Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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4
6
2 Neutrinos within the
S
tandard Model
the asymptotic form of the original and scattered waves reads
(
Fermi, 1949
;
Sakurai, 1967; Raffelt, 1996
)
Ψ
f
Ψ
Ψ
(
z
)
=e
i
kz
+
2
π
∞
0
Nδ
f
(
θ
)
e
i
kr
r
ρ
d
ρ,
(
2.104
)
where ρ
=
x
2
+
y
2
an
d
θ
=
arctan
(
ρ
/
z
)
,an
d
N
is the number of th
e
s
catterers per unit vo
l
ume. Note t
h
a
t
ρ
d
ρ
=
r
d
r
h
o
ld
s. Rep
l
acin
g
k
by
k
+i
wit
h
→
0
+
i
ntheintegrandinEq.
(
2.104
)
, one obtains
Ψ
f
ΨΨ
(
z
)
=e
i
kz
1
+
i
2
π
f
(
0
)
N
δ
k
.
(
2.105
)
F
or the slab with a finite thicknes
s
d
≡
mδ
,
wher
e
m
c
an be taken as the
n
umber o
f
the layers and
m
→
∞
holds, it is strai
g
ht
f
orward to obtain
Ψ
f
ΨΨ
(
z
)
= lim
m
→∞
e
i
kz
1
+i
2
π
f
(
0
)
N
d
mk
m
=e
i
kz
e
i2
π
f
(
0
)
Nd
/
k
,
(
2.106
)
which should be identified with the plane wave e
i
k
[
z
+(
n
−
1
)
d
]
.Th
e
r
e
fr
act
i
ve
i
ndex of neutrinos in matter turns out to be
(
Wolfenstein, 1978
)
n
=1+
2
π
k
2
Nf
(
0
)
.
(
2.107
)
N
ote that the energy dependence of the forward scattering amplitude
f
(
0
)
i
s implied.
A
nonzero refractive index means that the dispersion relation o
f
n
eutrinos is modified in matter
,
so is their flavor oscillation in matter. Thi
s
m
echanism has offered a natural solution to the longstanding solar neutrino
problem
(
Wolfenstein, 1978; Mikheyev and Smirnov, 1985
)
. The details o
f
n
eutrino oscillations in matter will be discussed in Cha
p
ter 5
.
Now t
h
at a re
l
ations
h
ip
b
etween t
h
e scattering amp
l
itu
d
e f
(
θ
)
and th
e
F
e
y
nman amp
l
itu
d
e
M
(
a
+
b
→
c
+
d
)
has been established, it is ready t
o
c
alculate the forward scattering amplitude
f
(
0
)
by using the same technique
s
i
n perturbative quantum
fi
eld theories. For instance, we shall comput
e
f
(
0
)
for
ν
e
-
e
−
s
catterin
g
in matter in
S
ection 2.3.2. Here let us
fig
ure out th
e
c
harged-current contribution to f
(
0
)
in the elasti
c
ν
e
-
e
−
s
cattering process
.
The relevant Feynman amplitude has been given in Eq.
(
2.59
)
.Onemayse
t
|
p
a
|
=
|
p
c
|
i
nEq.
(
2.103
)
for the elastic forward scattering under consider
-
a
tion. Using t
h
enorma
l
ization con
d
ition
s
u
†
(
p
,r
)
u
(
p
,s
)
=
2
Eδ
rs
for Dira
c
s
pinors, one can t
h
en arrive a
t
f
(
0
)
=
−
G
F
√
2
π
E.
(
2.108
)
This result, together with Eq.
(
2.107
)
, leads to the neutrino refractive index
n
−
1=
−
√
2
G
F
n
e
/E
,
wher
e
n
e
i
s the number density of electrons in matter
.
2.3 Neutrino Propagation in a Me
d
ium 47
The distribution o
f
electrons in matter is assumed to be isotropic, and thu
s
the avera
g
e of their three-momenta is vanishin
g
. One should take accoun
t
of both neutral- and charged-current contributions t
o
f
(
0
)
in a complet
e
a
nalysis o
f
the elastic
f
orward scatterin
g
o
f
neutrinos in matter, but only the
d
ifferences between the refractive indices of different neutrinos are relevan
t
to
n
eut
rin
o
fl
avo
r
co
n
ve
r
s
i
o
n
s.
2.
3
.2 The Effective Potentia
l
The neutrino waves scattered by the back
g
round particles may coherentl
y
i
nterfere in the forward direction. This matter-induced effect on the propaga
-
tion o
f
neutrinos can be described by an e
ff
ective potential.
S
ince neutrino
s
a
re massless in the SM, their dis
p
ersion relation in matter is modified t
o
|
p
|
=
n
E
,
w
h
ere
n
is just the refractive index. Since neutrinos only have
weak interactions with matter,
(
n
−
1
)
1
must
h
o
ld
.Int
h
is case one ma
y
d
efine the effective
p
otential of the neutrino beam in matter a
s
V
≡
E
−
|
p
|
=
−
(
n
−
1
)
E
=
−
2
π
N
E
f
(
0
)
,
(
2.109
)
where Eq.
