Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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36 6
Neutrinos
f
rom
S
tar
s
10
0
1
0
1
1
0
2
1
0
3
10
4
10
5
1
0
6
0
.
1
1
10
C
ross Section
[
10
−
46
cm
2
]
37
Cl
71
Ga
Neutrino Energy
[
MeV
]
Fig. 6.5 T
h
eneutrinoa
b
sorption cross sections o
n
37
C
land
7
1
G
aas
f
unctions o
f
the
energies of incident neutrinos
(
Bahcall
e
ta
l.
,
1996; Ba
h
ca
ll
, 1997. Wit
hp
ermissio
n
from the American Physical Society
)
trons
f
rom the electron-capture deca
y
o
f
3
7
A
r
(
i.e.
,
3
7
A
r+
e
−
→
3
7
C
l
+
ν
e
)
.
The main back
g
round is induced by the cascades of ener
g
etic muons fro
m
c
osmic rays. T
h
e casca
d
e partic
l
es, inc
l
u
d
ing pions, protons an
d
neutrons,
m
a
y
interact wit
h
3
7
C
l and produce
3
7
A
r. The back
g
round is estimated t
o
be
(
0
.
0
8
±
0
.
03
)
atoms of
37
Ar per day
(
Bahcall, 1989
)
.
Fig. 6.5 s
h
ows t
h
eneutrinoa
b
sorption cross section on t
h
ec
hl
orine. It is
easy to see t
h
at t
h
ec
hl
orine experiment is most sensitive to t
h
e
h
i
gh
-ener
gy
p
art o
f
8
B
neutrinos. The solar neutrino unit
(
SNU
)
has often been used t
o
express the neutrino capture rate
(
Bahcall, 1969, 1989
)
:
R
=
i
Φ
i
σ
∼
10
10
c
m
−
2
s
−
1
×
1
0
−
46
c
m
2
=
1
0
−
36
s
−
1
,
(
6.42
)
s
o1
S
NU corresponds to 10
−
36
interactions per tar
g
et atom per secon
d.
A
fter 25 years of measurements
(
from March 1970 to February 1994
)
,th
e
H
omestake experiment reported the following average solar neutrino rat
e
(
Cleveland
et al.
,
1998
):
R
e
x
p
C
l
=2
.
56
±
0
.
1
6
(
sta
t
.
)
±
0
.
16
(
syst
.
)
SNU =
2
.
5
6
±
0
.
23 SNU, which was onl
y
about 1
/
3
of the theoretical
p
rediction made
by Bahcall
(
Bahcall, 1989; 1997
)
. Current theoretical predictions are
R
th
Cl
=
8
.
4
6
+
0
.
87
−
0
.
88
S
NU based on the SSM of BPS
(
GS98
)
and R
th
C
l
=6
.
86
+0
.
69
−
0
.
70
S
NU
a
ccording to the SSM of BPS
(
AGS05
)(
Pena-Garay and Serenelli, 2008
)
.
There
f
ore, we conclude that the observed neutrino rate
g
iven above remains
a
bout one third o
f
the ex
p
ected one.
T
here were three other solar neutrino experiments using the radiochemical
m
ethod: GALLEX, GNO and SAGE, which detected solar neutrinos via the
i
nverse beta deca
y
o
f
71
G
a
(
Kuzmin, 1966
)
ν
e
+
7
1
Ga
→
7
1
Ge +
e
−
,
(
6.43
)
6.3 Experimental Detection o
fS
olar Neutrinos 23
7
w
h
ere t
h
eneutrinoener
g
yt
h
res
h
o
ld
i
s
E
th
ν
=
0
.
233
Me
V
.
S
uch a low valu
e
of
E
t
h
ν
m
akes it possible to detect solar neutrinos almost in the whole ener
g
y
range as shown in Fig. 6.4. In particular, a measurement of
pp
n
eutrino
s
f
r
o
m
t
h
e
pp
chain reactions is o
fg
reat importance because th
e
pp
n
eut
r
i
n
o
flux is closel
y
related to the solar luminosit
y
. The neutrino absorption cross
sect
i
o
n
o
n
71
G
a is shown in Fig. 6.5 as a function of the neutrino energ
y
(
from
0
.
24MeVto30MeV
)
. Let us briefly summarize the main results o
f
G
A
LLEX, GNO and S
A
GE ex
p
eriments.
(
1
)
The GALLium EXperiment
(
GALLEX
)
was done at the Laborator
i
N
azionali del Gran Sasso
(
LNGS
)
in Italy. Its detector contained 30
.
3
tons
of
71
G
a in the form of a concentrated
G
a
Cl
3
-HCl solution
,
and was ex-
posed to solar neutrinos
f
or a time period between three and
f
our weeks
.
The G
A
LLEX experiment ran from Ma
y
1991 to Januar
y
1997, with a
break in 1997
(
Hampel et al., 1999
)
, and was then followed by the Gal-
l
ium Neutrino Observatory
(
GNO
)
experiment. The GNO experiment use
d
the same detector of the G
A
LLEX ex
p
eriment but im
p
roved its extrac
-
tion equipment
(
Altmann
e
ta
l
., 2000, 2005
)
. A recent reanalysis of the
d
ata accumulated in the G
A
LLEX experiment
y
ields the solar neutrino rat
e
R
e
x
p
Ga
(
GALLEX
)
=73
.
