Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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1
96 5 Phenomenology o
f
Neutrino
O
scillations
c
oherent scatterin
g
e
ff
ects, which may dominate over the usual M
S
W matte
r
effects on neutrino flavor conversions;
(
2
)
there is no distinction between th
e
test and background neutrinos, since both of them contribute to the total flux
of supernova neutrinos which will be experimentall
y
detected.
A
reasonabl
e
wa
y
out is to consider the number densities of neutrinos for different flavors i
n
a
thermal ensemble. Such a picture is particularly suitable for the description
of neutrinos in the early Universe
(
Dolgov, 2002
).
For unmixed neutrinos and antineutrinos
,
it is sufficient to consider the
evolution of their occupation numbers
f
p
ff
a
n
d
f
p
fo
r
eac
hm
o
m
e
n
tu
mm
ode
p
.A
s shown in Chapter 2, the occupation number of a scalar boson
φ
i
s
gi
ven
by the ensemble average of the corresponding occupation number operator
a
†
p
a
p
;
i.e.,
f
p
f
f
=
a
†
p
a
p
,w
h
ere
a
†
p
a
n
d
a
p
a
r
ethec
r
eat
i
o
n
a
n
da
nni
h
i
lat
i
on
operators respectivel
y
. For mixed neutrinos and antineutrinos, their
fl
avor
s
s
hould be taken into account. In an analogous way one may define the density
m
atrices
(
Sigl and Raffelt, 1993
)
b
†
j
(
p
)
b
i
(
q
)
=
(2
π
)
3
δ
3
(
p
−
q
)
ρ
p
ij
,
d
†
i
(
p
)
d
j
(
q
)
=
(2
π
)
3
δ
3
(
p
−
q
)
ρ
p
ij
,
(
5.130
)
where b
†
i
a
nd
b
i
(
o
r
d
†
i
an
d
d
i
)
denote the creation and annihilation operator
s
of
n
eut
rin
os
ν
i
(
or antineutrinos
ν
i
)
. The diagonal matrix elements of
ρ
p
a
r
e
j
ust the occupation number densities
f
i
p
f
f
for neutrinos ν
i
,
and the neutrin
o
n
um
b
er
d
ensity
n
ν
i
is
g
iven by the inte
g
ration o
f
f
i
p
ff
ove
r
the
m
o
m
e
n
tum
sp
ace. Likewise
f
or the case o
f
antineutrino
s
ν
i
. Note that the
f
amil
y
indice
s
on the left-hand sides of two identities in Eq.
(
5.130
)
are in the opposite
order, which ensures the trans
f
ormation in the
fl
avor space to be the sam
e
f
o
r
ρ
p
a
nd
ρ
p
.
This point can be understood by notin
g
tha
t
b
i
(
p
)
and
d
†
i
(
p
)
a
re present in the quantization of the neutrino fiel
d
ν
i
(
x
)
,onwhichtheflavo
r
transformation is acting. On the other hand, we remark that Eq.
(
5.130
)
i
s
d
efined for the neutrino mass ei
g
enstates, because the neutrino flavor states
h
ave no definite masses and a construction of the Fock space for them is
problematic
(
Giunti
et al.
,
1992
).
For free neutrinos
,
the time evolution of their creation and annihilation
operators is simp
l
e:
b
k
(
p
,
t
)=
b
k
(
p
,
0
)e
−
i
E
k
t
a
n
d
d
k
(
p
,
t
)=
d
k
(
p
,
0
)
e
−
i
E
k
t
,
where
E
k
=
|
p
|
2
+
m
2
k
i
s the neutrino energy. Hence the free Hamiltonian
operator can
b
e written a
s
ˆ
H
0
=
d
3
p
(
2
π
)
3
3
k
=
1
b
†
k
(
p
)
b
k
(
p
)
+
d
†
k
(
p
)
d
k
(
p
)
E
k
.
(
5.131
)
L
et us define the operators
(
Sigl and Raffelt, 1993
)
D
j
k
(
p
,t
)
≡
b
†
k
(
p
,t
)
b
j
(
p
,t
)
,
D
j
k
(
p
,
t
)
≡
d
†
j
(
p
,
t
)
d
k
(
p
,t
)
.
(
5.132
)
5
.3 Density Matrix Formu
l
ation 197
A
s a result, the densit
y
matrices can be expressed a
s
D
jk
(
p
,t
)
=
ρ
p
a
n
d
D
j
k
(
p
,t
)
=
ρ
p
, where Eq.
(
5.130
)
has been used and the factor
(
2
π
)
3
δ
3
(
0
)
h
as been consistentl
y
set to unit
y
. Now we look at the time evolution o
f
th
e
d
ensity matrices. The evolution of the operators defined in Eq.
(
5.132
)
i
s
d
etermine
db
yt
h
e Heisen
b
erg equations
˙
D
jk
(
p
,
t
)
=
i
ˆ
H
,
D
j
k
(
p
,
t
)
,
˙
D
jk
(
p
,t
)
=i
ˆ
H
,
D
jk
(
p
,t
)
,
(
5.133
)
whe
r
e
ˆ
H
=
ˆ
H
0
+
ˆ
H
i
n
t
is t
h
e tota
l
Hami
l
tonian operator. T
h
eevo
l
ution equa-
t
i
o
n
sof
ρ
p
a
n
d
ρ
p
c
an be obtained by takin
g
the expectation values on both
s
ides of Eq.
