Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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306 8 Ultrahigh-energy
C
osmic Neutrinos
whe
r
e
P
(
ν
α
→
ν
β
ν
)
an
d
P
(
ν
α
→
ν
β
)
stand respectively for the probabilities of
ν
α
→
ν
β
ν
a
nd
ν
α
→
ν
β
o
scillations.
A
s the cosmolo
g
ical distances far exceed
t
h
eo
b
serve
d
neutrino osci
ll
ation
l
engt
h
s
,
P
(
ν
α
→
ν
β
ν
)
and
P
(
ν
α
→
ν
β
)
are
a
ctually avera
g
ed over many oscillations and take a very simple
f
orm:
P
(
ν
α
→
ν
β
ν
)
=
P
(
ν
α
→
ν
β
)
=
i
|
V
αi
VV
|
2
|
V
βi
V
V
|
2
,
(
8.7
)
a
s already shown in Eq.
(
5.6
)
,where
V
αi
VV
a
n
d
V
βi
VV
(
for
i
=
1
,
2
,
3
an
d
α,
β
=
e, μ,
τ
)
denote the elements of the neutrino mixing matri
x
V
.Eqs.
(
8.6
)
and
(
8.7
)
lead us to a straightforward relation between
φ
α
an
d
φ
T
β
:
φ
T
β
=
α
i
|
V
αi
VV
|
2
|
V
βi
VV
|
2
φ
α
.
(
8.8
)
The unitarity o
f
V
assures a sum rul
e
φ
T
e
+
φ
T
μ
+
φ
T
τ
=
φ
e
+
φ
μ
+
φ
τ
≡
φ
0
.
For most o
f
the astrophysical sources which are able to produce UH
E
n
eutrinos via the decays of charged pions
(
and kaons
)
followed by the decay
s
of muons, we ex
p
ec
t
φ
e
:
φ
μ
:
φ
τ
=
1:2:
0
to hold. This flavor ratio is
only approximate in practice, not only because the production o
f
ν
τ
a
n
d
ν
τ
events is not purel
y
suppressed but also because the sizes o
f
φ
e
a
n
d
φ
μ
m
a
y
d
epend on the very details of a source model
(
e.g., the energy spectrum of
U
HE neutrinos at the source
)
. A careful estimate yields
φ
e
:
φ
μ
:
φ
τ
≈
1:
1
.
8
6:
x
w
ith
x
0
.
001
(
Lipar
i
e
tal
.
,
2007
;
Pakvas
a
e
tal
.
,
2008
)
.Forth
e
postulated neutron beam source
(
Anchordoqui et a
l.
,
2004
;
Crocker
e
ta
l.
,
2
005
)
, however
,
φ
e
:
φ
μ
:
φ
τ
=
1:0:0mi
gh
t approximate
l
y
h
o
ld
;an
d
f
or a possible muon-damped source
(
Rachen and Meszaros, 1998; Kashti an
d
Waxman, 2005
)
, one has
φ
e
:
φ
μ
:
φ
τ
=
0:1:0asagoo
d
approximation
.
A
s for the “GZK neutrinos” produced via the dominan
t
p
+
γ
C
MB
→
Δ
+
→
n
+
π
+
p
rocess, one ex
p
ec
t
s
φ
e
:
φ
μ
:
φ
τ
=1:0:0whentheneutrino
energy is
b
e
l
ow a
b
out 100 PeV an
d
t
h
eneutron
d
ecays
d
ominate t
h
eneutrino
pro
d
uction; o
r
φ
e
:
φ
μ
:
φ
τ
=1:2:0w
h
en t
h
eneutrinoener
g
yisa
b
ove
a
bout 100 PeV and the pion decays dominate the neutrino production
(
Enge
l
et al.
,
2001; Pakvasa, 2008
)
. To universally describe the flavor content of UH
E
c
osmic neutrinos at di
ff
erent sources, we
p
ro
p
ose a sim
p
le
p
arametrization o
f
their initial fluxes as
(
Xing and Zhou, 2006
)
φ
e
=
φ
0
s
in
2
ξ
cos
2
ζ, φ
μ
=
φ
0
cos
2
ξ
cos
2
ζ
,φ
τ
=
φ
0
s
i
n
2
ζ,
(
8.9
)
w
h
e
r
e
ξ
∈
[
0
,
π
/
2]
an
d
ζ
∈
[0
,
π
/
2
]
,and
φ
0
=
φ
e
+
φ
μ
+
φ
τ
de
n
otes t
h
e tota
l
flux. Then the conventional flux ratio
φ
e
:
φ
μ
:
φ
τ
=
1:2:0corres
p
ond
s
to
ξ
=arctan
(1
/
√
2
)
≈
35
.
3
◦
a
n
d
ζ
=0
◦
.
Of course
,
ξ
a
n
d
ζ
ta
k
ed
iff
e
r
e
n
t
va
l
ues fo
r
ot
h
e
r
afo
r
e
m
e
n
t
i
o
n
ed sou
r
ces
.
At a neutrino telesco
p
ethreeworking
o
bservables, which are largely fre
e
f
rom the systematic uncertainties associated with the measurements o
f
φ
T
α
8
.
3
Flavor Distribution o
fU
HE
C
osmic Neutrinos
307
(
fo
r
α
=
e
,
μ
,
τ
)
, can be defined as
R
α
≡
φ
T
α
/
(
φ
T
β
+
φ
T
γ
)
,wher
e
α
,
β
,
γ
run
o
v
e
r
e
,μ,
τ
c
y
clicall
y
.SinceR
e
,
R
μ
a
nd R
τ
satisf
y
R
e
1+
R
e
+
R
μ
1+
R
μ
+
R
τ
1+
R
τ
=1
,
(
8.10
)
only two of them are independent. Without loss of generality, we choose
R
e
a
n
d
R
μ
as two t
y
pica
l
o
b
serva
bl
es an
dd
erive t
h
eir exp
l
icit re
l
ations wit
h
ξ
a
nd
ζ
.
Substituting Eqs.
(
8.8
)
and
(
8.9
)
into Eq.