4
+7
.
1
−
7
.
3
S
NU
(
Kaethe
r
et al
.
,
2010
)
. The two gallium ex
-
periments at LNGS totally lasted 12 years, and their joint result has bee
n
fou
n
dtobe
R
e
x
p
G
a
(
GALLEX + GNO
)
=6
9
.
3
+5
.
5
−
5
.
5
SNU
(
Altman
n
et al.
, 2005
).
(
2
)
The Soviet-American Gallium Experiment
(
SAGE
)
was located i
n
the Baksan Neutrino Observatory of the Russian Academy of Sciences i
n
the northern Caucasus mountains
(
Abazo
v
et al.
, 1991;
A
bdurashito
v
et al
.
,
1
994, 2002, 2006
)
. The detection method used in the SAGE experiment is
the same as that used in the GALLEX and GNO experiments. After runnin
g
f
rom Januar
y
1990 to December 2001, the S
A
GE experiment reported th
e
a
verage solar neutrino rat
e
R
e
x
p
Ga
(
SAGE
)
=70
.
8
+6
.
5
−
6
.
1
S
NU
(
Abdurashito
v
et
al.
,
2002
)
. This result is in good agreement with the one obtained from the
G
A
LLEX and GNO ex
p
eriments.
T
heoretical
p
redictions for the neutrino ca
p
ture rate o
n
71
G
aar
e
R
th
Ga
=
1
2
7
.
9
+8
.
1
−
8
.
2
S
NU based on the SSM of BPS
(
GS98
)
an
d
R
th
G
a
=
120
.
5
+
6
.
9
−
7
.
1
S
N
U
a
ccording to the SSM of BPS
(
AGS05
)(
Pena-Garay and Serenelli, 2008
)
.
E
ither of them is nearly twice the result obtained from the GALLEX
/
GNO
or S
A
GE experiment. In other words, there is a discrepanc
y
of more than
5
σ
between theoretical
p
redictions and ex
p
erimental data in this res
p
ect.
6
.
3
.2 Water
C
herenkov Detector
s
I
f
the velocities o
f
char
g
ed particles exceed the speed o
f
li
g
ht in a medium
,
the
y
can radiate photons. This phenomenon is known as
C
herenkov radiation
(
Landau and Lifshitz, 1984
)
. So an important neutrino detection method is to
m
easure
C
herenkov radiation emitted
f
rom ultra-relativistic char
g
ed leptons
which are produced
f
rom neutral- and char
g
ed-current neutrino interactions.
The
C
herenkov detector allows a real-time measurement of the direction of
2
38 6
Neutrinos
f
rom
S
tar
s
c
har
g
ed leptons, and thus it can determine the direction o
f
incident neutri-
n
os. Since the refractive index of li
g
ht in water is
n
≈
1
.
33
,
the Cherenkov
radiation forms a light cone with an open angle given by cos
θ
=1
/
(
n
v
)
.
Ta
k
in
g
t
h
eve
l
ocity
v
≈
c
f
or ultra-relativistic char
g
ed leptons, one obtain
s
θ
≈
4
1
◦
. The axis of the light cone is just the track of the running charge
d
l
epton. A water Cherenkov detector usually consists of many photomultiplier
tubes
(
PMTs
)
used to measure Cherenkov radiation. Both the energy an
d
d
irection of incident neutrinos can in
p
rinci
p
le be reconstructed
.
All the
C
herenkov detectors used to detect solar neutrinos make use of
water as the medium. Let us brie
fly
summarize the main results o
f
a
f
e
w
i
mportant solar neutrino experiments of this type
.
(
1
)
The Kamioka Nucleon Decay Experiment
(
Kamiokande
)
,whichorig
-
i
nall
y
aimed
f
or the detection o
f
nucleon deca
y
s, was completed in 1983. Its
d
etector was placed 1000 m underground in the Kamioka mine, about 200 k
m
west o
f
Tokyo. It had a cylindrical water tank containin
g
3000 tons o
f
pure
water.
A
bout 1000 PMTs were attached to the inner surface of the tank. I
n
1
985, t
h
e
d
etector was upgra
d
e
d
to Kamio
k
an
d
e-II in or
d
er to o
b
serve so
l
a
r
8
B
neutrinos. T
h
ere
l
evant process is t
h
e neutrino-e
l
ectron e
l
astic scatterin
g
ν
α
+
e
−
→
ν
α
+
e
−
,
(
6.44
)
whe
r
e
α
=
e
,
μ
or
τ
.
S
ince the cross section
f
or electron neutrinos is about six
times larger than those for muon and tau neutrinos, as shown in Eq.
(
2.79
)
,
the reaction in Eq.
(
6.44
)
is most sensitive to electron neutrinos. The energ
y
threshold
f
or recoil electrons was lowered
f
rom
7
.
5 MeV in Kamiokande
-
II
to
7
.
0 MeV in Kamiokande-III
(
Fukuda
e
tal.
