Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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56
2 Neutrinos within the
S
tandard Model
w
i
th
μ
b
ein
g
the chemical potential. In the absence o
f
the chemical potential
a
nd in the rest frame, one can easily reproduce the propagator in Eq.
(
2.142
)
.
F
or instance, we look at the background-dependent term in the neutrino self-
energy from Fig. 2.7
(
a
)(
N¨otzold and Raffelt, 1988; Nieves, 1992
)
:
−
i
Σ
a
(
k
)=
i
g
√
2
2
d
4
p
(
2
π
)
4
γ
μ
P
L
PP
−
i
(
p
−
k
)
2
−
M
2
W
MM
˜
S
11
F
(
p
)
γ
μ
P
L
PP
,
(
2.145
)
w
h
ere
˜
S
11
F
singles out the temperature-dependent part of the full electron
propa
g
ator,
P
L
PP
≡
(1
−
γ
5
)
/
2isthele
f
t-handed pro
j
ection operator, and th
e
terms suppressed b
y
m
2
e
/M
2
W
M
M
have been neglected. Assuming
(
p
−
k
)
2
M
2
W
MM
a
n
d
using t
h
ei
d
entit
y
γ
μ
/
pγ
//
μ
=
−
2
/
p
//
,
we simplify Eq.
(
2.145
)
to
Σ
a
(
k
)=
g
2
M
2
W
M
M
d
4
p
(2
π
)
3
δ
(
p
2
−
m
2
e
)
/
pP
//
L
P
P
η
(
p
·
u
)
.
(
2.146
)
Given its Lorentz structure, the above integration should uniquely depend
o
n
/
u
//
.
In order to
f
urther identi
fy
the coe
ffi
cient, we work in the rest
f
rame o
f
the medium wit
h
u
μ
=
(
1
,
0
)
.ThenEq.
(
2.146
)
become
s
Σ
a
(
k
)
|
u
μ
=
(
1
,
0
)
=
g
2
M
2
W
MM
γ
0
P
L
PP
d
4
p
(2
π
)
3
p
0
δ
(
p
2
−
m
2
e
)
η
(
p
0
)
=
g
2
M
2
W
MM
γ
0
P
L
PP
d
3
p
(2
π
)
3
+
∞
−∞
p
0
δ
(
p
2
−
m
2
e
)
η
(
p
0
)d
p
d
0
,
(
2.147
)
and the integration over d
p
d
0
can be evaluated as follows
:
+
∞
−∞
p
0
1
2
|
p
0
|
δ
(
p
0
−
E
p
)
+
δ
(
p
0
+
E
p
)
η
(
p
0
)d
p
d
0
=
1
2
+
∞
0
δ
(
p
0
−
E
p
)
Θ
(
p
0
)d
p
d
0
e
β
(
p
0
−
μ
)
+1
−
1
2
0
−∞
Θ
(
−
p
0
)
δ
(
p
0
+
E
p
)d
p
d
0
e
−
β
(
p
0
−
μ
)
+
1
=
1
2
1
e
β
(
E
p
−
μ
)
+1
−
1
e
β
(
E
p
+
μ
)
+1
(
2.148
)
w
i
th
E
p
≡
|
p
|
2
+
m
2
e
. Substituting Eq.
(
2.148
)
into Eq.
(
2.147
)
, we find
Σ
a
(
k
)
|
u
μ
=
(
1
,
0
)
=
g
2
4
M
2
W
MM
γ
0
P
L
PP
(
n
e
−
−
n
e
+
)
,
(
2.149
)
where
n
e
−
(
or
n
e
+
)
denotes the number density of electrons
(
or positrons
)
:
n
e
∓
≡
g
e
∓
d
3
p
(2
π
)
3
·
1
e
β
(
E
p
∓
μ
)
+1
(
2.150
)
w
i
th
g
e
∓
=
2bein
g
an internal de
g
ree o
ff
reedom. The
g
eneral covariant
f
or
m
of the neutrino self-energy in Eq.
(
2.146
)
turns out to b
e
2.3 Neutrino Propagation in a Me
d
ium 57
Σ
a
(
k
)=
√
2
G
F
(
n
e
−
−
n
e
+
)
/
uP
//
L
PP
.