(
5.133
)
. These equations will be closed if their right-hand sides
ca
n
be
r
educed to
ρ
p
a
n
d
ρ
p
themselves. In the absence o
f
any interaction
s
(
i.e.
,
ˆ
H
=
ˆ
H
0
)
, we substitute the Hamiltonian operator in Eq.
(
5.131
)
int
o
E
q.
(
5.133
)
and then calculate its first identity as follows:
˙
D
jk
(
p
,t
)
=
i
d
3
q
(2
π
)
3
3
l
=1
E
l
b
†
l
(
q
)
b
l
(
q
)
,b
†
k
(
p
,t
)
b
j
(
p
,t
)
=i
E
k
−
E
j
E
D
jk
(
p
,t
)
,
(
5.134
)
where the anti-commutation relation
s
{
b
j
(
p
,
t
)
,b
†
k
(
q
,t
)
}
=
(
2
π
)
3
δ
3
(
p
−
q
)
a
n
d
{
b
j
(
p
,t
)
,b
k
(
q
,t
)
}
=
{
b
†
j
(
p
,
t
)
,b
†
k
(
q
,t
)
}
=
0
have been used.
G
iven the
r
elat
i
o
n
s
b
k
(
p
,
t
)
=
b
k
(
p
,
0
)
e
−
i
E
k
t
a
n
d
d
k
(
p
,t
)
=
d
k
(
p
,
0
)e
−
i
E
k
t
,
i
t
i
seas
y
t
o
obtain the result in Eq.
(
5.134
)
. This treatment is instructive for dealing with
t
h
ecasew
h
er
e
ˆ
H
in
t
is present. Ta
k
ing t
h
e expectation va
l
ues on
b
ot
h
si
d
e
s
of Eq.
(
5.134
)
, we get the equation of motion
˙
ρ
j
k
(
p
,t
)
=
i
E
k
−
E
j
E
ρ
j
k
(
p
,t
)
(
5.135
)
f
or the densit
y
matrices o
f
neutrinos. In a similar wa
y
we can obtain
˙
ρ
jk
(
p
,t
)
=
i
E
j
E
−
E
k
ρ
j
k
(
p
,t
)
(
5.136
)
f
or the densit
y
matrices of antineutrinos. Let us translate the above results
f
rom the mass basis into the flavor basis
,
whereitismoreconvenientt
o
accou
n
tfo
rn
eut
rin
o
in
te
r
act
i
o
n
s
.B
ecause of
ν
α
=
V
α
VV
1
ν
1
+
V
α
VV
2
ν
2
+
V
α
VV
3
ν
3
(
fo
r
α
=
e
,μ,τ
)
, the elements of the density matrice
s
ρ
p
(
t
)
an
d
ρ
p
(
t
)
tur
n
out to be
ρ
αβ
(
p
,t
)
=
j
k
V
αj
V
V
ρ
jk
(
p
,t
)
V
∗
βk
VV
,
ρ
αβ
(
p
,t
)
=
j
k
V
αj
V
V
ρ
j
k
(
p
,t
)
V
∗
βk
VV
.
(
5.137
)
In the flavor basis we make use of Eq.
(
5.135
)
to derive the equation of motio
n
f
or the density matrix elements ρ
α
β
(
p
,t
):
1
98 5 Phenomenology o
f
Neutrino
O
scillations
˙
ρ
α
β
=i
j
k
V
αj
VV
ρ
jk
E
k
V
∗
βk
VV
−
V
αj
VV
E
j
E
ρ
jk
V
∗
βk
VV
=i
j
k
l
γ
(
V
αj
VV
ρ
jl
V
∗
γl
VV
)(
V
γk
VV
E
k
V
∗
βk
VV
)
−
(
V
αj
VV
E
j
E
V
∗
γj
VV
)(
V
γl
VV
ρ
l
k
V
∗
βk
VV
)
=
i
γ
ρ
αγ
Ω
0
γβ
−
Ω
0
αγ
ρ
γβ
=
−
i
Ω
0
,
ρ
αβ
,
(
5.138
)
where
Ω
0
αβ
(
p
)
≡
|
p
|
2
δ
αβ
+
3
i
=
1
V
αi
VV
m
2
i
V
∗
βi
V
V
1
/
2
≈
Eδ
α
β
+(
H
v
)
αβ
(
5.139
)
i
nt
h
e approximation
|
p
|
≈
E
m
i
,
an
d
H
v
has been given in Eq.
(
5.40
).
The approximation made in Eq.
(
5.139
)
allows us to rewrite Eq.
(
5.138
)
a
s
5
˙
ρ
p
(
t
)=
−
i
H
v
,ρ
p
(
t
)
,
(
5.140
)
which is formally the same as that given in Eq.
(
5.100
)
. As far as antineutrino
s
a
re concerne
d
, one may ana
l
ogous
l
yo
b
tai
n
˙
ρ
p
(
t
)
=+i
H
v
,
ρ
p
(
t
)
.
