Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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36
2 Neutrinos within the
S
tandard Model
whe
r
e
Ψ
i
Ψ
Ψ
(
for
i
=
1
,
2
,
3
,
4
)
are Dirac spinors with the Greek subscripts bein
g
the spinor indices. This product should obviousl
y
be equivalent t
o
P
=
−
1
6
s
=1
ˆ
g
s
(
Γ
s
)
α
λ
(
Γ
s
Γ
)
ηβ
Ψ
1
ΨΨ
α
(
Ψ
4
Ψ
)
λ
Ψ
3
ΨΨ
η
(
Ψ
2
ΨΨ
)
β
,
(
2.63
)
where the minus sign comes from the exchange of fermion field operators
.
Identifying the coefficients in front of the fermion fields in these two equations,
16
s
=1
ˆ
g
s
(
Γ
s
)
αλ
(
Γ
s
Γ
)
η
β
=
−
16
r
=
1
g
r
(
Γ
r
)
αβ
(
Γ
r
ΓΓ
)
ηλ
,
(
2.64
)
a
nd multiplying both sides of Eq.
(
2.64
)
by
Γ
p
ΓΓ
λ
α
(
Γ
q
)
βη
,
we o
b
tai
n
ˆ
g
p
=
−
1
16
16
r
=1
Tr
Γ
r
ΓΓ
Γ
p
ΓΓ
Γ
r
Γ
p
g
r
,
(
2.65
)
where the normalization relation in Eq.
(
2.61
)
has been used. In view o
f
L
orentz invariance, one usuall
y
needs to treat the interactions wit
h
Γ
I
(
for
I
=
S, V, T, A,
P
)
defined in Eq.
(
2.60
)
. In this case one may find the relations
between ˆ
g
I
a
n
d
g
I
b
y using Eq.
(
2.65
):
⎛
⎜
⎛
⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜
⎜
ˆ
g
S
ˆ
g
V
ˆ
g
T
ˆ
g
A
ˆ
g
P
⎞
⎟
⎞
⎞
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎠
⎟
⎟
=
−
1
4
⎛
⎜
⎛
⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜
⎜
1
4641
1
−
2
0
2
−
1
10
−
20 1
12 0
−
2
−
1
1
−
4
6
−
41
⎞
⎟
⎞
⎞
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎠
⎟
⎟
⎛
⎜
⎛
⎛
⎜
⎜⎜
⎜
⎜⎜
⎜
⎜⎜
⎝
⎜
⎜
g
S
g
V
g
T
g
A
g
P
⎞
⎟
⎞
⎞
⎟
⎟⎟
⎟
⎟⎟
⎟
⎟⎟
⎠
⎟
⎟
.
(
2.66
)
F
or the
(
V
−
A
)
-type interaction, it is straightforward to ge
t
Ψ
1
ΨΨ
γ
μ
(1
−
γ
5
)
Ψ
2
ΨΨ
Ψ
3
ΨΨ
γ
μ
(1
−
γ
5
)
Ψ
4
Ψ
=
Ψ
1
ΨΨ
γ
μ
(1
−
γ
5
)
Ψ
4
Ψ
Ψ
3
ΨΨ
γ
μ
(
1
−
γ
5
)
Ψ
2
ΨΨ
.
(
2.67
)
A
pplying this Fierz identity to Eq.
(
2.59
)
, we can simplify the overall ampli-
tu
de
M
=
M
n
c
+
M
cc
f
or the elastic neutrino-electron scattering:
M
(
s
ν
,
s
e
;
s
ν
,s
e
)=
G
F
√
2
u
(
p
,s
ν
)
γ
μ
(1
−
γ
5
)
u
(
p
,
s
ν
)
u
(
q
,s
e
)
γ
μ
×
(
1+
C
l
L
CC
)(
1
−
γ
5
)+
C
l
R
CC
(
1
+
γ
5
)
u
(
q,
s
e
)
,
(
2.68
)
where
M
2
Z
M
cos
2
θ
w
=
M
2
W
MM
h
as been used
,
and
G
F
=
g
2
/
(
4
√
2
M
2
W
MM
)
is th
e
F
ermi constant. Note t
h
a
t
|
M
|
2
should be summed over the spins of the final
s
tate and avera
g
ed over the spins o
f
the initial state
:
1
2
s
p
i
n
|
M
|
2
=
G
2
F
4
L
−
μν
(
p
,
p
)
1+
C
l
L
CC
2
L
−
μν
(
q
,
q
)
2.2
S
tandard Interactions o
f
Neutrinos
37
+16
m
2
e
g
μ
ν
(
1
+
C
l
L
CC
)
C
l
R
CC
+
C
l
R
CC
2
L
+
μν
(
q
,
q
)
,
(
2.69
)
whe
r
e
L
±
μν
(
p
,
p
)
≡
Tr
/
pγ
//
μ
(1
±
γ
5
)
/
p
//
γ
ν
(1
±
γ
5
)
=
8
p
μ
p
ν
+
p
ν
p
μ
−
p
·
p
g
μν
±
i
ε
μνρ
σ
p
ρ
p
σ
(
2.70
)
h
ave been defined. Note also that neutrinos are always left-handed in the SM,
s
oanavera
g
eoverthespinso
f
the initial state is only actin
g
on the electron
.