(
2.151
)
N
ow let us examine how the above self-ener
g
y in the rest frame of the mediu
m
a
ffects the dispersion relation of neutrinos. The latter is just the solution to
t
h
e equation
0
=Det
[
/
k
/
/
−
Σ
a
(
k
)]
=Det
E
−
c
2
k
·
σ
−
k
·
σ
c
2
−
E
=
(
E
−
c
2
)
2
−
|
k
|
2
2
,
(
2.152
)
whe
r
e
c
2
=
√
2
G
F
(
n
e
−
−
n
e
+
)
. It becomes evident that the dispersion relatio
n
fo
rm
ass
l
ess
n
eut
rin
os
i
s
c
2
=
E
−
|
k
|
,
s
o
c
2
is
j
ust the e
ff
ective potential
V
d
efined in Eq.
(
2.109
)
. This result is consistent with our previous calculation
s
by means o
f
the coherent scatterin
g
amplitude and the e
ff
ective Hamiltonia
n
i
n the absence o
f
positrons in matter.
O
ne ma
yf
ollow a similar procedur
e
to com
p
ute the effective
p
otentials or the refractive indices of neutrinos in
-
d
uced by the processes in Fig. 2.7
(
b
)
and Fig. 2.7
(
c
)
. The final results for the
effective potentials are summarized below
(
N¨otzold and Raffelt, 1988
).
(
1
)
In th
e
e
±
b
ac
k
groun
d
:
V
e
=
V
cc
e
+
V
n
c
e
V
=
1
√
2
G
F
1
+4
si
n
2
θ
w
(
n
e
−
−
n
e
+
)
,
V
μ
V
=
V
τ
VV
=
V
n
c
μ
V
=
V
nc
τ
VV
=
−
1
√
2
G
F
1
−
4s
i
n
2
θ
w
(
n
e
−
−
n
e
+
)
.
(
2.153
)
(
2
)
In the nucleon background:
V
p
α
V
=
1
√
2
G
F
1
−
4s
i
n
2
θ
w
n
p
−
n
p
,
V
n
α
V
=
−
1
√
2
G
F
(
n
n
−
n
n
)
,
(
2.154
)
whe
r
e
α
=
e
,
μ
o
r
τ
.
(
3
)
In the neutrino background:
V
ν
α
=
√
2
G
F
1+
δ
α
β
n
ν
β
−
n
ν
β
,
(
2.155
)
w
h
e
r
e
ν
β
ν
(
o
r
ν
β
)
denote the background neutrinos
(
or antineutrinos
).
For the antineutrino beam in a given medium, the corresponding effective
potentia
l
ta
k
es t
h
e opposite si
g
n. Note t
h
at on
l
yt
h
e
l
ea
d
in
g
-or
d
er resu
l
t
s
h
ave been presented here. More
g
eneral results includin
g
the next-to-leadin
g-
order corrections can be found in some literature
(
Botella et a
l
.
,
1987
;
N¨otzo
ld
a
nd Ra
ff
elt, 1988; Nieves, 1992; Mirizz
i
et al
., 2009
)
.
58 2 Neutrinos within the
S
tandard Model
R
eference
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3
N
eutrinos be
y
ond the
S
tandard Mode
l
Some salient and model-inde
p
endent
p
ro
p
erties of massive neutrinos will b
e
i
ntroduced in this chapter. First of all, we shall summarize current experimen
-
tal evidence
f
or neutrino oscillations, which point to
fi
nite neutrino masse
s
a
nd lepton flavor mixin
g
an
g
les. Then we shall describe how to write out th
e
m
ass terms of Dirac and Majorana neutrinos, analyze their C, P and T prop-
erties, and discuss their electroma
g
netic properties. The phenomenolo
g
yo
f
l
epton flavor mixing and CP violation, together with the running behaviors
of neutrino mass parameters from one energy scale to another, will also b
e
d
i
scussed
in
so
m
edeta
i
l
.
