Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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1
66 5 Phenomenology o
f
Neutrino
O
scillations
0
θ
π
/
4
to
g
et
h
er wit
h
positive or ne
g
ativ
e
Δ
m
2
;
(
2
)
0
θ
π
/
2to
g
et
h
er
with
p
ositive
Δ
m
2
. We shall take the second one, as commonl
y
taken in the
l
iterature
,b
ecauseitisa
l
so consistent wit
h
Δ
m
2
21
>
0
extracted from curren
t
experimental data on solar neutrino oscillations. Then Eq.
(
5.18
)
leads u
s
to
θ
m
→
θ
w
hen the matter effect is negligible
(
i.e.
,
A
→
0)
,or
θ
m
→
π
/
2
when the matter effect is overwhelming
(
i.e.,
A
→∞
)
. If the matter densit
y
h
appens to satis
fy
A
=
Δ
m
2
cos 2
θ
,onewi
ll
simp
ly
arrive at
θ
m
=
π
/
4
f
r
om
E
q.
(
5.18
)
. This resonant effect is particularly striking because it gives ris
e
to s
in
2
2
θ
m
= 1 in matter even for a very small value of si
n
2
2
θ
i
n
vacuu
m
(
Barge
r
et al
., 1980
)
.
A more interesting and realistic case is the slowly-varying matter density
;
i
.e., t
h
ee
l
ectron
d
ensity un
d
er consi
d
eration is time-
d
epen
d
ent. For a spe
-
c
i
fi
c neutrino source, such as the
S
un, the time de
p
endence o
f
n
e
s
h
ou
l
dbe
u
nderstood as the distance de
p
endence o
f
n
e
i
n the assum
p
tion that solar
n
eutrinos are re
l
ativistic. Note t
h
at t
h
e instantaneous neutrino mass ei
g
en
-
s
tates in matter are linked to the neutrino
fl
avor ei
g
enstates as
f
ollows:
|
ν
e
|
ν
μ
=
U
m
UU
|
˜
ν
1
|
˜
ν
2
≡
cos
θ
m
sin
θ
m
−
sin
θ
m
cos
θ
m
|
˜
ν
1
|
˜
ν
2
,
(
5.20
)
where
U
m
UU
denotes the effective
2
× 2 neutrino mixing matrix in matter, an
d
i
ts mixing ang
l
e
θ
m
varies wit
h
time. Expressing t
h
e state vector in t
h
ein-
sta
n
ta
n
eous
m
ass bas
i
sas
|
Ψ
(
t
)
=
a
1
(
t
)
|
˜
ν
1
+
a
2
(
t
)
|
˜
ν
2
a
n
d
in
t
h
eflavo
r
bas
i
s
as
|
Ψ
(
t
)
=
a
e
(
t
)
|
ν
e
+
a
μ
(
t
)
|
ν
μ
,
we are then able to establish the following
re
l
ations
h
ip:
a
e
(
t
)
a
μ
(
t
)
=
cos
θ
m
si
n
θ
m
−
s
i
n
θ
m
cos
θ
m
a
1
(
t
)
a
2
(
t
)
.
(
5.21
)
On the other hand, the Schr¨odinger-like equation in Eq.
(
5.14
)
can be pu
t
i
nto a simpler
f
orm:
i
d
d
t
a
e
(
t
)
a
μ
(
t
)
=
1
2
E
U
m
UU
˜
m
2
1
0
0˜
m
2
2
U
†
m
UU
a
e
(
t
)
a
μ
(
t
)
.
(
5.22
)
With the help of Eq.
(
5.21
)
, one may convert Eq.
(
5.22
)
into
i
d
d
t
a
1
(
t
)
a
2
(
t
)
=
1
4
E
−
Δ
˜
m
2
−
i4
E
˙
θ
m
(
t
)
i4
E
˙
θ
m
(
t
)
Δ
˜
m
2
a
1
(
t
)
a
2
(
t
)
(
5.23
)
i
nt
h
e instantaneous mass
b
asis, w
h
ere
˙
θ
m
(
t
)
≡
d
θ
m
(
t
)
d
t
i
sde
fi
ned, an
d
Δ
˜
m
2
≡
˜
m
2
2
−
˜
m
2
1
d
e
p
ends on
t
t
oo. The evolution e
q
uations o
f
two instan
-
taneous mass eigenstates are therefore entangled with each other, unless the
ad
i
abat
i
cco
n
d
i
t
i
o
n
|
˙
θ
m
(
t
)
|
|
Δ
˜
m
2
|
/
(
4
E
)
is satisfied. In such an adiabati
c
a
pproximation we simply ne
g
lect the o
ff
-dia
g
onal terms and then inte
g
rat
e
E
q.
(
5.23
)
.Thesolutionsare
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
67
a
1
(
t
)
=
a
1
(
t
0
)
ex
p
2
+i
t
t
0
Δ
˜
m
2
(
t
)
4
E
d
t
6
,
a
2
(
t
)
=
a
2
(
t
0
)
ex
p
2
−
i
t
t
0
Δ
˜
m
2
(
t
)
4
E
d
t
6
,
(
5.24
)
w
h
ere
Δ
˜
m
2
(
t
)=
[
A
(
t
)
−
Δ
m
2
c
os
2
θ
]
2
+
(
Δ
m
2
s
in 2
θ
)
2
is time-
d
epen
d
ent
because of
A
(
t
)
=2
√
2
G
F
En
e
(
t
)
. As a result, the amplitude for the fina
l
s
tate to be electron
(
muon
)
neutrinos is given b
y
a
e
(
t
)
=co
s
θ
m
(
t
)
a
1
(
t
)
+si
n
θ
m
(
t
)
a
2
(
t
)
,
a
μ
(
t
)
=co
s
θ
m
(
t
)
a
2
(
t
)
−
s
i
n
θ
m
(
t
)
a
1
(
t
)
.