(
2.107
)
has been used. This relationship links the effective potentia
l
V
to t
h
e
r
ef
r
act
i
ve
in
dex
n
,
or equivalently, to the
f
orward scatterin
g
ampli-
tud
e
f
(
0
)
. If only the charged-current interactions are taken into account,
then the effective potential of electron neutrinos in an electron background
c
an
b
e
g
iven
by
V
=
−
2
π
N
E
f
(
0
)
=
√
2
G
F
n
e
,
(
2.110
)
where Eq.
(
2.108
)
has been used. Hence a dense medium is possible to affect
the behavior o
f
neutrino propa
g
ation, in particular the oscillation o
f
neutrin
o
fl
avors. Because neutrinos are massless and le
p
ton
fl
avors are conserved in th
e
SM, the
p
henomenon of neutrino oscillations is im
p
ossible to take
p
lace either
i
n
vacuu
m
o
rinm
atte
r
u
n
less
n
o
n-
sta
n
da
r
d
n
eut
rin
o
in
te
r
act
i
o
n
sa
r
e
in
t
r
o
-
d
uced
(
Wolfenstein, 1978
)
. Although three neutrinos may have finite masses
beyond the SM, the precision measurements of electroweak interactions hav
e
i
ndicated that their interactions with other
S
M particles can still be described
by the SM to an excellent degree of accuracy
(
Nakamura
e
tal
.
, 2010
)
. Hence
the neutrino refractive indices calculated in the SM are applicable, at least
i
n the leadin
g
-order approximation, to those reasonable extensions o
f
the
S
M
which can naturall
y
accommodate massive neutrinos
.
Now we consider the ordinary matter which is made of electrons an
d
n
uc
l
eons
b
ut e
l
ectrica
lly
neutra
l
as a w
h
o
l
e. In t
h
is case t
h
enum
b
er
d
ensities
of electrons and protons are equal
(
i.e.
,
n
e
=
n
p
)
. The forward scattering
o
f
ν
e
n
eutrinos off electrons is caused by both charged- and neutral-current
i
nteractions, while that o
f
ν
μ
or
ν
τ
n
eutrinos o
ff
electrons can onl
y
be caused
4
8
2 Neutrinos within the
S
tandard Model
b
y
neutral-current interactions. The Fe
y
nman amplitude o
f
ν
e
+
e
−
→
ν
e
+
e
−
s
cattering has been given in Eq.
(
2.68
)
and can be rewritten a
s
M
(
ν
e
e
−
)
= −
G
F
√
2
u
ν
(
p
)
γ
μ
(
1
−
γ
5
)
u
ν
(
p
)
u
e
(
q
)
γ
μ
(
g
e
V
−
g
e
A
γ
5
)
u
e
(
q
)
,
(
2.111
)
w
h
e
r
e
g
e
V
≡
1+
C
l
V
CC
a
n
d
g
e
A
≡
1+
C
l
A
C
have been defined in Eq.
(
2.75
)
.
The Feynman amplitude o
f
ν
μ
+
e
−
→
ν
μ
+
e
−
scattering o
r
ν
τ
+
e
−
→
ν
τ
+
e
−
scattering can directly be read off from Eq.
(
2.111
)
by replacing th
e
coeffic
i
e
n
ts
g
e
V
a
n
d
g
e
A
w
i
th
C
l
V
CC
a
n
d
C
l
A
C
g
iven in Table 2.2.
O
ne may construct
a
low-energy effective Hamiltonian for elastic ν
e
-
e
−
scattering, which should
be able to reproduce the exact Feynman amplitude in Eq.
(
2.111
)
. Such
a
requirement is a common criterion for constructin
g
the effective Hamiltonians
i
n quantum field theories.
T
he
f
act that neutrinos propa
g
atin
g
in matter may interact with the back-
g
round particles can be described by means of the picture of free neutrino
s
traveling in an effective potential. The first step to derive the effective poten
-
tial is to write down the e
ff
ective Hamiltonian at an ener
g
y scale much lowe
r
than the masses of
W
±
and Z
0
b
osons. The effective Hamiltonian densit
y
responsible for the interactions of
ν
e
n
eut
rin
os w
i
th elect
r
o
n
s
i
s
H
e
ff
(
x
)
=
G
F
√
2
ν
e
(
x
)
γ
μ
(1
−
γ
5
)
ν
e
(
x
)
e
(
x
)
γ
μ
(
g
e
V
−
g
e
A
γ
5
)
e
(
x
)
.
(
2.112
)
A
ssuming the background electrons to be thermally distributed and unpo-
l
arized, we should average the effective Hamiltonian over the electron states.