3
.1 Ex
p
erimental Evidence
f
or Neutrino Masses
The mass o
f
an elementary particle represents its inertial ener
g
y when it exists
a
t rest. Hence a mass
l
ess partic
l
e
h
asnowaytoexistatrest—instea
d
,it
m
ust always move at the speed of li
g
ht.
A
massive fermion must exist in bot
h
l
eft-handed and ri
g
ht-handed states, because the field operators responsibl
e
f
or the non-vanishing mass of a fermion have to be bilinear products of the
s
pinor fields which flip the fermion’s handedness or chirality
(
Xing, 2004
)
.
T
he standard model
(
SM
)
of electroweak interactions contains three neu
-
trinos
(
ν
e
,
ν
μ
,
ν
τ
)
which are purely left-handed and massless. Note that th
e
m
asslessness o
f
the photon in the
S
Mis
g
uaranteed by the electroma
g
neti
c
U
(
1
)
Q
gauge symmetry.
A
lthough the masslessness of three neutrinos cor
-
respon
d
stot
h
e
l
epton num
b
er conservation
1
,
t
h
e
l
atter is an acci
d
enta
l
sy
mmetr
y
rather than a
f
undamental s
y
mmetr
y
o
f
the
S
M. There
f
ore, man
y
1
I
tisactua
lly
t
he
B
−
L
s
y
mmetr
y
t
h
at ma
k
es neutrinos exact
ly
mass
l
ess in t
h
e
SM
,
wher
e
B
=baryonnumberan
d
L
= lepton number. The reason is simply that
a
n
eut
rin
oa
n
da
n
a
n
t
in
eut
rin
o
h
ave d
iff
e
r
e
n
tva
l
ues o
f
B
−
L
(
Witten, 2001
)
.Thus
the naive ar
g
ument
f
or massless neutrinos is valid to all orders in perturbation and
non-perturbation theories, if B
−
L is an exact symmetry.
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
62 3 Neutrinos beyond the
S
tandard Mode
l
p
h
ysicists stron
gl
y
b
e
l
ieve
d
t
h
at neutrinos s
h
ou
ld b
e massive even
l
on
gb
e
-
f
ore some incontrovertible ex
p
erimental evidence for massive neutrinos wer
e
a
ccumulated. A good reason for this belief is that neutrinos are more natura
l
to be massive than to be massless in some grand unified theories
(
GUTs
),
s
uch as the
SO
(
10
)
theory
(
Fritzsch and Minkowski, 1975; Georgi, 1975
)
,
which try to unify the electromagnetic, weak and strong interactions on the
one
h
an
d
an
d
t
h
e
l
eptons an
d
quar
k
sont
h
eot
h
er
h
an
d
.
I
f neutrinos are massive and their masses are non-degenerate, it will in
general be impossible to find a flavor basis in which the coincidence be
-
tween flavor and mass eigenstates holds both for charged leptons
(
e
,
μ
,τ
)
a
nd for neutrinos
(
ν
e
,
ν
μ
,ν
τ
)
. In other words, the phenomenon of flavor mix
-
i
n
g
is natura
ll
y expecte
d
to appear
b
etween t
h
ree c
h
ar
g
e
dl
eptons an
d
t
h
ree
m
assive neutrinos, just like the phenomenon o
ffl
avor mixin
g
between thre
e
u
p-type quarks
(
u,
c
,t
)
and three down-type quarks
(
d,
s
,b
)(
Cabibbo, 1963
;
K
obayashi and Maskawa, 1973
).
T
he neutrino oscillation, or the spontaneous periodic chan
g
e
f
rom one
n
eutrino flavor eigenstate to another, is a wonderful quantum phenomenon
which can naturall
y
take place i
f
neutrinos are massive and lepton
fl
avors are
m
ixed. In a simple two-neutrino mixin
g
scheme, the neutrino flavor ei
g
en
-
st
a
t
e
s
ν
α
an
d
ν
β
ν
are linear combinations of the neutrino mass eigenstate
s
ν
a
a
n
d
ν
b
(
Mak
i
et al.
