Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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286 7 Neutrinos
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8
U
ltrahigh-energy
C
osmic Neutrino
s
Man
y
astroph
y
sical ob
j
ects in the Universe are expected to produce cosmi
c
n
eutrinos with very high energies
(
1Te
V
E
ν
1
PeV
)
, ultrahigh energies
(
E
ν
1PeV
)
or extremely high energies
(
E
ν
100 EeV
)
1
.
S
uch ener
g
eti
c
n
eutrinos may serve as a unique cosmic messen
g
er and provide us with useful
i
nformation about the cosmos that cannot be extracted from the measure
-
m
ents o
f
cosmic rays and
g
amma rays. The burnin
g
questions in neutrin
o
a
stronomy include where ultrahigh-energy
(
UHE
)
cosmic neutrinos originat
e
f
rom and how they can be detected. In this chapter we shall first describ
e
s
ome possible sources o
f
UHE cosmic neutrinos and then outline a
f
ew pos
-
s
ible wa
y
s to detect them. The flavor distribution and oscillations of UH
E
c
osmic neutrinos, toget
h
er wit
h
t
h
eir sensitivities to new p
h
ysics, wi
ll
a
l
s
o
be discussed. We shall
fi
nally hi
g
hli
g
ht the importance o
f
multi-messen
g
er
a
stronomy by illustrating the interplay between UHE cosmic neutrinos and
c
osmic rays,
g
amma rays or
g
ravitationa
l
waves
.
8
.1 Possible
S
ources o
fU
HE
C
osmic Neutrinos
The sources o
f
UHE cosmic neutrinos are presumabl
y
related to those o
f
U
HE cosmic rays. In fact, most of the models for the origin of UHE cos
-
m
ic rays predict the existence of UHE cosmic neutrinos. Such models can
be classi
fi
ed into two cate
g
ories: the top-down and bottom-up scenarios. I
n
the top-down scenarios it is assumed that UHE cosmic rays originate fro
m
the annihilation or decays of some superheavy particles, which must be th
e
1
N
ote that the terms VHE
(
very high energy
)
, UHE
(
ultrahigh energy
)
and
EHE
(
extremely high energy
)
are highly relative and taste-dependent. Some useful
units of energy in neutrino astronomy are keV = 1
0
3
eV
,
MeV = 1
0
6
e
V
,
GeV =
10
9
e
V
,Te
V
=1
0
12
e
V
,
P
e
V
=1
0
1
5
eV
,Ee
V
=1
0
18
e
V
,
Z
e
V
=1
0
21
e
V
an
d
Ye
V
=
10
2
4
e
V,where“k”,“M”,“
G
”, “T”, “P”, “E”, “Z” and “Y” denote kilo, me
g
a
,
giga, tera, peta, exa, zetta and yotta, respectively
.
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
290 8 Ultrahigh-energy
C
osmic Neutrinos
relics o
f
the Bi
g
Ban
g
. In the bottom-up scenarios it is assumed that UH
E
c
osmicraysori
g
inate in some cosmic accelerators. One may find a lon
g
list o
f
the hitherto-proposed sources of UHE cosmic neutrinos
(
Halzen and Hooper
,
2
002
)
. Some of them will be briefly introduced in this section
.
8
.1.1 The
G
ZK
C
utoff and
U
HE Neutrino
s
Charged cosmic rays consist mostly of protons and heavier nuclei with life
-
t
im
es of
O
(
1
0
6
)
years or longer. The “primary” cosmic rays are referred to as
those particles accelerated at the astroph
y
sical sources, and the “secondar
y
”
c
osmic rays are those particles produced in the interactions of the primar
y
c
osmic rays with the interstellar
g
as. The existence o
f
cosmic rays was dis-
c
overed b
y
Victor Hess in 1912, and the extensive air showers initiated b
y
the primary cosmic rays were first observed by Pierre Auger in 1938
(
Auge
r
et al.
