Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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206 5 Phenomenology o
f
Neutrino
O
scillations
A
detailed description of the “standard” neutrino factory setup, to
g
ether
with its physics potential, has been presented by the ISS Physics Workin
g
Group
(
Bandyopadhyay
e
ta
l.
, 2009
)
. We hope that a neutrino factory, if it
c
an be built in the
f
uture, will deepen our understandin
g
o
f
the dynamics o
f
n
eutrino oscillations and open a promising window towards some new physic
s
which might be relevant to the origin of neutrino masses or to the unrevealed
i
n
te
r
act
i
o
n
sbetwee
nn
eut
rin
os a
n
d
m
atte
r
.
5.4.2 Pros
p
ects o
f
Reactor Antineutrino Ex
p
eriments
It is well known that Wolfgang Pauli’s neutrino hypothesis was first verified in
1
956 thanks to the Savannah River reactor antineutrino experiment
(
Cowan
et al
.
, 1956
)
. It is also known that very convincing evidence for the electron
a
ntineutrino oscillations was first achieved in 2
003
thanks to the KamLAN
D
reactor antineutrino experiment
(
Eguch
i
et al
., 2003
)
. These two examples
a
re sufficient for illustration of the im
p
ortance of reactor antineutrino ex
p
er-
i
ments in the study of neutrino properties and neutrino interactions. Hence
one expects t
h
at a
h
i
gh
-intensity reactor antineutrino osci
ll
ation experiment
c
an help pin down the smallest neutrino mixing angle
θ
13
in the near future
.
Th
e present upper
b
oun
d
on
θ
13
was actually obtained from the CHOOZ
reactor antineutrino experiment
(
Apollonio
et al.
,
1998
)
. A lot of attention
h
as recently been paid to the next generation of reactor antineutrino exper-
i
ments based on the observation that the performance of the CHOOZ ex
-
periment can be remarkabl
y
improved i
f
the s
y
stematics is better controlle
d
a
nd the statistics is significantly enhanced
(
Anderso
n
e
tal
.
,
2004
)
. Differen
t
f
rom the accelerator neutrino oscillation experiments, the reactor antineu
-
trino oscillation experiments are
f
ree o
f
the parameter de
g
eneracy proble
m
a
nd able to determine or constrain the magnitude of
θ
13
in a direct and clea
n
way. In the three-
fl
avor neutrino mixin
g
scheme, the probability o
f
reactor
ν
e
→
ν
e
o
scillations at a baseline ran
g
eo
f
1 km to 2 km can be simpli
fi
ed t
o
a
two-flavor oscillation pattern to a good degree of accuracy
:
P
(
ν
e
→
ν
e
)
=
1
−
4
i<j
|
V
ei
VV
|
2
|
V
ej
VV
|
2
sin
2
1
.
27
Δ
m
2
ji
L
E
≈
1
−
4
|
V
e
VV
3
|
2
1
−|
V
e
VV
3
|
2
s
in
2
1
.
2
7
Δ
m
2
3
1
L
E
=1
−
sin
2
2
θ
1
3
sin
2
1
.
2
7
Δ
m
2
31
L
E
,
(
5.152
)
where
|
Δ
m
2
32
|
≈
|
Δ
m
2
31
|
Δ
m
2
21
a
n
d
|
V
e
V
V
3
|
2
=
s
in
2
θ
13
1
have been used.
So far a number of projects, including Double Chooz in France
(
Ardellier
et
al.
,
2006
)
and Daya Bay in China
(
Gu
o
et al
.
,
2007
)
, have been proposed t
o
p
robe how smal
l
θ
13
c
ould be
(
Goodman, 2009
).
5.4 Future Long-baseline Neutrino
O
scillation Facilities 207
Fig. 5.5 The layout of the Day Bay reactor antineutrino experiment. Two pairs o
f
reactor cores
(
Daya Bay and Ling Ao
)
are running and the third one
(
Ling Ao II
)
will run soon. Four detector modules are deplo
y
ed at the far site, and two detector
modules are deployed at each of the two near sites
(
Guo
et al.