,
1962
)
:
ν
α
=
ν
a
cos
θ
+
ν
b
sin
θ
a
n
d
ν
β
ν
=
ν
b
cos
θ
−
ν
a
sin
θ
,
where
θ
d
enotes the flavor mixin
g
an
g
le. Then the probabilities of neutrin
o
osci
ll
ations are governe
db
ytwoc
h
aracteristic parameters: one is t
h
eneutrin
o
m
ass-squared di
ff
erence
Δ
m
2
≡
m
2
b
−
m
2
a
(
in units of eV
2
)
and the other i
s
the flavor mixing factor si
n
2
2
θ
.
Corresponding to the “disappearance” and
“appearance
”
neutrino experiments, t
h
e surviva
l
an
d
conversion pro
b
a
b
i
l
itie
s
o
f
aneutrino
fl
avor ei
g
enstat
e
ν
α
c
an exp
l
icit
ly b
e expresse
d
as
P
(
ν
α
→
ν
α
)
=1
−
sin
2
2
θ
si
n
2
1
.
27
Δ
m
2
L
E
(
3.1
)
a
nd
P
(
ν
α
→
ν
β
ν
)
=
1
−
P
(
ν
α
→
ν
α
)
with
β
=
α
,
where
E
is the neutrino beam
energy
(
in units of GeV
)
,an
d
L
de
n
otes the d
i
sta
n
ce betwee
n
the
n
eut
rin
o
s
ource and the neutrino detector
(
in units of km
)
. A lot of data, includin
g
those from solar, atmos
p
heric, reactor and accelerator neutrino oscillatio
n
experiments, have been analyzed by using Eq.
(
3.1
)
as a good approximation.
3.1.1
A
tmospheric Neutrino Oscillation
s
The first compellin
g
and model-independent evidence for neutrino oscilla
-
tions was achieved from the Super-Kamiokande
(
SK
)
experiment
(
Fukud
a
et al
., 1998b
)
on atmospheric neutrinos, which are produced in the Earth’
s
a
tmosphere by cosmic rays and are detected in an underground detector. I
f
t
h
ere were no neutrino osci
ll
ation, t
h
e atmosp
h
eric
ν
e
(
o
r
ν
e
)
an
d
ν
μ
(
or
ν
μ
)
n
eutrinos enterin
g
and exitin
g
the detector should have an almost per
f
ect
3.1 Experimental Evidence
f
or Neutrino Masses 6
3
s
p
h
erica
l
symmetry. In ot
h
er wor
d
s, t
h
e
d
ownwar
d
-
g
oin
g
an
d
upwar
d
-
g
oin
g
n
eutrino fluxes should be e
q
ual to each other
:
Φ
α
(
θ
z
)
=
Φ
α
(
π
−
θ
z
)
versus
t
h
e zenit
h
ang
le
θ
z
(
for
α
=
e
or
μ
)
. The SK Collaboration has observed
a
n approximate up-down
fl
ux s
y
mmetr
yf
or atmospheri
c
ν
e
n
eutrinos
(
or
ν
e
a
ntineutrinos
)
and a significant up-down flux asymmetry for atmospheri
c
ν
μ
n
eutrinos
(
or
ν
μ
a
ntineutrinos
)
. For example
(
Ashi
e
et al
.
,
2005
),
Φ
e
(
−
1
cos
θ
z
−
0
.
2
)
Φ
e
(
+
0
.
2
cos
θ
z
+1
)
=0
.
961
+
0
.
086
−
0
.
07
9
±
0
.
016
,
Φ
μ
(
−
1
cos
θ
z
−
0
.
2
)
Φ
μ
(
+
0
.
2
cos
θ
z
+1
)
=0
.
551
+
0
.
035
−
0
.
033
±
0
.
0
0
4
,
(
3.2
)
f
or the multi-
G
e
V
e
-
likeeventsand
f
or the sin
g
le-rin
g
multi-
G
eV
μ
-
l
i
ke eve
n
ts
plus partiall
y
-contained events, respectivel
y
. This result can well be inter
-
p
reted in the assum
p
tion o
f
ν
μ
→
ν
τ
neutrino oscillations.