,
1996
)
, implying that th
e
n
eutrino ener
g
y threshold decreased
f
ro
m
E
ν
=7
.
2
M
e
V
t
o
E
ν
=6
.
7M
e
V.
The measurements made b
y
the Kamiokande experiment
f
rom Januar
y
1987
to February 1995 led us to the flux of
8
B
neutrino
s
Φ
8
B
(
Kamiokande
)
=
(
2
.
80
±
0
.
19
±
0
.
3
3
)
×
10
6
cm
−
2
s
−
1
(
Fukud
a
et al
., 1996
)
. This result is onl
y
a
bout 1
/
2ofth
e
8
B neutrino flux predicted b
y
the SSM in Table 6.7.
(
2
)
The Super-Kamiokande
(
SK
)
experiment has a 50 kiloton water
C
herenkov detector, which is also located in the Kamioka mine, about 500
m
f
rom the cavit
y
occupied b
y
the Kamiokande detector. The SK detector con
-
s
ists of a stainless steel cylindrical water tank with about 11000 PMTs i
n
the inner detector and about 1900 PMTs in the outer detector
(
Fukud
a
et
a
l
.
,
2003
)
. The first phase of the SK experiment started in 1996 and ended
i
n 2001. The energy threshold for recoil electrons is 6
.
5
Me
V
for the first 2
80
d
a
y
san
d5
.
0MeV
f
or the remainin
g
time. The correspondin
g
neutrino ener
gy
thresholds ar
e
E
ν
=
6
.
2
MeV an
d
E
ν
=
4
.
7
MeV, respectivel
y
. The detecte
d
8
B
n
eut
rin
o
fl
u
x
was
Φ
8
B
(
SK
)
=
(
2
.
35
±
0
.
02
±
0
.
08
)
×
10
6
c
m
−
2
s
−
1
(
Hosaka
et al
., 2006
)
. In particular, the angular distribution of neutrino events was
m
easured in the SK experiment. It provided a convincing evidence that the
observed neutrinos really came from the Sun. Althou
g
h the SK experiment
s
earched
f
or the time variation o
f
solar neutrino events, the data were consis
-
tent with a null day-night asymmetry and the measured seasonal variation
6.3 Experimental Detection o
fS
olar Neutrinos 23
9
was compatible with the chan
g
e in the Sun-Earth distance.
A
fter an unfor-
tunate accident, the second
p
hase of the SK ex
p
eriment ran from Decembe
r
2
002 to October 2005 with a reduced PMT coverage and a higher energ
y
threshold
(
Craven
s
et al.
,
2008
)
.Someeffortsweremadeinthethirdphas
e
of this experiment, which ran from
A
ugust 2006 to
A
ugust 2008, to cut the
s
ystematic errors by a factor of two, to reduce the low-energy backgroun
d
a
n
d
to ac
h
ieve a
l
ow ener
g
yt
h
res
h
o
ld
E
ν
=4
.
5MeV
(
Yan
g
et al
.
,
2009
)
.
(
3
)
The Sudbury Neutrino Observatory
(
SNO
)
has a Cherenkov detector
l
ocated in the Creighton mine, near Sudbury, Ontario
(
Boger
et al.
,
2000
).
The detector consists o
f
a transparent acr
y
lic sphere
fi
lled with 1000 tons o
f
h
eavy water
D
2
O. The Cherenkov light is detected by about 9500 inward-
l
ookin
g
PMTs. The
S
N
O
experiment can measure solar neutrinos via char
g
ed
-
c
urrent
(
CC
)
, neutral-current
(
NC
)
and elastic scattering
(
ES
)
reactions:
CC :
ν
e
+D
→
p
+
p
+
e
−
,
N
C:
ν
α
+D
→
p
+
n
+
ν
α
,
E
S: ν
α
+
e
−
→
ν
α
+
e
−
,
(
6.45
)
w
h
ere
α
=
e
,
μ
o
r
τ
. The CC reaction in the SNO experiment has a threshol
d
o
f
5
.
5MeV
f
or the recoil ener
g
yo
f
the
fi
nal-state electron, correspondin
g
to
a
neutrino energy threshol
d
E
ν
=
6
.
9
MeV. Hence it is sensitive to sola
r
8
B
n
eutrinos. The N
C
reaction is sensitive to all three neutrino flavors with a
n
ener
g
yt
h
res
h
o
ld
E
ν
=
2
.
224 MeV, so it is ver
y
important to measure t
h
e
total flux of
8
B
neutrinos via this
p
rocess. The ES reaction has a neutrin
o
energy t
h
res
h
o
ld
E
ν
=5
.
7
MeV in the SNO experiment, and thus it is only
se
n
sitive to
8
B
n
eut
rin
os
.
