Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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276 7 Neutrinos
f
rom
S
upernovae
˙
P
j
=
ω
j
B
×
P
j
+
√
2
G
F
V
J
×
P
j
,
(
7.52
)
whe
r
e
ω
j
≡
Δ
m
2
/
(2
p
j
)
. If the vacuum oscillation term is switched off, the
n
P
j
p
recesses around
J
.A
fter switchin
g
on the vacuum oscillation term and
s
umming over the inde
x
j
o
n both sides of Eq.
(
7.52
)
,weobtai
n
˙
J
=
N
ν
j
=
1
ω
j
B
×
P
j
,
(
7.53
)
w
hi
c
hi
s
n
ot
in
t
h
ec
l
osed fo
rm
fo
r
J
.I
ft
h
e
n
eut
rin
ose
l
f
-in
te
r
act
i
o
n
do
min
ates
over the vacuum oscillation term
,
the
n
P
j
rotates around
J
rapidl
y
and
thus its transverse components are averaged to zero. We are finally left wit
h
P
j
=
ˆ
J
(
P
j
·
ˆ
J
)
with
ˆ
J
≡
J
/
|
J
|
.
So Eq.
(
7.53
)
can be rewritten as
˙
J
=
B
×
ˆ
J
N
ν
j
=
1
ω
j
(
P
j
·
ˆ
J
)
≡
ω
syn
B
×
J
,
(
7.54
)
where the gyromagnetic rati
o
ω
syn
i
sgivenb
y
ω
s
y
n
=
1
|
J
|
N
ν
j
=1
ω
j
(
P
j
·
ˆ
J
)
.
(
7.55
)
If all the initial neutrinos are
p
re
p
ared in a s
p
ecific flavor state, then the
po
l
arization vectors
P
j
a
re a
l
i
g
ne
d
wit
h
J
a
n
dthus
|
J
|
=
N
ν
N
N
a
n
d
P
j
·
ˆ
J
=
1,
l
eadin
g
t
o
ω
syn
=(
ω
1
+
ω
2
+
···
+
ω
N
ν
)
/N
ν
N
N
(
i.e., an average value of the
vacuum oscillation frequencies
)
. If the neutrino self-interaction is dominant
over t
h
e vacuum osci
ll
ation term, we conc
l
u
d
et
h
at a
ll
t
h
epo
l
arization vectors
P
j
a
re a
l
i
g
ne
d
wit
h
J
w
hich precesses around the weak ma
g
netic
fi
el
d
B
w
i
th
the frequenc
y
ω
syn
g
iven in Eq.
(
7.55
)
. For neutrino oscillations in vacuum,
a
broad distribution o
f
neutrino ener
g
ies will induce the
fl
avor decoherence
s
uch that the oscillations will be damped. In contrast, these ener
g
ymodes
will oscillate coherently with a common frequency
ω
sy
n
p
rovided the neutrino
s
el
f
-interaction is stron
g
enou
g
h. This collective phenomenon is just the so
-
c
alled synchronized neutrino oscillations
(
Pasto
r
et al
., 2002
)
.
I
n more realistic cases
,
one should take into account the effects of an-
tineutrinos. The equations o
f
motion
f
or both neutrinos and antineutrinos
a
re given by
(
Pasto
r
et al., 2002
)
˙
P
j
=+
ω
j
B
×
P
j
+
√
2
G
F
V
J
−
J
×
P
j
,
˙
P
k
=
−
ω
k
B
×
P
k
+
√
2
G
F
V
J
−
J
×
P
k
,
(
7.56
)
where the total
p
olarization vector
J
is t
h
esu
m
of
P
k
(
for
k
=
1
,
2
,
···
,N
ν
NN
)
wit
h
N
ν
N
N
being the number of antineutrinos in the ensemble. Analogous t
o
7
.4
C
ollective Neutrino Flavor
C
onversions 2
77
t
h
ecasewit
h
on
ly
neutrinos,
b
ot
h
P
j
a
n
d
P
k
w
i
ll b
ea
l
i
g
ne
d
wit
h
t
h
e
gl
o
b
a
l
vec
t
or
I
≡
J
−
J
w
hich precesses around the weak ma
g
netic fiel
d
B
.
The
relevant equation of motion is
˙
I
=
ω
s
y
n
B
×
I
,
w
h
er
e
ω
sy
n
=
1
|
I
|
⎡
⎣
N
ν
j
=
1
ω
j
P
j
·
ˆ
I
+
N
ν
k
=
1
ω
k
P
k
·
ˆ
I
⎤
⎦
(
7.57
)
wit
h
ˆ
I
≡
I
/
|
I
|
.
Therefore, the neutrino-antineutrino system behaves in a wa
y
s
imi
l
ar to t
h
epureneutrinosystem.Notet
h
at a
l
ar
g
e asymmetry
b
etwee
n
n
eutrinos and antineutrinos is required to assure the s
y
nchronized oscilla-
tions, ot
h
erwis
e
I
=
J
−
J
wou
l
dva
ni
s
h
a
n
dt
h
e
n
eut
rin
ose
lf-in
te
r
act
i
o
n
wou
ld b
ea
b
sent. In a supernova environment,
h
owever, t
h
es
y
nc
h
ronize
d
n
eutrino oscillations are suppressed by either the high matter density or the
l
arge neutrino fluxes themselves
(
Duan
et al.