,
1938
)
. After about a century of measurements, the energy spectrum o
f
c
osmicraysisnowdeterminedtoa
g
ood de
g
ree of accuracy. It spans more
than ten orders of magnitude and exhibits an approximate power-law shape.
F
i
g
. 8.1 shows the “all-particle” spectrum o
f
cosmic rays, in which the di
f-
f
erential flu
x
F
(
E
)
≡
d
Φ
d
E
has been multiplied b
y
E
2
.
7
so as to make the
visibility of its steep structure much better
(
Nakamur
a
et al.
, 2010
)
.Theim
-
pressive steepenin
g
t
h
at occurs
b
etween
E
=
10
1
5
e
Van
d
E
=10
1
6
eV is
the so-called “knee” of the s
p
ectrum. A structure known as the “ankle” of
t
h
e spectrum appears aroun
d
E
=
10
19
e
V. I
f
thecosmicraysuptoener
g
ie
s
o
f
1
0
1
7
e
Vto1
0
18
eV ori
g
inate in our
G
alaxy, those around and beyond the
a
nkle are most likely to have an extragalactic origin. In fact, the origin o
f
U
HE cosmic rays is not known at all, nor are the reasons
f
or either the kne
e
or the ankle of the spectrum of cosmic ra
y
s.
S
oon after Arno Penzias and
R
obert
W
ilson discovered the cosmic mi
-
c
rowave background
(
CMB
)
radiation
(
Penzias and Wilson, 1965
)
,itwa
s
pointed out by Kenneth Greisen and independently by Geor
g
iy Zatsepin
a
nd Vadim Kuzmin that this radiation would have a strong attenuating ef-
f
ect on the propa
g
ation o
f
cosmic rays with ener
g
ies hi
g
her than about 10
20
eV
(
Greisen, 1966; Zatsepin and Kuzmin, 1966
)
. Today we expect that th
e
Greisen-Zatsepin-Kuzmin
(
GZK
)
cutoff takes place in the spectrum of cosmic
ra
y
saroun
d
E
∼
5
×
10
19
e
V, as a result o
f
the onset o
f
inelastic interac
-
tions of UHE cosmic ra
y
s with the CMB photons. The dominant reactio
n
l
eading to the GZK cutoff is the pion photoproduction on the
Δ
+
r
eso
n
a
n
ce
;
i
.e.,
p
+
γ
CM
B
→
Δ
+
(
1232
)
→
p
+
π
0
or
n
+
π
+
,wherethe
fi
nal-state par-
ticles roughly have only half of the energy of the initial proton. If the
Δ
+
resonance is unavailable, a weakened version o
f
the
G
ZK cuto
ff
may resul
t
f
rom the non-resonant photoproduction of one or more pions
(
Coleman and
Glashow, 1998
)
:
p
+
γ
C
MB
→
p
+
n
π
o
r
n
+
n
π
,
where
n
i
s a positive intege
r
a
nd the total electric char
g
eo
f
n
π
should be co
n
s
i
ste
n
tw
i
th the co
n
se
r
vat
i
o
n
of electric char
g
es for this multi-pion channel.
A
simple kinematic analysis
8
.1 Possible
S
ources o
fU
HE
C
osmic Neutrinos 2
91
Grigoro
v
J
AC
EE
MGU
T
ienSha
n
T
ib
et0
7
Ak
en
o
C
ASA/MIA
H
e
g
r
a
F
l
y
s E
ye
A
g
asa
H
i
Res1
H
i
Res2
A
uger S
D
A
u
g
er h
y
brid
Ka
sc
a
de
E
[eV]
E
E
2
.
7
F
(
FF
E
((
)
[Ge
V
1.7
m
−
2
s
−
1
sr
−
1
]
Ankle
Knee
2nd Knee
10
4
1
0
5
10
3
10
14
10
1
5
10
13
10
16
10
1
7
10
18
10
19
10
20
Fig. 8.1 The all-particle spectrum of cosmic rays from the air shower measure-
ments, w
h
ere t
h
es
h
a
d
e
d
area s
h
ows t
h
eran
g
ecovere
db
yt
h
e
d
irect measurements
(
Nakamur
a
e
ta
l
.