(
5.25
)
I
f
there are onl
y
the electron neutrinos at
t
=
t
0
,
then we have
a
e
(
t
0
)
=
1
a
nd
a
μ
(
t
0
)
= 0, leading to
a
1
(
t
0
)
=cos
θ
m
(
t
0
)
an
d
a
2
(
t
0
)
=sin
θ
m
(
t
0
)
. These
i
nitial conditions allow us to work out the survival probability o
f
electron
n
eutrinos as follows
:
P
(
ν
e
→
ν
e
)=
1
2
[
1+cos2θ
m
(
t
)
cos 2
θ
m
(
t
0
)]
+
1
2
s
in
2
θ
m
(
t
)
sin
2
θ
m
(
t
0
)
co
s
t
t
0
Δ
˜
m
2
(
t
)
2
E
d
t
.
(
5.26
)
If neutrinos are produced in a very dense medium, where
n
e
(
t
0
)
→
∞
i
sa
g
oo
d
approximation, t
h
einitia
l
neutrino mixin
g
an
gl
es
h
ou
ld be
θ
m
(
t
0
)
=
π
/
2. Because the detection of neutrinos is usuall
y
performed b
y
means of a
n
u
nderground detector below the Earth’s surface, the final neutrino mixin
g
a
n
g
le should take its value in vacuum:
θ
m
(
t
)=
θ
.Furthermore,onl
y
the
probability averaged over the energy spectrum and detection time is relevant
f
or a realistic experiment. Hence one should drop the last term in Eq.
(
5.26
)
a
nd then arrive at the avera
g
ed probability
P
(
ν
e
→
ν
e
)
=
1
2
(1
−
cos
2
θ
)
=sin
2
θ
.
(
5.27
)
T
h
is resu
l
tcan
b
eintuitive
l
yun
d
erstoo
d
.T
h
ee
l
ectron neutrin
o
ν
e
i
sini
-
tially produced almost in the mass eigenstate ˜
ν
2
because of
θ
m
(
t
0
)=
π
/
2
a
rising from the high matter density.
A
s the matter density decreases in a
n
extremely slow way and the adiabatic condition is satisfied, the evolution o
f
the neutrino state will keep in the same mass ei
g
enstate.
S
o we end up wit
h
the mass eigenstate
ν
2
in vacuum, implying that the probability of finding i
t
a
st
h
ee
l
ectron neutrino is just sin
2
θ
. The key point here is the validity o
f
the ad
i
abat
i
cco
n
d
i
t
i
o
n
|
˙
θ
m
(
t
)
|
|
Δ
˜
m
2
|
/
(4
E
)
. Taking account of Eq.
(
5.18
)
together with the expression o
f
Δ
˜
m
2
given below Eq.
(
5.24
)
,weobtai
n
˙
θ
m
(
t
)
=
Δ
m
2
si
n
2
θ
2(
Δ
˜
m
2
)
2
˙
A
(
t
)=
√
2
G
F
Δ
m
2
E
si
n2
θ
(
Δ
˜
m
2
)
2
˙
n
e
(
t
)
.
(
5.28
)
1
68 5 Phenomenology o
f
Neutrino
O
scillations
The adiabatic condition can then be written as
(
Kuo and Pantaleone, 1989a
)
√
2
G
F
d
n
e
(
t
)
d
t
4
π
2
λ
λ
3
m
s
in
2
θ
,
(
5.29
)
whe
r
e
λ
m
≡
4
π
E
/
Δ
˜
m
2
a
n
d
λ
≡
4
π
E
/
Δ
m
2
are t
h
e neutrino osci
ll
ation
l
engt
h
s
i
n matter and in vacuum, respectivel
y
. Note that the mass-squared di
ff
erenc
e
Δ
˜
m
2
r
eaches its minimum at the resonant
p
oin
t
A
=
Δ
m
2
cos 2θ
,
at whic
h
t
h
e osci
ll
ation
l
engt
h
λ
m
b
ecomes maxima
l
,sot
h
ea
d
ia
b
atic con
d
ition mig
h
t
probabl
y
be violated at this point.
O
n the resonance one ma
y
de
fi
ne the
a
diabaticity parameter
(
Kuo and Pantaleone, 1989a
)
γ
≡
1
4
E
Δ
˜
m
2
˙
θ
m
(
t
)
t
=
t
r
=
Δ
m
2
s
in
2
2
θ
2
E
cos 2
θ
1
n
e
(
t
)
d
n
e
(
t
)
d
t
−
1
t
=
t
r
,
(
5.30
)
w
h
ere
t
r
is t
h
e resonant point; i.e.,
A
(
t
r
)
=
Δ
m
2
cos
2
θ
.T
h
ea
d
ia
b
atic con
d
i
-
tion given in Eq.
(
5.29
)
can now be expressed as
γ
1. W
h
en t
h
ea
d
ia
b
aticit
y
p
arame
t
e
r
γ
is of
O
(
1
)
, the off-diagonal matrix elements in Eq.