Give
n
p
+
q
=
p
+
q
,
Eq.
(
2.69
)
is simplified t
o
1
2
s
pin
|
M
|
2
=64
G
2
F
1
+
C
l
L
CC
2
(
p
·
q
)
2
+
C
l
R
CC
2
(
p
·
q
)
2
−
1
+
C
l
L
CC
C
l
R
CC
m
2
e
(
p
·
p
)
.
(
2.71
)
In the laborator
yf
rame where the electron is at rest, we hav
e
q
=
m
e
,
0
,
0
,
0
,p
=
E
ν
,
p
,
(
2.72
)
a
n
d
E
ν
=
|
p
|
. Denoting the energy of the outgoing electron as
E
e
and that o
f
the scatte
r
ed
n
eut
rin
oas
E
ν
,
we express the total cross section o
f
the elastic
n
eutrino-electron scattering as follows
:
σ
E
S
(
ν
e
e
−
)
=
1
4
E
ν
m
e
d
3
p
(
2
π
)
3
2
E
ν
d
3
q
(2
π
)
3
2
E
e
(2
π
)
4
δ
4
(
p
+
q
−
p
−
q
)
×
64
G
2
F
m
2
e
E
2
ν
1+
C
l
L
CC
2
+
C
l
R
CC
2
1
−
T
E
ν
2
−
1
+
C
l
L
CC
C
l
R
CC
(
1
−
cos
θ
)
E
ν
E
ν
,
(
2.73
)
where the kinetic ener
g
yo
f
the recoil electron is de
fi
ned as
T
=
E
e
−
m
e
,
a
n
dcos
θ
i
s the cosine o
f
the an
g
le between the momenta o
f
the scattered
a
nd incident neutrinos. With the help of Eq.
(
2.73
)
, it is easy to obtain the
d
ifferential cross section
(
’t Hooft, 1971c; Marciano and Parsa, 2003
)
d
σ
ES
d
T
(
E
ν
,T
)
=
G
2
F
m
e
2
π
2
+
+
2
−
1
−
T
E
ν
2
−
+
−
m
e
T
E
2
ν
,
(
2.74
)
w
h
ere
±
≡
g
e
V
±
g
e
A
an
d
g
e
V
≡
1+
C
l
V
C
C
,
g
e
A
≡
1
+
C
l
A
C
.
(
2.75
)
A
measurement of this differential cross section allows one to pin down th
e
coeffic
i
e
n
ts
g
e
V
a
n
d
g
e
A
,
so as to test the
S
M
.
38
2 Neutrinos within the
S
tandard Model
ν
e
e
−
ν
e
Z
0
ν
e
ν
e
e
−
e
−
e
−
W
−
Fig. 2.3 Feynman diagrams for the elastic antineutrino-electron scattering process
ν
e
+
e
−
→
ν
e
+
e
−
i
nthe
SM
If
the ener
g
yo
f
the incident neutrino is much lar
g
er than the electro
n
m
ass
(
i.e.,
E
ν
m
e
)
, then the total cross section given in Eq.
(
2.73
)
can
s
a
f
el
y
approximate to
σ
E
S
(
ν
e
e
−
)
=
2
G
2
F
π
m
e
E
ν
1
+
C
l
L
CC
2
+
1
3
C
l
R
CC
2
.
(
2.76
)
One may similarly figure out the total cross section of the elastic antineutrino
-
e
l
ectron scatterin
g
,w
h
ic
h
ta
k
es p
l
ace via t
h
eFeynman
d
ia
g
rams s
h
own i
n
F
ig. 2.3. The result i
s
σ
E
S
(
ν
e
e
−
)=
2
G
2
F
π
m
e
E
ν
1
3
1+
C
l
L
CC
2
+
C
l
R
CC
2
.