(
8.10
)
, we obtain
R
e
=
P
ee
PP
s
i
n
2
ξ
+
P
μe
P
cos
2
ξ
+
P
τe
PP
tan
2
ζ
sec
2
ζ
−
P
ee
PP
sin
2
ξ
+
P
μe
P
cos
2
ξ
+
P
τe
PP
tan
2
ζ
,
R
μ
=
P
eμ
PP
sin
2
ξ
+
P
μμ
P
cos
2
ξ
+
P
τμ
PP
ta
n
2
ζ
sec
2
ζ
−
P
eμ
PP
s
i
n
2
ξ
+
P
μμ
P
cos
2
ξ
+
P
τμ
PP
t
a
n
2
ζ
,
(
8.11
)
where
P
αβ
P
≡
|
V
α
V
V
1
|
2
|
V
β
VV
1
|
2
+
|
V
α
VV
2
|
2
|
V
β
VV
2
|
2
+
|
V
α
VV
3
|
2
|
V
β
VV
3
|
2
(
for α,
β
=
e, μ,
τ
)
.
The source
fl
avor parameter
s
ξ
a
n
d
ζ
t
urn out to be
(
Xing and Zhou, 2006
)
sin
2
ξ
=
r
e
(
P
τμ
PP
−
P
μμ
P
)
−
r
μ
(
P
τe
PP
−
P
μe
P
)
+
(
P
μμ
P
P
τe
PP
−
P
μe
P
P
τμ
PP
)
(
r
e
−
P
τe
PP
)(
P
eμ
PP
−
P
μμ
P
)
−
(
r
μ
−
P
τμ
PP
)(
P
ee
PP
−
P
μe
P
)
,
tan
2
ζ
=
r
e
(
P
μμ
P
−
P
eμ
PP
)
−
r
μ
(
P
μe
P
−
P
ee
PP
)
+
(
P
eμ
P
P
P
μe
P
−
P
ee
PP
P
μμ
P
)
(
r
e
−
P
τe
PP
)(
P
eμ
PP
−
P
μμ
P
)
−
(
r
μ
−
P
τμ
PP
)(
P
ee
PP
−
P
μe
P
)
,
(
8.12
)
whe
r
e
r
e
≡
φ
T
e
/φ
0
=
R
e
/
(
1
+
R
e
)
and
r
μ
≡
φ
T
μ
/
φ
0
=
R
μ
/
(
1
+
R
μ
)
have been
u
sed to simplify the expressions. One may in principle choose either
(
R
e
,
R
μ
)
or
(
r
e
,r
μ
)
as a set of working observables to inversely determin
e
ξ
a
nd
ζ
.
In the standard parametrization of
V
shown in Eq.
(
3.107
)
, the magnitudes
of
P
αβ
P
d
epend on three mixing angles
(
θ
1
2
,θ
1
3
,θ
2
3
)
and one CP-violatin
g
phase
(
δ
)
.S
o
R
e
a
nd
R
μ
totally rely on six parameters. One may make two
a
r
g
uments on
fl
avor physics at a neutrino telescope:
•
If
θ
12
,
θ
13
,
θ
23
an
d
δ
are known to a
g
ood de
g
ree of accuracy, then a mea-
su
r
e
m
e
n
tof
R
e
a
n
d
R
μ
w
ill help determine or constrain the initial flavor
composition o
f
UHE cosmic neutrinos
f
or a
g
iven astrophysical sourc
e
(
Xing and Zhou, 2006; Choubey and Rodejohann, 2009; Lai et al.
,
2009
).
•
I
f the production mechanism and flavor distribution of UHE neutrino
s
at an astroph
y
sical source have been understood, then a measurement o
f
R
e
an
d
R
μ
will help probe the neutrino mixing pattern and CP violation
(
Serpico and Kachelrieß, 2005; Serpico, 2006; Xing, 2006; Winter, 2006;
Rodejohann, 2007; Xin
g
, 2007; Blum
et al
., 2007;
C
hoube
y
et al.
,
2008
).
In the following we illustrate the second point by taking a simple example.
Let us
f
ocus on a conventional astroph
y
sical source o
f
UHE cosmic neu
-
trinos with the ideal flavor rati
o
φ
e
:
φ
μ
:
φ
τ
=
1 : 2 : 0. Because of neutrino
oscillations, the flavor ratio at a neutrino telescope can be calculated b
y
308 8 Ultrahigh-energy
C
osmic Neutrinos
m
eans of Eq.
(
8.8
)
. It is easy to show that the flavor distribution at the tele-
s
cope ma
y
have a democratic pattern
φ
T
e
:
φ
T
μ
:
φ
T
τ
=
1:1:1,
p
rovided th
e
c
on
d
itio
n
|
V
μi
VV
|
=
|
V
τi
VV
|
(
for
i
=1
,
2
,
3
)
or equivalently the
μ
-
τ
s
ymmetry i
s
s
atisfied
(
Xing and Zhou, 2008
)
. This condition corresponds to eithe
r
θ
13
=0
a
nd θ
23
=
π
/
4
(
CP invariance
)
or
δ
=
±
π
/
2
an
d
θ
23
=
π
/
4
(
CP violation
)
i
n the standard parametrization of
V
.
C
urrent neutrino oscillation data in-
d
i
cate that
θ
13
i
sver
y
sma
ll
an
d
θ
2
3
i
scloseto
π
/
4
,an
d
t
h
us
V
e
x
h
i
b
i
ts an
app
rox
i
mat
e
μ
-
τ
sy
mmetr
y
.Inthiscaseweusetwosmallparametersθ
13
a
nd
ε
≡
θ
23
−
π
/
4
to desc
ri
be t
h
es
m
a
ll
e
ff
ects o
f
μ
-
τ
symmetry
b
rea
k
ing
:
⎡
⎣
|
V
e
VV
1
|
2
|
V
e
VV
2
|
2
|
V
e
V
V
3
|
2
|
V
μ
V
V
1
|
2
|
V
μ
VV
2
|
2
|
V
μ
VV
3
|
2
|
V
τ
V
V
1
|
2
|
V
τ
VV
2
|
2
|
V
τ
VV
3
|
2
⎤
⎦
=
1
2
⎡
⎣
2
c
2
12
2
s
2
12
0
s
2
1
2
c
2
1
2
1
s
2
12
c
2
12
1
⎤
⎦
+
ε
⎡
⎣
0
00
−
s
2
1
2
−
c
2
1
2
1
s
2
1
2
c
2
1
2
−
1
⎤
⎦
+
θ
13
2
si
n2
θ
1
2
cos
δ
⎡
⎣
000
1
−
10
−
11
0
⎤
⎦
+
···
,
(
8.13
)
where
c
12
≡
cos
θ
12
,
s
12
≡
sin
θ
12
,
and higher-order terms of θ
13
and
ε
h
av
e
been omitted. Substituting Eq.