, 2007
)
O
f the currentl
y
-proposed reactor antineutrino experiments, the Double
Chooz experiment has the opportunity to obtain physical results first. Thi
s
experiment emplo
y
s two almost identical medium-size detectors. The near
d
etector is located 2
80
mfromthe
C
hooz nuclear cores and can be used t
o
c
ontrol the systematics. The
f
ar detector, which is situated in the same caver
n
a
s the CHOOZ detector
(
i.e., 1.05 km from the Chooz nuclear cores
)
, will b
e
a
ble to detect
5
×
10
5
ν
e
e
vents in three years of operation with a relative
s
ystematic error o
f0
.
6
%
. Hence the Double Chooz experiment is likely to
d
etermine or limit si
n
2
2
θ
13
to the ran
g
eof0
.
0
22 to
0
.
030
(
for
|
Δ
m
2
31
|
v
ary
i
n
g
f
rom
3
.
5
×
10
−
3
e
V
2
d
own to 2.
5
×
10
−
3
eV
2
)
within an unrivaled time scale
a
nd a modest cost
(
Ardellier
et al.
, 2006
).
T
he Da
y
aBa
y
experiment is the first neutrino oscillation experiment be
-
i
ng done in China. In this experiment the electron antineutrino beam take
s
i
ts source at t
h
eDa
y
aBa
y
nuc
l
ear power comp
l
ex w
h
ic
h
is current
ly
com
-
posed of two pairs of reactor cores
(
Daya Bay and Ling Ao
)
separated b
y
ab
out 1.1
k
m, as i
ll
ustrate
d
in Fig. 5.5. T
h
e comp
l
ex can now generate 11.6
G
Wo
f
thermal power, which will increase to 17.4
G
Wb
y
earl
y
2011 a
f
ter
the third pair of reactor cores
(
Ling Ao II
)
is put into operation
(
Guo et al.
,
2
007
)
. The location of this complex is adjacent to a mountainous terrain i
n
208 5 Phenomenology o
f
Neutrino
O
scillations
10
-2
1
0
-1
0.
50.5
11
1
.
51.5
22
2.
52.5
33
3.
5
3.5
44
4
.
5
4.5
55
si
n
2
2
θ
13
Δ
m
2
(
×
1
0
-3
e
V
2
)
Chooz
D
aya Bay
3
y
Fig. 5.6 The expected sensitivity of the Daya Bay experiment to si
n
2
2
θ
13
at
the 90
%
confidence level with three
y
ears of data, in comparison with the present
CHOOZ limit on sin
2
2
θ
1
3
(
Wang, 2010. With permission from the American
Institute of Physics
)
. The shaded region corresponds to the allowed interval o
f
Δ
m
2
≡
|
Δ
m
2
31
|
at the 90
%
confidence leve
l
S
henzhen, southern
C
hina, and thus it is ideal
f
or sittin
g
under
g
round de
-
tectors that are well shielded from the cosmogenic contamination. The basi
c
l
ayout of the Daya Bay experiment consists of three underground labora-
tories with a tetrad o
ff
ar detector modules and two pairs o
f
near detector
m
odules, which are linked through a few horizontal tunnels. Fig. 5.5 show
s
t
h
e
d
etector
d
ep
l
oyment at t
h
ese sites. T
h
e Daya Bay near
d
etector is 36
3
m
from the Daya Bay reactor cores, while the Lin
gA
o near detector is 481
m
from the Ling Ao reactor cores and 526 m from the Ling Ao II reacto
r
c
ores. They can monitor the rate and ener
g
y distribution o
f
ν
e
eve
n
ts a
n
d
reduce the systematic uncertainties in detectin
g
si
n
2
2
θ
13
.Th
ed
i
sta
n
ces of
the far detector from the Daya Bay, Ling Ao and Ling Ao II reactor cores ar
e
1
985 m, 1618 m and 1613 m, respectivel
y
.There
f
ore, the
fi
rst
ν
e
→
ν
e
osc
i
l
-
l
ation maximum can be observed at the far detector to ensure the maximal
experimenta
l
sensitivity to si
n
2
2
θ
1
3
.
T
he sensitivit
y
o
f
the Da
y
aBa
y
experiment to sin
2
2
θ
1
3
h
as
b
een ana
ly
ze
d
by using the single parameter raster scan method
(
Guo et al
.
, 2007
)
. Fig. 5.
6
sh
ows t
h
e sensitivity contours in t
h
esi
n
2
2
θ
1
3
ve
r
sus
Δ
m
2
≡
|
Δ
m
2
31
|
p
lane fo
r
three
y
ears o
f
operation.