,
2010. With permission from the Institute of Physics
)
yields a threshold ener
g
yo
f
the proton in either case:
E
th
E
E
=
⎧
⎪
⎧⎧
⎪
⎪⎪
⎪
⎪⎪
⎨
⎪⎪
⎪
⎨
⎨
⎪
⎪⎪
⎪
⎪⎪
⎩
⎪⎪
m
2
Δ
−
m
2
p
2
E
γ
E
≈
3
.
19
E
γ
E
×
10
17
eV
2
(
on resonance
)
,
m
π
2
m
p
+
m
π
γ
2
E
γ
E
≈
1
.
36
E
γ
E
×
1
0
1
7
eV
2
(
no resonance
)
,
(
8.1
)
where
E
γ
E
denotes the average CMB photon energ
y
2
,
and only a single pio
n
(
n
=
1
)
has been taken into account in the non-resonant case. An input o
f
E
γ
E
≈
6
.
3
5
×
1
0
−
4
eV leads t
o
E
th
EE
≈
5
.
0
×
1
0
20
GeV
(
or 2
.
1
×
1
0
20
GeV
)
f
or the resonant
(
or non-resonant
)
pion photoproduction. This naive result
,
w
h
ic
h
is somew
h
at
l
ar
g
er t
h
an
E
G
Z
K
∼
5
×
10
19
eV, implies that Eq.
(
8.1
)
c
an only be used to
g
ive a ballpark estimate of the GZK cutoff. The reaso
n
i
s simply that many details of the inelastic photon-proton interactions in the
CMB need to be carefully dealt with
(
Stecker, 1968
).
Fi
g
. 8.1 shows that the observed spectrum of cosmic rays has a dip struc
-
ture at the GZK-cutoff energy
E
G
ZK
. This significant steepening has bee
n
revealed mainly in the HiRes and Auger experiments
(
Abbasi
et al.
, 2008
;
A
braha
m
e
tal.
,
2008
)
, and it is apparently inconsistent with the AGAS
A
m
easurement
(
Takeda
et al.
,
2003
)
. Such a cutoff implies that the Univers
e
i
s opaque to EHE cosmic rays, with an absorption len
g
th o
f
λ
p
+
γ
C
MB
=
2
G
iven the
C
MB tem
p
eratur
e
T
γ
TT
=
2
.
7
25 K, it is easy to
g
et t
h
ep
h
oton ener
gy
E
γ
=2.
3
4
8
×
10
−
4
eV. The average value o
f
E
γ
at
T
γ
TT
is
E
γ
T
γ
T
≈
6
.
35
×
1
0
−
4
e
V.
292 8 Ultrahigh-energy
C
osmic Neutrinos
n
−
1
C
MB
σ
−
1
p
+
γ
CM
B
∼
8
Mpc
f
o
r
n
C
M
B
≈
4
11
c
m
−
3
a
n
d
σ
p
+
γ
C
MB
∼
10
−
28
cm
2
(
Halzen and Hooper, 2002
)
. In other words, the GZK effect suppresses EH
E
c
osmic rays coming from distances larger than a few tens of Megaparsecs.
The
A
u
g
er experiment is expected to provide a solid observational basis fo
r
the existence or non-existence of the GZK cutoff by detectin
g
a sufficientl
y
l
arge number of events at and above
E
G
ZK
in t
h
e coming years.
A
ssociated with the GZK cutoff of cosmic rays, cosmo
g
enic neutrinos can
be produced from the decays of charged pions
(
or neutrons
)(
Berezinsky and
Zatsepin, 1969
)
:
π
+
→
μ
+
+
ν
μ
a
n
d
μ
+
→
e
+
+
ν
μ
+
ν
e
(
o
r
n
→
p
+
e
−
+
ν
e
)
.