Thiscanbedoneinthe
f
ollowin
g
way:
H
eff
(
x
)=
1
2
s
e
d
3
p
(2
π
)
3
g
e
f
(
p,
T
)
e
(
p
,
s
e
)
|H
eff
(
x
)
|
e
(
p
,s
e
)
,
(
2.113
)
whe
r
e
g
e
= 2 denotes an internal de
g
ree o
ff
reedom
,
f
(
p,
T
)
is the distribution
f
unction of electrons in the
p
hase s
p
ace, and
|
e
(
p
,
s
e
)
=
2
E
p
b
†
e
(
p
,s
e
)
|
0
s
tands
f
or the one-particle electron state with the momentum
p
an
d
t
h
espi
n
ei
g
envalue
s
e
.
Recallin
g
the quantization o
f
the electron
fi
eld
,
e
(
x
)
=
d
Π
q
Π
s
b
e
(
q
,s
)
u
e
(
q
,
s
)e
−
i
qx
+
d
†
e
(
q
,s
)
v
e
(
q
,
s
)e
i
qx
(
2.114
)
wit
h
t
h
ep
h
ase e
l
ement
d
Π
q
Π
≡
d
3
q
/
[(
2
π
)
3
2
E
q
]
, one may figure out th
e
a
veraged effective Hamiltonian in Eq.
(
2.113
)
:
H
eff
(
x
)
=
G
F
√
2
n
e
g
e
V
ν
e
(
x
)
γ
0
(
1
−
γ
5
)
ν
e
(
x
)
,
(
2.115
)
where the inte
g
ration over the distribution
f
unction
g
ives rise to the number
d
ensity of electrons
n
e
,
and the three-momentum
p
is taken to be isotro
p
ic
.
2.3 Neutrino Propagation in a Me
d
ium 49
There
f
ore, the electrons in matter contribute an e
ff
ective potential
f
or th
e
l
eft-handed electron neutrinos
:
V
e
=
d
3
p
d
3
x
(2
π
)
3
2
E
p
ν
e
L
(
p
,s
)
|
H
eff
(
x
)
|
ν
e
L
(
p
,
s
)
=
√
2
G
F
n
e
g
e
V
,
(
2.116
)
w
h
ere we
h
ave intro
d
uce
d
t
h
e one-partic
l
e state
|
ν
e
L
(
p
,s
)
correspon
d
ing to
t
h
e
l
eft
-h
a
n
ded
n
eut
rin
ofie
l
d
ν
e
L
(
x
)
. Note that
(
1
−
γ
5
)
ν
e
L
(
x
)
=
2
ν
e
L
(
x
)
and
the normalization relation of the one-
p
article state
s
ν
e
L
(
p
,
s
)
|
ν
e
L
(
q
,s
)
=
2
E
p
(2
π
)
3
δ
3
(
p
−
q
)
δ
ss
s
hould be carefully taken into account in the above cal-
c
ulation. For this reason, a similar calculation done in the
fi
nite-volume quan-
tization scheme is somewhat more transparent and simpler
(
Linder, 2005
;
Giunti and Kim, 2007
)
.
I
n comparison, the e
ff
ective Hamiltonian responsible
f
o
r
ν
μ
or
ν
τ
neut
r
i
n
os
propagating in matter only arises from neutral-current interactions and can
t
h
e
r
e
f
o
r
ebew
ri
tte
n
as
H
e
ff
(
x
)=
G
F
√
2
ν
α
(
x
)
γ
μ
(1
−
γ
5
)
ν
α
(
x
)
e
(
x
)
γ
μ
(
C
l
V
CC
−
C
l
A
C
γ
5
)
e
(
x
)
,
(
2.117
)
whe
r
e
α
=
μ
o
r
τ
. Following a way similar to the treatment of electron
n
eutrinos, one may easily
fig
ure out the e
ff
ective potentials o
f
muon and ta
u
n
eutrinos in normal matter:
V
μ
V
=
V
τ
VV
=
√
2
G
F
n
e
C
l
V
CC
.
(
2.118
)
N
ote that the equalit
y
V
μ
V
=
V
τ
VV
w
ill n
o
m
o
r
e
h
o
l
d
i
ft
h
e
r
ad
i
at
i
ve co
rr
ect
i
o
n
s
a
re taken into account
(
Botell
a
et al
.
,
1987
)
. The difference between
V
μ
V
a
nd
V
τ
VV
is likely to impact the flavor conversions of supernova neutrinos
(
Esteban
-
P
retel et al., 2008
)
.