(
5.23
)
will no
l
onger
b
eneg
l
igi
bl
e. In t
h
is non-a
d
ia
b
atic case a transition
b
etween t
h
ein
-
s
tantaneous mass e
ig
enstate
s
|
˜
ν
1
and ˜
ν
2
m
ust ta
k
ep
l
ace, so t
h
e osci
ll
ation
probabilities will inevitably receive very significant corrections
.
5
.1.
3
Non-adiabatic Neutrino
O
scillations in Matte
r
A
s we have shown above, the electron neutrino
ν
e
p
ro
d
uce
d
in an extreme
ly
dense medium stays in the instantaneous mass eigenstate ˜
ν
2
.
I
t
r
e
m
a
in
s
i
n
this state if the adiabatic condition
γ
1 is fulfilled. If this condition i
s
violated, however, the neutrino state a
f
ter crossin
g
the resonant point should
be a linear combination of ˜
ν
1
and ˜
ν
2
.In
suc
h
acaseo
n
e
h
as to dea
l
w
i
th
the transition between the instantaneous mass eigenstates. This is typicall
y
one of the level-crossing problems
(
Landau, 1932; St¨uckelberg, 1932; Zener
,
1
932
)
and has been extensively discussed in connection with solar neutrin
o
oscillations
(
Bethe, 1986; Barger et a
l.
,
1986
;
Parke
,
1986
;
Rosen and Gelb
,
1
986; Haxton, 1986, 1987; N¨otzo
ld
, 1987; Tos
h
ev, 1987; Kim
et al
.
,
1987; Dar
et al
.
,
1987; Petcov, 1988; Kuo and Pantaleone, 1989a, 1989b; Bru
gg
en
e
tal.
,
1
995; Friedland, 2001; Kachelrieß and Tom`as, 2001
).
Here let us consider a monotonically decreasin
g
matter density pro
fi
le
.
The electron neutrinos are
p
roduced a
t
t
=
t
0
i
n the highest density region,
a
n
d
t
h
en t
h
ey pass t
h
roug
h
t
h
e resonant point aroun
d
t
=
t
r
a
n
dco
m
e
to the vacuu
m
at
t
=
t
f
.
In a region far above
(
t
=
t
−
t
r
)
or belo
w
(
t
=
t
+
t
r
)
the resonant point, the adiabatic approximation should be
valid. In between these two adiabatic regions one may introduce the followin
g
transition probability between the instantaneous mass eigenstates
(
Kuo and
P
antaleone, 1989a, 1989b
):
P
c
P
P
≡
˜
ν
1
(
t
+
)
|
˜
ν
2
(
t
−
)
2
.
(
5.31
)
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
69
The survival probabilities
f
or the ei
g
enstate
s
|
˜
ν
i
af
ter crossin
g
the resonant
re
gi
on are 1
−
P
c
P
P
.
A
s a result, one obtains the avera
g
ed survival probability
of electron neutrinos
(
Parke, 1986
)
:
P
(
ν
e
→
ν
e
)
=
cos
2
θ
m
(
t
0
)
si
n
2
θ
m
(
t
0
)
1
−
P
c
PP
P
c
PP
P
c
P
P
1
−
P
c
P
P
cos
2
θ
m
(
t
f
)
si
n
2
θ
m
(
t
f
)
=
1
2
1
+
(
1
−
2
P
c
PP
)
cos
2
θ
m
(
t
0
)
cos
2
θ
m
(
t
f
)
.
(
5.32
)
This result can reproduce Eq.
(
5.27
)
in the adiabatic limit with
P
c
PP
=
0
,
i
f
the initial and final neutrino mixing angles are identified a
s
θ
m
(
t
0
)
=
π
/
2
a
n
d
θ
m
(
t
f
)
=
θ
.
Note that we have used the classical probabilities
f
or
fi
ndin
g
the electron neutrinos in the mass eigenstates, or vice versa, in the adiabati
c
re
g
ions. The reason is simply that a
fi
nite chan
g
eo
f
the matter density occurs
i
n the distance much lon
g
er than the neutrino oscillation len
g
th
.
I
n order to calculate the probabilit
y
P
c
PP
,
one may solve the coupled first-
order differential equations in Eq.
(
5.14
)
with a given time-dependent matter
d
ensity profile. The strate
g
y is to convert the equation array into one second
-
order differential equation of the amplitude
a
e
(
t
)
. For some special matte
r
d
ensit
y
pro
fi
les, such as the exponential and linear
f
unctions, the exact solu-
tions can be obtained
(
Kuo and Pantaleone, 1989b
)
. Then one should expand
t
h
eso
l
utions aroun
d
t
h
epro
d
uction an
dd
etection points, an
d
average t
h
em
over the phase factor.
A
different method based on the inte
g
ration in th
e
c
omplex-time plane was pointed out by Lev Landau, who suggested tha
t
one could find the approximate solutions to such differential equations in
the quasi-classical limit
(
Landau, 1932; Dykhne, 1962; Landau and Lifshitz,
1
977
)
. Note that the level-crossing point should be at
Δ
˜
m
2
=0
,
where th
e
mass eigenvalues ˜
m
2
1
and ˜
m
2
2
a
reexactlyequal.Thisispossibleonlyifth
e
f
ollowin
g
condition is satis
fi
ed:
A
(
t
)
=
Δ
m
2
(
cos
2
θ
±
i
s
i
n
2
θ
)=
Δ
m
2
e
±
2i
θ
.