(
5.141
)
Thanks to these equations of motion, a geometrical representation of neutrin
o
oscillations is also applicable. The difference between the density matrice
s
i
ntroduced in
S
ection 5.3.2 and discussed here is that the
y
work separatel
y
f
or the one-
p
article state and an ensemble of neutrinos. In the two-flavor
s
c
h
eme one may expan
d
ρ
p
,
ρ
p
a
n
d
H
v
i
n
te
rm
so
f
t
h
e
P
au
li m
at
ri
ces as
ρ
p
=
1
2
f
p
ff
+
P
p
·
σ
,
ρ
p
=
1
2
f
p
+
P
p
·
σ
,
(
5.142
)
a
n
d
H
v
=
1
2
Σ
+
ω
p
B
·
σ
.
(
5.143
)
Of course, neutrino flavor conversions can also be described by the evolution
o
f
the polarization vectors
P
p
(
t
)
an
d
P
p
(
t
)
. The above four equations yiel
d
˙
P
p
(
t
)
=+ω
p
B
×
P
p
(
t
)
,
˙
P
p
(
t
)=
−
ω
p
B
×
P
p
(
t
)
,
(
5.144
)
where
ω
p
≡
Δ
m
2
/
(
2
|
p
|
)
and
B
=(
sin 2
θ,
0
,
−
cos
2
θ
)
wit
h
θ
b
ein
g
the neu
-
trino mixin
g
an
g
le in vacuum. This result indicates that
P
p
(
t
)
an
d
P
p
(
t
)
rotate aroun
d
B at t
h
e same spee
db
ut in t
h
e opposite
d
irections.
5
N
ote t
h
at we
h
ave use
d
ρ
ij
(
p
,
t
)
an
d
ρ
α
β
(
p
,t
)
to denote the elements of th
e
density matrix
ρ
p
(
t
) in the mass and flavor bases, respectively
.
5
.3 Density Matrix Formu
l
ation 199
I
n the presence o
f
neutrino interactions, we take the expectation values
on both sides of Eq.
(
5.133
)
and then obtain
˙
ρ
p
(
t
)=
−
i
Ω
0
p
,ρ
p
(
t
)
+i
ˆ
H
i
n
t
(
t
)
,
D
p
(
t
)
,
(
5.145
)
a
nd a similar equation
f
or
ρ
p
(
t
)
. In the assumption that neutrinos are no
t
c
orrelated with the background, it is possible to decompose the expectatio
n
value into a medium part and a neutrino part. In this case Eq.
(
5.145
)
and
i
ts counterpart
f
or antineutrinos can be reduced to the equations includin
g
onl
y
ρ
p
(
t
)
and
ρ
p
(
t
)
. Taking account of the neutrino interactions in the SM,
one may wor
k
out t
h
e non-
l
inear evo
l
ution equations in t
h
e
l
ea
d
in
g
-or
d
er
a
pproximation
(
Sigl and Raffelt, 1993; Hannesta
d
et al.
,
2006
):
˙
P
p
=+
ω
p
B
+
λ
L
+
√
2
G
F
d
3
q
(2
π
)
3
(
P
q
−
P
q
)
g
(
θ
pq
)
×
P
p
,
˙
P
p
=
−
ω
p
B
−
λ
L
−
√
2
G
F
d
3
q
(2
π
)
3
(
P
q
−
P
q
)
g
(
θ
pq
)
×
P
p
,
(
5.146
)
where
g
(
θ
pq
)
≡
1
−
cos
θ
pq
w
ith θ
pq
being the angle between the neutrino
mo
m
e
n
ta
p
a
n
d
q
.
The ordinary matter effects are represented by the term
λ
L
w
i
th
λ
=
√
2
G
F
n
e
. The integral terms in the brackets of Eq.
(
5.146
)
com
e
f
rom the neutrino-neutrino coherent scattering, which makes the evolutio
n
equations o
f
the polarization vectors non-linear
.
T
o illustrate the non-linear e
ff
ects induced b
y
the neutrino-neutrino co
-
h
erent scattering, we consider a simple ensemble which initially contains pur
e
elect
r
o
nn
eut
rin
os a
n
da
n
t
in
eut
rin
os
.T
he
ir n
u
m
be
r
de
n
s
i
t
i
es a
r
e assu
m
ed to
be equal
(
n
ν
e
=
n
ν
e
)
, and the flavor polarization vector can be normalized
to unity. Hence Eq.
(
5.146
)
is simplified to
˙
P
=
+
ω
B
+
λ
L
+
μ
(
P
−
P
)
×
P
,
˙
P
=
−
ω
B
+
λ
L
+
μ
(
P
−
P
)
×
P
,
(
5.147
)
w
h
e
r
e
μ
≡
√
2
G
F
n
ν
e
is the stren
g
th o
f
neutrino-neutrino interactions. Two
additional assumptions will be made: (1) only one single momentum mode
e
i
s considered, so the vacuum oscillation
f
requenc
y
ω
i
su
ni
ve
r
sa
l
fo
r
both
n
eutrinos and antineutrinos;
(
2
)
the neutrino density dominates over the
e
l
ectron
d
ensit
y
n
ν
e
n
e
, so the matter-effect term in Eq.
(
5.147
)
can be
n
e
g
lected. We
f
ocus on the non-linear e
ff
ects
g
overned by the equations
˙
P
=
+
ω
B
+
μ
(
P
−
P
)
×
P
,
˙
P
=
−
ω
B
+
μ
(
P
−
P
)
×
P
.