(
8.13
)
into Eq.
(
8.8
)
, we arrive at
(
Xing, 2006
)
φ
T
e
:
φ
T
μ
:
φ
T
τ
=(1
−
2
Δ
)
:
(
1
+
Δ
)
:
(
1
+
Δ
)
,
(
8.14
)
w
h
ere
Δ
=(2
ε
sin
2
2
θ
1
2
−
θ
1
3
s
in 4
θ
1
2
cos
δ
)
/
4
compati
bl
ewit
h
t
h
e approxima
-
tion made in Eq.
(
8.13
)
. It is obvious that the naive flavor democracy is broken
a
s a direct conse
q
uence o
f
μ
-
τ
s
ymmetry breakin
g
.Given30
◦
<
θ
12
<
3
8
◦
,
θ
1
3
<
10
◦
(
≈
0
.
17
)
an
d
|
ε
|
<
9
◦
(
≈
0
.
1
6
)
as indicated by current experimental
d
ata
(
see Section 3.1.5
)
, we find −
0
.
1
Δ
+0
.
1b
ya
ll
owin
g
δ
to var
yf
ro
m
0to
2
π
. The working observable
s
R
e
,
R
μ
and R
τ
d
efined above Eq.
(
8.10
)
f
or a neutrino telescope turn out to be
R
e
≈
1
2
−
3
2
Δ, R
μ
≈
R
τ
≈
1
2
+
3
4
Δ
.
(
8.15
)
So
R
e
is most sensitive to the effect o
f
μ
-
τ
s
ymmetry
b
rea
k
ing.
O
ne may also consider to probe the breakin
g
o
f
μ
-
τ
sy
mmetr
ybyd
e-
tectin
g
th
e
ν
e
flux from a distant astrophysical source throu
g
h the Glashow
resonance
(
GR
)
channe
l
ν
e
+
e
→
W
−
→
anything
(
Glashow, 1960
)
.Th
e
l
atter can actua
ll
yta
k
ep
l
ace over a narrow ener
g
yinterva
l
aroun
d
t
h
e
ν
e
ener
g
y
E
GR
ν
e
≈
M
2
W
MM
/
2
m
e
≈
6
.
3PeV.
A
neutrino telesco
p
eisin
p
rinci
p
le
possible to measure both the
G
R-mediated
ν
e
e
vents
(
N
G
R
ν
NN
e
)
and the
ν
μ
+
ν
μ
events of charged-current
(
CC
)
interactions
(
N
CC
ν
N
N
μ
+
ν
μ
)
in the vicinity o
f
E
GR
ν
e
.
Their ratio
,
defined a
s
R
G
R
≡
N
GR
ν
NN
e
/N
CC
ν
N
N
μ
+
ν
μ
,
can be related to the ratio of
ν
e
even
t
s
to
ν
μ
a
nd
ν
μ
e
vents entering the detector,
R
0
≡
φ
T
ν
e
/
(
φ
T
ν
μ
+
φ
T
ν
μ
)
.
O
ne obtain
s
R
GR
=
aR
0
w
it
h
a
≈
30
.
5
(
Bhattacharjee and Gupta, 2005
)
by
c
onsi
d
erin
g
t
h
emuoneventswit
h
containe
d
vertices in a water- or ice-
b
ase
d
8
.
3
Flavor Distribution o
fU
HE
C
osmic Neutrinos
309
d
etector
(
Beacom
et al
., 2003a
)
. Provided the neutrinos are initially produce
d
via the decays of char
g
ed pions created from hi
g
h-ener
g
y
pp
or
p
γ
collision
s
a
t an optically thin source, their flavor composition can be expressed as fol
-
lows
:
{
φ
ν
e
,φ
ν
e
,φ
ν
μ
,
φ
ν
μ
,
φ
ν
τ
,φ
ν
τ
}
=
{
1
/
6
,
1
/
6
,
1
/
3
,
1
/
3
,
0
,
0
}
φ
0
(
for
pp
c
ollisions
)
o
r
{
1
/
3
,
0
,
1
/
3
,
1
/
3
,
0
,
0
}
φ
0
(
for
pγ
collisions
)
. In either case
φ
e
:
φ
μ
:
φ
τ
= 1 : 2 : 0 holds. Due to neutrino oscillations
,
th
e
ν
e
flux detecte
d
a
t a neutrino telescope reads
(
Xing, 2006, 2008
)
φ
T
ν
e
(
pp
)
=
φ
0
6
(1
−
2
Δ
)
,
φ
T
ν
e
(
pγ
)
=
φ
0
12
s
in
2
2
θ
1
2
−
4
Δ
,
(
8.16
)
whe
r
e
Δ
h
as been given below Eq.
(
8.14
)
.Inaddition,wehave
φ
T
ν
μ
+
φ
T
ν
μ
≈
φ
0
(
1
+
Δ
)
/
3. It is then strai
g
ht
f
orward to obtain
R
0
(
pp
)
≈
1
2
−
3
2
Δ, R
0
(
pγ
)
≈
si
n
2
2
θ
12
4
−
4
+si
n
2
2
θ
12
4
Δ.
(
8.17
)
This result indicates that R
0
(
pγ
)
is very sensitive to the value of si
n
2
2
θ
12
(
Anchordoqu
i
e
ta
l.
