
β
ν
→
ν
α
)
an
d
P
(
ν
α
→
ν
β
ν
)
measures T violation
(
Cabibbo, 1978
)
. They must be equal to each other
(
Xing, 2004
)
,
Δ
P
≡
P
(
ν
α
→
ν
β
ν
)
−
P
(
ν
α
→
ν
β
)
=
P
(
ν
α
→
ν
β
ν
)
−
P
(
ν
β
ν
→
ν
α
)
=
1
6
J
γ
α
β
γ
sin
Δ
m
2
21
L
4
E
si
n
Δ
m
2
31
L
4
E
sin
Δ
m
2
32
L
4
E
,
(
5.12
)
a
s dictated b
yC
PT invariance. The
C
PT theorem also ensures that there is
n
o difference between the
p
robabilities o
f
ν
α
→
ν
α
a
nd
ν
α
→
ν
α
oscillations
.
1
64 5 Phenomenology o
f
Neutrino
O
scillations
H
ence it is possible to discover
C
P or T violation onl
y
in the “appearance
”
n
eutrino oscillation experiments.
Ag
eometric description of CP violation i
n
the language of leptonic unitarity triangles has been given in Section 3.5.3.
S
o
f
ar we have onl
y
calculated the probabilities o
f
neutrino-neutrino and
a
ntineutrino-antineutrino oscillations. Here let us explain wh
y
it is extremel
y
d
ifficult to realize Pontecorvo’s original idea and do a neutrino-antineutrino
oscillation experiment. We consider a beam o
f
ν
α
antineutrinos pro
d
uce
d
f
rom the vertex of standard charged-current interaction
s
α
+
+
W
−
→
ν
α
.
A
fter traveling a distance
L
it w
i
ll be detected at a detecto
r
v
i
atheve
r
te
x
o
f
standard char
g
ed-current interaction
s
ν
β
ν
→
β
−
+
W
+
. The amplitude o
f
ν
α
→
ν
β
ν
oscillations can then be written as
A
(
ν
α
→
ν
β
ν
)
=
K
3
j
=
1
V
αj
VV
V
βj
VV
m
j
E
e
xp
2
−
i
m
2
j
2
E
L
6
,
(
5.13
)
where
E
is the beam energy and
K
d
enotes a kinematic factor
(
Schechter
a
nd Valle, 1981; de Gouve
a
et al
., 2003
)
. Different from normal
ν
α
→
ν
β
ν
or
ν
α
→
ν
β
o
sci
ll
ations, t
h
e
ν
α
→
ν
β
ν
oscillations involve a suppression
f
acto
r
m
j
/
E
i
n their amplitude. It reflects the fact that the incomin
g
α
+
leads t
o
a
n antineutrino
ν
α
in a
d
ominant
l
yrig
h
t-
h
an
d
e
dh
e
l
icity state, w
h
ereas t
h
e
s
tan
d
ar
d
c
h
ar
g
e
d
-current interactions t
h
at pro
d
uce t
h
eout
g
oin
g
β
−
would
p
refer the incident neutrino
ν
β
ν
b
eing in a left-handed state
(
Kayser, 2005
).
B
ecause o
f
m
j
1
e
V
an
d
E
1 MeV in a rea
l
istic experiment, t
h
is
h
e
l
icity
s
uppression factor
(
i.e.
,
m
j
/
E
1
0
−
6
)
makes it impossible to observe the
phenomenon of neutrino-antineutrino oscillations.
5
.1.2 Adiabatic Neutrino
O
scillations in Matte
r
In 1978, Lincoln Wol
f
enstein
p
ointed out that the coherent
f
orward scat-
tering of neutrinos off matter could significantly change the neutrino flavo
r
c
ontent
(
Wolfenstein, 1978, 1979
)
. It was later on realized that the effectiv
e
n
eutrino mixin
gf
actor sin
2
2
θ
m
in m
atte
r
cou
l
deve
n
be
m
a
xim
a
ln
o
m
atter
h
ow small its counter
p
art sin
2
2
θ
i
n vacuum was,
p
rovided the resonance con
-
d
ition was satisfied
(
Barge
r
et al
.
,
1980
)
. In 1985, Stanislav Mikheyev and
A
lexei Smirnov found that the adiabatic evolution of massive neutrinos in
m
atter cou
ld
provi
d
eane
l
egant so
l
ution to t
h
e
l
ongstan
d
ing so
l
ar neutrino
puzzle
(
i.e., the observed flux of sola
r
ν
e
n
eut
rin
os was
m
uch lowe
r
tha
n
the
one predicted by the standard solar model
)(
Mikheyev and Smirnov, 1985
)
.