C
urrent neutrin
o
osci
ll
ation experiments
h
ave convincin
gl
yru
l
e
d
out t
h
e possi
b
i
l
ity t
h
at t
h
eat-
m
ospheric neutrino anomal
y
is dominantl
y
attributed t
o
ν
μ
→
ν
e
o
r
ν
μ
→
ν
s
osci
ll
ations
,
w
h
er
e
ν
s
stands for a “sterile” neutrino which does not take part
i
n
the
n
o
rm
al elect
r
oweak
in
te
r
act
i
o
n
s
.
T
he SK Collaboration studied the
ν
μ
d
isappearance probabilit
y
as a func
-
tion of the neutrino flight length
L
over t
h
eneutrinoenerg
y
E
a
n
d
c
l
aime
d
t
h
efi
r
st
direct
e
vidence for atmospheric neutrino oscillations
(
Ashi
e
et al
.
,
2
004
)
.Adipinth
e
L/
E
d
istribution was observed in the ex
p
erimental data
(
see Fig. 3.1 for illustration
)
, as predicted from the sinusoidal flavor transi-
tion probabilit
y
o
f
neutrino oscillations. We observe that the exotic neutrino
d
eca
y
and neutrino decoherence scenarios are disfavored b
y
the data, becaus
e
t
h
ey are una
bl
etorepro
d
uce t
h
e
d
ip. In a
dd
ition, t
h
e no-osci
ll
ation case i
s
s
tron
g
ly dis
f
avored by the data at lar
ge
L
/E
.
3
.1.2 Accelerator Neutrino
O
scillation
s
I
f
the atmospheri
c
ν
μ
d
isappearance is attri
b
ute
d
t
o
ν
μ
→
ν
τ
o
sci
ll
ations
,
then a si
g
nificant fraction of the accelerator-produced
ν
μ
events should als
o
d
isappear on their way to a sufficiently distant detector. This expectatio
n
h
as been confirmed by the long-baseline K2K
(
Ah
n
et al.
, 2003
)
and MINOS
(
Michae
l
et al.
,
2006
)
experiments. As for the K2K experiment, the
ν
μ
b
ea
m
was produced at the KEK accelerator, first measured by a near detecto
r
be
f
ore an
y
oscillation was developed, and then measured 250 km awa
y
b
y
th
e
SK detector at the Kamiokande. In comparison, the baseline length of th
e
MINOS experiment is 735 km, from Fermilab
(
thesourceo
f
ν
μ
n
eutrinos
)
to northern Minnesota
(
the location of the far detector
)
. Both experiment
s
h
ave observed a reduction of the
ν
μ
fl
ux and a distortion of th
e
ν
μ
energy
s
pectrum, which must take place in the presence o
f
ν
μ
→
ν
τ
osc
i
llat
i
o
n
s
.
T
he atmos
p
heric neutrino data
f
rom the
S
Kex
p
eriment and the accelera
-
tor neutrino data from the K2K and MINOS experiments can simultaneously
64 3 Neutrinos beyond the
S
tandard Mode
l
0
.
2
0
.
4
0
.
6
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
1
10 10
2
10
3
10
4
L/
E
(
km
/G
eV
)
D
ata/Prediction
(
null osc.
)
Fi
g
.3.1
A
ratio of the experimental data to the Monte Carlo events without
neutrino oscillations
(
points
)
as a function of the reconstructed L
/E
to
g
et
h
er wit
h
the best-fit expectation for
ν
μ
→
ν
τ
oscillations
(
solid line
)
. As a comparison,
the best-fit expectations for the neutrino decay
(
dashed line
)
and the neutrino
decoherence
(
dotted line
)
are also shown in this plot
(
Ashi
e
e
ta
l
.
,
2004. With
permission from the American Physical Society
)
be inter
p
reted as the standar
d
ν
μ
→
ν
τ
o
scillations. Fig. 3.2 shows the allowe
d
region o
f
Δ
m
2
a
nd sin
2
2
θ
c
onstrained by these data
(
Adamson et al
.
, 2008
)
.
O
ne can see that the best-
fi
t point o
f
the MIN
OS
measurement corresponds
to
|
Δ
m
2
|
=2
.
4
3
×
1
0
−
3
e
V
2
a
n
ds
in
2
2
θ
=1
.
0
.