The ener
g
ies o
f
such
G
ZK neutrinos should be close t
o
E
G
Z
K
,b
ecause rou
ghl
y
equal energies go into the secondary nucleons and pions in the photoproduc-
tion processes discussed above. A measurement of UHE cosmic neutrino
s
w
i
th
E
ν
∼
E
G
Z
K
can there
f
ore serve
f
or a promisin
g
way to probe the
G
Z
K
c
utoff of UHE cosmic ra
y
s. Of course, the flux of GZK neutrinos is so low
(
10
−
26
M
e
V
−
1
cm
−
2
s
−
1
s
r
−
1
)
that it will be very challenging to observ
e
them even at an un
p
recedented k
m
3
-scale detector
(
Spiering, 2009
)
.
8.1.2 Astrophysical Sources o
f
UHE Neutrino
s
The expected astrophysical sources of UHE neutrinos include the active galac
-
tic nuclei
(
AGN
)
, gamma ray bursts
(
GRBs
)
and other sources associate
d
with compact stellar ob
j
ects, such as the supernova remnants, X-ra
y
bina
-
ries and microquasars
(
Halzen and Hooper, 2002; Torres and Anchordoqui,
2
004
)
. Here we give a brief introduction about two typical examples, AG
N
a
nd GRBs, together with a rough estimate of the UHE neutrino fluxes from
t
h
ose optica
ll
yt
h
in sources
.
(
1
)
UHE cosmic neutrinos from AGN
. The AGN should presumably be-
NN
l
ong to the most promising UHE neutrino sources, but the detailed mecha
-
n
ism of neutrino production in the AGN remains unclear. One may follow
a
c
onservative wa
y
, which is independent o
f
an
y
speci
fi
c neutrino-production
m
odels, to describe the AGN according to their geometric properties an
d
electromagnetic emission
(
Achterberg
et al
., 2006
)
. For instance, the obser-
vational differences among a variety of AGN
(
e.g., blazars, radio galaxies
a
nd quasars
)
can be partially interpreted with the help of a geometrically
a
xisymmetric scheme as the result o
f
their di
ff
erent inclination an
g
les whic
h
a
re defined as the angles between the lines of sight and the AGN axis
(
Urr
y
a
nd Padovani, 1995
)
. Fig. 8.2 shows some basic ingredients of such an ax
-
i
symmetric scheme of the
A
GN. It consists of a rotatin
g
supermassive blac
k
h
ole, two jets with matter outflowing along the rotation axis and an accretio
n
d
isk of matter perpendicular to the rotation axis
.
T
he radio emission of
A
GN is assumed to ori
g
inate mostly in relativisti
c
j
ets; namely, it is caused by the synchrotron radiation of electrons movin
g
a
long the jet. The energy spectrum of the photons emitted from the AGN
a
nd observed on the Earth may ran
g
e
f
rom radio waves to TeV
g
amma rays
.
8
.1 Possible
S
ources o
fU
HE
C
osmic Neutrinos 2
93
bl
azar
FR−I
/
II
a
lax
y
radio
ga
S
SR
Q
Fig. 8.2 A pictorial scheme of the AGN with a black hole in the center and an
accretion disk perpendicular to the direction o
f
two jets alon
g
its rotation axis. The
di
ff
erent inclination an
g
les o
f
the lines o
f
si
g
ht with respect to the jet
f
or blazars,
steep spectrum radio quasars
(
SSRQs
)
and radio galaxies are indicated by arrows
(
Urry and Padovani, 1995; Achterberg
et al
.