Today we know that the MSW matter effects on neutrino flavor conversions
a
re a
l
so important in t
h
e
l
on
g
-
b
ase
l
ine neutrino osci
ll
ation experiments an
d
c
rucial for the su
p
ernova neutrino detection.
As discussed in Section 2.3, the coherent forward scattering of neutrinos i
n
ordinar
y
matter can be described b
y
the neutrino re
f
ractive index, or equiva-
l
entl
y
the effective potentia
l
V
=
√
2
G
F
n
e
w
ith
n
e
being the number density
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
65
o
f
electrons. In the two-
fl
avor mixin
g
scheme the state vector o
f
the neutrin
o
sy
stem can be written as
|
Ψ
(
t
)
=
a
e
(
t
)
|
ν
e
+
a
μ
(
t
)
|
ν
μ
,
so the correspondin
g
Schr¨odinger-like equation reads
i
d
d
t
a
e
(
t
)
a
μ
(
t
)
=
1
2
E
U
m
2
1
0
0
m
2
2
U
†
+
A
0
00
a
e
(
t
)
a
μ
(
t
)
,
(
5.14
)
w
h
e
r
e
A
≡
2
√
2
G
F
n
e
E
arises
f
rom the e
ff
ective
p
otential
V
,and
U
is t
h
e
2
×
2
n
eutrino mixing matrix linking the flavor eigenstates to the mass eigenstates
:
|
ν
e
|
ν
μ
=
cos
θ
s
i
n
θ
−
s
i
n
θ
cos
θ
|
ν
1
|
ν
2
(
5.15
)
wit
h
θ
b
eing the mixing angle in vacuum. Let us recast the effective Hamil
-
tonian in Eq.
(
5.14
)
into a more instructive form:
H
m
=
1
4
E
(
Σ
+
A
)
1
+
A
−
Δ
m
2
cos
2
θ
Δ
m
2
si
n2
θ
Δ
m
2
sin
2
θ
−
A
+
Δ
m
2
c
os
2
θ
,
(
5.16
)
whe
r
e
Δ
m
2
≡
m
2
2
−
m
2
1
a
n
d
Σ
≡
m
2
1
+
m
2
2
w
i
th
m
i
(
for
i
=1
,
2)
being th
e
n
eutrino mass eigenvalues. Note that the
(
Σ
+
A
)
term, which is proportional
to the identity matrix and thus irrelevant to the neutrino flavor conversion
,
c
an actually be ne
g
lected in calculatin
g
the physical quantities o
f
matter
effects. For simplicit
y
, we first consider the case of
n
e
= constant; namel
y
,th
e
m
atter density is independent of time and space. Analogous to the treatmen
t
o
f
neutrino oscillations in vacuum, the e
ff
ective Hamiltonian
H
m
should be
d
ia
g
onalized so as to fi
g
ure out both the effective neutrino masses and th
e
effective neutrino mixing angle in matter:
˜
m
2
1
=
1
2
Σ
+
A
−
(
A
−
Δ
m
2
cos 2
θ
)
2
+(
Δ
m
2
s
in 2
θ
)
2
,
˜
m
2
2
=
1
2
Σ
+
A
+
(
A
−
Δ
m
2
cos 2
θ
)
2
+(
Δ
m
2
si
n2
θ
)
2
;
(
5.17
)
a
n
d
ta
n
2
θ
m
=
Δ
m
2
s
in
2
θ
Δ
m
2
cos 2
θ
−
A
.
(
5.18
)
N
ote t
h
at on
ly
Δ
˜
m
2
≡
˜
m
2
2
−
˜
m
2
1
, which is independent of
(
Σ
+
A
)
, affects th
e
p
robabilities of neutrino oscillations in matter:
˜
P
(
ν
e
→
ν
μ
)=
˜
P
(
ν
μ
→
ν
e
)
=sin
2
2
θ
m
sin
2
1
.
2
7
Δ
˜
m
2
L
E
,
˜
P
(
ν
e
→
ν
e
)=
˜
P
(
ν
μ
→
ν
μ
)
=1
−
s
in
2
2
θ
m
s
i
n
2
1
.
2
7
Δ
˜
m
2
L
E
,
(
5.19
)
w
h
e
r
e
Δ
˜
m
2
,
L
a
n
d
E
a
re in units o
f
eV
2
,kmand
G
eV, respectivel
y
. Note als
o
that there exist two equivalent parameter-space conventions in vacuum:
(
1
)