(
2.77
)
F
or the sake of completeness, we also consider the neutral-current scatterin
g
pr
ocesses
ν
x
+
e
−
→
ν
x
+
e
−
a
n
d
ν
x
+
e
−
→
ν
x
+
e
−
(
fo
r
x
=
μ
o
r
τ
)
.Their
F
eynman
d
ia
g
rams are
g
iven in Fi
g
.2.4,an
d
t
h
eir tota
l
cross sections rea
d
σ
ES
(
ν
x
e
−
)
=
2
G
2
F
π
m
e
E
ν
C
l
L
C
C
2
+
1
3
C
l
R
CC
2
,
σ
ES
(
ν
x
e
−
)
=
2
G
2
F
π
m
e
E
ν
1
3
C
l
L
CC
2
+
C
l
R
CC
2
,
(
2.78
)
w
h
ere x
=
μ
or
τ
.Theratioo
f
σ
ES
(
ν
x
e
−
)
t
o
σ
E
S
(
ν
e
e
−
)
turns out to be
σ
ES
(
ν
x
e
−
)
σ
ES
(
ν
e
e
−
)
=
3
C
l
L
CC
2
+
C
l
R
CC
2
3
1
+
C
l
L
CC
2
+
C
l
R
CC
2
≈
0
.
155
,
(
2.79
)
whe
r
e
C
l
L
CC
a
n
d
C
l
R
CC
can be found in Table 2.2, and si
n
2
θ
w
≈
0
.
2381
(
Porse
v
et
al.
,
2009
)
has been input. It is worth stressing that gauge coupling constant
s
d
epend on the energy scale where the interactions take place. At the elec
-
t
r
oweak scale we have s
in
2
θ
w
=0
.
23119
±
0
.
00014
(
Nakamur
a
et al.
, 2010
)
,
which is sli
g
htly smaller than the values measured at extremely low ener
g
ies
.
2.2
S
tandard Interactions o
f
Neutrinos
39
ν
x
ν
e
−
ν
x
ν
Z
0
Z
0
ν
x
e
−
e
−
ν
x
e
−
Fig. 2.4 Feynman diagrams for the elastic neutrino-electron scattering processes
ν
x
+
e
−
→
ν
x
+
e
−
a
n
d
ν
x
+
e
−
→
ν
x
+
e
−
(
fo
r
x
=
μ
o
r
τ
)
in the SM
ν
α
ν
ν
α
ν
Z
0
Z
0
ν
α
ν
ν
α
ν
ν
α
ν
ν
α
ν
ν
α
ν
ν
α
ν
Fi
g
. 2.5 Feynman dia
g
rams
f
or the elastic neutrino-neutrino scatterin
g
processes
ν
α
+
ν
α
→
ν
α
+
ν
α
(
fo
r
α
=
e
,
μ
,τ
)
in the SM
2.2.2 Neutrino-neutrino
S
cattering
Contrary to the elastic neutrino-electron scatterin
g
, the elastic neutrino
-
n
eutrino scattering seems less important in the development of particl
e
p
hy
sics. However, it
h
as recent
ly b
een notice
d
t
h
at t
h
eco
h
erent neutrino
-
n
eutrino scatterin
g
processes in supernovae are essential to the flavor con
-
versions of supernova neutrinos
(
Dighe, 2008; Duan and Kneller, 2009
)
.
N
eutrino-neutrino interactions are also expected to be si
g
ni
fi
cant in the
n
eutrino-decoupling era of the early Universe. In this subsection we shal
l
briefly discuss the total cross sections of elastic neutrino-neutrino scattering
a
n
d
n
eut
rin
o
-
a
n
t
in
eut
rin
oa
nni
h
i
lat
i
o
n
.