W
e proceed to consider the effective Hamiltonian responsible for neutrino
-
n
ucleon interactions. The above results indicate that an e
ff
ective potential o
f
n
eutrinos propa
g
atin
g
in isotropic matter is independent of the pseudovec-
tor couplings. In addition, the conservation of vector currents implies that
the vector couplin
g
so
f
nucleons can simply be taken as a sum over the vec
-
tor couplings of valence quarks. Both neutrino-proton and neutrino-neutron
s
cattering processes are me
d
iate
db
yt
h
eneutra
l
gauge
b
oson
Z
0
,
so t
h
eir
effect
i
ve
H
a
mil
to
ni
a
n
ta
k
es t
h
efo
rm
H
eff
(
x
)=
G
F
√
2
ν
α
(
x
)
γ
μ
(
1
−
γ
5
)
ν
α
(
x
)
N
(
x
)
γ
μ
C
N
V
CC
−
γ
5
C
N
A
C
N
(
x
)
,
(
2.119
)
w
h
e
r
e
C
N
V
CC
,
A
≡
2
C
u
V
CC
,
A
+
C
d
V
CC
,
A
f
or
p
roton
s
N
(
x
)=
p
(
x
)
,o
r
C
N
V
CC
,
A
≡
2
C
d
V
CC
,
A
+
C
u
V
C
C
,
A
for neutron
s
N
(
x
)
=
n
(
x
)
. One can then work out the effective poten-
tials o
f
neutrinos induced by the back
g
round protons and neutrons in matter:
V
p
α
V
=
√
2
G
F
n
p
2
C
u
V
CC
+
C
d
V
CC
=
1
√
2
G
F
n
p
1
−
4
si
n
2
θ
w
,
50
2 Neutrinos within the
S
tandard Model
V
n
α
V
=
√
2
G
F
n
n
2
C
d
V
CC
+
C
u
V
CC
=
−
1
√
2
G
F
n
n
,
(
2.120
)
whe
r
e
α
=
e
,
μ
o
r
τ
,
n
p
a
n
d
n
n
stand respectivel
yf
or the number densitie
s
o
f
protons and neutrons, and the values o
f
C
u
V
CC
a
n
d
C
d
V
CC
g
iven in Ta
bl
e2.
3
h
ave been used. Note that the neutral-current contributions are universal for
a
ll three types of neutrinos. Note also that the total effective potential for
elect
r
o
nn
eut
rin
os
i
s
V
e
V
+
V
p
e
V
+
V
n
e
=
√
2
G
F
n
e
+
V
n
α
V
, but the total e
ff
ectiv
e
potential for muon
(
or tau
)
neutrinos is
V
μ
V
+
V
p
μ
V
+
V
n
μ
V
=
V
n
α
V
,
where
n
e
=
n
p
i
s taken as a consequence of the electric neutrality of matter. The differenc
e
between the potentials of electron and muon
(
or tau
)
neutrinos turns out to
b
e
Δ
V
=
V
e
V
−V
μ
V
=
√
2
G
F
n
e
.
A
nim
p
ortant comment should be made on
the effective potentials of antineutrinos in matter. The effective Hamiltonia
n
o
f
neutrinos is also applicable to the correspondin
g
antineutrinos. But a di
f
-
f
erence appears when evaluating the effective potential in Eq.
(
2.116
)
.For
n
eutrinos, we are left with a unique non-vanishing matrix element o
f
H
eff
(
Linder, 2005
):
ν
α
(
p
)
|
H
eff
(
x
)
|
ν
α
(
p
)
∝
0
|
b
ν
(
p
)
d
3
q
b
†
ν
(
q
)
b
ν
(
q
)
b
†
ν
(
p
)
|
0
;
(
2.121
)
while for antineutrinos, the corresponding matrix element o
f
H
e
ff
reads
ν
α
(
p
)
|
H
eff
(
x
)
|
ν
α
(
p
)
∝
0
|
d
ν
(
p
)
d
3
q
d
ν
(
q
)
d
†
ν
(
q
)
d
†
ν
(
p
)
|
0
=
−
0
|
d
ν
(
p
)
d
3
q
d
†
ν
(
q
)
d
ν
(
q
)
d
†
ν
(
p
)
|
0
,
(
2.122
)
where the quantization of neutrino fields is the same as that in Eq.
(
2.114
)
.
H
ence we conclude that the effective
p
otentials res
p
onsible for neutrinos an
d
a
ntineutrinos propa
g
atin
g
in matter
h
ave t
h
eoppositesi
g
ns. For t
h
e same
reason, we can see from Eq.
(
2.113
)
that the effective potentials of neutri
-
n
os will change their signs if the background particles are replaced by the
c
orrespon
d
in
g
antipartic
l
es
.
Finally, let us turn to the coherent forward scatterin
g
of neutrinos off
t
h
eir am
b
ient neutrinos — a p
h
enomenon w
h
ic
h
can
b
eextreme
l
yimportant
f
or the
fl
avor conversions o
f
supernova neutrinos. It is obvious that onl
y
the
n
eutral-current interactions contribute to this process. The Fe
y
nman ampli
-
tude of
ν
α
+
ν
α
→
ν
α
+
ν
α
scattering has been given in Eq.