(
5.33
)
T
h
e time varia
bl
es correspon
d
ingtot
h
ese two possi
b
i
l
ities are
d
enote
d
as
t
+
c
a
n
d
t
−
c
, which are located respectivel
y
in the upper and lower hal
f
-planes o
f
the complex time. Eq.
(
5.33
)
indicates that the levels never cross for the real
time, and the level-crossin
g
problem in question is de
fi
nitely o
f
the quantum
n
ature.
A
fter extendin
g
the Hamiltonian to the complex-time plane analyt-
i
cally, one may choose a continuous path in the upper or lower half-plane.
This path, which connects the real-time points
(
t
−
,
t
+
)
and circles around
the sin
g
ular poin
t
t
+
c
o
r
t
−
c
,
can be chosen
f
ar awa
yf
ro
m
t
±
c
suc
h
t
h
at t
h
e
a
diabatic condition is always valid along it. It has been generally proved that
t
h
ea
d
ia
b
atic evo
l
ution a
l
on
g
t
h
epat
h
in t
h
e comp
l
ex-time p
l
ane
d
etermines
the non-adiabatic transition probability along the real-time axis
(
Hwang and
P
echukas, 1977
)
. For non-adiabatic neutrino oscillations in matter, one ha
s
(
Kuo and Pantanleone, 1989a
)
1
70 5 Phenomenology o
f
Neutrino
O
scillations
P
c
PP
=
ex
p
2
−
Im
t
±
c
t
r
Δ
˜
m
2
(
t
)
E
d
t
6
,
(
5.34
)
whe
r
e
t
+
c
a
n
d
t
−
c
c
orrespond to the increasin
g
and decreasin
g
density pro
fi
les
,
respectivel
y
. Given a linear matter densit
y
profile, the well-known Landau
-
Zener formul
a
P
LZ
c
PP
=
e
x
p
{
−
π
γ/
ππ
2
}
wit
h
γ
b
eing t
h
ea
d
ia
b
aticity parameter
d
efinedinEq.
(
5.30
)
can be obtained
(
Parke, 1986
).
T
o explain the complex integration in Eq.
(
5.34
)
, let us examine the ana
-
l
ytical properties of the integrand in the neighborhood of the singular points
t
±
c
.Heretherelevant
f
unction is the e
ff
ective mass-squared di
ff
erenc
e
Δ
˜
m
2
(
t
)
given below Eq.
(
5.24
)
. Redefining Δ
˜
m
2
(
t
)
≡
f
(
t
)
,wehav
e
f
(
t
)
=
[
A
(
t
)
−
Δ
m
2
cos 2
θ
]
2
+(
Δ
m
2
s
in 2
θ
)
2
.
(
5.35
)
Combining Eq.
(
5.35
)
with Eq.
(
5.33
)
, we find tha
t
f
(
t
±
c
)
=0holds.Bu
t
the first derivative o
f
f
(
t
)
is in general nonzero at the crossing point; i.e.,
˙
f
(
t
±
c
)
= 0. With the help of the Taylor expansion of
f
(
t
)
with respect t
o
t
±
c
,
one can obtain
(
Davis and Pechukas, 1976
)
Δ
˜
m
2
(
t
)=
˙
f
(
t
±
c
)
t
−
t
±
c
1+
β
(
t
)(
t
−
t
±
c
)
,
(
5.36
)
where
β
(
t
)
is analytic and single-valued aroun
d
t
±
c
. Because of the s
q
uare-
root function, it is straightforward to observe tha
t
t
±
c
a
re actua
ll
yt
h
e
b
ranc
h
points o
f
Δ
˜
m
2
(
t
)
. Along the circle around
t
±
c
,
the
f
unction
Δ
˜
m
2
(
t
)
will change
its sign, implying an exchange of the mass eigenvalues ˜
m
2
1
and ˜
m
2
2
.Tobe
exp
l
icit, we consi
d
er a circ
l
earoun
d
t
±
c
a
n
d
parametrize it a
s
t
−
t
±
c
=
ρ
e
i
φ
.
It then follows from Eq.
(
5.36
)
tha
t
Δ
˜
m
2
(
ρ
,
φ
)
=
−
Δ
˜
m
2
(
ρ
,
φ
+2
π
)
. Hence
the square-root branch points are essential for the level crossing, and th
e
i
nte
g
ration path alon
g
the branch cut
f
ro
m
t
r
to
t
+
c
(
or
t
−
c
)
should be chose
n
to keep the imaginary part in Eq.
(
5.34
)
positive.
T
he matter density profile of the Sun, which can significantly affect th
e
behaviors o
f
solar neutrino oscillations, is expected to be exponential:
n
e
(
t
)=
n
0
e
xp
(
−
t/r
s
)
with
0
.
2
3 R
<r<R
,
wher
e
n
0
i
s a normalization constant,
R
denotes the radius of the Sun
,
an
d
r
s
=
0
.
09
2
R
re
p
resents the scal
e
h
eight
(
Pizzochero, 1987; Bahcall, 1989
)
. In this case the zero points of
f
(
t
)
c
an be explicitly determined from Eq.
(
5.33
)
:
t
±
c
=
−
r
s
l
n
Δ
m
2
2
√
2
G
F
E
n
0
∓
2i
r
s
(
θ
+
k
π
)
,
(
5.37
)
where
k
is an arbitrary integer
(
Pizzochero, 1987
)
2
. Then the non-adiabati
c
transition pro
b
a
b
i
l
ity is given
by
2
If
there exist several level-crossin
g
points in the complex-time plane, one should
choose the one nearest to the real-time axis (Landau and Lifshitz, 1977).