, 2005
)
, and the effect o
f
μ
-
τ
s
ymmetry
b
rea
k
ing s
h
ows
u
pin
b
ot
h
R
0
(
pγ
)
an
d
R
0
(
pp
)
.
8.3.2 Flavor Effects in New Physics Scenarios
Some “exotic” scenarios of new physics, such as quantum decoherence, CP
T
violation, neutrino decays, sterile neutrinos and non-unitarity o
f
the
3
×
3
n
eutr
i
no m
i
x
i
n
g
matr
ix
V ,ma
y
also a
ff
ect the
fl
avor oscillations o
f
UH
E
c
osmic neutrinos for a given source
.
(
1
)
Q
uantum decoherence.
I
nsomemodelso
f
quantum
g
ravity the phe
-
n
omenon of spacetime “foam” may be incorporated and the sin
g
ular micro
-
s
copic fluctuations of the metric give the ground state of quantum gravity the
s
tructure of a “stochastic medium”
(
Barenboim
et al
., 2006
)
. Such a mediu
m
h
as the profound effect of CPT-violatin
g
decoherence of quantum matter as
i
t propagates. A striking consequence of gravitational decoherence on neu
-
trino oscillations is the exponential dampin
g
o
f
oscillatory terms with tim
e
or distances. In a generic Lindblad decoherence model the probabilities o
f
ν
α
→
ν
β
ν
o
scillations can be written as
(
Barenboi
m
et al.
,
2006
)
P
(
ν
α
→
ν
β
ν
)
=
1
3
+
i
O
i
e
−
D
i
L
,
(
8.18
)
whe
r
e
O
i
d
enotes an oscillator
y
term modi
fi
ed b
y
quantum decoherence
,
D
i
i
s the corresponding decoherence parameter which can be either a constan
t
or a
f
unction o
f
ener
g
y, and
L
r
epresents the travel distance o
f
neutrinos.
F
or UHE cosmic neutrinos from a very distant source
(
i.e.,
L
→
∞
)
,oneis
l
eft wit
h
P
(
ν
α
→
ν
β
ν
)
=
1
/
3 under com
p
lete
q
uantum decoherence and thu
s
310 8 Ultrahigh-energy
C
osmic Neutrinos
φ
T
β
=
φ
0
/
3
or
φ
T
e
:
φ
T
μ
:
φ
T
τ
=
1 : 1 : 1 at a neutrino te
l
escope re
g
ar
dl
es
s
of their initial flavor composition
(
Li
u
et al
.
,
1997;
A
hluwalia, 2001; Hoo
p
er
et a
l
., 2005
)
. This result mimics the one obtained in Section 8.3.1 from
a
s
peci
fi
castroph
y
sical source wit
h
φ
e
:
φ
μ
:
φ
τ
=1:2:
0
an
d
un
d
er t
h
eexact
μ
-
τ
sy
mmetr
y
. To search for possible evidence for quantum decoherence i
n
U
HE cosmic neutrino osci
ll
ations, one s
h
ou
ld
consi
d
er an astrop
h
ysica
l
sourc
e
which has a different flavor content
(
e.g.
,
φ
e
:
φ
μ
:
φ
τ
=
1:0:0
f
rom th
e
beta decay of neutrons
)(
Hoope
r
e
tal., 2005; Pakvasa, 2008
)
.
(
2
)
C
PT violation. In a theory with spontaneous CPT violation
(
Col-
l
aday and Kosteleck´y, 1997
)
, its Lagrangian may contain the vector- an
d
p
seudovector-like s
p
inor bilinear
s
ψ
γ
μ
ψ
an
d
ψγ
μ
γ
5
ψ
which are apparentl
y
CPT-odd as shown in Table 3.5. An effective CPT- and Lorentz-violatin
g
operator contributin
g
to the neutrino sector can be parametrized a
s
O
ν
=
ν
α
L
C
α
β
μ
C
γ
μ
ν
β
ν
L
,
where
α
an
d
β
run
o
v
er
e
,
μ
and
τ
,
an
d
C
αβ
μ
C
are real coeffi-
c
ients
(
Dighe and Ray, 2008
)
. This operator can modify the dispersion rela
-
tion o
f
neutrinos and the e
ff
ective Hamiltonian responsible
f
or their propa
g
a-
tion. Hence the behaviors of UHE cosmic neutrino oscillations get modified:
P
(
ν
α
→
ν
β
ν
)
=
P
(
ν
α
→
ν
β
)
=
i
|
V
αi
VV
|
2
|
V
βi
V
V
|
2
,
(
8.19
)
w
h
e
r
e
V
i
sthee
ff
ective neutrino mixin
g
matrix which depends on bot
h
V
a
nd
C
μ
C
(
Bustamant
e
e
tal.
,
2010
)
. Given a conventional source of UHE cosmic
neut
rin
os w
i
th
φ
e
:
φ
μ
:
φ
τ
=
1:2:0,
C
PT violation has a lar
g
eroomt
o
m
odi
fy
the
fl
avor democrac
y
φ
T
e
:
φ
T
μ
:
φ
T
τ
=1:1:1ataneutrinote
l
escope
(
Barenboim and Quigg, 2003; Bustamante et al.
,
2010
)
.
(
3
)
Unitarity vio
l
ation.
I
n
so
m
e seesaw
m
odels
in
wh
i
ch the
r
ee
xi
sts the
m
ixing between three light neutrinos and a few heavy degrees of freedom
(
e.g.
,
h
eavy Majorana neutrinos
)
as discussed in Chapter 4, the 3
×
3
neutrin
o
m
ixin
g
matri
x
V
m
ust be non-unitary. The stren
g
th o
f
unitarity violatio
n
ca
n
be of
O
(
1
0
−
2
)
if a seesaw model works at the TeV scale
(
Xing, 2009
),
a
n
d
t
h
us it may s
h
ow up in UHE cosmic neutrino osci
ll
ations. Fo
ll
owing t
h
e
parametrization o
f
V
g
iven in
S
ection 4.5.2, we consider a simple pattern o
f
V which sli
g
htly deviates from the tri-bimaximal neutrino mixin
g
pattern
V
0
VV
given in Eq.