O
ne can see that the sensitivit
y
o
f
this experiment is
a
ble to reach the challenging goal of si
n
2
2
θ
13
∼
1
%, provided both s
y
stematic
a
nd statistic uncertainties are well under control
(
Wang, 2006; Wang, 2010
)
.
R
e
f
erences 20
9
Th
e Daya Bay experiment is smoot
hl
y
g
oin
g
on. Its civi
l
constructio
n
i
s now underwa
y
, so is the deplo
y
ment of the near detectors.
A
fter the far
d
etector is deployed in 2011 or 2012, the Daya Bay Collaboration will star
t
to take data.
A
determination of the smallest neutrino mixin
g
an
g
le
θ
13
,o
r
a
t least a strin
g
ent constraint o
n
θ
13
i
f it is too small to be experimentall
y
a
ccessible, is highly expected from this experiment
7
.
I
tis
fi
nally worth mentionin
g
the REN
O
experiment located on the sit
e
of the Yonggwang power plant in the southwest of Korea
(
Ahn et al.
,
2010
)
.
This plant, which spans about 1.3 km, consists of six reactors lined up i
n
rou
g
hly equal distances and has a total thermal power o
f
16.4
G
W. There
a
re two identical detectors in the RENO experiment: they are at about 29
0
m(
near
)
and 1380 m
(
far
)
, respectively, from the reactor array. The design o
f
the REN
O
detectors is
q
uite similar to that o
f
the Double
C
hooz detectors
.
One expects that the RENO experiment may start data taking in the near
f
uture and reach the sensitivity si
n
2
2
θ
13
=0
.
0
19 a
f
ter
fi
ve years o
f
runnin
g
(
Mezzetto and Schwetz, 2010
)
.
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eference
s
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hn, J. K.
,
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)
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hn
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n
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)
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Δ
m
2
31
a
n
ddete
rmin
ethe
n
eut
rin
o
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ffi
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e
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6
N
eutrinos
f
rom
S
tars
Stars represent an important source of astroph
y
sical neutrinos, and the stud
y
of stellar neutrinos is an important branch of neutrino astronomy. In thi
s
c
hapter we shall
fi
rst introduce the basic theor
y
o
f
stellar evolution and then
d
escribe the basic
p
ro
p
erties of stellar neutrinos. To be ex
p
licit, we shall
c
oncentrate on the neutrinos generated from thermal nuclear reactions i
n
the solar core. The production processes, experimental detection and
fl
avo
r
c
onversions of solar neutrinos will be discussed in detail.
6
.1
S
tellar Evolution in a Nutshel
l
A
lthough there does not exist a complete theory of star formation, it is com
-
m
only believed that stars were
f
ormed
f
rom the clouds o
f
the primordial
g
ases
bounded by the gravity.
A
star can in general be defined as an astrophysica
l
object bounded by the gravity of itself and radiating energy from its inter
-
n
al thermal nuclear reactions
(
Prialnik, 2000
)
. In this section we outline the
q
uantities describing a star and the equations governing its evolution
.
6.1.1 D
i
stance, Lum
i
nos
i
ty and Mas
s
The
S
un is a t
y
pical main-sequence star best known to us. Let us
fi
rst sum
-
m
arize the observational characteristics of the Sun so as to get a ballpark
f
eeling of the main-sequence stars, which represent a long and static perio
d
i
n the evolution o
f
stars. The avera
g
e distance between the
S
un and the Eart
h
i
s defined as an astronomical unit: 1
A
U=1.49
6
×
1
0
13
c
m. The total mass
o
f
the
S
un is
M
=
1
.
989
×
10
33
g
,an
d
its ra
d
ius i
s
R
=6
.
955
×
10
10
c
m.
In com
p
arison, the mass and radius o
f
the Earth ar
e
M
⊕
=5
.
9
7
4
×
10
2
7
g
a
nd
R
⊕
=
6
.
378
× 1
0
8
c
m. Note that
R
(
or AU
)
serves as a standard scale
i
n determining the distances of planets
(
or stars
)
from the Earth within
(
or
beyond
)
the solar system. A common method for the distance determina
-
tion is the stellar parallax
(
Prialnik, 2000; Carroll and Ostlie, 2007
)
.The
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
214
6
Neutrinos
f
rom
S
tar
s
parallax is one hal
f
o
f
the an
g
le between the lines o
f
si
g
ht o
f
astar
f
rom
two different positions of the observers. B
y
a measurement of the paral
-
l
ax
p
,
the distance of a star from the solar system can be determined as
d
=(
1AU
)
/
ta
n
p
≈
(
1AU
)
/
p
,
w
h
ere
p
is assume
d
to
b
ever
y
sma
ll
an
d
its
u
nit is radian. In view of 1 radian = 57
.