Fig. 2.5 shows the Feynman diagrams for the elastic neutrino-neutrin
o
s
cattering process
ν
α
(
p
)
+
ν
α
(
q
)
→
ν
α
(
p
)
+
ν
α
(
q
)(
for
α
=
e
,
μ
,τ
)
. Although
onl
y
the neutral-current interactions contribute to this process, there are two
d
ifferent channels due to the exchange of the final-state neutrinos. The overal
l
a
mp
l
itu
d
ecan
b
e written a
s
M
=
−
G
F
√
2
u
(
p
)
γ
μ
(
1 −
γ
5
)
u
(
p
)
u
(
q
)
γ
μ
(1
−
γ
5
)
u
(
q
)
,
(
2.80
)
w
h
ere t
h
e Fierz i
d
entity
h
as
b
een use
d
to com
b
ine t
he
t
-
a
n
d
u
-
cha
nn
els
.T
he
tota
l
c
r
oss sect
i
o
ni
sfou
n
dtobe
σ
ES
(
ν
α
ν
α
)
=
G
2
F
s
π
,
(
2.81
)
4
0
2 Neutrinos within the
S
tandard Model
whe
r
e
s
≡
(
p
+
q
)
2
. It is wort
h
mentionin
g
t
h
at t
h
ea
b
ove resu
l
tmu
l
tip
l
ie
d
b
y
a factor
1
/
2
will be a
pp
licable to the elastic
ν
α
+
ν
β
ν
→
ν
α
+
ν
β
ν
reac
-
tions
(
fo
r
β
=
α
)
. The reason is simply that these processes only have th
e
t
-
c
h
a
nn
e
l
co
n
t
ri
but
i
o
n
sa
n
dt
h
e
ir
fi
n
a
l-
state
n
eut
rin
os a
r
e
n
ot
i
de
n
t
i
ca
l. F
u
r
-
thermore, one can immediately figure out the total cross sections of the elasti
c
a
ntineutrino-antineutrino scattering processes. The final results ar
e
σ
ES
(
ν
α
ν
α
)=
σ
ES
(
ν
α
ν
α
)
=
G
2
F
s
π
,
σ
ES
(
ν
α
ν
β
)=
σ
ES
(
ν
α
ν
β
ν
)=
G
2
F
s
2
π
,
(
2.82
)
where
α
=
β
. The two equalities obtained above are
g
uaranteed by the CP
T
theorem, which is always valid in the local and Lorentz-invariant field theorie
s
(
Schwinger, 1951; Pauli, 1955; Bell, 1955; L¨uders, 1957
)
. Following the elastic
n
eutrino-neutrino scatterin
g
, one may write out the amplitude for the elastic
n
eutrino-antineutrino scattering proces
s
ν
α
(
p
)
+
ν
α
(
q
)
→
ν
α
(
p
)+
ν
α
(
q
),
w
h
ose Feynman
d
ia
g
rams are
g
iven in Fi
g
.2.6
:
M
=
−
G
F
√
2
u
(
p
)
γ
μ
(1
−
γ
5
)
u
(
p
)
v
(
q
)
γ
μ
(1
−
γ
5
)
v
(
q
)
.
(
2.83
)
D
i
ffe
r
e
n
tf
r
o
m
t
h
ee
l
ast
i
c
ν
α
ν
α
s
catterin
g
,t
h
ee
l
asti
c
ν
α
ν
α
s
catterin
gd
oes
n
ot involve the s
y
mmetr
y
factor 1
/
2 originating from the identical final-state
particles. Therefore,
σ
E
S
(
ν
α
ν
α
)
=2
σ
E
S
(
ν
α
ν
α
)=
2
G
2
F
s
π
.
(
2.84
)
Note that the elast
i
c
ν
α
+
ν
β
→
ν
α
+
ν
β
s
cattering processes
(
fo
r
β
=
α
)
c
an onl
y
take place via the
t
-channel. Hence it is strai
g
ht
f
orward to obtai
n
σ
ES
(
ν
α
ν
β
)
=
σ
ES
(
ν
α
ν
β
ν
)
=
σ
ES
(
ν
α
ν
α
)
/
4
. Note also that one should
p
erfor
m
the thermal avera
g
e over the initial particles i
f
the
f
requent scatterin
g
occurs
i
n a thermal bath, such as in the earl
y
Universe or in the interior o
f
a massive
s
tar. In this case it is evident that the neutrino-antineutrino scattering process
s
hould be most important. Nevertheless, the total cross sections
f
or neutrin
o
i
nteractions are extremely small and can be ne
g
lected in most cases.
A
swe
s
hall show later, the coherent forward scattering of neutrinos off matter or
other neutrinos plays a crucial role in astrophysics and cosmology
(
e.g., i
t
h
as to be taken into account in understandin
g
solar neutrino oscillations and
s
upernova neutrino oscillations
)
.
2.2.
3
Neutr
i
no-nucleon Interact
i
on
s
The
fi
rst detection o
f
electron antineutrinos in 1956 made use o
f
a kind o
f
n
eutrino-nucleon interactions — the inverse beta decay
ν
e
+
p
→
e
+
+
n
2.2
S
tandard Interactions o
f
Neutrinos 41
ν
α
ν
ν
α
ν
α
ν
Z
0
ν
α
ν
ν
α
ν
ν
α
ν
α
ν
α
Z
0
Fig. 2.6 Feynman diagrams for the elastic neutrino-antineutrino scattering pro-
cesses
ν
α
+
ν
α
→
ν
α
+
ν
α
(
α
=
e
,
μ
,
τ
)
in the S
M
(
Cowan
e
tal.