T
he SNO experiment underwent three phases:
(
a
)
the
D
2
Op
hase ran fro
m
N
ovember 1999 to May 2001, and the neutron produced in the N
C
reactio
n
was detected via the
p
roces
s
n
+D
→
3
He
+
γ
;
(
b
)
the NaCl phase operated
f
rom July 2001 to August 2003, during which time about 2 tons of NaC
l
were added into heavy water to enhance the neutron detection efficiency
(
via
n
+
35
Cl
→
36
C
l
+
γ
’
s
)
and the ability to statistically separate the NC and C
C
s
ignals, leading to a significant improvement in the accuracy of the measure
d
ν
e
a
n
d
ν
x
(
fo
r
x
=
μ
o
r
τ
)
fluxes;
(
c
)
the final phase use
d
3
H
e to captur
e
the neutron, yieldin
g
a proton-tritium pair and thus an electronic pulse in
the counter wire
(
Aharmi
m
e
ta
l.
,
2008
)
. The fluxes of sola
r
8
B
neutrinos
obtained
f
rom di
ff
erent reactions in the Na
C
l phase o
f
the
S
N
O
experiment
a
re
(
Aharmi
m
e
tal
.
,
2005
)
Φ
S
N
O
CC
=
1
.
68
+
0
.
06
−
0
.
06
+0
.
08
−
0
.
09
×
10
6
c
m
−
2
s
−
1
,
Φ
S
N
O
E
C
=
2
.
35
+
0
.
22
−
0
.
22
+0
.
15
−
0
.
1
5
×
1
0
6
c
m
−
2
s
−
1
,
Φ
S
N
O
N
C
=
4
.
94
+0
.
21
−
0
.
21
+
0
.
38
−
0
.
34
×
1
0
6
c
m
−
2
s
−
1
.
(
6.46
)
24
06
Neutrinos
f
rom
S
tar
s
T
he
r
at
i
o
Φ
S
N
O
CC
/Φ
S
N
O
NC
=0
.
340
±
0
.
023
+
0
.
0
2
9
−
0
.
031
d
i
ff
ers
f
rom unit
y
b
y
1
7
σ
.
T
hese
results are crucial in establishin
g
neutrino oscillations as the best solution t
o
the solar neutrino problem, as one can see in Section 6.4.
6.3.3 Future
S
olar Neutrino Ex
p
eriments
We have summarized some solar neutrino ex
p
eriments based on the radio
-
c
hemical methods and Cherenkov detectors. There is another ongoing ex
-
periment, Borexino, w
h
ic
h
uses t
h
e
l
iqui
d
scinti
ll
ator
d
etector an
d
aims t
o
observe the low-energy components of solar neutrinos
(
Alimont
i
et al
., 2009a,
2
009b
)
. The Borexino detector is located at the underground LNGS in cen
-
tral Ital
y
.
S
olar neutrinos are detected via elasti
c
ν
e
-
e
−
s
catterin
g
,an
d
t
he
s
cintillator li
g
ht induced by the recoil electrons is observed by the PMTs
.
The Borexino Collaboration has recently reported the first result of
8
B
so-
l
ar neutrinos wit
h
an ener
g
yt
h
res
h
o
ld
E
ν
=3MeV
(
Bellin
i
et al.
,
2010a
)
:
Φ
8
B
(
Borexino
)
=
(2
.
4
±
0
.
4
±
0
.
1)
×
1
0
6
cm
−
2
s
−
1
, which is in
g
ood a
g
ree-
m
ent with the results of the SNO and SK experiments. In addition, a direct
measu
r
e
m
e
n
tof
7
B
eneutrinos
(
E
ν
=0
.
862 MeV
)
in the Borexino experi-
m
ent
y
ields the neutrino interaction rate 49
±
3
(
stat
.
)
±
4
(
syst.
)
counts pe
r
d
ay per 100 tons from an analysis of the data accumulated during the period
between May 2007 and
A
pril 2008, while the prediction
g
iven by the hi
g
h-
m
etallicit
y
SSM is 7
4
±
4 counts per day per 100 tons
(
Arpesell
a
e
tal.
,
2008
)
.
The observation o
f
such low-ener
g
y neutrinos is very important in testin
g
n
uclear reactions in the
S
un and matter e
ff
ects on solar neutrinos with di
f-
f
erent energies. It is worth mentioning that the Borexino Collaboration ha
s
al
so o
b
serve
dg
eoneutrinos — t
h
ee
l
ectron antineutrinos pro
d
uce
d
in t
h
e
b
eta
d
ecays of radioactive isotopes in the Earth
(
Bellin
i
et al.
,
2010b
)
.