(
5.148
)
Three-flavor oscillations can well a
pp
roximate to the two-flavor ones due t
o
t
h
e strong
h
ierarc
hy
|
Δ
m
2
3
2
|
≈
|
Δ
m
2
3
1
|
Δ
m
2
21
, as shown in Eq.
(
5.121
)
.I
n
200 5 Phenomenology o
f
Neutrino
O
scillations
t
h
is approximation on
l
yt
h
esma
ll
est neutrino mixin
g
an
gl
e
θ
13
i
s
r
eleva
n
t.
F
or the normal neutrino mass hierarch
y
,wehav
e
B
=(
sin
2
θ
13
,
0
,
−
c
os
2
θ
13
)
.
F
or t
h
einverte
d
neutrino mass
h
ierarc
h
y
,
θ
13
i
sc
l
ose t
o
π
/
2
an
d
t
h
us a
tiny an
gle
θ
1
3
≡
π
/
2
−
θ
13
ca
n
be defi
n
ed
.In
t
hi
scaset
h
e
z
-component o
f
the ma
g
netic field becomes positive
,
B
=(
sin
2
θ
13
,
0
,
cos 2
θ
13
)
. Defining th
e
d
ifference between the polarization vectors a
s
D
≡
P
−
P
a
n
dt
h
esu
m
o
f
the
m
as
S
≡
P
+
P
,
we transform Eq.
(
5.148
)
int
o
˙
S
=
ω
B
×
D
+
μ
D
×
S
,
˙
D
=
ω
B
×
S
.
(
5.149
)
In view of
˙
B
=0
an
d
B × B =0
,
we further define
Q
=
S
−
ω
B
/μ
an
d
t
h
en
rewrite Eq.
(
5.149
)
as the differential equations of
Q
a
n
d
D
:
˙
Q
=
μ
D
×
Q
,
˙
D
=
ω
B
×
Q
.
(
5.150
)
G
iven the initial conditions
P
(
0
)=
P
(
0
)
=
(
0
,
0
,
1)
,whichleadtoD
(
0
)
=0
a
n
d
S
(
0
)
=
(0
,
0
,
2
)
,Eq.
(
5.150
)
can be solved for the
z
-
components o
f
Q
a
n
d
D
.A
saconse
q
uence, we find
P
z
PP
=
S
z
+
D
z
2
=
1
2
Q
z
+
ω
μ
B
z
+
D
z
,
P
z
=
S
z
−
D
z
2
=
1
2
Q
z
+
ω
μ
B
z
−
D
z
.
(
5.151
)
Th
e
n
u
m
e
ri
ca
lr
esu
l
ts for
P
z
P
P
a
n
d
P
z
in
t
h
ecaseoft
h
e
in
ve
r
ted
n
eut
rin
o
m
ass hierarchy are shown in Fi
g
.5.4,where
θ
13
=0
.
01 has been in
p
ut
.
A
ssuming the neutrino-neutrino interaction strength
μ
t
o be much larger
than the vacuum oscillation
f
requency
ω
, we can o
b
serve t
h
at
P
z
PP
a
n
d
P
z
keep their initial values
(
i.e.
,
P
z
PP
(
0
)=
P
z
(
0
)
=+1
)
unchanged for som
e
time, and then rapidly fall down to abou
t
−
1
. This behavior im
p
lies that the
i
nitial electron neutrinos can almost completely trans
f
orm into the muon or
tau neutrinos. For the normal neutrino mass hierarch
y,
P
z
PP
an
d
P
z
s
li
g
htly
fluctuate around their initial values as a result of the tiny value o
f
θ
13
.T
h
i
s
i
nterestin
g
evolution o
f
P
z
PP
a
n
d
P
z
,w
h
ic
h
is sometimes ca
ll
e
d
t
h
e“
b
ipo
l
a
r
flavor conversion”, can be understood by convertin
g
the equation of motion
i
nEq.
(
5.150
)
into that for the pendulum in a vertical plane
(
Hannestad
et al.
, 2006
)
. In the small-mixing-angle limit, the evolution associated with
the normal neutrino mass hierarchy corresponds to the pendulum swingin
g
a
round the stable point where the potential energy is minimal. As far as
t
h
einverte
d
neutrino mass
h
ierarc
hy
is concerne
d
,
h
owever, t
h
epen
d
u
l
um is
i
nitially placed at the highest point which is unstable. Hence it is ready to fal
l
d
own to the lowest point, causing the complete neutrino flavor conversion.
I
n a realistic case
f
or supernova neutrino oscillations, one should care
f
ull
y
take into account the neutrino energy spectra, the asymmetry between th
e
n
umber densities o
f
neutrinos and antineutrinos, the matter e
ff
ects and th
e
n
eutrino trajectories
(
Duan and Kneller, 2009
)
. We shall discuss the flavor
c
onversions of su
p
ernova neutrinos in more detail in Cha
p
ter 7.
5.4 Future Long-baseline Neutrino
O
scillation Facilities 201
0
2
4
6
8
−
1
−0
.
5
0
0
.