,
1956; Reines and Cowan, 1956
)
. In the SM both the beta decay
a
nd the inverse beta decay take place via the charged-current interactions of
q
uarks and leptons. It remains di
ffi
cult to quantitativel
y
describe how quark
s
i
nteract and form hadrons at low ener
g
ies, so one has to find out a consistent
way to convert t
h
e neutrino interactions wit
h
quar
k
sintot
h
ose wit
h
nuc
l
eons.
The e
ff
ective La
g
ran
g
ian responsible
f
or the inverse beta decay is
g
iven by
L
eff
=
G
F
√
2
J
l
μ
J
J
qμ
+h
.
c
.
=
G
F
√
2
V
ud
VV
l
α
γ
μ
(1
−
γ
5
)
ν
α
uγ
μ
(
1 −
γ
5
)
d
+h
.
c
.,
(
2.85
)
whe
r
e
J
l
μ
J
a
n
d
J
q
μ
J
s
tand respectively
f
or the char
g
ed currents o
f
leptons and
q
uarks, an
d
V
ud
VV
i
sanelemento
f
the
C
KM matri
x
V
.
The am
p
litude
f
or th
e
q
uasi-elastic scattering
ν
e
(
k
)
+
p
(
q
)
→
e
+
(
k
)
+
n
(
q
)
can therefore be read
off f
r
o
m
L
eff
:
M
=
G
F
√
2
v
ν
(
k
)
γ
μ
(
1 −
γ
5
)
v
e
(
k
)
n
(
q
)
|
J
q
†
μ
J
|
p
(
q
)
,
(
2.86
)
where we have replaced the charged current of quarks with the corresponding
n
ucleon matrix element.
A
lthou
g
h protons and neutrons are not elementary
particles, they must obey the Fermi-Dirac statistics and satisfy the Dira
c
equation. It is instructive to
d
ivi
d
et
h
ec
h
ar
g
e
d
current
J
q
μ
J
i
n
to the vector
c
om
p
onen
t
v
q
μ
=
uγ
μ
d
a
nd the
p
seudovector com
p
onen
t
a
q
μ
=
u
γ
μ
γ
5
d
,such
tha
t
J
q
μ
J
=
V
ud
VV
(
v
q
μ
−
a
q
μ
)
. With the help of Lorentz invariance and Dira
c
equations, the nucleon matrix element can be written as
(
Llewellyn Smith
,
1
971
;A
lberic
o
e
tal
.
,
2002
)
n
(
q
)
|
v
q
μ
(
0
)
|
p
(
q
)
=
u
n
(
q
)
γ
μ
F
1
FF
(
Q
2
)+
i
σ
μν
q
ν
−
2
m
N
F
2
FF
(
Q
2
)
u
p
(
q
)
,
(
2.87
)
a
nd
n
(
q
)
|
a
q
μ
(
0
)
|
p
(
q
)
=
u
n
(
q
)
γ
μ
γ
5
G
A
(
Q
2
)+
q
−
μ
m
N
γ
5
G
P
(
Q
2
)
u
p
(
q
)
,
(
2.88
)
w
h
e
r
e
q
−
≡
q
−
q
a
n
d
Q
2
≡−
q
2
−
a
re de
fi
ned, and
m
p
≈
m
n
≡
m
N
i
sta
k
e
n.
L
et us give some comments on the form factors in Eqs.
(
2.87
)
and
(
2.88
).
42 2 Neutrinos within the
S
tandard Model
F
irst, the time reversal invariance makes all the
f
orm
f
actors rea
l
3
.
S
econd,
the isospin symmetry in stron
g
interactions makes the second-class current
s
vanishing
(
Weinberg, 1958
)
. Among the four form factors,
F
1
FF
a
n
d
F
2
FF
a
r
e
a
ssociated with the electroma
g
netic properties o
f
nucleons, while
G
A
a
n
d
G
P
a
re related with the weak interactions.
A
ll of them can be extracted from th
e
relevant experimental results
(
Cheng and Li, 1988
)
. Combining Eqs.
(
2.86
)
,
(
2.87
)
and
(
2.88
)
, one obtains the differential cross section in the laboratory
f
rame
(
Llewellyn Smith, 1971
):
d
σ
(
ν
e
p
)
d
Q
2
=
G
2
F
|
V
ud
V
V
|
2
m
4
N
8
π
(
k
·
q
)
2
C
1
−
C
2
CC
s
−
u
m
2
N
+
C
3
C
(
s
−
u
)
2
m
4
N
,
(
2.89
)
whe
r
e
C
i
CC
(
fo
r
i
=
1
,
2
,
3
)
are the analytical functions of
Q
2
d
epen
d
in
g
on t
he
f
orm factors defined above
,
an
d
s
=(
k
+
q
)
2
,
t
=(
q
−
q
)
2
a
nd
u
=
(
k
−
q
)
2
a
re the Mandelstam invariants. The results of
C
i
C
a
re summarized as follow
s
(
Llewellyn Smith, 1971
)
:
C
1
=4
κ
(
1+κ
)
G
2
A
−
(
1
−
κ
)
F
2
1
FF
−
κF
2
2
FF
+4
κF
1
FF
F
2
FF
,
C
2
CC
=4
κG
2
A
,
C
3
C
=
1
4
G
2
A
+
F
2
1
FF
+
κF
2
2
FF
,
(
2.90
)
whe
r
e
κ
≡
Q
2
/
(
4
m
2
N
)
, and the terms involvin
g
m
2
e
/m
2
N
h
ave
b
een neg
l
ecte
d.