T
he future solar neutrino experiments will extensively make use of th
e
s
cintillator detectors, and nearl
y
all o
f
them have other scienti
fi
c purposes
s
uch as a search for dark matter, neutrinoless double-beta deca
y
s and proto
n
d
ecays
(
Klein, 2008
)
. The main goal is to measure low-energ
y
pp
,
pep
,
7
Be an
d
C
N
O
neutrinos by lowerin
g
the relevant back
g
rounds. The Borexino experi-
m
ent has made an important step forward in measurin
g
7
Be neutrinos
,
which
a
re crucial for understanding neutrino oscillations in matter and physics in
the solar core. The KamL
A
ND experiment plans to reduce its back
g
rounds
b
y
means of scintillator purification, such that
7
B
e neutrinos can be observed
via the elastic neutrino-electron scattering reaction. The SNO+ experimen
t
is also based o
n
the sc
in
t
i
llato
r
detecto
r
a
n
d the elast
i
c
ν
e
-
e
−
s
catter
i
n
g
pro-
c
ess
,
and it will be sensitive t
o
p
e
p
a
nd CNO neutrinos. The
p
ro
p
osed LEN
S
experiment uses t
h
ein
d
ium-
d
ope
d
scinti
ll
ator an
d
t
h
us is sensitive to t
he
c
har
g
ed-current interactio
n
ν
e
+
1
1
5
In
→
e
−
+
1
1
5
S
n
+2
γ
.The
f
ull LEN
S
d
etector is expected to measure the energy spectrum of
pp
neutrinos and t
o
d
etect the si
g
nals o
f
7
B
e,
C
N
O
an
d
pep
n
eutrinos
(
Raghavan, 2002
)
.Th
e
CLE
A
NandXM
A
SS ex
p
eriments use the li
q
uid neon and xenon, res
p
ec-
tively, to observe the scintillator light in order to detect dark matter. The
6
.4
S
olar Neutrino
O
scillations 24
1
l
ow-ener
g
y thresholds o
f
these experiments allow a measurement o
f
sola
r
pp
n
eutrinos
(
McKinsey and Coakley, 2005
)
.
6
.4
S
olar Neutrino
O
scillations
It has been seen that all the solar neutrino experiments pointed to a ver
y
s
i
g
ni
fi
cant discrepancy between the measured neutrino
fl
ux and the expecte
d
one from the SSM. This is the well-known solar neutrino problem
(
Davis,
1
994
)
. In this section we first describe this problem and then show that its
best solution comes
f
rom neutrino oscillations to
g
ether with the Mikheyev
-
Smirnov-Wolfenstein
(
MSW
)
matter effects
(
Wolfenstein, 1978; Mikheye
v
a
nd Smirnov, 1985, 1986
)
. It is also possible to place strict constraints o
n
the li
f
etimes and electroma
g
netic properties o
f
neutrinos
f
rom solar neutrin
o
oscillation ex
p
eriments
.
6.4.1 The
S
olar Neutrino Proble
m
F
ig. 6.6 offers a comparison between the rates of solar neutrinos predicte
d
by the SSM of BPS
(
GS98
)
200
8
3
a
n
d
m
easu
r
ed
in
the sola
rn
eut
rin
oe
x
-
periments
(
Pena-Garay and Serenelli, 2008
)
. To examine whether the sola
r
n
eutrino problem is real or not, one should
fi
rst o
f
all make the theoretica
l
u
ncertainties o
f
the
SS
M itsel
f
as small as
p
ossible. Note that the
SS
Mis
c
onstructed based on some important input parameters
(
e.g., the chemica
l
c
omposition, radiative opacity and nuclear reaction rates
)
which may affec
t
i
ts predictions for solar neutrino fluxes
(
Bahcall, 1989
)
. Some comments o
n
these parameters are in order
(
Pena-Garay and Serenelli, 2008
)
.
(
1
)
The abundances of heavy elements have a great impact on the calcula
-
tions of the CNO neutrino fluxes. Hence the solar com
p
osition dominates th
e
u
ncertainties in the fluxes of
13
N
(
13
%)
,
15
O
(
12
%)
and
17
F(
17
%)
neutrinos
(
as one can see in Table 6.7
)
, but it is not dominant for other neutrinos
.
(
2
)
Fo
r
8
Band
7
Be neutrinos, the main uncertainties
(
6
.
8
%
and 3
.
2
%
)
c
ome from the opacity, which is closely related to solar metallicity
(
especiall
y
the abundance of iron
)
. The reason is that the fluxes of
8
Ba
n
d
7
B
e
n
eut
rin
os
a
re highly sensitive to the temperature, determined mainly by the opacity
.
(
3
)
The astrophysica
l
S
factors cause some subleading uncertainties i
n
evaluatin
g
the solar neutrino
fl
uxes. For example
,
S
33
a
n
d
S
34
c
orrespon
d
in
g
to
3
He
+
3
He
→
4
H
e
+
2
p
and
3
He
+
4
H
e
→
7
B
e
+
γ
react
i
ons g
i
ve r
i
s
e
to 2
.
5%
and
2
.
8
%
uncertainties for
7
Be neutrinos; and the
f
acto
r
S
1
7
f
r
om
t
h
e
r
eact
i
o
n
7
Be
+
p
→
8
B+
γ
i
nduces a 3
.
8% uncertaint
y
. Thermal and
gravitational diffusion effects may lead to comparable uncertainties
.
3
U
se
f
ul numerical tables and
p
lotsbasedonthe
SS
Mcanbe
f
ound
f
rom the
website http://www.mpa-garching.mpg.de/
∼
a
ldos/.