(
2.80
)
,which
c
orresponds to the
f
ollowin
g
e
ff
ective Hamiltonian:
H
eff
(
x
)
=
G
F
2
√
2
ν
α
(
x
)
γ
μ
(1
−
γ
5
)
ν
α
(
x
)
ν
α
(
x
)
γ
μ
(1
−
γ
5
)
ν
α
(
x
)
.
(
2.123
)
It is straightforward to reproduce Eq.
(
2.80
)
from Eq.
(
2.123
)
. An average over
the back
g
round neutrinos receives contributions
f
rom both spinor bilinear
s
2.3 Neutrino Propagation in a Me
d
ium 5
1
i
nEq.
(
2.123
)
and thus gives rise to a factor 2. Taking account of this factor,
one can work out the effective
p
otential for
ν
α
neutrinos propa
g
atin
g
in the
background of
ν
α
neutrinos
:
V
ν
α
=2
√
2
G
F
n
ν
α
,
(
2.124
)
where
n
ν
α
is the number density of the background neutrinos. One should
keep in mind that neutrinos are always le
f
t-handed in the
S
M when doin
g
the
a
bove calculation. One may analo
g
ously find that the effective potential for
ν
α
n
eutrinos traveling in the background o
f
ν
β
ν
n
eutrinos
(
fo
r
β
=
α
)
is simply
V
ν
α
=
√
2
G
F
n
ν
β
. Hence the effective potential of neutrino-neutrino scatter
-
i
n
g
can be recast into a much more compact form:
V
ν
α
=
√
2
G
F
(
1+
δ
α
β
)
n
ν
β
.
T
h
eangu
l
ar corre
l
ation
b
etween t
h
einci
d
ent an
db
ac
k
groun
d
neutrinos i
s
very important in the case o
f
supernova neutrinos. This e
ff
ect is ne
g
lecte
d
h
ere but it will be discussed in detail in Cha
p
ter 7
.
2.3.3 Neutrino Sel
f
-ener
g
y Approach
In the astrophysical and cosmolo
g
ical environments, such as in the core o
f
a
supernova or in the earl
y
Universe, there ma
y
exist neutrinos in therma
l
equi
l
i
b
rium wit
h
ot
h
er partic
l
es. In t
h
is case it is more reasona
bl
eto
d
ea
l
with the s
y
stem b
y
means o
f
quantum
fi
eld theories at
fi
nite temperatures
(
Landsman and van Weert, 1987; Kapusta, 1989; Quiros, 1999
)
. On the other
h
an
d
, it is certain
l
y inappropriate to treat neutrinos in t
h
eear
l
yUniversea
s
a
beam of free particles.
A
proper strate
g
y is to calculate the self-ener
g
ies
of neutrinos in the presence of background particles and then extract the
effective potential by using the dispersion relation. The dispersion relatio
n
c
an be derived from
(
N¨otzold and Raffelt, 1988
)
D
e
t
/
p
//
−
Σ
(
/
p
//
)
=
0
,
(
2.125
)
whe
r
e
Σ
(
/
p
//
)
denotes the self-energy of a fermion. For a free fermion with mas
s
m
,we
h
av
e
Σ
(
/
p
//
)=
m
. Then an explicit calculation o
f
the determinant i
n
E
q.
(
2.125
)
yields the dispersion relation
E
2
=
|
p
|
2
+
m
2
.Inthe
p
resence o
f
i
nteractions, the general form of the neutrino self-energy in the SM can be
w
r
itte
n
as
Σ
(
/
p
//
)
=
{
c
1
/
p
//
+
c
2
/
u
//
+
c
3
[
/
p,
[[
//
/
u
//
]
}
P
L
PP
w
i
th
u
μ
b
ein
g
the
f
our-velocity o
f
the medium
(
Weldon, 1982
)
. A calculation of the coefficients
c
i
(
fo
r
i
=
1
,
2
,
3
)
i
nvolves the Feynman rules in quantum
fi
eld theories at
fi
nite temperatures,
which will be brie
fl
y discussed in the
f
ollowin
g
.
C
onsider a thermal system at rest with the Hamiltonia
n
H
and the con
-
s
erve
d
c
h
ar
g
e
Q
.T
h
eequi
l
i
b
rium states are t
h
en
d
escri
b
e
db
yt
h
e
g
ran
d-
c
anonical densit
y
operato
r
5
5
Th
e
g
ran
d
canonica
l
ensem
bl
ecanexc
h
an
g
et
h
e ener
g
yan
d
partic
l
es wit
h
t
h
e
thermal reservoir, but its temperature, volume and chemical potential are fixed.