,
2007
).
7.4.3 Bipolar Flavor Conversion
s
The bipolar
fl
avor conversions may take place when the stren
g
th o
f
neutrino-
n
eutr
i
no
i
nteract
i
ons
μ
≡
√
2
G
F
n
ν
is com
p
arable with the vacuum oscillation
f
requenc
y
ω
≡
Δ
m
2
/
(
2
p
)
.I
f
μ
ω
h
o
ld
s, t
h
esync
h
ronize
d
osci
ll
ations wi
ll
occur, as discussed in the previous subsection. Let us
fi
rst o
f
all explai
n
the bipolar s
y
stem, which has alread
y
been introduced in Section 5.3. The
s
implest bipolar system initially consists of pure
ν
e
a
n
d
ν
e
wit
h
equa
ld
ensities
a
nd ener
g
ies. The correspondin
g
equations o
f
motion
f
o
r
P
a
n
d
P
a
re
gi
ven
in
E
q.
(
5.148
)
, or equivalently fo
r
D
≡
P
−
P
an
d
Q
≡
S
−
ω
B
/μ
w
ith
S
≡
P
+
P
giveninEq.
(
5.150
)
. The numerical result for the survival probability in th
e
c
ase o
f
the inverted neutrino mass hierarchy has been shown in Fi
g
.5.4,
where one can observe the com
p
lete flavor conversions of both neutrinos an
d
a
ntineutrinos. In the case o
f
the normal neutrino mass hierarchy, only the
oscillations with extremely small mixin
g
an
g
les are allowed and thus th
e
flavor content is almost unchanged
.
Th
is
b
ipo
l
ar system can
b
ewe
ll
un
d
erstoo
db
yana
l
o
g
ywit
h
t
h
esp
h
eri
-
c
al pendulum
(
Hannestad
e
tal., 2006
)
. Given the initial condition
s
P
(
0
)
=
P
(
0
)
=
(
0
,
0
,
1)
and thus
D
(
0
)
= 0 and S
(
0
)
=
(0
,
0
,
2)
, it is possible to get
Q
(
0
)
=
(
−
ω
si
n2
θ/
μ
,
0
,
2+
ω
cos 2
θ
/
μ
)
because o
f
B
=(
sin
2
θ,
0
,
−
cos 2
θ
)
wit
h
θ
b
ein
g
the mixin
g
an
g
le. In view of the equations of motion shown in
E
q.
(
5.150
)
, one may observe that
B
an
d
Q
lie in the
(
x, z
)
-plane and D is
al
on
g
t
he
y
-axis. In addition, the len
g
th o
f
Q
is conserve
d
; i.e.,
Q
≡
|
Q
|
=
4
+
ω
μ
2
+4
ω
μ
cos 2
θ
1
/
2
.
(
7.58
)
I
f
Q
=
Q
(
si
n
ϕ,
0
,
cos
ϕ
)
an
d
D
=
D
(
0
,
1
,
0
)
wit
h
D
≡
|
D
|
a
re ta
k
en, t
he
e
q
uations o
f
motion o
f
Q
a
n
d
D
i
nEq.
(
5.150
)
can be simplified to those o
f
ϕ
a
nd
D
:
278 7 Neutrinos
f
rom
S
upernovae
˙
ϕ
=
μ
D,
˙
D
=
−
ω
Q
s
in
(
ϕ
+2
θ
)
.
(
7.59
)
E
q.
(
7.59
)
allows us to obtain a second-order differential equation of the tilt
a
n
g
le ϕ
(
Hannesta
d
et al.
,
2006
):
¨
ϕ
=
−
κ
2
s
in
(
ϕ
+2
θ
)
,
(
7.60
)
whe
r
e
κ
≡
√
ωμQ
√√
d
enotes the intrinsic
f
requenc
y
. This result implies that
Q
i
n the flavor space behaves like a spherical pendulum. Note that one ma
y
a
lso derive Eq.
(
7.59
)
from the Hamiltonian
H
(
ϕ
,
D
)
=
κ
2
μ
[
1
−
c
os
(
ϕ
+2
θ
)]
+
1
2
μD
2
,
(
7.61
)
if o
n
e
i
de
n
t
i
fies
ϕ
as a
g
eneral coordinate and
D
a
s the canonicall
y
con
j
u
-
gate momentum. Therefore, the first and second terms in Eq.
(
7.61
)
stan
d
respectively
f
or the potential and kinetic ener
g
ies. Let us de
fi
ne the potentia
l
energy as
(
Hannesta
d
et al
.
,
2006
)
V
(
ϕ
)
=
κ
2
[1
−
cos
(
ϕ
+2
θ
)]
,
(
7.62
)
which determines whether the system is stable or not
f
or a
g
iven initial valu
e
of
ϕ
.
In the case of the normal neutrino mass hierarch
y
, we can expand the
potential in terms of small tilt and mixing angles
(
i.e.,
ϕ
1
a
n
d
θ
1)
:
V
(
ϕ
)
=
κ
2
2
(
ϕ
+2
θ
)
2
.