5
1
P
z
PP
,
P
z
t
Fig. 5.4 The evolution o
f
P
z
PP
a
n
d
P
z
f
or the inverted neutrino mass hierarchy
,
w
h
er
e
θ
13
=
0
.
0
1an
d
ω
=1to
g
et
h
er wit
h
μ
=
5
(
dot-dashed line
),
μ
=10
(
solid
line
)
and
μ
=15
(
dashed line
)
have been input
(
Hannestad et a
l
.
,
2006. With
permission from the American Physical Society
)
5
.4 Future Lon
g
-baseline Neutrino
O
scillation Facilities
Current solar, atmos
p
heric, accelerator and reactor neutrino ex
p
eriments
h
ave provided us with very compelling evidence for neutrino oscillations,
f
r
o
m
w
hi
c
h
Δ
m
2
21
,
|
Δ
m
2
32
|
,
θ
12
a
n
d
θ
23
a
re
d
etermine
d
to a reasona
bl
y
g
oo
d
d
e
g
ree of accuracy. The on
g
oin
g
and future neutrino oscillation experiment
s
a
re expected to answer the following three questions:
(
a
)
what is the sign of
Δ
m
2
32
?(
b
)
how small is the smallest neutrino mixing angl
e
θ
1
3
?(
c
)
how larg
e
i
s the CP-violating phas
e
δ
?
In this section we shall first give a brief overview
of the prospects of some high-intensity long-baseline neutrino oscillation ex
-
periments, an
d
t
h
en
g
ive a s
l
i
gh
t
l
y
d
etai
l
e
d
intro
d
uction a
b
out t
h
eDayaBay
reactor antineutrino oscillation ex
p
eriment in China
.
5.4.1 Pros
p
ects of Accelerator Neutrino Ex
p
eriment
s
Three types o
ff
acilities have so
f
ar been proposed to provide hi
g
h-intensity
n
eutrino beams: the super-beam facilit
y
, the beta-beam facilit
y
and the neu
-
trino factory
(
Gonzalez-Garcia and Maltoni, 2008; Bandyopadhya
y
et al.
,
2
009
)
. The physics performance of each facility can be evaluated by a set of
observables, including the number of degrees of freedom which are used t
o
d
escri
b
et
h
e experimenta
l
outputsaswe
ll
as t
h
e sensitivities t
o
θ
13
,
δ
,
t
h
e
s
i
g
no
f
Δ
m
2
32
, and a deviation o
f
θ
23
f
r
o
m
π
/
4
.
S
uch
p
recision measurement
s
a
re extremely important to pin down the hierarchy of the neutrino mas
s
202 5 Phenomenology o
f
Neutrino
O
scillations
s
pectrum, the accurate pattern o
f
the
3
×
3 neutrino mixin
g
matrix an
d
t
he
s
tren
g
th of leptonic CP violation in neutrino oscillations.
(
1
)
T
he super-beam facilitie
s
.C
onventional accelerator-based neutrin
o
beams are produced
f
rom the primary decays o
f
char
g
ed pions and have bee
n
u
sed to do either
ν
μ
→
ν
μ
and
ν
μ
→
ν
μ
d
isa
pp
earance ex
p
eriments, such as
the K2K and MINOS experiments
(
Ah
n
et al.
,
2006; Mic
h
ae
l
et al
., 2006
),
o
r
ν
μ
→
ν
τ
a
n
d
ν
μ
→
ν
τ
a
ppearance experiments, such as the OPER
A
experiment
(
Kodama
e
tal.
,
1999
)
. They can be optimized by designing
a
h
ig
h
-power proton acce
l
erator w
h
ic
hd
e
l
ivers more intense proton
b
eams t
o
t
h
etar
g
et,soastopro
d
uce
h
i
gh
-intensity
b
ut
l
ow-ener
gy
ν
μ
a
n
d
ν
μ
bea
m
s
f
o
r
ν
μ
→
ν
e
an
d
ν
μ
→
ν
e
appearance experiments.
A
super-beam is usuall
y
re
f
erred to as a conventional neutrino beam driven by the proton driver with
a
beam power in the range of 2 MW to 5 MW
(
Bandyopadhyay
et al
., 2009
)
.
T
he technology required for a super-beam facility is well known. But th
e
s
ensitivity o
f
a super-beam experiment to
ν
μ
→
ν
e
a
n
d
ν
μ
→
ν
e
osc
i
llat
i
o
n
s
i
s limited b
y
the
f
act that the primar
y
π
±
deca
y
s, the subsequen
t
μ
±
d
eca
ys
a
nd the kaon decays in the decay pipe of the facility can produce a smal
l
f
r
act
i
o
n
of
ν
e
a
n
d
ν
e
eve
n
ts
.T
h
i
s
in
t
rin
s
i
cco
n
ta
min
at
i
o
nin
c
r
eases w
i
th the
beam ener
g
y and thus has to be kept as low as possible. One may desi
gn
a
n appropriate beam line configuration to suppress th
e
ν
e
or
ν
e
b
ac
k
groun
d
f
rom the kaon deca
y
s.