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
7
1
ln
P
c
PP
=
−
1
E
Im
t
r
+
2
i
θ
r
s
t
r
s
[
A
(
t
)
−
Δ
m
2
cos 2
θ
]
2
+(
Δ
m
2
si
n2
θ
)
2
d
t
.
(
5.38
)
The above integration can be simplified by changing the variable in this way:
t
=
t
r
+i
y
a
n
d
x
=
−
Δ
m
2
e
xp
(
−
i
y
/
r
s
)
/
(2
E
)
. The final result is found to b
e
(
Pizzochero, 1987; Kuo and Pantaleone, 1989a
)
P
c
PP
=e
x
p
−
π
2
γ
1
−
t
an
2
θ
,
(
5.39
)
where
γ
=
r
s
Δ
m
2
s
in
2
2
θ/
(
2
E
cos
2
θ
)
is derived from Eq.
(
5.30
)
by taking
the exponential density pro
fi
le
n
e
(
t
)=
n
0
e
xp
(
−
t
/r
s
)
. Note that Eq.
(
5.39
)
r
educes to t
h
esta
n
da
r
d
L
a
n
dau
-Z
e
n
e
r
fo
rm
u
l
a
in
t
h
e
θ
→
0
limit, because th
e
c
orresponding resonance is so narrow that the linear and exponential profile
s
a
re essentially identical. The analytical results o
f
the transition probability
P
c
PP
for a variety of interestin
g
matter density profiles can be found in the
l
iterature
(
Kuo and Pantaleone, 1989a
).
5.1.4 The
3
×
3
Neutrino Mixin
g
Matrix in Matte
r
N
ow we concentrate on terrestrial matter effects on lon
g
-baseline neutrin
o
oscillations. For simplicity, we assume a constant matter density profile
(
i.e.
,
n
e
= constant
)
. This assumption is reasonably good for most of the ongoin
g
or proposed lon
g
-baseline neutrino oscillation experiments. Note, however,
that the changes o
f
n
e
s
hould be carefully taken into account when a neutrin
o
beam travels through the Earth’s core
(
Mocioiu and Shrock, 2000; Akhmedov
et al.
,
2008; Liao, 2008
).
In matter we define the effective neutrino masses as ˜
m
i
(
fo
r
i
=1
,
2
,
3)
and
the e
ff
ective
3
×
3neutr
i
no m
i
x
i
n
g
matr
i
xas
˜
V
.Th
e effect
i
ve
H
a
mil
to
ni
a
n
s
responsible for the propagations of neutrinos in vacuum and in matter ar
e
H
v
=
1
2
E
V
⎛
⎝
⎛⎛
m
2
1
0
0
0
m
2
2
0
00
m
2
3
⎞
⎠
⎞⎞
V
†
,
H
m
=
1
2
E
˜
V
⎛
⎝
⎛⎛
˜
m
2
1
00
0˜
m
2
2
0
00˜
m
2
3
⎞
⎠
⎞⎞
˜
V
†
=
1
2
E
V
⎛
⎝
⎛⎛
m
2
1
00
0
m
2
2
0
00
m
2
3
⎞
⎠
⎞
⎞
V
†
+
⎛
⎝
⎛⎛
a
00
0
0
0
000
⎞
⎠
⎞⎞
.
(
5.40
)
w
h
e
r
e
E
i
s the neutrino beam ener
g
y, an
d
a
=
√
2
G
F
n
e
sta
n
ds fo
r
t
h
ete
r-
restrial matter effect. Analogous to Eq.
(
5.7
)
, the probabilities of neutrino
osci
ll
ations in matter are
g
iven
by
˜
P
(
ν
α
→
ν
β
ν
)=
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
,
(
5.41
)
1
72 5 Phenomenology o
f
Neutrino
O
scillations
whe
r
e
Δ
˜
m
2
ji
≡
˜
m
2
j
−
˜
m
2
i
are the e
ff
ective neutrino mass-squared di
ff
erences,
a
nd
˜
J
i
s the effective Jarlskog parameter defined throug
h
Im
(
˜
V
αi
VV
˜
V
βj
VV
˜
V
∗
αj
VV
˜
V
∗
βi
VV
)
=
˜
J
γ
α
βγ
k
ij
k
.
(
5.42
)
To reveal the behaviors o
f
neutrino oscillations in matter, one needs to work
out the relations between ( ˜
m
2
i
,
˜
V,
˜
J
)
and
(
m
2
i
,V,
J
)
. This can be done by
c
omparin
gb
etwee
n
H
m
a
n
d
H
v
i
nEq.
(
5.40
)
. A straightforward calculatio
n
yields
(
Barger
e
tal
.
,
1980; Zaglauer and Schwarzer, 1988; Xing, 2000
)
˜
m
2
1
=
m
2
1
+
1
3
x
−
1
3
x
2
−
3
y
z
+
3
(1
−
z
2
)
,
˜
m
2
2
=
m
2
1
+
1
3
x
−
1
3
x
2
−
3
y
z
−
3
(1
−
z
2
)
,
˜
m
2
3
=
m
2
1
+
1
3
x
+
2
3
z
x
2
−
3
y
,
(
5.43
)
whe
r
e
x
=
Δ
m
2
2
1
+
Δ
m
2
3
1
+
A
,
y
=
Δ
m
2
21
Δ
m
2
31
+
A
Δ
m
2
21
1
−|
V
e
VV
2
|
2
+
Δ
m
2
31
1
−|
V
e
V
V
3
|
2
,
z
=cos
1
3
a
r
ccos
2
x
3
−
9
xy
+
2
7
A
Δ
m
2
21
Δ
m
2
3
1
|
V
e
VV
1
|
2
2(
x
2
−
3
y
)
3
/
2
(
5.44
)
wit
h
A
=
2
a
E
=2
√
2
G
F
E
n
e
an
d
Δ
m
2
ji
≡
m
2
j
−
m
2
i
.