(
7.63
)
H
ence the minimum o
f
the potential is reached at
ϕ
m
i
n
=
−
2
θ
.
O
n the othe
r
h
and, one may
g
et si
n
ϕ
(
0
)=
−
[
ω
/
(
μQ
)]
sin 2
θ
or
ϕ
(
0
)
≈−
2
θ
[
ω
/
(
μQ
)]
>
−
2
θ
f
ro
m
Q
(
0
)
=
(
−
ω
s
in 2
θ/
μ
,
0
,
2
+
ω
cos
2
θ
/μ
)
. So the initial valu
e
ϕ
(
0
)
is in th
e
vicinit
y
o
f
the stable point
ϕ
m
i
n
,
an
d
t
h
es
y
stem wi
ll
osci
ll
ate aroun
d
ϕ
m
i
n
with the frequenc
y
κ
and the am
p
litude
2
θ
[1
−
ω
/
(
μQ
)]
.Thatiswhythere
i
s no significant flavor conversion for the normal neutrino mass hierarchy.
T
h
einverte
d
neutrino mass
h
ierarc
hy
correspon
d
st
o
θ
∼
π
/
2, an
dh
ence t
h
e
e
q
uations of motion are the same as those in the case of the normal mas
s
h
ierarc
h
y
b
ut wit
h
θ
b
eing rep
l
ace
db
y
θ
≡
π
/
2
−
θ
.Int
h
is case t
h
e potentia
l
i
nEq.
(
7.62
)
canbeexpandeda
s
V
(
ϕ
)=
−
κ
2
2
(
ϕ
−
2
θ
)
2
,
(
7.64
)
whose
m
a
xim
u
mi
sach
i
eved at
ϕ
m
ax
=2
θ
.
Similarly, the initial value o
f
ϕ
i
s
g
iven by si
n
ϕ
(
0
)
= −
[
ω
/
(
μQ
)]
sin
2
θ
w
i
th
Q
≈|
2
−
ω/
μ
|
i
n
t
h
e
θ
1
l
imit. Depending on the value of
ω/
μ
,
the initial tilt angl
e
ϕ
(
0
)
varies from
0
to
2
θ
−
π
.
There
f
ore
,
ϕ
i
nitia
ll
y
l
ies aroun
d
t
h
e maximum point an
d
i
t
s
hould
fi
nall
y
roll down to the minimu
m
ϕ
m
i
n
=
−
π
.
P
z
P
P
a
n
d
P
z
at
ϕ
m
i
n
a
r
e
c
onsequently given b
y
7
.4
C
ollective Neutrino Flavor
C
onversions 2
79
Fi
g
. 7.4 The relative
fl
uxes o
f
ν
e
a
n
d
ν
e
eve
n
ts
in
the
n
eut
rin
o
-
a
n
t
in
eut
rin
o
asymmetric scenario
(
Hannestad et a
l
.
,
2006. With
p
ermission from the American
Physical Society
)
:
(
1
)
the initial conditions
P
z
P
P
(
0
)
=1an
d
P
z
(
0
)
=
0
.
8 are taken
;
(
2
)
the profile of the neutrino self-interaction strengt
h
μ
(
r
)
≈
0
.
3
×
10
5
km
−
1
r
−
4
10
is
assumed
,
where
r
1
0
is the radius
r
in units of 10 km
(
i.e., the radius of the neutrin
o
sphere
)
;
(
3
)
the mixing angle is given by sin
2
θ
=
0
.
00
1 in the case of the inverte
d
neutrino mass
h
ierarc
hy
P
z
PP
|
ϕ
min
=
P
z
ϕ
min
≈
1
2
ω
μ
−
2
−
ω
μ
.
(
7.65
)
So the complete flavor conversions
(
i.e.
,
P
z
PP
=
P
z
=
−
1)
are possible only
i
f the neutrino self-interaction strength is large enough
(
i.e.,
ω/μ
1
)
.For
i
ntermediate values o
f
ω
/
μ, the flavor conversions will be
p
artial
.
Fi
g
. 5.4 has shown that the complete
fl
avor conversion is periodic, imply
-
i
n
g
that the trans
f
ormed neutrinos will be converted back into the ori
g
ina
l
flavor state. This is not the case, however, if the variation of the strength o
f
n
eutrino-neutrino interactions is taken into account.
S
inc
e
μ
(
r
)
∝
r
−
4
,t
he
i
nteraction stren
g
th decreases as neutrinos propa
g
ate outwards. Hence th
e
k
inetic energy
μD
2
/
2 of the pendulum given in Eq.
(
7.61
)
becomes smalle
r
af
ter some oscillations such that the pendulum is unable to reach its initia
l
h
ei
g
ht. Fi
g
. 7.4 illustrates the relative neutrino fluxes in the bipolar system
wit
h
asymmetri
c
ν
e
an
d
ν
e
e
vents. These results follow from a numerical
s
olution to the equations o
f
motion with two t
y
pical inputs
f
or realistic su-
pernova neutrinos: the neutrino-antineutrino as
y
mmetr
y
wit
h
P
z
PP
(
0
)
= 1 and
P
z
P
P
(
0
)
=
0
.