O
ne ma
y
also produce an o
ff
-axis neutrino beam
,
which is arran
g
ed to be tilted by a few de
g
rees with respect to the vecto
r
pointing from the source to the far detector
(
McDonald, 2001
)
, to suppres
s
the
ν
e
o
r
ν
e
b
ac
kg
roun
d
in
d
uce
dby
π
±
d
eca
y
s. The kinematics o
f
atwo
-
body pion decay assures that all the pions above a given momentum produc
e
n
eutrinos of almost the same energy at a given angle
θ
= 0 with respect to
the axis.
S
otheo
ff
-axis technique can yield a low-ener
g
y neutrino beam wit
h
a
small energy spread, which has several advantages over the broad-band on
-
a
xis neutrino beam. For example, the o
ff
-axis neutrino beam allows ener
gy
c
uts to be applied to reduce back
g
rounds and allows the L
/
E
p
arameter o
f
a
n experiment to be tuned to the oscillation maximum, although its flux i
s
muc
h
s
m
a
ll
e
r
t
h
a
n
t
h
eo
n-
a
xi
s
n
eut
rin
oflu
x.
T
he ongoing T2K experiment is just an off-axis experiment
(
Ito
w
et al.
,
2
001
)
. Its facility consists of a conventional neutrino beam driven by the
protons of 30 GeV from the J-P
A
RC proton s
y
nchrotron at a beam power
of 0.75 MW, two near detectors
(
one on-axis and the other off-axis
)
locate
d
2
80 m away from the neutrino source, and the far detector which is just th
e
Super-Kamiokande
(
SK
)
neutrino detector. Its main goal is to observe th
e
app
earance of
ν
e
events in an almost
p
ure
ν
μ
n
eutrino beam after travelin
g
a
d
istance of 2
95
km from the
J
-PA
RC
accelerator center to the
S
K detector.
The o
ff
-axis beam an
g
le o
f
this experiment has been chosen to maximize the
s
ens
i
t
i
v
i
t
y
t
o
θ
13
.
A
n upgrade to the power of the J-P
A
RC proton synchrotro
n
i
s under consideration in order to provide a 50 GeV proton beam at the power
o
f
4 MW. In this case the construction o
f
a Mton-scale water
C
herenko
v
5.4 Future Long-baseline Neutrino
O
scillation Facilities 203
d
etector
(
the so-called Hyper-Kamiokande detector
)
at a distance of abou
t
1
000 km awa
y
from the J-P
A
RC accelerator center would allow one to do
a
h
igh-intensity off-axis long-baseline neutrino oscillation experiment, which is
expected to be sensitive to both the si
g
no
f
Δ
m
2
32
and the
C
P-violatin
g
phase
δ
(
Ishitsuk
a
e
tal., 2005; Ka
j
it
a
e
tal
.
,
2007; Hagiwara
e
tal
.
,
2007
)
.
T
he N
O
ν
A
facility, which is now under construction, will also do an off
-
a
xis neutrino experiment optimized
f
or searchin
gf
o
r
ν
μ
→
ν
e
osc
i
llat
i
o
n
sw
i
th
a
sensitivity ten times better than the ongoing MINOS experiment
(
Ayres
et al
.
,
2005
)
. It employs an upgraded version of the already existing NuM
I
beamline at a power o
f
0.7 MW in the Fermilab. The neutrino beam ener
g
y
i
s about 2 GeV, and the baseline length is in the range of 700 km to 90
0
km
(
from the Fermilab to northern Minnesota
)
with a far detector sited 1
2
km
(
equivalent t
o
θ
∼
0
.
8
5
◦
)
off-axis. Both the near and far detectors ar
e
the “totally active” liquid scintillator detectors. Given a 15-kton far detector
a
nd a 5-year run, the N
O
ν
A
experiment is likely to achieve a sensitivity t
o
sin
2
2
θ
13
comparable to that o
f
the T2K experiment. In particular, the lon
g
baseline of the N
O
ν
A experiment will allow one to probe the sign of
Δ
m
2
32
with the help o
f
terrestrial matter e
ff
ects
.
T
2K and N
O
ν
A
belon
g
to the first-
g
eneration super-beam experiments.
The second-generation ones, such as T2HK and CERN-SPL, are under con
-
s
ideration
(
Gonzalez-Garcia and Maltoni, 2008; Bandyopadhyay
et al.
,
2009
).
(
2
)
T
he beta-beam
f
acilities
.A
beta-beam is
p
roduced from the booste
d
ra
d
ioactive-ion
d
ecays an
d
t
h
us is a pur
e
ν
e
or
ν
e
b
eam
(
Zucchelli, 2002
)
.I
t
ca
n
be used to do e
i
ther
ν
e
→
ν
e
a
n
d
ν
e
→
ν
e
d
isappearance experiments
o
r
ν
e
→
ν
μ
(
o
r
ν
τ
)
an
d
ν
e
→
ν
μ
(
or
ν
τ
)
appearance experiments. There are
three variables that determine the properties of a beta-beam facility
(
Bandy
-
opa
dhy
a
y
et al
., 2009
)
: the type of ions to be used
(
especially the endpoin
t
kinetic energy of the electron in the beta decay,
E
0
)
; the relativistic
γ
f
ac
-
tor
(
equal to energy divided by mass
)
of the ion; and the baseline length
L
.
O
nce these parameters are
fi
xed, the neutrino
fl
ux can be precisel
y
calculate
d
because the kinematics of the beta decay is well known
(
Burguet-Castel
l
et
al.