(
3.110
)(
Xing and Zhou, 2008
)
:
V
≈
V
0
V
V
−
⎛
⎜
⎛⎛
⎝
⎜⎜
2
√
6
W
1
WW
1
√
3
W
1
WW
0
1
√
6
(
2
X
−
W
2
WW
)
1
√
3
(
X
+
W
2
WW
)
1
√
2
W
2
WW
1
√
6
(2
Y
−
Z
+
W
3
W
W
)
1
√
3
(
Y
+
Z
−
W
3
W
W
)
1
√
2
(
Z
+
W
3
W
W
)
⎞
⎟
⎞⎞
⎠
⎟⎟
,
(
8.20
)
whe
r
e
W
i
WW
=(
s
2
i
4
+
s
2
i
5
+
s
2
i
6
)
/
2
(
for
i
=1
,
2
,
3
)
,
X
=ˆ
s
14
ˆ
s
∗
2
4
+ˆ
15
ˆ
s
∗
2
5
+ˆ
1
6
ˆ
s
∗
2
6
,
Y
=ˆ
s
1
4
ˆ
s
∗
34
+ˆ
1
5
ˆ
s
∗
35
+ˆ
1
6
ˆ
s
∗
36
a
n
d
Z
=ˆ
s
24
ˆ
s
∗
34
+ˆ
25
ˆ
s
∗
35
+ˆ
26
ˆ
s
∗
36
w
i
th
s
ij
≡
si
n
θ
ij
and ˆ
ij
≡
e
i
δ
ij
s
i
j
bein
g
de
fi
ned. The mixin
g
an
g
les
θ
ij
ca
n
at
m
ost be of
O
(0
.
1
)
, but the CP-violating phase
s
δ
ij
are entirel
y
unrestricted. Now w
e
l
ook at the oscillations of UHE cosmic neutrinos coming from a conventiona
l
8
.
3
Flavor Distribution o
fU
HE
C
osmic Neutrinos
3
11
a
strop
hy
sica
l
source wit
h
φ
e
:
φ
μ
:
φ
τ
=
1 : 2 : 0. With the help of Eq.
(
8.8
)
,
we arrive at
(
Xing and Zhou, 2008
)
φ
T
e
≡
φ
0
3
1
−
4
9
(7
W
1
WW
+
2
W
2
WW
)
+
4
9
Re
X
,
φ
T
μ
≡
φ
0
3
1
−
4
9
(
W
1
WW
+
8
W
2
WW
)
−
2
9
R
e
X
,
φ
T
τ
≡
φ
0
3
1
−
2
9
(
2
W
1
+7
W
2
WW
+9
W
3
WW
)
−
2
9
Re
X
,
(
8.21
)
a
t a neutrino telescope. The flavor democrac
y
o
f
φ
T
α
(
for
α
=
e, μ,
τ
)
is clearly
broken, and the effect of this symmetry breaking can be as large as several
percent. Because o
f
the non-unitarit
y
o
f
V
,
the total
fl
ux o
f
UHE cosmic
n
eutrinos at the telesco
p
e is not e
q
ual to that at the source:
α
φ
T
α
=
φ
0
1
−
2
3
(
2
W
1
WW
+3
W
2
WW
+
W
3
WW
)
.
(
8.22
)
This sum approximatel
y
amounts to
0
.
9
6
φ
0
if
W
i
∼
0
.
01
(
for
i
=1
,
2
,
3)
.
The non-unitarity of
V
i
s very small and certainly difficult to be observed i
n
a
realistic experiment, but Eqs.
(
8.21
)
and
(
8.22
)
can at least illustrate how
s
ensitive a neutrino telescope should be to this kind of new ph
y
sics
.
(
4
)
Neutrino
d
ecays. Now that neutrinos have finite masses and lepto
n
fl
avors are mixed, the heavier neutrinos are in
g
eneral expected to decay int
o
the lighter ones via the flavor-changing channels. But one may wonder
(
a
)
whether the lifetimes of neutrinos are short enough to be phenomenologicall
y
i
nteresting and
(
b
)
what the dominant decay modes are
(
Pakvasa, 2000
)
.Ra
-
d
iative neutrino decays are less interesting in this sense, whereas the “exotic”
d
ecay
s
ν
i
→
ν
j
ν
+
χ
w
i
th
χ
b
eing a Majoron
(
Chikashige
et al
., 1981; Gelmini
a
nd Roncadelli, 1981
)
or unparticle
(
Zhou, 2008
)
might be suggestive and
i
nteresting. Regardless of any details of model building and other astrophys-
i
cal implications, here we
f
ocus on the
fl
avor issues o
f
unstable UHE cosmi
c
n
eutrinos.
G
iven a neutrino mass ei
g
enstate
ν
i
,
a characteristic
f
eature o
f
its
d
ecay is the strong energy dependence: exp
(
−
t/τ
lab
ττ
)
wit
h
τ
lab
ττ
=
τ
0
τ
τ
E
/
m
i
,
whe
r
e
τ
0
τ
τ
,
m
i
,
E
,
t
a
n
d
L
de
n
ote t
h
e
r
est
-
f
r
a
m
e
li
fet
im
eof
ν
i
,
its mass, it
s
energy, its travel time and its travel distance, respectively
(
Beaco
m
et al.
,
2
003b
)
. For simplicity, we only consider the case that the decays are com
-
plete
(
i.e., such exponential factors vanish
)
for UHE cosmic neutrinos o
n
their wa
y
from a distant astroph
y
sical source to a neutrino telescope on th
e
E
arth. In this case only the lightest neutrinos arrive at the Earth
(
Farzan
a
nd Smirnov, 2002
)
. If the neutrino mass spectrum has a normal hierarchy,
the li
g
htest neutrino i
s
ν
1
=
V
∗
e
V
V
1
ν
e
+
V
∗
μ
V
V
1
ν
μ
+
V
∗
τ
V
V
1
ν
τ
a
nd thus the flavor ratio
a
tt
h
ete
l
escope is
φ
T
e
:
φ
T
μ
:
φ
T
τ
=
|
V
e
V
V
1
|
2
:
|
V
μ
V
V
1
|
2
:
|
V
τ
V
V
1
|
2
,
(
8.23
)
312 8 Ultrahigh-energy
C
osmic Neutrinos
i
ndependent o
f
the initial
fl
avor composition at the source.