242
6
Neutrinos
f
rom
S
tar
s
Fig. 6.6 The rates of solar neutrinos from theoretical predictions and experimental
data
(
Pena-Garay and Serenelli, 2008
)
I
tiso
b
vious t
h
at t
h
e uncertainties
g
iven in Ta
bl
e6.7an
d
Fi
g
.6.6can-
n
ot accommodate the remarkable discrepancy between theoretical predictions
a
nd experimental results. In particular, the NC events
(
sensitive to all thre
e
n
eutrino flavors
)
measured in the SNO experiment show that the total flux
of
8
B
neutrinos is consistent with that predicted by the SSM. So the sola
r
n
eutrino problem should be resolved by reexaminin
g
the basic properties o
f
n
eutrinos themselves instead o
f
modi
f
yin
g
the
SS
M. The most convincin
g
s
olution to this problem turns out to be neutrino oscillations. On the other
h
an
d
,so
l
ar neutrino osci
ll
ation experiments are now a
bl
etoma
k
eimportant
c
ontributions to the theory of stars
(
Serenelli, 2010; Basu and Antia, 2008
)
.
6
.4.2 The M
S
W Matter Effects
The experimental results illustrated in Fi
g
. 6.6 indicate that the
fl
uxes o
f
s
olar electron neutrinos must have chan
g
ed durin
g
their journey from the
s
olar core to the detector, and a proper solution to the observed deficit mus
t
h
ave an ener
g
y-dependent
f
eature. The standard solution to this problem i
s
the oscillation of solar neutrinos, a
p
ure
q
uantum
p
henomenon which allow
s
electron neutrinos to transform into muon and
(
or
)
tau neutrinos when they
travel
f
rom the solar core to the Earth. Let us explain the key points by takin
g
the two-flavo
r
ν
e
↔
ν
μ
o
scillations for example. The survival probability of
i
nitia
l
e
l
ectron neutrinos is
g
iven
b
y
P
(
ν
e
→
ν
e
)
=
1
−
s
in
2
2
θ
si
n
2
Δ
m
2
L
4
E
,
(
6.47
)
6
.4
S
olar Neutrino
O
scillations 24
3
whe
r
e
θ
is t
h
e neutrino mixin
g
an
gl
e in vacuum, an
d
Δ
m
2
≡
m
2
2
−
m
2
1
de
n
otes
the mass-squared difference between two neutrino mass ei
g
enstate
s
ν
1
an
d
ν
2
.
Since the energy spectrum of solar neutrinos is continuous, the probability
P
(
ν
e
→
ν
e
)
should be averaged over the spectrum. This treatment lead
s
u
stotheavera
g
ed probability
P
(
ν
e
→
ν
e
)
=1
−
0
.
5
s
i
n
2
2
θ
(
Gribov and
P
ontecorvo, 1969
)
. Hence the minimal value of this survival probability i
s
1
/
2whe
n
θ
=
π
/
4
holds.
C
onsiderin
g
the
S
N
O
measurements o
f
sola
r
8
B
n
eutrinos
(
i.e., the measured neutrino flux is only about one third of the
total neutrino flux
)
, one immediately observes that the above treatment is
n
ot applicable.
A
correct interpretation of all the solar neutrino experiment
s
h
as to take into account the matter effects on neutrino oscillations.
Neutrino oscillations in matter have been described in
C
hapter 5. Her
e
we appl
y
the main results to the case o
f
solar neutrino oscillations. First, w
e
estimate the effective
p
otential relevant to ν
e
→
ν
μ
o
scillations in the solar
c
ore, w
h
ere t
h
e matter
d
ensity is a
b
out
ρ
C
= 150 g
cm
−
3
a
n
d the elect
r
o
n
n
umber densit
y
i
s
n
e
=
100
N
A
N
c
m
−
3
w
i
th
N
A
N
=6
.
02
2
×
1
0
23
b
ein
g
th
e
A
vogadro constant
(
Bahcall, 1989
)
. So the effective potential reads
V
=
√
2
G
F
n
e
≈
7
.
6
×
10
−
12
eV
.
(
6.48
)
F
or the electron neutrinos with a typical energ
y
E
∼
10 MeV
,
the reso-
na
n
ce co
n
d
i
t
i
on
A
=2
V
E
=
Δ
m
2
cos 2
θ
ca
n
be sat
i
sfied
i
f
Δ
m
2
cos 2
θ
=
1
.
5
×
10
−
4
eV
2
holds. In
f
act, the electron number densit
y
o
f
the
S
un de-
c
reases from its core to its surface. One has an approximate exponential la
w
n
e
(
r
)
=
n
0
e
−
r
/
r
0
,
w
h
er
e
n
0
=
2
45
N
A
N
cm
−
3
a
n
d
r
0
=0
.
1
R
i
nt
h
eran
g
e
of 0
.
1
<r
/R
<
0
.
9. Therefore
,
whether the neutrinos can encounter th
e
resonance point depends on their energy and mixing parameters
(
Mikheyev
a
nd Smirnov, 1985; Kuo and Pantaleone, 1989
).
Next
,
we assume ta
n
2
θ
∼
0
.
5an
d
Δ
m
2
∼
8
×
1
0
−
5
e
V
2
. In this case on
e
c
an see t
h
at t
h
e resonance energy at t
h
eso
l
ar core is given
by
E
r
e
s
=
Δ
m
2
cos
2
θ
2
V
≈
1
.
75
Me
V
,
(
6.49
)
where the value o
f
V
obtained in Eq.