,
2004
)
. To set up a proper beta-beam, the isotope should be sufficiently
l
on
g
-lived to avoid stron
g
losses in the acceleration phase, but it must deca
y
f
ast enough to produce an intensive neutrino beam. Two ion species, whos
e
l
i
f
etimes are both around 1 s, have been identi
fi
ed as the optimal candidates
(
Zucchelli, 2002
)
:
6
H
ewit
h
E
0
=
3506
.
7
keV for
g
eneratin
g
ν
e
e
vents and
18
N
ewit
h
E
0
=3
42
3
.
7
keV for producin
g
ν
e
events.
S
ome other ions wit
h
l
ar
g
e
r
E
0
,
w
h
ic
h
are a
bl
etopro
d
uce more ener
g
etic neutrino
b
eams at t
h
e
sa
m
e
γ/L
, have also been considered
(
Donini and Fernandez-Martinez, 2006
;
R
u
bb
ia et a
l
., 2006
)
.
O
n the other hand, an optimization o
f
th
e
γ
f
actor and the baseline len
g
th
L
s
hould take account of several physics requirements
(
Bandyopadhyay
et
al.
,
2009
)
:
(
a
)
the value of L
/
E
ν
s
hould satisfy sin
2
(1
.
2
7
Δ
m
2
3
2
L
/E
ν
)
∼
1
(
the first oscillation maximum
)
such that the oscillation signatures can be as
204 5 Phenomenology o
f
Neutrino
O
scillations
l
arge as possible;
(
b
)
the neutrino beam energ
y
E
ν
should be above the
m
uon
p
roduction threshold such that the
ν
μ
or
ν
μ
a
pp
earance can be searched fo
r
i
n the far detector;
(
c
)
E
ν
s
hould be large enough for a measurement of th
e
s
pectra
ld
istortion to
b
euse
d
to reso
l
ve t
h
e intrinsic parameter
d
e
g
enerac
y
problem;
(
d
)
L
should be as lon
g
as possible to determine the si
g
no
f
Δm
2
32
with the help of terrestrial matter effects; and
(
e
)
the event rate should be as
l
ar
g
e as possible — increasin
g
the value o
f
γ
fo
r
afi
x
ed
i
o
n
flu
x
ca
n
e
nh
a
n
ce
E
ν
and thus the number of events because the cross sections of neutrino
-
nucleo
nin
te
r
act
i
o
n
s
in
c
r
ease w
i
th
E
ν
. All these requirements point to th
e
co
n
clus
i
o
n
that the
γ
f
actor should be enhanced as much as possible and th
e
baseline lengt
h
L
s
hould be tuned to sit near an oscillation
p
eak.
T
he beta-beams can rou
g
hly be classi
fi
ed into the low- and hi
g
h-ener
gy
ones.
A
low-ener
g
y beta-beam wit
h
E
ν
in the sub-
G
eV ran
g
ematchesth
e
d
istanc
e
L
= 130 km from CERN to the Modane laboratory in the Freju
s
tunnel
(
Autin
et al.
, 2003
)
. In comparison, a high-energy beta-beam wit
h
E
ν
i
ntheran
g
eo
f
1
G
eV to 1.5
G
eV matches the distance
L
∼
7
00 km be-
tween CERN and Canfranc
,
CERN and Gran Sasso
,
or Fermilab and Soudan
(
Bandyopadhyay
et al.
, 2009
)
. A higher-energy beta-beam experiment would
require the Lar
g
e Hadron Collider at CERN to accelerate the ions to produc
e
ν
e
e
vents o
f
E
ν
∼
5G
e
V
an
d
ν
e
e
vents o
f
E
ν
∼
7
.
5 GeV together with
a
very
l
on
gb
ase
l
in
e
L
∼
3
000 km
(
Gonzalez-Garcia and Maltoni, 2008
).
(
3
)
The neutrino
f
actor
y
. The technolo
g
y for producin
g
an intensive muon
beam was initially explored for the purpose of building a muon collider at the
TeV ener
g
y scale. This technolo
g
y is more
f
easible
f
or buildin
g
the neutrin
o
f
actory at much lower energies. In a neutrino factory muons are accelerate
d
f
rom a high-intensity source to energies of several tens of GeV and the
n
i
njected into a storage ring with long straight sections
(
Geer, 1998; De Rujula
et al., 1999
)
. Therefore, bot
h
ν
μ
(
or
ν
μ
)
an
d
ν
e
(
o
r
ν
e
)
events can be produce
d
f
rom the well-known muon decay modes
μ
−
→
e
−
+
ν
e
+
ν
μ
a
n
d
μ
+
→
e
+
+
ν
e
+
ν
μ
. These neutrino beams, whose ener
g
ies are up to the muo
n
energy itself, allow one to do the following neutrino oscillation experiments
:
(
1
)
ν
e
→
ν
e
ν
e
→
ν
e
Disa
pp
earance
(
2
)
ν
e
→
ν
μ
ν
e
→
ν
μ
Appearance
(
golden channel
)
(
3
)
ν
e
→
ν
τ
ν
e
→
ν
τ
Appearance
(
silver channel
)
(
4
)
ν
μ
→
ν
μ
ν
μ
→
ν
μ
Disappearance
(
5
)
ν
μ
→
ν
e
ν
μ
→
ν
e
Appearance
(
challenging
)
(
6
)
ν
μ
→
ν
τ
ν
μ
→
ν
τ
Appearance
(
interesting
)
S
ome care
f
ul studies about the desi
g
no
f
aneutrino
f
actory have been carried
out in Europe, Japan and the USA
(
Bandyopadhyay
e
tal.