295
8
◦
= 206265
,
one ma
y
define th
e
d
istance unit as parallax second
(
i.e., parsec or pc for short
)
.Thenanastro
-
p
hy
sica
ld
istance can
b
e written a
s
d
=(1
/p
)
pc with 1 pc =
3
.
086
×
10
1
8
cm
,
where the unit of
p
i
s arc second. Another conventional unit is light year
(
i.e.
,
l
y for short
)
, defined as the distance travelled by the light in a Julian year.
N
ame
ly
,1
ly
=9
.
4
6
1
×
10
1
7
c
man
d
t
h
en 1 pc
≈
3
.
2
62
ly
.T
h
este
ll
ar para
ll
a
x
i
s always smaller than
1
.
For instance, Proxima Centauri
(
the nearest sta
r
other than the Sun
)
has a parallax angl
e
∼
0
.
77
(
Carroll and Ostlie, 2007
)
.
The Earth’s atmosphere makes a
g
round-based telescope di
ffi
cult to reach
a
p
recision better than 0
.
03
,
implying that it is impossible to measure a dis
-
tance
f
arther than 30 pc. To overcome such an obstacle, the European
S
pac
e
Ag
ency launched the satellite Hipparcos in 1989. This mission was able t
o
m
easure the parallax angles with an accuracy in the range
(0
.
7
∼
0
.
9
)
×
10
−
3
a
rc seconds
f
or more than 1
0
5
s
tars
(
Perrymann
et al
.
,
1997
)
.
T
he observation of a star is to measure its apparent luminosit
y
l
,
which
d
epen
d
sont
h
ea
b
so
l
ute
l
uminosity
L
a
n
d
t
h
e
d
istanc
e
d
.Thevalueso
f
L
a
n
d
l
characterize the ener
g
ies emitted
f
rom the star per second and receive
d
by the detector per second per square centimeter, respectively. If the ener
gy
re
l
ease
db
y a star is isotropic an
d
not a
b
sor
b
e
d
on t
h
ewaytot
h
e
d
etector, we
have
l
≡
L/
(4
π
d
2
)(
Carroll and Ostlie, 2007
)
. Historically, the apparent mag-
n
itud
e
m is used to describe the brightness of stars. The Greek astronomer
H
ipparchus was the first to assign
m
=1tot
h
e
b
rig
h
test star an
d
m
=
6
to the dimmest one in the sky
(
Krisciunas, 2001
)
. It was later realized tha
t
the human eye actually responded to a difference in the logarithm of th
e
bri
g
htness. For instance, a di
ff
erence o
f
5ma
g
nitudes corresponds to a
f
acto
r
o
f
100 in bri
g
htness; i.e., one ma
g
nitude di
ff
erence is equivalent to a
f
actor
of 1
00
1
/
5
i
n the apparent luminosity. One may therefore obtain
l
∝
10
−
2
m/
5
,
i
mp
l
yin
g
l
2
l
1
=
10
2
(
m
1
−
m
2
)
/
5
.
(
6.1
)
T
h
ea
b
so
l
ute ma
g
nitu
de
M
i
sde
fi
ned as the apparent ma
g
nitude that a sta
r
would have if it were located at 10 pc from the Earth. So one ma
y
express
the distance of a star in terms of its apparent and absolute magnitudes
:
d
=10
(
m
−
M
+5
)
/
5
,
(
6.2
)
whe
r
e
d
is measure
d
in parsecs, an
d
m
−
M
=5log
[
d
/
(
10 pc
)]
is called the
s
tar’s radius modulus. The absolute luminosit
y
of the Sun is
L
=
3
.
839
×
10
33
e
rg s
−
1
.
The apparent luminosity of the Sun can then be given b
y
l
=
L
4
π
(
1AU
)
2
=1
.
365
×
10
6
er
g
cm
−
2
s
−
1
,
(
6.3
)
6
.1
S
tellar Evolution in a Nutshell 21
5
wh
i
ch
i
sk
n
ow
n
as the sola
r
co
n
sta
n
t
1
.