F
or low-energy neutrinos
(
E
ν
m
N
)
in the laboratory frame, where the
n
ucleon is at rest, the total cross section of the inverse beta deca
y
read
s
σ
(
ν
e
p
)
≈
G
2
F
|
V
ud
VV
|
2
π
1+3
g
2
A
E
2
ν
,
(
2.91
)
where the vector and pseudovector couplin
g
s of nucleons ar
e
g
V
≡
F
1
FF
(
0
)
≈
1
a
n
d
g
A
≡
G
A
(
0
)
≈
1
.25. The total cross section of the quasi-elastic scatterin
g
ν
e
+
n
→
e
−
+
p
,
de
fi
ned a
s
σ
(
ν
e
n
)
, is found to be equal t
o
σ
(
ν
e
p
)
in the
s
ame approximation. Takin
g
G
F
=
1
.
1
6
6
×
1
0
−
5
G
eV
−
2
an
d
|
V
ud
VV
|
=
0
.
9
7
3
(
Nakamur
a
e
ta
l.
,
2010
)
, we obtain
σ
(
ν
e
p
)
=
σ
(
ν
e
n
)
≈
9
.
1
×
10
−
44
E
ν
M
e
V
2
cm
2
.
(
2.92
)
It is worth mentionin
g
that the
fi
rst theoretical estimation o
f
the total cross
s
ection o
f
the neutrino-nucleon scatterin
g
was done by Hans Bethe and Rudol
f
P
eierls in 1934
(
Bethe and Peierls, 1934
)
, soon after Enrico Fermi formulated
h
is effective theory for the beta decay
(
Fermi, 1933
)
. Their result is amazingl
y
c
onsistent with the one given in Eq.
(
2.92
).
3
T
he
C
, P and T trans
f
ormations and their im
p
lications in
q
uantum
fi
eld the-
ories will be discussed in Section 3.3.
2.3 Neutrino Propagation in a Me
d
ium 43
W
e
h
ave
d
iscusse
d
t
h
e neutrino-nuc
l
eon scatterin
g
processes me
d
iate
dby
weak char
g
ed-current interactions. Note that the elastic scatterin
g
processe
s
ν
α
+
N
→
ν
α
+
N
(
for
α
=
e
,μ,τ
)
can take place via weak neutral-curren
t
i
nteractions, an
d
t
h
ey
h
ave
b
een use
d
to extract t
h
ewea
k
mixin
g
an
gl
eint
h
e
SM. To study the hi
g
h-ener
g
y neutrino interactions with nucleons, however,
one has to consider the fine structure of a proton or a neutron
(
e.g., its parton
d
istributions
)(
Llewellyn Smith, 1971
).
2
.3 Neutrino Propagation in a Mediu
m
In 1978, Lincoln Wol
f
enstein pointed out that the coherent
f
orward scatterin
g
of neutrinos off matter could be very important in studying the behavior o
f
n
eutrino flavor conversions in a dense medium
(
Wolfenstein, 1978
)
.Thisob
-
s
ervation has been verified in understandin
g
solar neutrino oscillations, since
the Sun itself is just a dense medium in which the propagation of solar ν
e
n
eutrinos ma
y
be modi
fi
ed b
y
their interactions with the ambient electrons
a
nd nucleons. The main effects induced b
y
a medium can be described b
y
a
n
effective potential, or equivalently a neutrino refractive index. In this sectio
n
we present three di
ff
erent approaches
f
or calculatin
g
the e
ff
ective potential
of neutrinos propagating in a medium:
(
1
)
to compute the coherent forward
s
cattering amplitudes and then derive neutrino refractive indices;
(
2
)
to eval-
u
ate the effective potential directly from the effective Hamiltonian;
(
3
)
t
o
c
alculate the neutrino self-energy in a medium and then extract the effectiv
e
potentia
l
via t
h
e
d
ispersion re
l
ation. We s
h
ow t
h
at t
h
ese t
h
ree met
h
o
d
sar
e
equivalent and lead to the same result. We also appl
y
the result to two inter-
esting cases — neutrinos propagating in a normal matter environment and i
n
ade
n
se
n
eut
rin
oe
n
v
ir
o
nm
e
n
t
.Th
edeta
il
ed d
i
scuss
i
o
n
sof
m
atte
r
effects on
n
eutrino
fl
avor conversions will be
g
iven in
C
hapter 5, while the application
s
to solar and su
p
ernova neutrinos can be found in Cha
p
ters 6 and 7.