It is eas
y
to observe the
f
ollowing sum rule:
3
i
=
1
˜
m
2
i
=
3
i
=
1
m
2
i
+
A
.
(
5.45
)
O
n the other hand, the matrix elements o
f
˜
V
a
n
d
V
a
r
e
r
elated to each othe
r
through
(
Xing, 2000
)
˜
V
αi
VV
=
N
i
N
N
D
i
V
αi
VV
+
A
D
i
V
ei
VV
˜
m
2
i
−
m
2
j
V
∗
ek
VV
V
αk
V
V
+
˜
m
2
i
−
m
2
k
V
∗
ej
VV
V
αj
VV
,
(
5.46
)
where the
G
reek subscript
α
r
uns over
(
e
,
μ
,
τ
)
, the Latin subscripts
i
=
j
=
k
run over
(1
,
2
,
3
)
,and
N
i
NN
=
˜
m
2
i
−
m
2
j
˜
m
2
i
−
m
2
k
−
A
˜
m
2
i
−
m
2
j
|
V
ek
V
V
|
2
+
˜
m
2
i
−
m
2
k
|
V
ej
VV
|
2
,
D
2
i
=
N
2
i
N
N
+
A
2
|
V
ei
VV
|
2
˜
m
2
i
−
m
2
j
2
|
V
ek
VV
|
2
+
˜
m
2
i
−
m
2
k
2
|
V
ej
VV
|
2
.
(
5.47
)
Of
course,
A
= 0 immediately leads to ˜
m
2
i
=
m
2
i
a
n
d
˜
V
αi
VV
=
V
αi
VV
.
T
he above
exact
f
ormulas clearly show how the neutrino mixin
g
matrix and neutrino
m
asses get corrected in the presence of terrestrial matter effects
.
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
73
Th
ere
l
ations
h
ip
b
etwee
n
˜
J
a
n
d
J
c
an be established b
y
means o
f
amuch
m
ore interesting sum rule
(
Xing, 2001a, 2001b
):
3
i=
1
˜
m
2
i
˜
V
αi
VV
˜
V
∗
βi
VV
=
3
i
=1
m
2
i
V
αi
VV
V
∗
βi
VV
,
(
5.48
)
which can be derived from the equality between the off-diagonal elements
of
H
m
a
n
d
H
v
in Eq.
(
5.40
)
.For
α
=
e
,
μ
a
n
d
τ
,t
h
ree equa
l
ities can
be
obtained from Eq.
(
5.48
)
and their product is also an equality. Therefore
,
3
i
=
1
3
j
=
1
3
k
=1
˜
m
2
i
˜
m
2
j
˜
m
2
k
Im
(
˜
V
ei
VV
˜
V
μj
VV
˜
V
τk
VV
˜
V
∗
ek
VV
˜
V
∗
μi
VV
˜
V
∗
τj
VV
)
=
3
i
=
1
3
j
=
1
3
k
=
1
m
2
i
m
2
j
m
2
k
Im
(
V
ei
VV
V
μj
VV
V
τk
VV
V
∗
ek
VV
V
∗
μi
VV
V
∗
τj
VV
)
.
(
5.49
)
This relation can be simpli
fi
ed by usin
g
the de
fi
nitions o
f
J
a
n
d
˜
J
to
g
et
h
er
with the unitarit
y
conditions o
f
V
a
n
d
˜
V
.A
fter a strai
g
htforward al
g
ebrai
c
exercise, we are left with
(
Xing, 2001a, 2001b
)
˜
J
Δ
˜
m
2
21
Δ
˜
m
2
31
Δ
˜
m
2
32
=
J
Δ
m
2
21
Δ
m
2
31
Δ
m
2
32
.
(
5.50
)
The same result can also be obtained in different ways
(
Naumov, 1992; Har
-
rison and Scott, 2000
)
. We observe that
˜
J
=
J
h
olds if
A
=
0
,
an
d
˜
J
=
0
ho
l
ds
i
f
J
=
0. In other words,
C
P or T violation in neutrino oscillations i
n
m
atter is
g
overned by
J
v
ia
˜
J
.
I
n many cases it is more convenient to parametrize
˜
V
i
n
te
rm
so
f
t
h
e
effective neutrino mixin
g
an
g
les and CP-violatin
g
phases.
A
nalo
g
ous to th
e
s
tandard
p
arametrization o
f
V
in vacuum, as given in Eq.