I
ns
p
ite of a careful search for th
e
ν
τ
app
earance from atmos
p
heric neu
-
trino oscillations, no si
g
ni
fi
cant evidence has been established by the
S
K
C
ol
-
l
aboration
(
Ab
e
e
tal., 2006
)
. It is expected that the
ν
τ
a
pp
earance can b
e
observed from ν
μ
→
ν
τ
oscillations in the OPERA experiment
(
Acquafredd
a
et al
.
,
2006
)
, which possesses a much higher neutrino beam energy
(
E
∼
5
GeVto30GeV
,
above th
e
ν
τ
→
τ
production threshold
)
and a suitabl
e
baseline length
(
L
=
730 km, from CERN to Gran Sasso
).
3
.1.
3S
olar Neutrino
O
scillations
The lon
g
-standin
g
problem associated with solar neutrinos is that the
fl
u
x
of solar
ν
e
n
eutrinos measured in almost all ex
p
eriments, such as the S
K
(
Fukuda
et al.
, 1998a
)
and SNO
(
Ahmad
et al.
, 2002
)
experiments as well
a
s the earlier Homestake
(
Clevelan
d
et al
.
,
1998
)
, GALLEX
/
GNO
(
Hampe
l
et a
l
.
,
1999
;
Altman
n
e
ta
l.
,
2000
)
and SAGE
(
Abdurashitov
e
ta
l.
, 2002
)
3.1 Experimental Evidence
f
or Neutrino Masses 6
5
)
θ
(
2
2
s
in
0
.
60
.7
0
.
80
.
91
1
1.5
2
2.5
3
3.5
4
−3
10×
)
2
eV
−3
e
e
|
(
10
2
m
Δ
|
1.
0
11
1.51.5
2
.
22
2.52.5
3
.
33
3.53.5
4
.
44
M
IN
OS
90
%
M
INOS 68%
M
INOS 2006 90%
M
IN
OS
best oscillation fi
t
Super−K 90%
Su
p
er−K L/E 90
%
K2K 90
%
Fi
g
. 3.2 Contours for the oscillation fit to the MINOS data taken in 2008.
A
lso
shown are the results from the SK and K2K measurements (Adamson
e
ta
l.
,
2008.
With permission from the American Physical Society
)
experiments, is significantly smaller than that predicted by the standard so-
l
ar model
(
SSM
)(
Bahcal
l
et al.
,
2001
)
. The deficits of solar neutrinos are
n
ot the same in di
ff
erent experiments, implyin
g
that the relevant physica
l
effects are energy-dependent. Before the SNO experiment, it had been hy
-
pothesized that the solar neutrino anomaly was due to the conversion o
f
so
l
ar
ν
e
neutrinos into other active or sterile neutrinos on their wa
yf
ro
m
thecoreofthe
S
un to the detectors on the Earth. The
S
N
O
measuremen
t
h
as mo
d
e
l
-in
d
epen
d
ent
l
y
d
emonstrate
d
t
h
a
t
ν
e
→
ν
μ
a
n
d
ν
e
→
ν
τ
t
r
a
n
s
i-
tions are dominantl
y
responsible
f
or the solar neutrino de
fi
cit. It is actuall
y
the
ν
e
→
ν
μ
o
scillation with a large mixing angle that dominates the sola
r
n
eutrino
fl
avor chan
g
e.
W
hat the SNO Collaboration has measured is the flux of solar
8
Bneutri-
n
os via the charged-current
(
CC
)
, neutral-current
(
NC
)
and elastic-scattering
(
ES
)
reactions
(
Ahma
d
et al.
,
2002
):
ν
e
+D
→
e
−
+
p
+
p
,
ν
α
+D
→
ν
α
+
p
+
n
a
nd
ν
α
+
e
−
→
ν
α
+
e
−
(
for
α
=
e
,
μ
o
r
τ
)
. In the presence of flavor conversion,
the observed neutrino fluxes in different reactions satisfy
φ
CC
=
φ
e
,φ
N
C
=
φ
e
+
φ
μ
τ
,φ
E
S
=
φ
e
+
σ
μ
σ
e
φ
μ
τ
≈
φ
e
+0
.
1
5
5
φ
μτ
,
(
3.3
)