5
2 2 Neutrinos within the
S
tandard Model
ρ
=
1
N
e
−
β
(
H
−
μQ
)
,
(
2.126
)
where
N
≡
T
r
e
−
β
(
H
−
μ
Q
)
is the normalization factor
,
μ
d
enotes the chemi-
c
a
l
potentia
l
,an
d
β
=
1
/
T
w
i
th
T
b
ein
g
t
h
e temperature is intro
d
uce
d
.Her
e
the normalization Tr
[
ρ
]
= 1 is implied. The thermal average of an operator
O
i
s then defined a
s
O
=Tr
[
ρ
O
]
.
(
2.127
)
F
or simplicity, we temporarily i
g
nore the chemical potential
μ
a
nd discus
s
the case of real scalar particles. In the Heisenberg picture one may write th
e
s
calar
fi
eld operator as
φ
(
x
)
=
e
i
Ht
φ
(
0
,
x
)
e
−
i
H
t
,
w
h
ere t
h
etim
e
x
0
=
t
is
to be anal
y
ticall
y
continued to the complex plane.
A
s usual, the two-point
Green function is defined as the thermal average of the time-ordered fiel
d
operators:
G
C
(
x
−
y
)
≡
T
C
TT
[
φ
(
x
)
φ
(
y
)]
≡
Θ
C
x
0
−
y
0
φ
(
x
)
φ
(
y
)
+
Θ
C
y
0
−
x
0
φ
(
y
)
φ
(
x
)
,
(
2.128
)
whe
r
ethe
T
C
TT
-orderin
g
indicates that the
fi
elds should be ordered alon
g
th
e
p
at
h
C
in the complex-time plane. We explicitl
y
parametrize the path a
s
t
=
z
(
τ
)
wit
h
τ
b
ein
g
area
l
parameter, t
h
en t
h
eor
d
erin
g
i
n
t
i
sdefi
n
ed as
t
h
at
i
n
τ
.
In addition, we have the ste
pf
unction
Θ
C
(
t
)=
Θ
(
τ
)
and the Dira
c
d
elta function
δ
C
(
t
)
=
d
z
d
τ
−
1
δ
(
τ
)
. The functional formalism in quantu
m
field theories is also applicable, so one may similarly define the generating
f
unctional o
fG
reen
f
unctions and that o
f
the one-particle irreducible
G
reen
f
unctions
(
Landsman and van Weert, 1987
)
. The two-point Green function
d
efinedinEq.
(
2.128
)
can be rewritten a
s
G
C
(
x
−
y
)
≡
Θ
C
(
x
0
−
y
0
)
G
+
(
x
−
y
)
+
Θ
C
(
y
0
−
x
0
)
G
−
(
x
−
y
)
,
(
2.129
)
where
G
+
(
x
−
y
)
≡
φ
(
x
)
φ
(
y
)
a
n
d
G
−
(
x
−
y
)
=
G
+
(
y
−
x
)
.Nowweevaluate
G
+
(
x
−
y
)
at the poin
t
x
=
y
= 0 by implementing the complete set o
f
e
ig
enstates
|
n
of the Hamiltonian
(
i.e.,
H
|
n
=
E
n
|
n
)
. More explicitly
,
G
+
(
x
0
−
y
0
)=
1
N
n
n
|
e
−
βH
e
i
H
x
0
φ
(
0
)e
−
i
H
(
x
0
−
y
0
)
φ
(
0
)e
−
i
Hy
0
|
n
=
1
N
m
,n
|
m
|
φ
(
0
)
|
n
|
2
e
−
i
E
m
(
x
0
−
y
0
)
e
i
E
n
(
x
0
−
y
0
+i
β
)
.
(
2.130
)
To assure the summation to be convergent, we requir
e
−
β
I
m
(
x
0
−
y
0
)
0
to
g
et
h
er wit
h
Θ
C
(
x
0
−
y
0
)
=0forIm
(
x
0
−
y
0
)
>
0. A similar analysis
of
G
−
(
x
0
−
y
0
)
yields the condition
0
I
m
(
x
0
−
y
0
)
β
to
g
ether with
Θ
C
(
y
0
−
x
0
)
=0forIm
(
x
0
−
y
0
)
<
0
. Consequently, the convergence of the
2.3 Neutrino Propagation in a Me
d
ium 53
G
reen
f
unction in the ran
ge
−
β
I
m
(
x
0
−
y
0
)
β
r
est
ri
cts us to defi
n
e
Θ
C
(
t
)
=0forIm
(
t
)
>
0
. The latter implies that the point alon
g
the chose
n
pat
h
C
s
h
ou
ld h
ave a constant or monotonica
ll
y
d
ecreasing imaginary par
t
(
Quiros, 1999
)
. Starting with the definition of the two-point Green function
,
we can further derive
G
+
(
t
−
i
β
,
x
)
≡
φ
(
t
−
i
β
,
x
)
φ
(
0
)
=
1
N
Tr
e
−
βH
e
i
H
(
−
i
β
)
φ
(
t,
x
)e
−
i
H
(
−
i
β
)
φ
(
0
)
=
1
N
Tr
e
−
βH
φ
(
0
)
φ
(
t,
x
)
=
G
+
(
−
t,
−
x
)=
G
−
(
t,
x
)
,
(
2.131
)
where the c
y
clic invariance o
f
the trace and the identit
y
G
−
(
x
−
y
)
=
G
+
(
y
−
x
)
h
ave been used. This is just the famous Kubo-Martin-Schwinger relatio
n
(
Kubo, 1957; Martin and Schwinger, 1959
).
