Flavor Composition and Energy Spectrum of Astrophysical Neutrinos
Paolo Lipari,1, ∗ Maurizio Lusignoli,1, † and Davide Meloni1, ‡
INFN and Dipartimento di Fisica, Università di Roma I, P.A.Moro 2, 00185 Roma, Italy
The measurement of the flavor composition of the neutrino fluxes from astrophysical sources has
been proposed as a method to study not only the nature of their emission mechanisms, but also
the neutrino fundamental properties. It is however problematic to reconcile these two goals, since a
sufficiently accurate understanding of the neutrino fluxes at the source is needed to extract informa-
tion about the physics of neutrino propagation. In this work we discuss critically the expectations
for the flavor composition and energy spectrum from different types of astrophysical sources, and
comment on the theoretical uncertainties connected to our limited knowledge of their structure.
PACS numbers: 95.85.Ry, 96.40.Tv, 14.60.Pq
I. INTRODUCTION
There is the expectation that in the near future we will see the opening of the new field of observational high
energy neutrino astronomy [1, 2, 3]. Theoretically there are very robust reasons to expect the existence of high energy
(Eν >∼ 1012 eV) neutrino sources. The strongest motivation is the observation of a cosmic ray flux, mostly composed of
protons and fully ionized nuclei that extends in energy up to E ∼ 1020 eV. These hadronic particles can interact inside
or near their acceleration sites or during their propagation in interstellar or intergalactic space. These interactions
produce weakly decaying particles (such as π± and kaons) that generate neutrinos. These “astrophysical neutrinos”
are intimately connected with the high energy photons that are created in the decay of π◦ and η particles produced
in the same hadronic primaries interactions, or in the radiation processes of relativistic electrons and positrons co–
accelerated in the sources. A rich variety of high energy gamma ray sources has been observed with detectors on
satellites [4] and ground–based [5] Cherenkov telescopes, suggesting several possible neutrino sources.
Observations with neutrinos have the potential to give us unique information about their astrophysical sources,
and hopefully could also result in the discovery of new classes of sources. On the other hand the possibility to use
these observations to obtain information about the fundamental properties of the neutrinos has been widely discussed
[1, 8, 9, 10, 11, 12, 13]. Astrophysical neutrinos travel pathlengths of order 10 Kpc (∼ 3 × 1022 cm) for galactic
sources, and as large as ∼ 1 Gpc (∼ 3× 1027 cm) for extragalactic sources. These remarkably long baselines allow the
study of phenomena such as ν flavor transitions or ν decay in a range of parameters that is unaccessible with other
methods. Only observations with SuperNova neutrinos, very likely only observable from galactic sources, but having
much smaller energy (Eν ∼ 10 MeV) could provide larger L/Eν .
In order to extract information about the neutrino fundamental properties from the observations one needs to have
sufficiently good understanding of the properties of the neutrino emission at the source. The common assumption in
several studies of this type is that the properties of the ν emission, and in particular the flavor composition at the
source can be robustly predicted. There is of course a well known historical precedent for the successful use of this
concept in the discovery of neutrino oscillations with atmospheric neutrinos. In fact, the first hints of the existence
of the now solidly established νµ ↔ ντ transitions were obtained with the Kamiokande [6] and IMB [7] detectors as
the measurement of a µ/e ratio for contained events smaller than the expectations. It is natural to try to make use
of new, very distant neutrino sources (when they will be discovered) to perform additional studies of the properties
of neutrino propagation.
Several “exotic” processes, beyond standard flavor oscillations, could reveal themselves only in the propagation of
neutrinos over very long distances. For example, it has been suggested that some neutrinos could decay into a lighter
neutrino and a majoron [14], if the lifetime is sufficiently long this phenomenon could be only detectable for neutrinos
propagating over astronomical distances [15, 16]. A second interesting possibility is that neutrinos are pseudo–Dirac
states [17] where each generation is actually composed of two maximally-mixed Majorana neutrinos separated by a
tiny mass difference. If the pseudo-Dirac splittings are sufficiently small, the phenomenology of oscillations on short
baselines remains unchanged, however when Eν/L becomes comparable or smaller than the pseudo–Dirac splittings
∗Electronic address: paolo.lipari@roma1.infn.it
†Electronic address: maurizio.lusignoli@roma1.infn.it
‡Electronic address: davide.meloni@roma1.infn.it
http://arxiv.org/abs/0704.0718v1
mailto:paolo.lipari@roma1.infn.it
mailto:maurizio.lusignoli@roma1.infn.it
mailto:davide.meloni@roma1.infn.it
new transitions become possible, and can in principle be detectable with astrophysical neutrinos [18]. More in general,
oscillations into sterile states that are quasi degenerate to the active neutrinos can in principle be investigated down to
very small squared mass splittings [19]. Several other mechanisms such as quantum decoherence [20], violations of the
equivalence principle [21, 22], neutrinos with varying masses [23, 24] could leave their signature on the propagation
of astrophysical neutrinos.