G
iven the inverted
n
eutrino mass hierarchy, the li
g
htest neutrino i
s
ν
3
=
V
∗
e
VV
3
ν
e
+
V
∗
μ
VV
3
ν
μ
+
V
∗
τ
VV
3
ν
τ
a
nd hence the flavor ratio at the telescope is given by
φ
T
e
:
φ
T
μ
:
φ
T
τ
=
|
V
e
VV
3
|
2
:
|
V
μ
VV
3
|
2
:
|
V
τ
VV
3
|
2
.
(
8.24
)
Ta
k
in
g
V
=
V
0
VV
t
o be the tri-bimaximal neutrino mixin
g
pattern,
f
or example
,
we obta
in
φ
T
e
:
φ
T
μ
:
φ
T
τ
=4:1:1
(
normal hierarchy
)
or 0 : 1 : 1
(
inverted
h
ierarchy
)
.I
f
|
V
e
VV
3
|
=
0, t
h
enumerica
l
resu
l
twi
ll d
epen
d
on t
h
ree mixing an
-
g
les and the
C
P-violatin
g
phas
e
δ
(
Beaco
m
et al
.
,
2004; Meloni and
O
hlsson
,
2
007; Xing and Zhou, 2008; Maltoni and Winter, 2008
)
. Of course, the fla-
vor ratio of UHE cosmic neutrinos may also get modified in the existence of
i
ncomplete decays, magnetic fields and “exotic” new physics
(
Pakvasa, 2008
).
(
5
)
Sterile neutrinos. Toda
y
one seems not to be well motivated to con
-
s
ider the existence of very light sterile neutrinos
5
,w
h
ic
h
may ta
k
epartint
he
oscillations of active neutrinos
(
i.e.
,
ν
e
,
ν
μ
a
n
d
ν
τ
)
and modify the standar
d
i
nter
p
retation of solar, atmos
p
heric, reactor and accelerator neutrino oscil
-
l
ation data
(
Xing, 2008
)
. If one or more sterile neutrinos exist, their mixin
g
with active neutrinos has to be su
ffi
cientl
y
suppressed; otherwise, neutrin
o
oscillations would bring sterile neutrinos in equilibrium with active neutrino
s
be
f
ore neutrino decouplin
g
— the resultant excess in ener
g
y density would
endan
g
er the standard scheme
f
or the Bi
g
Ban
g
nucleosynthesis o
f
li
g
ht el
-
ements
(
Barbieri and Dolgov, 1990; Kalliom¨ak
i
e
tal., 1999
)
.Using
V
αi
VV
a
nd
S
α
j
t
o
d
enote t
h
e active-active an
d
active-steri
l
e neutrino mixin
g
matrix e
l
e-
m
ents, respectively, we can express the avera
g
ed probabilities o
f
UHE cosmi
c
n
eutrino oscillations as
(
Xing and Zhou, 2008
)
P
(
ν
α
→
ν
β
ν
)
=
P
(
ν
α
→
ν
β
)
=
3
i
=1
|
V
αi
VV
|
2
|
V
βi
VV
|
2
+
n
j
=
1
|
S
αj
|
2
|
S
βj
S
|
2
,
(
8.25
)
whe
r
e
α
a
n
d
β
r
u
n
ove
r
e
,
μ
a
n
d
τ
,an
d
3
i
=
1
|
V
αi
VV
|
2
+
n
j
=
1
|
S
αj
|
2
=
1
.
(
8.26
)
E
q.
(
8.26
)
shows the direct unitarity violation o
f
V
in
d
uce
db
y
l
ig
h
tsteri
l
e
n
eutrinos. Two observations have been achieved in the literature
(
Athar
et
a
l
.
,
2000
;
Ker¨ane
n
et al
.
,
2003; Awasthi and Choubey, 2007
)
:
(
a
)
for small
a
ctive-sterile neutrino mixing, the effect of non-unitarity o
f
V
at
n
eut
rin
o
telescopes is very small and quite similar to that obtained in Eq.
(
8.21
)
;
(
b
)
f
or relatively large hitherto-unconstrained mixing between active and sterile
n
eutrino species, the flavor distribution of UHE cosmic neutrinos at neutrino
telescopes mi
g
ht be si
g
ni
fi
cantly modi
fi
ed
.
5
A
steri
le
neutrino is b
y
de
fi
nition a h
y
pothetical neutrino that does not interact
with other known particles via electroweak or strong interactions except gravity
.
8
.4 Neutrinos and Multi-messenger Astronomy 313
W
e admit that the present versions o
f
neutrino telescopes are mainl
y
d
esi
g
ned to be a discovery tool for UHE cosmic neutrinos.
A
determination o
f
the flavor distribution of UHE cosmic neutrinos actually requires the precision
m
easurements. Usin
g
neutrino te
l
escopes to pro
b
enewp
h
ysics is certain
ly
m
ore challenging and could only be done in the long run
.
8
.4 Neutrinos and Multi-messen
g
er
A
stronom
y
A
lthou
g
h most of what we have known about the cosmos derives from th
e
observation of photons, we are learning some further things beyond the solar
s
ystem by observin
g
cosmic rays and neutrinos. Di
ff
erent astronomical mes
-
s
engers, such as gamma rays, protons, neutrinos and even gravitational waves,
a
re expecte
d
to comp
l
ement one anot
h
er in pro
b
ing t
h
e
h
ig
h
-energy Universe
.