8, an
d
μ
(
r
)
=0
.
3
×
10
5
k
m
−
1
r
−
4
10
w
i
th
r
1
0
b
eing t
h
era
d
ius
r
i
n units o
f
10 km. Just above the neutrino sphere
r
=
10
k
m, t
h
eneutrino
s
elf-interaction strength
μ
i
s10
5
times larger than the vacuum oscillation
f
requenc
y
ω
.
Significant flavor conversions start aroun
d
r
≈
4
5
k
m, an
d
t
h
e
be
h
av
i
o
r
between
r
≈
1
0kmand
r
≈
4
5 km is due to the s
y
nchronized oscil-
l
ations which are guaranteed by the large neutrino-antineutrino asymmetry
a
nd the lar
g
e value o
f
μ
in this re
g
ion. However, the
fl
avor conversions i
n
this re
g
ion are essentially ne
g
li
g
ible just because the e
ff
ective mixin
g
an
g
le is
h
ighly suppressed
.
280 7 Neutrinos
f
rom
S
upernovae
E
(MeV)
0 1020304050
Flux (a.u.
)
0.0
0
.2
0
.
4
0
.
6
0
.
8
1
.
0
e
ν
x
ν
E (MeV
)
0
1
0
2
030
4
050
e
ν
x
ν
Initi
a
l n
eu
trin
o
a
n
d
a
ntin
eu
trin
o
fl
u
x
es
Fig. 7.5 The initial fluxes
(
in arbitrary units
)
at the neutrino spher
e
r
=1
0k
m
for different neutrino flavors as functions of the neutrino energy
(
Fogli
e
ta
l.
,
2007
.
With permission from the Institute of Physics)
Fi
g
.7.4s
h
ows t
h
at t
h
esync
h
ronize
d
osci
ll
ations ta
k
ep
l
ace at sma
ll
r
(
e.g.
,
1
0k
m
r
45 km
)
or for large
μ
,
and the ordinar
y
vacuum oscillations occu
r
a
t
l
arg
e
r
(
e.g.,
r>
2
00 km
)
or for smal
l
μ
.
T
h
us t
h
e
b
ipo
l
ar osci
ll
ations are
expecte
d
to
h
appen at interme
d
iat
e
r
o
r
fo
rm
odest
μ
.Int
h
e
g
enera
l
cas
e
with a neutrino-antineutrino as
y
mmetr
y
P
(
0
)
= α
P
(
0
)(
for 0
α
1),
the condition for the bipolar neutrino oscillations turns out to be
ω
μ<
4
(
1+
α
)
ω/
(1
−
α
)
2
(
Dua
n
et al.
,
2006a; Hannesta
d
et al
., 2006
)
.
7.4.4 Neutrino Spectral Split
s
A
n intriguing phenomenon observed in the numerical simulations of super-
n
ova neutrino
fl
avor conversions is the existence o
f
acriticalener
gy
E
c
wh
i
ch
sp
lits the trans
f
ormed neutrino s
p
ectra into
p
arts o
f
almost
p
ure but di
ff
er-
ent flavors
(
Duan
e
tal.
,
2006b; Fogli
e
tal
.
,
2007
)
. Let us first specify th
e
i
nitial
fl
uxes o
f
neutrinos and antineutrinos. For simplicity, the relevant neu-
t
rin
os a
n
da
n
t
in
eut
rin
os a
r
e assu
m
ed to fo
ll
ow t
h
et
h
e
rm
a
l
d
i
st
ri
but
i
o
n
at
t
h
eneutrinosp
h
ere
r
=
R
ν
=
10
k
m
;
i.e.
,
φ
i
(
E
)=
2
3
ζ
(
3
)
T
3
i
TT
·
E
2
e
E/T
i
T
+
1
,
(
7.66
)
w
h
ere
ζ
(
3
)
≈
1
.
2
02 is a Riemann zeta function, and the subscript
“
i
”
stan
d
s
for
ν
e
,
ν
e
,
ν
x
or
ν
x
.
The e
ff
ective temperatur
e
T
i
TT
ca
n
be dete
rmin
ed such
that the avera
g
eneutrinoener
g
ies are
E
ν
e
=
10 MeV
,
E
ν
e
= 15 MeV and
E
ν
x
=
E
ν
x
=24MeV.T
h
estan
d
ar
dd
e
l
aye
d
-exp
l
osion supernova mo
d
e
l
i
ndicates that the luminosities o
f
di
ff
erent neutrino s
p
ecies are almost e
q
ual
,
s
o the number fluxes of neutrinos per unit energy in any direction are
7
.4
C
ollective Neutrino Flavor
C
onversions 2
81
E
(
MeV
)
0 1020304050
F
lux
(
a.u.
)
0.0
0
0
.
2
0
.4
0
.
6
0
.
8
1
.
0
x
ν
e
ν
E
(
MeV
)
0102030
4
0
5
0
x
ν
ν
Final
f
luxes in inverted hierarchy
(
single-angle
)
E
(
MeV
)
0 1020304050
F
lux
(
a.u.