(
6.48
)
has been input. This estimate
i
mplies that matter e
ff
ects are insi
g
ni
fi
cant
f
or those solar neutrinos wit
h
energies below
E
r
es
;
in other words, such low-energy neutrinos mainly undergo
vacuum oscillations. The Borexino experiment has measured the flux o
f
7
Be
n
eutrinos, w
h
ose ener
g
ies are we
ll b
e
l
ow t
h
e resonance ener
g
y
E
re
s
.
S
oit
i
smorea
pp
ro
p
r
i
ate to us
e
ν
e
→
ν
e
o
scillations in vacuum to inter
p
ret the
experimental data of
7
Be neutrinos
(
i.e., the observed neutrino flux is about
0.6 of the expected one, as shown in Fig. 6.6
)
.Give
n
P
(
ν
e
→
ν
e
)
=
1
−
0
.
5
sin
2
2
θ
≈
0
.
6, a straightforward calculation leads us to
θ
≈
32
◦
,
which is
co
n
s
i
ste
n
tw
i
t
h
t
h
eva
l
ues e
x
t
r
acted f
r
o
m
ot
h
e
r
so
l
a
rn
eut
rin
oosc
ill
at
i
o
n
data.
I
f
the ener
g
ies o
f
solar neutrinos are above
E
r
e
s
, the resonant
fl
avor conversion
m
ay take place a
t
r
=
r
r
es
,
where the resonance condition
A
=
Δ
m
2
cos 2
θ
is
244
6
Neutrinos
f
rom
S
tar
s
s
atisfied. In this case the adiabaticity parameter defined in Eq.
(
5.30
)
can b
e
estimated as follows:
γ
≡
Δ
m
2
s
in
2
2
θ
2
E
cos 2
θ
1
n
e
(
r
)
d
n
e
(
r
)
d
r
−
1
r
=
r
r
es
∼
2M
e
V
E
×
10
5
,
(
6.50
)
whe
r
eta
n
2
θ
∼
0
.
5
an
d
Δ
m
2
∼
8
×
10
−
5
e
V
2
h
ave
b
een input. T
h
is resu
lt
i
mplies that the resonance point is adiabatically crossed. Hence the avera
g
e
d
s
urvival probability is determined by Eq.
(
5.32
)
with the vanishing Laudau
-
Zener pro
b
a
b
i
l
it
y
P
c
P
P
=
0:
P
(
ν
e
→
ν
e
)
=
1
2
[
1+cos2θ
m
(
r
i
)
cos
2
θ
]
,
(
6.51
)
whe
r
e
θ
m
(
r
i
)
is the effective neutrino mixing angle in matter at the neutrin
o
pro
d
uction poin
t
r
i
.F
o
r
8
B neutrinos with ener
g
ies
f
ar abov
e
E
r
e
s
,one
h
as
A
Δ
m
2
cos 2
θ
a
nd thus θ
m
(
r
i
)
≈
π
/
2
as indicated by Eq.
(
5.18
)
.Wear
e
s
imply left with
P
(
ν
e
→
ν
e
)
≈
s
in
2
θ
∼
1
/
3(
for tan
2
θ
∼
0
.
5
)
, which is jus
t
t
h
eobse
r
ved
r
ate of
8
B neutrinos in the
S
N
O
experiment
.
Now that matter effects depend on both the neutrino energies and fla
-
vor mixin
g
parameters, a
f
ull determination o
f
the solar neutrino mixin
g
a
n
g
le and mass-squared di
ff
erence requires a
g
lobal analysis o
f
all the avail
-
a
ble experimental data on solar neutrino oscillations
(
with different energy
thresholds
)
. The earlier data from chlorine, gallium, Kamiokande and S
K
experiments indicated four possible regions in the
(
tan
2
θ,
Δ
m
2
)
parameter
s
pace, but only the large-mixing-angle MSW solution was singled out by the
SNO experiment and confirmed by the KamLAND experiment
(
Eguch
i
et
a
l
.
,
2003
)
. In the three-flavor framework it is possible to determine or con-
s
train three neutrino mixing angles
(
θ
12
,
θ
13
an
d
θ
23
)
and two mass-square
d
d
ifferences
(
Δ
m
2
21
a
n
d
Δ
m
2
32
)
from a global analysis of current data on solar,
a
tmospheric, reactor and accelerator neutrino oscillations
(
Gonzalez-Garcia
a
nd Maltoni, 2008
)
. A brief summary of the present values of these five pa
-
rameters
h
as
b
een
g
iven in Ta
bl
es 3.1 an
d
3.2
.
6.4.3
C
onstraints on Neutrino Pro
p
ertie
s
In the early days some interesting ideas other than neutrino oscillations,
s
uch as the spin-flavor oscillation
(
Voloshin and Vysotskii, 1986; Voloshi
n
et
a
l
.
,
1986a, 1986b
)
and neutrino decays
(
Bahcall
e
tal
.
,
1972
;A
cker
e
tal.
,
1
991; Acker and Pakvasa, 1994
)
, were proposed to solve the solar neutrin
o
problem. Toda
y
we believe that the spin-
fl
avor oscillation or neutrino deca
ys
c
an at most belon
g
to the subleadin
g
effects as compared with solar neutrin
o
oscillations. One may constrain the magnetic dipole moments or lifetimes of
n
eutrinos
b
yusin
g
more accurate so
l
ar neutrino osci
ll
ation
d
ata
.