H
ence some lon
g
standin
g
problems in astronomy may
fi
nally be solved in th
e
era of multi-messenger astronomy. In this section we shall briefly describ
e
t
h
einterp
l
ay
b
etween UHE cosmic neutrinos an
d
cosmic rays, gamma rays
or
g
ravitational waves in a
g
iven astrophysical source. The interaction o
f
U
HE cosmic neutrinos with the cosmic neutrino background
(C
ν
B)
via th
e
Z
-r
eso
n
a
n
ce w
i
ll also be d
i
scussed
.
8.4.1
C
osmic Neutrinos and
Z
-
bu
r
sts
Theexistenceo
f
the
C
ν
B
,com
p
rised o
f
relic neutrinos
f
rom the e
p
och o
f
d
ecoupling of weak interactions about one second after the Big Bang, is a
robust prediction of the standard Big Bang cosmology
(
Kolb and Turner,
1
990; Peebles, 1993
)
. Today’s temperature and number density of cosmi
c
background neutrinos can be calculated in terms of the observed temperatur
e
a
nd number density o
f
cosmic back
g
round photons
:
T
ν
T
T
i
=
T
ν
T
T
i
=
4
11
1
/
3
T
γ
T
T
≈
1
.
9
4
5K
,
n
ν
i
=
n
ν
i
=
3
22
n
γ
≈
56 c
m
−
3
,
(
8.27
)
whe
r
e
ν
i
(
o
r
ν
i
)
denotes a neutrino
(
or antineutrino
)
species with the definite
mass
m
i
(
fo
r
i
=1
,
2
,
3
)
,
T
γ
TT
=
2
.
7
25 K an
d
n
γ
=4
11
c
m
−
3
(
Nakamura
et al.
,
2
010
)
have been input. As a result, the average three-momentum of each reli
c
neut
rin
o
i
s
r
athe
r
s
m
all
:
p
ν
i
=
p
ν
i
=
3
T
ν
TT
i
≈
5
.
835 K
≈
5
.
028
×
10
−
4
eV,
implying that at least two mass eigenstates of the relic neutrinos are non-
i
relativistic today
(
i.e.,
p
ν
i
=
p
ν
i
m
i
)
no matter how the neutrino mas
s
s
pectrum
l
oo
k
s
l
i
ke
6
.
T
h
ese cosmic
b
ac
kg
roun
d
neutrinos are su
b
ject to t
h
e
6
I
nviewo
f
Δ
m
2
21
≈
7
.
6
×
10
−
5
eV
2
an
d
|
Δm
2
31
|≈
2
.
4
×
1
0
−
3
eV
2
in Table 3.1, on
e
may
h
av
e
m
2
≈
8
.
7
×
10
−
3
e
V
an
d
m
3
≈
4
.
9
×
10
−
2
e
V
(
normal mass hierarchy
);
o
r
m
1
≈
4
.
9
×
10
−
2
e
Van
d
m
2
≈
5
.
0
×
10
−
2
e
V
(
inverted mass hierarchy
)
;o
r
m
1
∼
m
2
∼
m
3
|
Δ
m
2
31
|≈
4
.
9
×
10
−
2
e
V (near mass degeneracy)
.
314 8 Ultrahigh-energy
C
osmic Neutrinos
gravitational clustering on massive structures
(
e.g., cold dark matter
)
.Henc
e
there ma
y
exist some local overdensities in the
C
ν
B
relative to the standard
value given in Eq.
(
8.27
)
, and the corresponding momentum distribution o
f
relic neutrinos is possible to deviate
f
rom the one
g
iven by the homo
g
eneous
a
nd isotropic relativistic Fermi-Dirac distribution
(
Ringwald, 2009
)
.
An interesting observation is that UHE cosmic neutrinos, when propa
-
g
atin
g
over cosmolo
g
ical distances, may interact with the
C
ν
Bv
i
athe
Z
boson and thus get absorbed
(
Weiler, 1982
)
. Although the cross sections o
f
ν
UH
E
+
ν
C
ν
B
→
Z
∗
→
had
r
o
n
sa
n
d
ν
UHE
+
ν
C
ν
B
→
Z
∗
→
had
r
o
n
sa
r
e
n
e
g
li
g
ibly small in most cases, they can be si
g
ni
fi
cantly enhanced when th
e
reactions take
p
lace on th
e
Z
r
esonance. In this special case the energy of an
UH
Eneutrino
ν
i
h
as to satis
fy
M
2
Z
M
≈
(
E
ν
i
+
m
i
)
2
−
E
2
ν
i
≈
2
m
i
E
ν
i
;
name
l
y,
E
ν
i
≈
M
2
Z
M
2
m
i
≈
4
.
2
×
eV
m
i
×
10
21
e
V
.
(
8.28
)
S
uch extremely ener
g
etic cosmic neutrinos mi
g
ht ori
g
inate
f
rom the super
-
h
eav
y
DM or TDs in the ver
y
earl
y
Universe. Given the width of th
e
Z
boso
n
Γ
Z
Γ
=(
2
.
4
95
2
±
0
.
0023
)
GeV
(
Nakamura
et al
.
,
2010
)
,th
e
Z
-r
eso
n
a
n
ce e
n-
h
ancement is more
g
enerally achievable
f
or
E
ν
i
o
r
E
ν
i
to
l
ie in t
h
eran
ge
M
2
Z
M
/
(2
m
i
)
±
Γ
Z
Γ
. The hadrons produced from th
e
Z
decay form a highly
-
c
ollimated
fi
nal state, because th
e
Z
boso
nin
t
h
e
r
est f
r
a
m
eof
r
e
li
c
n
eut
rin
os
h
as a Lorentz boost factor characterized b
y
E
ν
i
/M
Z
M
∼
1
0
11
f
o
r
m
i
∼
O
(0
.
1
)
eV. If th
e
Z
-
mediated interaction between ν
U
HE
(
o
r
ν
UHE
)
and ν
C
ν
B
(
o
r
ν
C
ν
B
)
takes place at a distance of less than 50 Mpc from the Earth, the at-
tenuation should be small such that the UHE cosmic rays arisin
gf
rom th
e
Z-bursts may induce air showers in the Earth’s atmosphere and give rise to
s
omeeventsabovetheGZKcutoff
(
Fargio
n
et al.