)
0.0
0
0
.
2
0
.4
0
.
6
0
.
8
1
.
0
x
ν
e
ν
E
(
MeV
)
0102030
4
0
5
0
x
ν
ν
F
inal
f
luxes in inverted hierarchy
(
multi-angle
)
Fi
g
.7.6 The
fi
nal
fl
uxes at the radiu
s
r
=
200 km
f
or di
ff
erent neutrino
fl
avor
s
as functions of the neutrino energy, where the dotted lines represent the initial
fl
uxes. The upper and lower panels are
f
or the single- and multi-angle calculations,
respectively
(
Fogl
i
et a
l.
, 2007. With permission from the Institute of Physics
)
j
i
(
E
)
=
L
ν
4
π
2
R
2
ν
·
φ
i
(
E
)
E
i
,
(
7.67
)
w
h
ere t
h
e common neutrino
l
uminosity i
s
L
ν
=10
51
e
rg
s
−
1
.
T
he
ini
t
i
al
n
eutrino spectra are plotted in Fi
g
.7.5,inwhichthe
fl
uxes are proportional
to
φ
i
(
E
)
/
E
i
and properly normalized. In the single-angle approximation,
the neutrino distribution
f
unctions are
g
iven b
y
f
i
ff
(
r,
E
)
=
2
π
D
(
r
)
j
i
(
E
)
wit
h
D
(
r
)
being the geometrical function given in Eq.
(
7.49
)
. With the help of
these initial conditions and the matter density profile, one may solve th
e
equationsofmotioninEq.
(
7.47
)
in the single- and multi-angle scenarios
.
Th
efi
n
a
lr
esu
l
ts of
n
eut
rin
oflu
x
es at t
h
e
r
ad
i
us
r
=
200 km are shown i
n
F
ig. 7.6 and summarized as follows
(
Fogl
i
e
tal
.
,
2007
).
282 7 Neutrinos
f
rom
S
upernovae
(
1
)
In the final neutrino spectra, there exists a critical energ
y
E
c
7
Me
V.
In the ener
g
yre
g
ion belo
w
E
c
,
the neutrino flavors remain the same
;
bu
t
ab
ove
E
c
,
the complete flavor conversions take place. As for antineutrinos
,
t
h
ecritica
l
ener
g
yis
E
c
4
Me
V
.
(
2
)
The split in the antineutrino spectra disappears in the multi-angle
c
alculations, but the one for neutrinos survives. The neutrino spectral split
s
onl
y
occur in the case o
f
the inverted neutrino mass hierarch
y.
Note that the matter densit
y
profile used in the above calculations is
h
igh enough to make sure that the MSW resonances are absent in the regio
n
of collective neutrino oscillations
(
Fogli
et al.
,
2007
)
. If the shallow matte
r
profiles are taken, the spectral splits may exist in both normal and inverted
n
eutrino mass hierarchies
(
Duan
et al.
,
2006b, 2007
)
. In the following w
e
s
hall assume the decouplin
g
o
f
collective neutrino oscillations
f
rom the M
S
W
resonances and analytically understand the neutrino spectral splits
(
Raffel
t
a
nd Smirnov, 2007a, 2007b
).
W
e start with the equations of motion in Eq.
(
7.51
)
. Instead of usin
g
P
ω
,
one may extend the polarization vector
P
ω
t
o negative frequencies such tha
t
P
ω
=
P
−
ω
w
i
th
ω>
0. In this case the equations o
f
motion can be rewritte
n
i
n terms of onl
y
P
ω
;i
.e.
,
˙
P
ω
=(
ω
B
+
λ
L
+
μ
D
)
×
P
ω
,
(
7.68
)
where
D
=
7
+
∞
−∞
77
s
ω
P
ω
d
ω wit
h
s
ω
≡
s
ign
(
ω
)
= ω
/
|
ω
|
. Since the absence o
f
the MSW resonances has been assumed, it is possible to eliminate the matte
r
te
r
m
λ
L
b
y going into a rotating frame
(
Dua
n
et al.
,
2006a; Hannesta
d
et
a
l
.
,
2006
)
. After the elimination o
f
λ
L and the integration of Eq.
(
7.68
)
wit
h
s
ω
, we obtain the equation of motion o
f
D
as
˙
D
=
B
×
M
,
(
7.69
)
w
h
e
r
e
M
≡
7
+
∞
−∞
77
s
ω
ω
P
ω
.
Eq.
(
7.69
)
indicates that
d(
D
·
B
)
d
t
=
0 holds an
d
thus
D
z
=
B
·
D
is conserve
d
. For t
h
esma
ll
mixin
g
an
gl
e, one approximate
ly
h
as
B
=(
0
,
0
,
−
1)
o
r
B
=
(0
,
0
,
1
)
in the case of the normal or inverte
d
n
eutrino mass
h
ierarc
h
y. Because
D
z
d
enotes the excess of the
ν
e
fl
ux
,
the
co
n
se
r
vat
i
o
n
of
D
z
implies that the collective e
ff
ects onl
y
induce the pai
r
transformation ν
e
+
ν
e
→
ν
x
+
ν
x
.