,
2006. With permission from Elsevier
)
Theluminousener
g
yofthe
A
GN is presumably supplied by the
g
ravitational
energy of matter falling into the supermassive black hole in the center of their
c
ore. In this case the in
f
allin
g
matter
f
orms an accretion shock at a certai
n
d
istance
f
rom the central black hole, where protons can be accelerated b
y
th
e
first-order Fermi acceleration
(
Protheroe and Szabo, 1992; Tu, 2004
)
.Proton
s
a
n
d
ot
h
er c
h
ar
g
e
d
primaries can a
l
so
b
e acce
l
erate
d
in t
h
ere
l
ativistic jets o
r
i
n the hot spots
(
e.g., of the FR-II radio galaxies
)
to energies around or above
10
20
e
V
(
Rachen and Biermann, 1993
)
. Then a large amount of pions can b
e
produced
f
rom the interactions o
f
protons with the dense matter surroundin
g
thecoreoftheAGN
(
in particular, the intense photon field
)
. In this scenario
the photons and protons cannot escape from the source region
(
Stecker an
d
Salamon, 1996
)
, and thus the latter might only be visible by detecting the
U
HE neutrinos arising from the decays of charged pions.
Th
e acce
l
erate
d
protons may interact wit
h
t
h
eam
b
ient p
h
otons or matte
r
v
i
athe
r
eact
i
o
n
s
p
+
γ
→
Δ
+
→
n
+
π
+
a
n
d
p
+
X
→
π
±
+
Y
,l
ea
d
in
g
to
the production of UHE neutrinos through
π
+
→
μ
+
+
ν
μ
an
d
π
−
→
μ
−
+
ν
μ
d
ecays as we
ll
a
s
μ
+
→
e
+
+
ν
μ
+
ν
e
a
n
d
μ
−
→
e
−
+
ν
μ
+
ν
e
decays.
O
n
the other hand, the decays of neutral pions
(
i.e.,
π
0
→
2
γ
)
can produce
h
igh-energy gamma rays. The resultant neutrino and gamma-ray fluxes ar
e
expected to be of the same order of magnitude
(
Achterber
g
et al
.
,
2006
)
.Since
the proton and tar
g
et photon spectra
f
all steeply with the increase o
f
ener
g
ies
i
n most scenarios, t
h
emu
l
ti-pion pro
d
uction,
k
aon pro
d
uction an
dh
ig
h
e
r
resonances on
l
y represent a sma
ll
correction to t
h
esin
gl
epionpro
d
uction
.
Given the low plasma densit
y
in the
A
GN
j
ets, the proton-photon interaction
s
a
re likely to dominate over the proton-proton interactions
(
or generically, the
294 8 Ultrahigh-energy
C
osmic Neutrinos
proton
-
X
i
nteractions
)
unless the latter can take place at the inner edge of
the accretion disk
(
Nellen et al., 1993; Stecker and Salamon, 1996
)
.
Bl
azars, w
h
ic
h
emit
h
ig
h
-energy ra
d
iation in co
ll
imate
d
jets pointing a
t
the Earth, are a subclass of
A
GN. The
y
are characterized b
y
aflatradi
o
s
pectrum
(
Achterber
g
e
tal., 2006
)
, and appear less distant than the GRB
s
s
ince their lower luminosity makes distant sources difficult to observe
(
Halzen
a
nd Hooper, 2002
)
. Blazars have been identified to be prodigious gamma-ray
s
ources by a number of experiments in different energy regimes: infrared, ke
V
(
or X-ray
)
, GeV and TeV. Gamma rays in the multi-TeV range are observe
d
f
rom the most power
f
ul blazars. I
f
the emission o
f
hi
g
h-ener
g
y photons in th
e
blazar jets only involves the electromagnetic processes
(
i.e., electron acceler-
a
tion and inverse Compton scattering
)
, it is impossible to produce neutrinos
.
I
f
hi
g
hly-shocked protons are present in the blazar jets, however, they may in
-
teract with the ambient photons and matter to generate charged and neutra
l
pions w
h
ic
h
can t
h
en
d
ecay into UHE neutrinos an
d
p
h
otons
.