, 1999; Weiler, 1999
)
. Con-
s
iderin
g
that the avera
g
e multiplicity o
f
the reactio
n
e
+
e
−
→
Z
→
had
r
o
n
s
i
s about 30 and its final state is on average composed of about 2 nucleons,
1
0 neutral pions and 17 charged pions
(
Bilenky
et al.
,
2003
)
, one may es-
timate the avera
g
eener
g
yo
f
protons and pions produced
f
rom th
e
Z
-
bu
r
st
p
r
ocess
:
E
p
E
∼
E
π
∼
E
ν
i
/
3
0. Neutral and charged pions originate photon
s
a
nd neutrinos, respectively, via
π
0
→
γ
+
γ
a
nd
π
+
→
μ
+
+
ν
μ
f
ollowed by
μ
+
→
e
+
+
ν
μ
+
ν
e
(
o
r
π
−
→
μ
−
+
ν
μ
f
ollowed by
μ
−
→
e
−
+
ν
μ
+
ν
e
)
.Asdis
-
c
ussed in Section 8.1.3, about 1
/
4 of the charged pion’s energy is taken away
by each neutrino or antineutrino. So the average energies of UHE photon
s
a
n
d
n
eut
rin
os e
mi
tted f
r
o
m
t
h
e
Z
-
bu
r
sts a
r
e
7
E
γ
E
∼
E
ν
i
6
0
≈
7
.
0
×
e
V
m
i
×
10
19
eV
,
E
ν
∼
E
ν
i
1
20
≈
3
.
5
×
e
V
m
i
×
10
1
9
eV
.
(
8.29
)
7
N
ote that more ener
g
etic neutrinos may come
f
rom the direct
Z
→
ν
i
+
ν
i
decay with an average energ
y
E
ν
≈
E
ν
i
/2
≈
2.
1
×
(
e
V
/
m
i
)
×
10
21
eV
.
8
.4 Neutrinos and Multi-messenger Astronomy 315
In
the
Z
-b
urst scenario t
h
ese protons, p
h
otons an
d
neutrinos constitute t
h
e
primaries of UHE cosmic ra
y
s.
A
measurement of such UHE cosmic ra
ys
would in principle allow one to determine the neutrino masses
(
P¨as an
d
Wei
l
er, 2001; Fo
d
or
et al.
,
2002
).
T
he
Z
-burst mechanism provides an appealing opportunity to detect the
C
ν
B
in two approaches. The first one is to search for the “emission” feature of
Z
-bursts, characterized by a directional excess o
f
UHE cosmic rays,
g
amm
a
ra
y
s and neutrinos be
y
ond the GZK cutoff.
A
s discussed above, this scenari
o
probes the Universe delimited by the GZK sphere and is sensitive to the local
overdensities in the
C
ν
B
which may result
f
rom the
g
ravitational clusterin
g
on cold dark matter and baryonic structures
(
Van Elewyck, 2007
)
. The other
a
pproach is to search
f
or the “absorption”
f
eature in the UHE cosmic neutrino
fl
ux which would re
fl
ect their interactions with the
C
ν
Bv
i
at
h
e
Z
-
r
eso
n
a
n
ce
a
long their path
(
Weiler, 1982; Roulet, 1993; Yoshida, 1994
)
. Such an en-
er
g
etic neutrino
fl
ux arrivin
g
at the Earth is there
f
ore expected to exhibit
a
bsorption dips whose locations in the ener
g
y spectrum are determined b
y
the respective redshifted resonance energies of
ν
U
HE
+
ν
C
ν
B
→
Z
→
anyt
h
ing
a
n
d
ν
U
H
E
+
ν
C
ν
B
→
Z
→
a
nyt
h
in
g
processes
:
E
ν
i
≈
M
2
Z
M
/
[2
m
i
(
1
+
z
)]
,de-
pendin
g
on both the neutrino masses
m
i
and the source redshift
z
.The
observation of these dips would not only serve as the most direct evidenc
e
f
or the existence o
f
the
C
ν
B
but also help determine the absolute values o
f
relic neutrino masses and the distribution of UHE cosmic neutrino sources
(
Eberle
et al.
, 2004; Baren
b
oim
et al.
, 2005; D’Oliv
o
et al
.
,
2006; Ringwa
ld
a
nd Schrempp, 2006; Scholten and van Vliet, 2008
).
8.4.2
C
osmic Neutrinos and
G
amma Ray
s
Supernova remnants
(
SNRs
)
have been conjectured to be the accelerators
of galactic cosmic rays for a long time
(
Baade and Zwicky, 1934; Ginzburg
a
nd Syrovatskii, 1964
)
. In the past few years the HESS Collaboration ha
s
d
etermined the energy spectra of a few young SNRs, such as RX J1713.7
-
3
946 and RX J0852.0-4622
(
Vela Junior
)
, and demonstrated that they emi
t
gamma rays with energies above 10 TeV
(
Aharonia
n
e
tal.
,
2006a
,
2007a
,
2
007b
)
. But it is still impossible to conclusively pinpoint SNRs as the sources
o
fg
alactic cosmic rays by identi
f
yin
g
that the observed
g
amma rays ori
g
i
-
n
ate from the decays of neutral pions. Eliminating the possibility of a purely
electromagnetic origin of TeV gamma rays is challenging, and hence detect
-
i
n
g
their accompanyin
g
neutrinos would provide incontrovertible evidence
f
or
the acceleration of cosmic rays at the sources
(
Halzen, 2009
)
. The connection
between photon and neutrino fluxes in the case of galactic supernova shock
s
i
s clear: cosmic rays interact with the ambient matter
(
e.g., dense molecula
r
c
louds in the disk
)
and then produce equal numbers o
f
π
+
,
π
−
an
d
π
0
in
had
r
o
ni
c coll
i
s
i
o
n
s
p
+
p
→
n
π
+
+
π
−
+
π
0
+
X
w
i
th
n
1
b
ein
g
an inte-
g
er.
O
ne may there
f
ore estimate the neutrino
fl
ux expected
f
rom the sourc
e