The constant o
f
proportionalit
y
o
f
l
∝
1
0
−
2
m/
5
h
as been experimentall
y
determined to be
2
.
52
×
1
0
−
5
e
r
g
cm
−
2
s
−
1
(
Weinberg, 2008
)
. Given the value of
l
i
nEq.
(
6.3
)
, the apparent magnitud
e
o
f
the
S
un is
f
ound to be
m
S
un
=
−
26
.
83
. In view o
f
the Earth-
S
un distanc
e
d
=
1
A
U=4
.
8
4
8
×
1
0
−
6
pc, we obtain the absolute ma
g
nitude of the Sun
M
Sun
MM
=
m
S
u
n
−
5l
o
g
d
1
0
p
c
=
+
4
.
74
.
(
6.4
)
Since the absolute ma
g
nitude is defined at the fixed distance 10 pc, the ratio
of two different absolute magnitudes is the same as that of the corresponding
ab
so
l
ute
l
uminosities. T
h
ea
b
so
l
ute ma
g
nitu
de
M
of a sta
ri
st
h
e
r
efo
r
e
r
e
l
ated
to its absolute luminosit
y
L
as follows
:
M
=
M
Sun
MM
−
2
.
5
l
o
g
L
L
.
(
6.5
)
It seems natural for an observer to first measure the apparent magnitud
e
m
a
n
dthe
n
use the d
i
sta
n
ce
d
to probe the intrinsic properties o
f
astar,suc
h
as
M
v
ia Eq.
(
6.2
)
o
r
L
through Eq.
(
6.5
)
. If a star belongs to a special class
(
e.g., the pulsating variable stars
)
, however, it
s
L
a
n
d
M
ca
n
be dete
rmin
ed
i
nawa
y
independent o
f
d
(
Carroll and Ostlie, 2007
)
. In this case one ma
y
u
se Eq.
(
6.2
)
to work out the star’s distance
.
N
ote that both
M
a
n
d
L
have been de
fi
ned
f
or all the wavelen
g
ths o
f
the
l
i
g
ht from a star.
A
stronomers usually introduce the color ma
g
nitudes
M
U
MM
,
M
B
M
an
d
M
V
MM
f
or the ultraviolet
(
U
)
,blue
(
B
)
and visual
(
V
)
wavelengt
h
re
g
ions. T
h
e reason is simp
l
yt
h
at a measurement can on
l
y
b
e carrie
d
ou
t
i
n a finite wavelen
g
th ran
g
e. For a star with know
n
d
,
its absolute colo
r
m
agnitude can be fixed by the radius modulu
s
m
−
M
=5log
[
d
/
(
10 pc
)].
Th
eco
l
o
rin
de
x
defi
n
ed as
B
−
V
≡
M
B
M
−
M
V
M
M
is obviousl
y
independent o
f
the
d
istanc
e
d
.Fi
g
. 6.1 shows the color-ma
g
nitude dia
g
ram for more than 370
0
s
tars from the Hipparchos catalog. The main feature of this diagram is tha
t
the stars are not uni
f
orml
y
distributed but concentrate on the band denoted
a
s the main sequence, implying that it is the longest and most static perio
d
i
n
t
h
ew
h
o
l
e
lif
et
im
eo
f
sta
r
s
.
O
n the other hand, it has been
f
ound that the spectra o
f
starli
g
ht are
well consistent with that of the blackbody radiation. Hence the spectrum is
pea
k
e
d
at a wave
l
en
g
t
h
λ
max
,
correspondin
g
to an e
ff
ective temperatur
e
T
via Wien’s dis
p
lacement law
b
=
λ
m
a
x
T
=
0
.
2
9 cm K. The luminosit
y
L
of a
blackbody of area
A
and tem
p
eratur
e
T
i
sgivenby
L
=
A
σ
SB
T
4
,
where
σ
S
B
i
sthe
S
te
f
an-Boltzmann constant
g
iven in Table 6.1. For a spherical star wit
h
1
H
ere we adopt the centimeter-gram-second
(
cgs
)
unit system instead of the In-
ternational System of units
(
abbreviated to SI from its French name
)
. For example,
1J
ou
l
e=1
0
7
erg an
d
L
=3
.
839
×
10
2
6
W
. Some frequently-used fundamenta
l
constants in the
S
I units are listed in Table 6.1
,
and the conversion
f
actors between
the SI and cgs unit systems are given in Table 6.2.