2.3.1 Coherent Forward Scatterin
g
In quantum
fi
eld theories, the probability amplitude o
f
the scatterin
g
proces
s
α
→
β
is given by th
e
S
-matrix elemen
t
S
βα
S
≡
Φ
o
u
t
β
|
Φ
in
α
,
(
2.93
)
w
h
ere
Φ
i
n
α
a
n
d
Φ
o
u
t
β
a
re t
he
in
an
d
ou
t
s
tates, respective
l
y. Here t
h
esu
b-
s
cr
i
pt
s
α
a
n
d
β
s
tand
f
or the momentum, spin and other quantum numbers
c
haracterizin
g
the correspondin
g
particles. If there is no any interaction, the
i
n-state must be identical to the out-state such that the probability for
α
→
α
i
s unity. Hence the transition amplitude o
f
the scatterin
g
proces
s
α
→
β
(
for
α
=
β
)
can be defined throug
h
S
≡
1
+i
T
a
nd written a
s
44 2 Neutrinos within the
S
tandard Model
i
T
βα
T
=
(
2
π
)
4
i
δ
4
(
p
α
−
p
β
)
M
β
α
,
(
2.94
)
where the Fe
y
nman amplitud
e
M
β
α
i
s free of the delta functions. It i
s
s
traightforward to compute the differential cross section for the reactio
n
1
+
2
→
3+
4+
···
+
n
w
i
th
n
−
2
particles in the final state
(
Cheng an
d
L
i, 1988; Peskin and Schroeder, 1995
)
:
d
σ
=
(
2
π
)
4
δ
4
(
p
1
+
p
2
−
p
3
−···
−
p
n
)
4
(
p
1
·
p
2
)
2
−
m
2
1
m
2
2
|
M
|
2
n
j
=
3
d
3
p
j
(
2
π
)
3
2
E
j
E
,
(
2.95
)
whe
r
e
p
i
,
p
i
a
n
d
E
i
=
|
p
i
|
2
+
m
2
i
(
for
i
=1
,
2
,
···
,n
)
stand respectively for
the
f
our-momenta, three-momenta and ener
g
ies o
f
relevant particles wit
h
m
i
being their masses. Let us consider a two-body scattering proces
s
a
+
b
→
c
+
d
f
or example. Its di
ff
erential cross section involves the inte
g
ration over th
e
fi
nal-state
p
hase s
p
ace
(2
π
)
4
δ
4
(
p
a
+
p
b
−
p
c
−
p
d
)
d
3
p
c
(
2
π
)
3
2
E
c
·
d
3
p
d
(
2
π
)
3
2
E
d
.
(
2.96
)
In the center-of-mass frame we have
p
a
+
p
b
= 0. The total energy of th
e
i
nitia
l
partic
l
es i
s
E
=
E
a
+
E
b
.T
h
einte
g
ration ove
r
p
d
i
s equiva
l
ent t
o
d
roppin
g
the three-dimensional delta function
δ
3
(
p
c
+
p
d
)
and replacing the
t
h
ree-momentum
p
d
w
it
h
−
p
c
i
n the integrand. The phase-space factor in
E
q.
(
2.96
)
is therefore simplified t
o
1
16
π
2
E
c
E
d
δ
(
E
c
+
E
d
−
E
)
|
p
c
|
2
d
|
p
c
|
d
Ω
,
(
2.97
)
where
E
c
=
|
p
c
|
2
+
m
2
c
an
d
E
d
=
|
p
c
|
2
+
m
2
d
,
and
d
Ω
≡
sin
θ
d
θ
d
φ
i
sth
e
d
i
ff
erential element o
f
the solid an
g
le. The inte
g
ration o
f
the delta
f
unctio
n
over
d
|
p
c
|
r
equires us to find out the zero point of the ar
g
ument of the delta
f
unction. Namely,
|
p
c
|
2
+
m
2
c
+
|
p
c
|
2
+
m
2
d
=
E
(
2.98
)
leadsustothesolut
i
o
n
|
p
c
|
=
1
2
E
(
E
2
−
m
2
c
−
m
2
d
)
2
−
4
m
2
c
m
2
d
.