Therefore
,
the ν
e
excess from dele
p
toniza
-
t
i
o
ni
sco
n
se
r
ved.
T
he equation of motion in Eq.
(
7.68
)
canalsobewrittenas
˙
P
ω
=
H
ω
×
P
ω
wit
h
H
ω
≡
ω
B
+
μ
D
,
implying that each polarization vecto
r
P
ω
p
r
ecesses
a
r
ou
n
d
H
ω
.In
the ad
i
abat
i
cl
imi
t each
H
ω
m
oves s
l
ow
l
yascompare
d
wit
h
the
p
recession o
f
P
ω
, so the latter will
f
ollow the
f
ormer. We assume that
a
ll
P
ω
are initially prepared in a specific flavor and thus aligned with eac
h
ot
h
e
r. I
f
μ
i
svery
l
ar
g
e
,
P
ω
is essentia
ll
ya
l
i
g
ne
d
wit
h
H
ω
.There
f
ore
,
P
ω
keeps ali
g
ned with
H
ω
durin
g
the entire period o
f
evolution in the adiabati
c
l
imit.
P
ω
(
μ
)
=
ˆ
H
ω
(
μ
)
P
ω
P
P
with
ˆ
H
ω
=
H
ω
/
|
H
ω
|
an
d
P
ω
P
P
=
|
P
ω
|
i
s consequently
7
.4
C
ollective Neutrino Flavor
C
onversions 2
83
the solution to the equation o
f
motion.
O
n the other hand, al
l
H
ω
l
i
e
in
the
s
o-called corotatin
g
plane spanned by
B
an
d
D
,
and thus all
P
ω
an
d
M
s
t
a
y
i
nt
h
esamep
l
ane in t
h
ea
d
ia
b
atic
l
imit. We can now
d
ecompos
e
M
i
nt
o
M
=
b
B
+
ω
c
D
a
nd express Eq.
(
7.69
)
a
s
˙
D
=
ω
c
B
×
D
.
(
7.70
)
T
h
is resu
l
timp
l
ies t
h
at
b
ot
h
D
an
d
t
h
e corotating p
l
ane precess aroun
d
B
with the common
f
requenc
y
ω
c
.
Hence the s
y
stem experiences a
f
ast preces
-
s
ion aroun
d
B
w
ith
ω
c
(
μ
)
and the drift in the corotating plane due to the
va
ri
at
i
o
n
o
f
μ
(
t
)
. A convenient treatment is to go into the corotating frame
,
w
h
e
r
et
h
e effect
i
ve
H
a
mil
to
ni
a
nr
eads
H
ω
=
[
ω
−
ω
c
(
μ
)]
B
+
μ
D
.
(
7.71
)
Initia
ll
y
μ
→∞
,
t
h
e osci
ll
ations are sync
h
ronize
d
an
dh
ave
ω
c
(
μ
→∞
)=
ω
s
yn
;
i.e., all
P
ω
f
orm a total
p
olarization vector
P
.As
μ
decreases, th
e
P
ω
will spread out but remain in the corotating plane. Finall
y
μ
→
0, Eq.
(
7.71
)
i
mp
l
ies t
h
at a
ll
H
ω
o
r
all
P
ω
w
i
th
ω
>
ω
0
c
≡
ω
c
(
μ
=0
)
are aligned with
B
,
while the others are antiali
g
ned. In other words, the
fl
avor conversion
s
a
re complete for
ω<ω
0
c
a
nd absent fo
r
ω<ω
0
c
;
i.e., t
h
e spectra
l
sp
l
it is
i
n
ev
i
table at
ω
=
ω
0
c
.
S
ince the len
g
th o
f
P
ω
i
sco
n
se
r
ved a
n
dall
P
ω
a
r
e
finally ali
g
nedwitheither
+
B
or
−
B
,
the lepton number conservation
y
ields
(
Raffelt and Smirnov, 2007a, 2007b
)
D
z
=
0
−∞
P
ω
PP
d
ω
−
ω
0
c
0
P
ω
PP
d
ω
+
+
∞
ω
0
c
P
ω
PP
d
ω
.
(
7.72
)
H
ence t
h
ecritica
l
energ
y
E
c
=
|
Δ
m
2
|
/
(
2ω
0
c
)
is determined by the neutrino
s
pectrum
P
ω
PP
a
n
d
t
h
einitia
ll
epton num
b
e
r
D
z
.
T
h
ea
b
ove ar
g
uments c
l
ear
ly
explain the existence of neutrino spectral splits and reveal the underlyin
g
p
h
ysics
b
e
h
in
d
t
h
em — t
h
e
l
epton num
b
er conservation
.
M
uch pro
g
ress has recently been made in the study o
f
supernova neutrino
sp
ectral s
p
lits. It is found that there exist multi
p
le s
p
ectral s
p
lits for bot
h
no
rm
a
l
a
n
d
in
ve
r
ted
n
eut
rin
o
m
ass
hi
e
r
a
r
c
hi
es
if
t
h
e
n
eut
rin
o
fl
u
x
es
in
t
h
e
c
ooling phase are considered
(
Dasgupta
et al.