(
2
)
UHE cosmic neutrinos
f
rom
G
RBs
.G
RBs are the bri
g
htest
g
amma
ray sources in the Universe during a period of a few seconds or tens of sec
-
onds.
A
n enormous amount of ener
g
y, typically 10
51
e
r
g
to
10
5
4
er
g
per
s
econd, is released from a GRB. It can phenomenolo
g
ically be described a
s
a
fireball expanding with a highly relativistic velocity and powered by the
radiation pressure
(
Waxman, 2003
)
. The delayed low-energy emission or “af
-
terglow” of GRBs has been observed
(
Kulkarni
e
tal., 2000
)
and it indicate
s
that GRBs are predominantly generated in host galaxies and are likely th
e
result o
f
a stellar process — e.
g
., the “collapsar” scenario, where a supermas-
s
ive star undergoes the core collapse resulting in a failed supernova, is on
e
of the most popular models proposed for the fireball’s inner engine
(
Mac-
F
adyen and Woosley, 1999; Halzen and Hooper, 2002
)
. The dynamics of a
GRB fireball is quite similar to the physics of the early expanding Universe:
(
a
)
a radiation-dominated soup of leptons and photons
(
and few baryons
)
i
s initially present, and it is hot enou
g
hto
f
reely produce electron-positron
pairs;
(
b
)
the optical depth of photons is large enough that it is possible
to create a thermal plasma o
f
photons, electrons and positrons — a
fi
rebal
l
which can then expand and accelerate to relativistic velocities;
(
c
)
the firebal
l
expan
d
swit
h
increasing ve
l
ocity unti
l
it
b
ecomes optica
ll
yt
h
in, an
d
t
h
en t
h
e
photons ma
y
escape and the radiation is released in the visual displa
y
o
f
a
GRB
(
Halzen and Hooper, 2002; Waxman, 2003
)
. The observed hard photo
n
s
pectra are produced from the synchrotron radiation of electrons accelerate
d
by the shocks in the expandin
gfi
reball, because the latter is made up o
f
m
ulti
p
le shocks. The
p
rotons
p
resent in the fireball can also be accelerate
d
to very high energies in the same region, making the GRBs a good candidat
e
f
or the source o
f
UHE cosmic ra
y
s
.
A
nalogous to the case of
A
GN, pions can be produced from a GRB via
i
nteractions of the accelerated protons with the target photon field whic
h
i
s simply the hi
g
hly variable radiation
f
ormedinthe
G
RB blast wave and
8
.1 Possible
S
ources o
fU
HE
C
osmic Neutrinos 2
95
d
etected as the
G
RB. The most important pion photoproduction processes
a
r
e
p
+
γ
→
n
+
π
+
a
nd
p
+
γ
→
p
+
π
0
,
in which a lar
g
e fraction of the initial
proton energy is lost. In contrast, only a small fraction of the initial proton
ener
g
yis
l
ost in t
he
e
+
e
−
p
air p
h
otopro
d
uction process
p
+
γ
→
p
+
e
+
+
e
−
.
F
or the resonant pion photoproduction
(
i.e.
,
p
+
γ
→
Δ
+
→
p
+
π
0
o
r
n
+
π
+
),
h
igh-energy gamma rays may come from
π
0
→
2
γ
d
ecays an
dh
ig
h
-energ
y
neut
rin
os ca
n
a
ri
se f
r
om
π
+
→
μ
+
+
ν
μ
d
ecays to
g
et
h
er wit
h
μ
+
→
e
+
+
ν
μ
+
ν
e
d
eca
y
s. For the non-resonant multi-pion channe
l
p
+
γ
→
p
+
n
π
o
r
n
+
n
π
,
the production rate of
π
+
,
π
−
a
n
d
π
0
events is possi
bl
e to approac
h
1:1:1
(
Dermer and Atoyan, 2006
)
, implying that the ratio of
ν
μ
,
ν
μ
,
ν
e
,
ν
e
a
n
d
γ
events would be 2 : 2 : 1 : 1 : 2. The energies of cosmic neutrinos produce
d
f
rom the interactions o
f
shocked protons with the ambient photons in
G
RB
s
m
ay vary from TeV to EeV
(
Halzen and Hooper, 2002
)
3
.