,
2009
)
.Thecon
-
c
lusion is that an accelerator complex capable o
f
providin
g
about 10
21
muon
d
eca
y
sper
y
ear can be built, and the optimal detector should better be abl
e
to
p
erform both a
pp
earance and disa
pp
earance ex
p
eriments. The detectio
n
of
ν
α
(
or
ν
α
)
neutrinos
(
for
α
=
e
,
μ
,
τ
)
is to look for the
α
−
(
or
α
+
)
event
s
5.4 Future Long-baseline Neutrino
O
scillation Facilities 205
produced
f
rom the char
g
ed-current interactions o
f
ν
α
(
o
r
ν
α
)
neutrinos wit
h
the medium of the far detector.
Among twelve neutrino oscillation processes that can be measured at
a
n
eutrino
f
actory, the “
g
olden” channel is th
e
ν
e
→
ν
μ
t
r
a
n
sition
6
.
B
ecause
ν
e
n
eutrinos are produced from the deca
y
s of positive muons whil
e
ν
μ
n
eutr
i
no
s
pro
d
uce negative muons via t
h
ec
h
arge
d
-current interactions in t
h
e
d
etector,
the
ν
μ
appearance can simply be detected by lookin
gf
or the “wron
g
-si
g
n
”
m
uons which are unable to get contaminated in the dominant background
(
i.e., “right-sign” muons
)
. The full statistical sensitivity can therefore be ex
-
ploited to probe ver
y
small values o
f
θ
13
a
n
d
δ
(
Huber and Winter, 2003
).
T
he so-called “silver” channel at a neutrino factor
y
is the ν
e
→
ν
τ
t
ran
-
s
ition si
g
ni
fi
ed by the
ν
τ
appearance
(
or equivalently, th
e
τ
−
events
)
in th
e
f
ar detector.
A
careful comparison between the CP- or T-violatin
g
effects i
n
golden and silver channels would allow one to test whether leptonic CP viola
-
tion arises
f
rom the unique phas
e
δ
, because the
g
enuine
C
P- and T-violatin
g
terms in these two channels should have the opposite signs
(
Strumia an
d
Vissani, 2006
)
.
I
n fact, a determination of
(
θ
1
3
,
δ
)
from the measurement of a single neu
-
trino oscillation channel suffers from an ei
g
ht-fold ambi
g
uity; i.e., there are
i
n general eight different regions of the parameter space which can fit th
e
s
ame experimental data in the
(
θ
1
3
,δ
)
plane
(
Bandyopadhyay
et al.
, 2009
).
One may speculate two possible ways to resolve this parameter de
g
eneracy
problem: one is to measure the golden neutrino oscillation channel at differ
-
ent baselines
(
i.e., different values o
f
L
/E
)
; and the other is to make use of
the rich flavor content of neutrino beams by combining the measurements o
f
d
ifferent channels at a neutrino factory.
I
tisveryc
h
a
ll
en
g
in
g
to o
b
serve t
h
e
ν
μ
→
ν
e
transition,
b
ecause t
h
e
e
−
s
ignals associated with the ν
e
a
ppearance can easily get contaminated in the
f
ar detector. In this sense it mi
g
ht be a bit easier, at least not hopeless, t
o
sea
r
c
h
fo
r
t
h
e
e
+
s
i
g
nals associated with the
ν
e
a
ppearance arisin
gf
rom th
e
ν
μ
→
ν
e
t
ransition in the detector with a delicate detection techni
q
ue. We
s
tress that the study o
fC
P violation i
n
ν
μ
→
ν
τ
a
n
d
ν
μ
→
ν
τ
osc
i
llat
i
o
n
s
a
t a neutrino
f
actory is particularly interestin
g
, since they are insensitive to
the standard CP-violating effect characterized by the phas
e
δ
but sensitive
to some new
C
P-violatin
g
e
ff
ects induced by either the non-unitarity o
f
th
e
3
×
3
neutrino mixing matrix or the non-standard neutrino interactions
(
Mel
-
oni et a
l.
,
2010
)
. In the presence of terrestrial matter effects and
(
or
)
new
ph
y
sics, it is also use
f
ul to combine the measurements o
f
both appearanc
e
a
nd disa
pp
earance neutrino oscillation channels so as to
p
in down the wanted
parameters wit
h
muc
h
sma
ll
er uncertainties or am
b
iguities
.
6
I
ts CP-conjugate proces
s
ν
e
→
ν
μ
can also be regarded as a golden channel, but
one should keep in mind that the rate o
f
μ
+
s
igna
l
s associate
d
wit
h
t
h
e appearanc
e
o
f
ν
μ
e
ventsisabouttwotimeslowerthanthato
f
μ
−
s
i
g
na
l
s associate
d
wit
h
t
he
appearance of
ν
μ
e
vents in the far detector (Strumia and Vissani, 2006)
.