,
2009
)
. However, the three
-
flavor effects may smear one of the splits
(
Friedland, 2010; Dasgupta
e
tal.
,
2
010b
)
. Many open issues, including the kinematical decoherence of different
a
n
g
ular modes in the multi-an
g
le scenario and the criteria
f
or adiabaticity,
remain to be solved
(
Duan
e
tal
.
,
2010
).
7
.4.
5
Effects of Three Neutrino Flavor
s
S
o
f
ar we have discussed collective neutrino oscillations in the two-
fl
avor
a
pproximation. This approximation is well
j
usti
fi
ed in the supernova envi-
ronment:
(
a
)
the charged leptons
μ
a
nd
τ
aretooheavytobeproducedin
284 7 Neutrinos
f
rom
S
u
p
ernova
e
t
h
e supernova core, so
ν
μ
a
n
d
ν
τ
h
ave on
ly
neutra
l
-current interactions w
h
ic
h
c
annot distinguish between the muon and tau flavors;
(
b
)
for the same reason,
the total fluxes and average energies o
f
ν
μ
a
n
d
ν
τ
a
re equal;
(
c
)
because o
f
t
h
estron
gh
ierarc
hy
|
Δ
m
2
3
1
|
Δ
m
2
21
,
t
h
ere
l
evant neutrino mixin
g
parame-
ters for collective neutrino oscillations should be
Δ
m
2
31
an
d
θ
13
.
However
,
th
e
t
hr
ee
-fl
avo
r
e
ff
ects s
h
ou
l
dbeta
k
e
nin
to accou
n
t
f
o
r
a
f
ew
r
easo
n
s
.
(
1
)
For completeness, the analytical and numerical studies of collectiv
e
n
eutrino oscillations should be
p
erformed in the three-flavor framework in-
c
luding the CP-violating phase
(
Gava and Volpe, 2008
)
.
(
2
)
The three-flavor effects may trigger collective neutrino oscillation
s
even if the smallest neutrino mixing angle
θ
13
is exactly vanishing
(
Dasgupta
et al.
,
2010a
)
. As mentioned before, the bipolar system can be viewed a
s
a
spherical pendulum. In the case o
f
the inverted neutrino mass hierarch
y
,
the pendulum is initially placed around an unstable point and will rapidl
y
m
ove to a stable point, causin
g
almost complete
fl
avor conversions. This
trans
f
ormation is inde
p
endent o
f
θ
13
as lon
g
as it is small. I
f
θ
13
=
0exactl
y
h
olds, the system may continue to stay at the unstable point and thus flavor
c
onversions cannot take place.
A
lthou
g
h quantum fluctuations mi
g
ht tri
gg
er
the transformation
(
Hannesta
d
e
tal.
,
2006
)
, the three-flavor effects including
radiative corrections and small differences betwee
n
ν
μ
a
n
d
ν
τ
fluxes coul
d
practica
lly d
ot
h
is
j
o
b.
(
3
)
We stress that the generic features of collective neutrino oscillations
(
i.e., synchronized oscillations, bipolar flavor conversions and spectral splits
)
preserve in the three-
fl
avor case. However, some new
f
eatures may emer
g
e
a
nd quantitatively change the two-flavor predictions. For instance, there ex-
i
st two different splits in the neutrino spectra, which are determined by
two distinct conditions for lepton number conservation
(
Dasgupt
a
et al.
,
2
008
;
Duan
e
tal
.
,
2008
)
. To be explicit, let us use the Bloch-vector for
-
m
alism
g
iven in
S
ection 5.3 and decompose the vacuum Hamiltonian ter
m
i
n
to
H
ω
=
ω
(
B
H
+
ε
B
L
)
,wher
e
B
H
(
or
B
L
)
denotes the magnetic field asso-
c
iated with
Δ
m
2
31
(
o
r
Δ
m
2
21
)
an
d
ε
≡
Δ
m
2
21
/
|
Δ
m
2
31
|
is defined. With the hel
p
o
f
the equation o
f
motion o
f
the polarization vecto
r
P
,itisstrai
g
ht
f
orwar
d
to show
d(
B
H
·
P
)
d
t
=
d
(
B
L
·
P
)
d
t
=
0
(
i.e., the conditions for lepton number
c
onservation
)
. By virtue of the same arguments leading to Eq.
(
7.72
)
,on
e
c
an understand the
p
atterns of s
p
ectral s
p
lits in the three-flavor case.
Finally, it is worth mentionin
g
that the
fi
rst multi-an
g
le simulations o
f
n
eutrino
fl
avor conversions in the three-
fl
avor case have recentl
y
been don
e
f
or an O-Ne-Mg core-collapse supernova
(
Cherry
e
tal., 2010
)
. Although the
resu
l
ts are qua
l
itative
l
y consistent wit
h
t
h
ose in t
h
esin
gl
e-an
gl
e
l
imit, it is
f
ound that the multi-an
g
le e
ff
ects reduce the adiabaticity o
ffl
avor evolution
f
or the normal neutrino mass hierarchy. Further studies of collective neutrino
osci
ll
ations are nee
d
e
d
in or
d
er to un
d
erstan
d
t
h
eun
d
er
l
yin
g
p
h
ysics.