(
3.107
)
,
˜
V
=
⎛
⎝
⎛⎛
10
0
0˜
23
˜
s
23
0
−
˜
s
23
˜
c
23
⎞
⎠
⎞⎞
P
⎛
⎝
⎛⎛
˜
c
13
0˜
13
0
1
0
−
˜
s
13
0˜
1
3
⎞
⎠
⎞⎞
P
†
⎛
⎝
⎛⎛
˜
c
12
˜
s
12
0
−
˜
s
12
˜
c
12
0
001
⎞
⎠
⎞⎞
P
=
⎛
⎜
⎛
⎛
⎝
⎜⎜
˜
c
12
˜
c
13
˜
s
12
˜
c
13
˜
s
13
e
−
i
˜
δ
−
˜
s
12
˜
c
23
−
˜
c
12
˜
s
23
˜
s
13
e
i
˜
δ
˜
c
12
˜
c
23
−
˜
s
12
˜
s
23
˜
s
13
e
i
˜
δ
˜
s
23
˜
c
13
˜
s
12
˜
s
23
−
˜
c
12
˜
c
23
˜
s
13
e
i
˜
δ
−
˜
c
12
˜
s
23
−
˜
s
12
˜
c
23
˜
s
13
e
i
˜
δ
˜
c
23
˜
c
13
⎞
⎟
⎞⎞
⎠
⎟⎟
P
,
(
5.51
)
where ˜
ij
≡
cos
˜
θ
ij
and ˜
ij
≡
sin
˜
θ
ij
(
fo
r
i
j
=12
,
13
,
2
3
)
together with the
p
h
ase matrices
P
=Dia
g
{
1
,
1
,
e
i
˜
δ
}
a
n
d
P
=Dia
g
{
e
i
ρ
,
e
i
σ
,
1
}
.
S
ince
P
itse
l
f
pla
y
s no role in neutrino oscillations, its two Ma
j
orana phases
ρ
a
nd σ
a
r
e
c
ompletely insensitive to matter effects. Substituting Eqs.
(
3.107
)
and
(
5.51
)
i
nto Eqs.
(
5.46
)
and
(
5.47
)
, one may get the relations between
(
˜
θ
12
,
˜
θ
1
3
,
˜
θ
2
3
,
˜
δ
)
a
nd
(
θ
12
,
θ
13
,
θ
23
,δ
)
. But the exact analytical results are not simple
(
Zaglauer
a
nd Schwarzer, 1988; Xing, 2001c; Freund, 2001
)
. For exampl
e
3
,
3
H
ere we only present the next-to-leadin
g
order expression
f
or ta
n
˜
θ
23
/
tan
θ
23
,
because the exact result is too complicated to be instructive
.
1
74 5 Phenomenology o
f
Neutrino
O
scillations
t
an
˜
θ
12
t
an θ
12
=
D
1
D
2
·
N
2
NN
+
A
˜
m
2
2
−
m
2
1
s
2
13
+
˜
m
2
2
−
m
2
3
c
2
1
2
c
2
13
N
1
NN
+
A
[( ˜
m
2
1
−
m
2
2
)
s
2
13
+(˜
m
2
1
−
m
2
3
)
s
2
12
c
2
13
]
,
ta
n
˜
θ
2
3
ta
n
θ
23
=
1+
A
Δ
m
2
2
1
s
1
2
c
12
s
13
cos
δ
s
23
c
23
{
N
3
NN
−
A
[( ˜
m
2
3
−
m
2
1
)
s
2
12
+(˜
m
2
3
−
m
2
2
)
c
2
12
]
s
2
13
}
,
s
i
n
˜
θ
13
s
i
n
θ
1
3
=
N
3
NN
D
3
+
A
D
3
˜
m
2
3
−
m
2
1
s
2
12
+
˜
m
2
3
−
m
2
2
c
2
12
c
2
13
,
(
5.52
)
whe
r
e
D
i
a
n
d
N
i
NN
can be read off from Eq.
(
5.47
)
. In addition, one may relate
sin
˜
δ
to si
n
δ
b
y using Eq.
(
5.50
)
together wit
h
˜
J
=˜
c
1
2
˜
s
12
˜
c
2
1
3
˜
s
13
˜
c
23
˜
s
23
sin
˜
δ
a
nd the similar ex
p
ression o
f
J
.
Note, however, that both the
(
2,3
)
rotation
m
atrix in Eq.
(
5.51
)
and the phase matri
x
P
co
mm
ute w
i
th the
m
atte
r
-
i
nduced matrix in Eq.
(
5.40
)
. This observation allows one to derive an exact
relationship between
(
˜
θ
23
,
˜
δ
)
and
(
θ
23
,δ
)(
Toshev, 1991
)
:
s
in 2
˜
θ
2
3
s
in
˜
δ
=
s
in 2
θ
2
3
s
i
n
δ.
(
5.53
)
So
˜
δ
≈
δ h
o
l
ds as a
r
esu
l
tof
˜
θ
23
≈
θ
23
i
n the leadin
g
-order approximation
.
H
ence both θ
23
and
δ
are essentially insensitive to matter effects
.
T
he results obtained above are only valid
f
or neutrinos propa
g
atin
g
in
m
atter. For the case o
f
antineutrinos propa
g
atin
g
in matter, the correspond-
i
ng expressions can be easily obtained through the replacements
δ
=
⇒−
==
δ
(
o
r
J
=
⇒−J
==
)
and
A
=
⇒−
==
A
.
S
uch
f
ormulas should be ver
y
use
f
ul
f
or th
e
purpose of recastin
g
the fundamental parameters of lepton flavor mixin
g
from
the matter-corrected ones, which can be extracted from a variety of long- an
d
m
edium-baseline neutrino oscillation experiments in the near
f
uture
.
5.1.5 Lepton
i
cUn
i
tar
i
ty Tr
i
angles
i
n Matte
r
We proceed to explore some more properties of lepton flavor mixing an
d
C
P violation in matter. For simplicity, we are a
g
ain subject to a constant
m
atter densit
y
pro
fi
le. The e
ff
ective Hamiltonian
s
H
v
a
n
d
H
m
in Eq.