(
3
)
UHE neutrino fluxes from optically thin source
s
.BothA
G
Nand
GR
Bs
a
re pro
b
a
bl
yt
h
e cosmic acce
l
erators in w
h
ic
h
UHE cosmic rays may ori
g
i
-
n
ate. The accelerated electrons lose their ener
g
yinthema
g
netic
fi
eld as
a
result of their synchrotron radiation. The resultant photons provide a target
fi
eld
f
or protons and heavier nuclei to tri
gg
er the meson photoproduction and
photo-disinte
g
ration processes, respectively.
A
neutron produced in this way
m
ay diffuse out of the magnetically confining source before it undergoes th
e
beta deca
y
.
S
uch an astroph
y
sical source is re
f
erred to as the opticall
y
thi
n
s
ource, as shown in Fig. 8.3
(
left
)
. For an optically thick source in Fig. 8.
3
(
right
)
, however, some neutrons may scatter inelastically off the photon gas
before they escape from the region of magnetic confinement
(
Ahlers, 2007
).
F
ocusing on the optically thin source and assuming the resonant photopro-
duct
i
o
n
p
+
γ
→
Δ
+
→
n
+
π
+
t
o be the dominant process for the charged-pio
n
production, we shall estimate the UHE neutrino
fl
ux in the
f
ollowin
g
.
I
n the assum
p
tion made above, we obtain
ν
μ
,
ν
μ
an
d
ν
e
e
vents from
π
+
→
μ
+
+
ν
μ
a
n
d
μ
+
→
e
+
+
ν
μ
+
ν
e
decays. They constitute the
fl
ux o
f
U
HE neutrinos emitted
f
rom an optically thin source. Followin
g
the Waxman-
Bahcall treatment
(
Waxman and Bahcall, 1999
)
,wedefin
e
π
≡
E
π
/
E
n
a
saratioo
f
the avera
g
epionener
g
ytotheavera
g
e neutron ener
g
yinthe
r
eactio
n
p
+
γ
→
Δ
+
(
1232
)
→
n
+
π
+
. The inelasticit
y
o
f
this process is
kinematically determined by requiring equal boosts fo
r
n
a
nd
π
+
(
Stecker,
1
968
)
, giving
π
≈
0
.
28. The avera
g
eener
g
ies o
f
ν
μ
a
n
d
μ
+
e
mi
tted f
r
o
m
t
h
e
π
+
d
eca
y
ar
e
E
ν
μ
≈
γ
π
(
m
2
π
−
m
2
μ
)
/
(2
m
π
)
≈
E
π
(1
−
m
2
μ
/m
2
π
)
/
2
≈
0
.
22
E
π
a
n
d
E
μ
≈
E
π
−
E
ν
μ
≈
0
.
78
E
π
.B
ecause
μ
+
is
m
uch heav
i
e
r
tha
n
e
+
a
nd the neutrino masses are ne
g
li
g
ibly small, we assume that the ener
gy
of
μ
+
i
s equally distributed amon
g
e
+
,
ν
μ
a
nd
ν
e
i
nthe
μ
+
d
eca
y
; i.e.
,
3
L
ike a supernova, a
G
RB radiates the vast majority o
f
its initial ener
g
yas
thermal MeV neutrinos. On the other hand, neutrons may decouple from protons in
the expanding fireball. If their relative velocity is sufficiently large, their interactions
may produce charged pions
(
i.e.
,
p
+
n
→
p
+
p
+
π
−
a
n
d
p
+
n
→
n
+
n
+
π
+
)
which can then produce GeV neutrinos (Halzen and Hooper, 2002).