Now we calculate the thermal two-
p
oint
G
reen
f
unction
f
or a
f
ree real
s
calar field
,
whose Hamiltonian can be written as
H
=
d
3
p
(2
π
)
3
E
p
a
†
p
a
p
,
(
2.132
)
whe
r
e
E
p
=
|
p
|
2
+
m
2
.
N
ote that
a
†
a
n
d
a
stand respectivel
yf
or th
e
c
reation and annihilation operators, which satisf
y
the commutation relation
a
p
,a
†
q
=(2
π
)
3
δ
3
(
p
−
q
)
. We first consider a system with a single momentu
m
m
o
d
ean
d
t
h
eenergyω
.
Its Hami
l
tonian i
s
H
=
ω
a
†
a
,an
d
its comp
l
ete se
t
o
f
ei
g
enstates are denoted by
|
n
.
We
g
e
t
Tr
e
−
βH
=
∞
n=
0
n
|
e
−
βH
|
n
=
∞
n=
0
e
−
n
βω
=
1
1
−
e
−
βω
,
(
2.133
)
a
n
d
T
r
e
−
βH
a
†
a
=
∞
n
=
0
n
e
−
nβ
ω
=
e
−
βω
(
1
−
e
−
βω
)
2
.
(
2.134
)
Therefore, the thermal average of the number operator read
s
a
†
a
=
n
B
(
ω
)
a
n
d
aa
†
=1+
n
B
(
ω
)
,where
n
B
(
ω
)
≡
1
/
(
e
β
ω
−
1)
is the Bose-Einstein
d
istribution function. Then it is straightforward to obtain the results for an
y
mo
m
e
n
tu
mm
ode:
a
†
p
a
q
=
n
B
(
E
p
)(2
π
)
3
δ
3
(
p
−
q
)
,
a
p
a
†
q
=
1+
n
B
(
E
p
)
(
2
π
)
3
δ
3
(
p
−
q
)
.
(
2.135
)
R
ecalling the quantization of a real scalar field and using Eq.
(
2.135
)
,we
c
ompute the two-point Green function
(
Altherr, 1993
)
5
4 2 Neutrinos within the
S
tandard Model
G
C
(
x
−
y
)
=
"
Θ
C
(
x
0
−
y
0
)
a
†
p
a
q
e
i
(
px
−
q
y
)
+
a
p
a
†
q
e
−
i(
p
x
−
q
y
)
+
Θ
C
(
y
0
−
x
0
)
a
q
a
†
p
e
i(
p
x
−
qy
)
+
a
†
q
a
p
e
−
i
(
px
−
qy
)
#
=
d
4
p
(2
π
)
4
κ
(
p
)e
−
i
p
(
x
−
y
)
Θ
C
(
x
0
−
y
0
)
+
n
B
(
p
0
)
,
(
2.136
)
where the integration over the phase space in the first equality is implied,
a
nd the spectrum functio
n
κ
(
p
)
≡
2
π
δ
(
p
2
−
m
2
)
Θ
(
p
0
)
−
Θ
(
−
p
0
)
has bee
n
i
ntroduced. It is worth pointin
g
out that the speci
fi
c expression o
f
G
C
(
x
−
y
)
d
efinitel
y
depends on the choice of the pat
h
C
.
There exist two conven
-
tional options — the imaginary- and real-time formalisms
(
Landsman an
d
van Weert, 1987
)
. We shall concentrate on the real-time formalism, in whic
h
the
p
ath
C
is chosen in the complex-time plane as follows:
(
1
)
from the initial
r
eal t
im
e
t
i
alon
g
the real axis to the
fi
nalrealtime
t
f
;
(
2
)
from
t
f
to
t
f
−
i
σ
a
lon
g
the ima
g
inary axis with
0
<σ<
β
;
(
3
)
from
t
f
−
i
σ
to
t
i
−
i
σ
i
n
t
h
e
opposite direction of the real axis; and
(
4
)
fro
m
t
i
−
i
σ
t
o
t
i
−
i
β
.Inthis
f
ormalism four distinct propagators arise from Eq.