(
5.40
)
a
re related to each other via
H
m
=
H
v
+
⎛
⎝
⎛⎛
a
00
00
0
000
⎞
⎠
⎞⎞
,
(
5.54
)
w
h
e
r
e
a
=
A/
(2
E
)=
√
2
G
F
n
e
.
This equalit
y
actuall
y
allows us to obtain
a
s
um rule which is more general than the one given in Eq.
(
5.48
)
:
3
i
=
1
˜
m
2
i
˜
V
αi
VV
˜
V
∗
βi
VV
=
3
i
=
1
m
2
i
V
αi
VV
V
∗
βi
VV
+
Aδ
αe
δ
e
β
.
(
5.55
)
O
n the other hand
,
5
.1 Neutrino
O
scillations and Matter E
ff
ects 1
75
H
m
H
†
m
=
H
v
H
†
v
+
H
v
⎛
⎝
⎛⎛
a
00
00
0
000
⎞
⎠
⎞⎞
+
⎛
⎝
⎛⎛
a
00
0
00
000
⎞
⎠
⎞⎞
H
†
v
+
⎛
⎝
⎛⎛
a
2
00
000
000
⎞
⎠
⎞⎞
,
(
5.56
)
which leads us to a new sum rule
(
Xing and Zhang, 2005
)
3
i
=
1
˜
m
4
i
˜
V
αi
VV
˜
V
∗
βi
VV
=
3
i=1
m
4
i
+
A
m
2
i
(
δ
α
e
+
δ
eβ
)
V
αi
VV
V
∗
βi
VV
+
A
2
δ
α
e
δ
e
β
.
(
5.57
)
E
qs.
(
5.55
)
and
(
5.57
)
, together with the unitarity conditions
3
i
=
1
˜
V
αi
VV
˜
V
∗
βi
VV
=
3
i
=1
V
αi
VV
V
∗
βi
VV
=
δ
αβ
,
(
5.58
)
c
onstitute a full set of linear e
q
uations of
˜
V
αi
VV
˜
V
∗
βi
VV
(
for
i
=1
,
2
,
3
)
.Onema
y
m
ake use of these e
q
uations to derive the concrete ex
p
ressions of
˜
V
αi
VV
˜
V
∗
βi
VV
in
terms of m
2
i
,˜
m
2
i
and
V
αi
VV
V
∗
βi
VV
.
Let us first calculate the moduli of
˜
V
αi
VV
by using Eqs.
(
5.55
)
,
(
5.57
)
an
d
(
5.58
)
. Those equations are rewritten, in th
e
α
=
β
c
ase, as follows:
˜
X
⎛
⎝
⎛
⎛
|
˜
V
α
V
V
1
|
2
|
˜
V
α
V
V
2
|
2
|
˜
V
α
V
V
3
|
2
⎞
⎠
⎞
⎞
=(
X
+
2
A
δ
αe
Y
)
⎛
⎝
⎛
⎛
|
V
α
VV
1
|
2
|
V
α
VV
2
|
2
|
V
α
VV
3
|
2
⎞
⎠
⎞
⎞
+
⎛
⎝
⎛
⎛
0
A
A
2
⎞
⎠
⎞
⎞
δ
αe
,
(
5.59
)
whe
r
e
X
=
⎛
⎝
⎛
⎛
1
11
m
2
1
m
2
2
m
2
3
m
4
1
m
4
2
m
4
3
⎞
⎠
⎞
⎞
,Y
=
⎛
⎝
⎛
⎛
000
000
m
2
1
m
2
2
m
2
3
⎞
⎠
⎞
⎞
,
˜
X
=
⎛
⎝
⎛⎛
1
11
˜
m
2
1
˜
m
2
2
˜
m
2
3
˜
m
4
1
˜
m
4
2
˜
m
4
3
⎞
⎠
⎞⎞
.
(
5.60
)
With the help of Eq.
(
5.45
)
, we solve Eq.
(
5.59
)
and obtain the following
exact results
(
Xing and Zhang, 2005
):
|
˜
V
e
VV
1
|
2
=
ˆ
Δ
21
ˆ
Δ
31
˜
Δ
21
˜
Δ
31
|
V
e
VV
1
|
2
+
ˆ
Δ
11
ˆ
Δ
31
˜
Δ
21
˜
Δ
31
|
V
e
VV
2
|
2
+
ˆ
Δ
11
ˆ
Δ
21
˜
Δ
21
˜
Δ
31
|
V
e
VV
3
|
2
,
|
˜
V
e
VV
2
|
2
=
ˆ
Δ
22
ˆ
Δ
3
2
˜
Δ
12
˜
Δ
32
|
V
e
VV
1
|
2
+
ˆ
Δ
1
2
ˆ
Δ
32
˜
Δ
12
˜
Δ
32
|
V
e
VV
2
|
2
+
ˆ
Δ
12
ˆ
Δ
22
˜
Δ
12
˜
Δ
32
|
V
e
VV
3
|
2
,
|
˜
V
e
VV
3
|
2
=
ˆ
Δ
23
ˆ
Δ
33
˜
Δ
13
˜
Δ
23
|
V
e
VV
1
|
2
+
ˆ
Δ
13
ˆ
Δ
33
˜
Δ
13
˜
Δ
23
|
V
e
VV
2
|
2
+
ˆ
Δ
13
ˆ
Δ
23
˜
Δ
13
˜
Δ
23
|
V
e
VV
3
|
2
;
(
5.61
)
a
nd