R
e
f
erences 24
5
If
neutrinos have
fi
nite ma
g
netic dipole moments, their spins may b
e
flipped in stron
g
ma
g
netic fields.
A
s a consequence, the left-handed neu
-
trin
o
ν
L
wit
h
negative
h
e
l
icity can
b
e converte
d
into t
h
erig
h
t-
h
an
d
e
d
neu
-
t
r
i
n
o
ν
R
wit
h
positive
h
e
l
icit
y
.T
h
e
l
atter
d
oes not ta
k
epartint
h
estan
d
ar
d
weak interactions and thus esca
p
es the ex
p
erimental detection. Hence it was
proposed that the solar neutrino deficit could be ascribed to the spin pro-
c
ession effects
(
Cisneros, 1971; Pulido, 1992
)
. But the conversion rate fo
r
ν
L
→
ν
R
in the solar core was highly suppressed by matter effects
(
Voloshin
a
n
d
Vysots
k
ii, 1986; Vo
l
os
h
i
n
et al.
,
1986a, 1986b
)
. It was later realized that
the spin-
fl
avor oscillatio
n
ν
α
L
→
ν
β
ν
R
mi
gh
t
b
e resonant
l
yen
h
ance
d
in t
h
e
Sun if neutrinos had finite transition dipole moments
(
Akhmedov, 1988; Lim
a
nd Marciano, 1988
)
. If neutrinos are Majorana particles, the solar neutrino
fl
uxes are likel
y
to contain electron antineutrinos which ma
y
result
f
rom the
sp
in-flavor oscillation ν
e
→
ν
μ
a
nd the standard oscillation ν
μ
→
ν
e
.
By
m
easurin
g
the antineutrino component o
f
a solar neutrino
fl
ux via the in-
ve
r
se beta
r
eact
i
o
n
ν
e
+
p
→
n
+
e
+
,
the Kamiokande ex
p
eriment has set
s
ome restrictive limits on the solar
ν
e
flux
(
Barbier
i
e
ta
l.
,
1991
)
. More re
-
c
entl
y
,theKamL
A
ND experiment
y
ield
s
Φ
ν
e
<
3
.
7
×
10
2
c
m
−
2
s
−
1
fo
r
so
l
a
r
electron antineutrinos at the 90% confidence level
(
Eguch
i
et al., 2004
)
.Inth
e
c
ase of random magnetic fields inside the Sun, the above constraint o
n
Φ
ν
e
c
an be translated into a limit on the ma
g
netic dipole moment o
f
neutrinos
:
μ
ν
<
afew
×
10
−
12
μ
B
(
Miranda
e
tal
.
,
2004
).
I
f neutrinos are unstable, a heavier neutrino may decay into the light-
est one p
l
us a p
h
oton or an invisi
bl
enewpartic
l
e. In t
h
is case t
h
eneu
-
trino flux with an energ
y
E
over a distance L will be diminished b
y
a factor
exp
(
−
t/τ
lab
ττ
)
=exp
[
−
(
L
/E
)(
m
i
/τ
i
ττ
)]
,wher
e
m
i
a
n
d
τ
i
ττ
a
r
et
h
e
m
ass a
n
d
lif
e-
time o
f
the neutrino mass ei
g
enstat
e
ν
i
in the rest frame
(
Beacom and Bell
,
2
002
)
. Since radiative neutrino decays and their experimental constraint
s
h
ave been discussed in Chapter 3, here we only consider the
ν
i
→
ν
j
ν
+
φ
d
eca
y
wit
h
φ
being the Majoron
(
Chikashige
et al
.
,
1981;
G
elmini and Ron-
c
adelli, 1981
)
. Then the relevant mass eigenstate of solar neutrinos i
s
ν
2
a
nd
τ
2
τ
τ
/m
2
>
10
−
4
s
e
V
−
1
c
an be extracted from the experimental data
(
Beacom
a
nd Bell, 2002; Joshi
p
ura
et al.
, 2002; Band
y
opadh
y
a
y
et al.
, 2003
)
. For in-
visible neutrino decays, the best direct limit i
s
τ
1
/
m
1
>
1
0
5
s
e
V
−
1
o
btaine
d
f
rom the Supernova 1987A explosion. The future refinement of the SSM itsel
f
a
nd the precision measurement of solar neutrinos will definitel
y
improve th
e
c
onstraints on the intrinsic properties of neutrinos.
R
e
f
erence
s
A
bazov
,A
.I.
,
et a
l.
(
SAGE Collaboration
)
, 1991, Phys. Rev. Lett.
67
,
3332.
A
bdurashitov, J. N.,
et al.
(SAGE Collaboration), 1994, Phys. Lett. B
3
2
8
, 234.
A
bdurashitov
,
J. N.
,
et al
.
(
SAGE Collaboration
)
, 2002, JETP
95
, 181
.
A
bdurashitov
,
J. N.
,
et a
l.
(
SAGE Collaboration
)
, 2006, Astropart. Phys
.
2
5
,
349
.