R
e
f
erences 28
5
R
e
f
erence
s
Ag
lietta, M., et a
l.
, 1987, Europ
hy
s. Lett.
3
,
1315
.
A
hrens, J.,
et al.
,
2002, Astropart. Phys. 1
6
, 345
.
A
lexe
y
ev,E.N.,
et al.
, 1988, P
hy
s. Lett.
B
205
, 209.
A
ndo, S., and Sato, K., 2004, New J. Ph
y
s.
6
,
170.
A
ndreani, P., Ferlet, R., and Vidal-Madjar, A., 1987, Natur
e
326
,
77
0.
A
rnett, W. D., 1983,
A
stroph
y
s. J
.
263
,L
55.
A
rnett
,
W. D.
,
e
ta
l
.
1
989, Annu. Rev. Astron. Astroph
y
s.
2
7
,
629
.
B
ahcall, J. N., Spergel, D. N., and Press, W. H., 1988, i
n
S
upernova 1987A i
n
t
he Lar
g
eMa
g
ellanic
C
lou
d
,editedb
y
Kafatos, M., and Michalitsianos,
A
.
G
.
(
Cambridge University
)
,p.172
.
B
aldry, I. K., and Glazebrook, K., 2003, Astrophys. J
.
548
,
522
.
B
ar
b
ieri, R., an
d
Mo
h
apatra, R. N., 1988, P
hy
s. Rev. Lett.
61
,
2
7.
B
aron, E. A., and Cooperstein, J., 1990, Astrophys. J. 35
3
,
597
.
B
aron, E. A., Cooperstein, J., and Kahana, S., 1985a, Phys. Rev. Lett
.
55
, 126.
B
aron, E.
A
., Cooperstein, J., and Kahana, S., 1985b, Nucl. Ph
y
s.
A
4
40
,
7
44
.
B
eacom, J. F., 2010, Ann. Rev. Nucl. Part. Sci
.
60
, 439.
B
eacom, J. F., an
d
Vagins, M. R., 2004, P
h
ys. Rev. Lett.
93
,1
7
1101
.
B
ethe, H.
A
., 1990, Rev. Mod. Ph
y
s.
6
2
, 801
.
B
ethe, H. A., and Wilson, J. R., 1985, Astrophys. J
.
29
5
,
14
.
B
ethe, H. A.,
et al.
, 1979, Nucl. Phys.
A
3
2
4
,4
87
.
B
ionta
,R
.
M
.
,
et a
l
., 1987, P
hy
s. Rev. Lett.
5
8
,
1
4
9
4.
B
owers, R. L., and Wilson, J. R., 1982a, Astrophys. J. Suppl. 5
0
, 115.
B
owers, R. L., and Wilson, J. R., 1982b, Astrophys. J.
263
, 366.
B
ratton
,C
.B.
,
e
ta
l
.
,
1988, P
hy
s. Rev.
D
37
,
3361
.
B
rown, G. E., Bethe, H. A., and Baym, G., 1982, Nucl. Phys.
A
375, 481.
B
ueno, A., Gil-Botella, I., and Rubbia, A., 2003, arXiv:hep-ph
/
0307222.
B
urrows,
A
., 1988,
A
stroph
y
s. J.
3
3
4
,
891
.
C
handrasekhar, S., 1931a, Astrophys. J
.
74
,
81
.
C
handrasekhar, S., 1931b, Mon. Not. Ro
y
.
A
stron. Soc
.
91
,4
56
.
C
handrasekhar, S., 1935, Mon. Not. Ro
y
.
A
stron. Soc
.
9
5
,
207.
C
handrasekhar, S., 1938
,
A
n Introduction to the Study of Stellar Structur
e
(The
U
niversity of Chicago Press
)
.
C
handrasekhar,
S
., 1984, Rev. Mod. Ph
y
s
.
56
,
137.
C
herry, J. F.
,
et al
.
,
2010, Phys. Rev.
D
82
, 085025.
C
ol
g
ate, S.
A
., and Johnson, M. H., 1960, Phys. Rev. Lett
.
5
, 235.
C
olgate, S. A., and White, R. H., 1966, Astrophys. J
.
1
4
3
,
626.
C
ooperstein, J., 1988a, Phys. Rev. C
37
,
7
86.
C
ooperstein, J., 1988b, Ph
y
s. Rept
.
163
,95
.
C
oo
p
erstein, J., and Baron, E. A., 1990, i
n
S
u
p
ernovae, edited by Petschek, A
.
(
Springer, Berlin
)
, p. 213.
C
ooperstein, J., Bethe, H.
A
., and Brown, G. E., 1984, Nucl. Ph
y
s.
A
4
2
9
,
52
7.
D
asgupta, B., and Dighe, A., 2008, Phys. Rev. D
77
,
103002.
D
asgupta, B.,
et al
., 2008, Phys. Rev. D
77
,
11300
7.
D
as
g
upta,
B
.,
et al
., 2009, P
hy
s. Rev. Lett.
103
, 051105
.
D
asgupta, B., et a
l
., 2010b, Phys. Rev. D
8
1, 093008
.