(
2.99
)
It is then straightforward to see that the integration with the delta function
δ
(
E
c
+
E
d
−
E
)
contributes the facto
r
d
d
|
p
c
|
|
p
c
|
2
+
m
2
c
+
|
p
c
|
2
+
m
2
d
−
E
−
1
|
p
c
|
=
|
p
c
|
=
E
c
E
d
|
p
c
|
E
,
(
2.100
)
2.3 Neutrino Propagation in a Me
d
ium 45
whe
r
e
E
c
=
|
p
c
|
2
+
m
2
c
a
n
d
E
d
=
|
p
c
|
2
+
m
2
d
.
Combining Eqs.
(
2.96
),
(
2.97
)
,
(
2.99
)
and
(
2.100
)
with Eq.
(
2.95
)
, we arrive at the differential cros
s
s
ection of the two-body scattering proces
s
a
+
b
→
c
+
d
a
s follows:
d
σ
(
a
+
b
→
c
+
d
)
=
1
64
π
2
·
|
p
c
|
|
p
a
|
E
2
|M
(
a
+
b
→
c
+
d
)
|
2
d
Ω
(
2.101
)
i
n
t
h
ece
n
te
r-
of
-m
ass f
r
a
m
e
.Thi
s
r
esu
l
tw
ill
be used
l
ate
r
o
n
to d
i
scuss t
h
e
propagation of neutrinos in matter. When a neutrino beam propagates in
m
atter, the scattered neutrino waves may coherently interfere in the forwar
d
d
irection. We shall show that this coherent
f
orward scatterin
g
o
f
neutrinos o
ff
m
atter is ver
y
important in some astroph
y
sical environments where neutrinos
a
re copiously produced
(
Wolfenstein, 1978, 1979; Mikheyev and Smirnov,
1
985; Raffelt, 1996
)
.
Now let us consider that a neutrino beam propagating with momentum
k
i
n
t
h
ed
ir
ect
i
o
n
of t
h
e
z
-
a
xi
s
in
te
r
acts w
i
th a scatte
r
e
r
at the locat
i
o
n
r
=
0.
S
uch a neutrino beam is usuall
y
described b
y
the plane wav
e
Ψ
=
e
i
kz
,
an
d
the wave function of the system at extremely far distances after scattering
c
an be re
g
arded as a superposition o
f
the ori
g
inal and the scattered waves
(
Landau and Lifshitz, 1977
):
Ψ
f
ΨΨ
(
r
)
≈
e
i
kz
+
f
(
θ
)
e
i
k
r
r
,
(
2.102
)
where f
(
θ
)
is the scattering amplitude. The differential cross section of this
s
catterin
g
process is t
h
en
g
iven
b
y
d
σ
=
|
f
(
θ
)
|
2
d
Ω
.
Linkin
g
it to the di
ff
er-
ential cross section of a two-body scatterin
g
process
a
+
b
→
c
+
d
,
w
eca
n
esta
bl
is
h
t
h
ere
l
ation
b
etween t
h
e Feynman amp
l
itu
d
e
M
(
a
+
b
→
c
+
d
)
i
n
the center-o
f
-mass
f
rame and the scatterin
g
amplitud
e
f
(
θ
)
4
:
f
(
θ
)
=
1
8
π
E
!
|
p
c
|
|
p
a
|
M
(
a
+
b
→
c
+
d
)
.
(
2.103
)
In a perturbation theor
y
one ma
yfi
rst calculate the Fe
y
nman amplitude order
by order and then compute the forward scatterin
g
amplitud
e
f
(
0
)
,whichcan
be used to derive the refractive index of neutrinos propagating in matter
.
For t
h
is purpose, we consi
d
er t
h
eneutrino
b
eam to
b
eap
l
anewavee
i
kz
i
ncident on a slab which is located a
t
z
=
0and
p
er
p
endicular to th
e
z-ax
i
s
.
W
e assume that the slab has an infinitesimal thicknes
s
δ
a
nd is infinitely larg
e
i
n
both the
x
-
a
xi
sa
n
dthe
y
-axis.
A
fter traversin
g
the slab, the neutrino wav
e
i
s still a
p
lane wave at the distanc
e
z
2
π
/
k
b
ut it has attained a
p
hase
e
i
k
[
z
+(
n
−
1
)
δ
]
w
i
th
n
b
eing the neutrino refractive index. On the other hand
,
4
T
he sign of the scattering amplitude has been chosen in such a way tha
t
f
(
θ
)
is just the coe
ffi
cient o
f
the out
g
oin
g
wave in the solution o
f
the time-dependent
Schr¨odinger equation (Weinberg, 1995)
.