(
2.136
)
and they can be
ex
p
ressed in the momentum s
p
ace as
G
C
(
p
)=
G
11
(
p
)
G
12
(
p
)
G
21
(
p
)
G
22
(
p
)
=
U
(
β
,
p
)
Δ
(
p
)0
0
Δ
∗
(
p
)
U
(
β
,
p
)
,
(
2.137
)
where
U
(
β
,p
)
=
cosh
θ
p
s
in
h
θ
p
s
in
h
θ
p
cosh
θ
p
(
2.138
)
w
i
t
h
cosh
θ
p
≡
1
/
1
−
e
−
βE
p
,an
d
Δ
(
p
)
is just the usual Feynman propa-
gator at zero temperature. Although there are propagators associated wit
h
the superscript “2”, the ph
y
sical
fi
elds are onl
y
related t
o
G
11
(
p
)
. Explicitly,
we have
(
Bernard, 1974; Dolan and Jackiw, 1974; Weinberg, 1974; Landsma
n
a
nd van Weert, 1987
)
G
11
(
p
)=
i
p
2
−
m
2
+i
+
2
π
n
B
(
E
p
)
δ
(
p
2
−
m
2
)
.
(
2.139
)
O
ne ma
y
discuss the case o
ff
ermions in a similar wa
y
.The
f
ermionic
G
reen
f
unction can be written a
s
S
C
(
p
)
=
S
11
(
p
)
S
12
(
p
)
S
2
1
(
p
)
S
22
(
p
)
=
V
(
β
,
p
)
S
F
(
p
)0
0
S
∗
F
(
p
)
V
(
β
,
p
)
,
(
2.140
)
w
h
e
r
e
V
(
β
,
p
)=
cos
θ
p
s
i
n
θ
p
si
n
θ
p
cos
θ
p
(
2.141
)
2.3 Neutrino Propagation in a Me
d
ium 55
(
c
)
(b)
ν
α
ν
ν
α
ν
l
±
α
W
∓
f
Z
0
ν
α
ν
ν
α
ν
Z
0
ν
α
ν
(a
)
ν
α
ν
ν
α
ν
Fig. 2.7 The Feynman diagrams for the self-energy of neutrinos in the presence o
f
background particles:
(
a
)
charged-current contributions;
(
b
)
neutral-current contri-
butions with
f
being all the relevant particles in a medium; and (c) contribution
s
f
rom the background neutrinos
wit
h
co
s
θ
p
≡
[
Θ
(
p
0
)
−
Θ
(
−
p
0
)]
/
1
+
e
−
β
E
p
,
an
d
S
F
(
p
)
is just the fermioni
c
propagator at zero temperature. Eq.
(
2.140
)
allows us to obtain the physica
l
c
om
p
onent of the Green function:
S
11
(
p
)=
/
p
//
+
m
i
p
2
−
m
2
+i
−
2
π
n
F
(
E
p
)
δ
(
p
2
−
m
2
)
,
(
2.142
)
whe
r
e
n
F
(
E
p
)
≡
1
/
(
e
β
E
p
+
1
)
is the Fermi-Dirac distribution function. I
n
f
act, the Feynman dia
g
rams of a specific process in the real-time formalism
i
s the same as those in the zero-temperature field theory. But all the vertices
sh
ou
ld b
e
l
a
b
e
l
e
db
y“1
”
or “2
”
,an
d
t
h
ep
h
ysica
ll
e
g
smust
b
e assi
g
ne
d
as
“1”. The Fe
y
nman rules for the vertex “1” are the same as those in the zero
-
temperature field theory, and those for the vertex “2” are just the Hermitia
n
c
onjugate
(
Landsman and van Weert, 1987
).
Now we are ready to calculate the self-energy of neutrinos in the presenc
e
of background particles. To be more specific, we consider the ordinary mat
-
te
rin
wh
i
ch the
r
ea
r
e elect
r
o
n
sa
n
d
n
ucleo
n
so
rn
eut
rin
os the
m
selves
.T
he
F
eynman diagrams for possible contributions to the neutrino self-energy are
s
hown in Fig. 2.7, where the unitary gauge is implied. So far we have only
d
iscussed the propa
g
ators o
f
bosons and
f
ermions in a thermal system a
t
rest. It has been
p
ro
p
osed that all the calculations at finite tem
p
eratures ca
n
be per
f
ormed in a Lorentz-invariant way by introducin
g
the
f
our-velocit
y
u
μ
of the thermal bath
(
Weldon, 1982
)
. In this case the propagator of electro
n
fields is
(
Landsman and van Weert, 1987
)
S
11
F
(
p
)
=
/
p
//
+
m
e
i
p
2
−
m
2
e
+i
−
2
π
δ
(
p
2
−
m
2
e
)
η
(
p
·
u
)
,
(
2.143
)
w
h
e
r
ewe
h
ave defi
n
ed
η
(
p
·
u
)
≡
Θ
(
p
·
u
)
e
β
(
p
·
u
−
μ
)
+
1
+
Θ
(
−
p
·
u
)
e
−
β
(
p
·
u
−
μ
)
+1
(
2.144
)