A well known “naive” argument states that since the dominant source of astrophysical neutrinos is the decay of
charged pions, and each π±, after chain decays such as: π+ → µ+νµ → (νµ e+ νe) νµ, generates two muon neutrinos
and one electron neutrino, the flavor ratio at the source is Rµe = (νµ + νµ)/(νe + νe) ≃ 2. This naive argument
has some pedagogical value, but must be considered only as a first approximation. In fact, even if charged pions are
the only source of neutrinos, Rµe is only equal to 2 after integration over all ν energies, because the three neutrinos
generated in a charged pion decay have different energy spectra, and therefore in general the flavor ratio will depend
on the spectral shape of the ν signal, and will vary with the neutrino energy. An effect that can be of large importance
is the presence of sufficiently efficient energy loss mechanisms in the source. In this situation the particles that are
most affected are the muons. The presence of additional neutrino sources (such as kaons) can also be important for
the flavor ratio. In this work we will re–examine critically the uncertainties in the predictions of the spectra and flavor
composition of astrophysical neutrinos and discuss the implications for the extraction of information on the neutrino
properties.
This paper is organized as follows: in the next section we discuss how the observable neutrino flavor ratios carry
information at the same time about the flavor composition at the source and about the flavor transition probabilities.
In section 3 we outline the general structure of the calculation of the neutrino signal from an astrophysical source. The
different steps of these calculations are considered in more details in sections 4–7. In section 8, as an illustration, we
describe an explicit calculation of the neutrino emission from the fireballs of Gamma Ray Bursts following the model
of Waxman and Bahcall [25]. Section 9 contains a short discussion of the problem of the experimental determination
of the flavor composition and energy spectra of a neutrino signal. The last section presents our conclusions.
II. FLAVOR RATIOS OF ASTROPHYSICAL NEUTRINOS
The observable fluxes of astrophysical neutrinos from a source at distance L will be linear combinations of the fluxes
at the source. Leaving implicit the energy and distance dependences one can write:
φνα =
Pνβ→να φ
. (1)
The transition probabilities Pνβ→να have certainly a non trivial structure because of the known existence of “standard”
flavor oscillations, but might depend on additional “new physics” contributions, such as neutrino decay, that become
significant only for very long pathlengths. In the standard scenario the neutrino number is conserved and therefore
β Pα→β = 1. More in general, in the presence of a non negligible decay probability or of transitions to additional
sterile states, the sum can be less than unity.
The flavor, energy and pathlength dependences of the standard oscillation probabilities are well known, and are
determined by two squared mass differences (∆m212 ≃ 8.0 × 10−5 eV2, |∆m223| ≃ 2.5 × 10−3 eV2) and the neutrino
mixing matrix. The oscillation lengths (λij = 4πEν/|∆m2ij |) of the standard flavor transitions are short with respect
to the typical astrophysical distances and, in most cases, it is a good approximation to consider only the probability
averaged over distance. In this case the probability becomes independent from Eν and L and takes the form:
〈P standardνα→νβ 〉L =
|Uαj |2 |Uβj |2 (2)
where U is the unitary mixing matrix that relates the neutrino flavor and mass eigenstates. This matrix can be
written in terms of three mixing angles and one CP violating phase. From a global fit to all existing neutrino data
[26] one can extract the best fit values: θ12 ≃ 34◦, θ23 ≃ 45◦, θ13 ≃ 0, and 99% C.L. intervals for the mixing angles:
θ12 ∈ [30◦, 38◦], θ23 ∈ [36◦, 54◦], θ13 ≤ 10◦; the phase δ remains completely undetermined.
Assuming to know sufficiently well the properties of the source, the observed flavor ratios can give information on
the flavor transition probabilities. This can be used to help in the determination of the “standard” parameters in
the flavor oscillations, or more ambitiously to investigate the possible existence of additional phenomena in neutrino
propagation.
As an illustration in fig. 1 and 2 we show the expectations for the two independent flavor ratios Reµ = (νe/νµ)obs
and Rµτ = (νµ/ντ )obs, calculated for different assumptions for the source emission, in the presence of simple standard
oscillations, or also including neutrino decay (with two different assumptions) [16]. The distributions of the flavor
ratio for a given model are determined only by the present uncertainties on the neutrino mixing parameters. A
source model is defined by the relative intensity of the emission for the three neutrino flavors. For our illustration
we have considered three source models. The first model is the emission of form: [νe, νµ, ντ ]source = [1, 1.86, 0]. This
corresponds (as we will discuss more extensively in the following) to pion dominated emission from a thin source (no
significant energy loss for secondaries) with a power law energy spectrum of slope 2. Several classes of ν sources of
this type (such as Supernova Remnants and Gamma Ray Bursts) have been predicted. Note that the flavor relative
abundances in this model are close but not identical to the naive expectation [1,2,0]. The deviations are a simple
consequence of the shape of the energy spectra of the neutrinos in pion chain decay. The observable Reµ ratio for
this model takes value in the interval ≃ (0.7, 1.2) with a most likely value close to unity, while the other independent
ratio Rµτ has a narrow distribution sharply peaked at the value 1, with a tail extending up to ≃ 1.3.
Figures 1 and 2 also show the predicted flavor ratios for emission with (summing over ν and ν) flavor abundances
[0, 1, 0], and [1, 0, 0], that is pure muon or electron neutrino emission. The motivation for using these two models
is that, to a very good approximation, they can be considered the two extreme models for the emission from a
“standard” (not involving new physics) source, since a significant production of ντ in an astrophysical environment
is extraordinary unlikely. Sources emitting pure fluxes of νµ or νe are in principle possible, and have been in fact
advocated in the literature. They correspond to a source dominated by pions where muons lose all their energy before
decay, or to a pure neutron source, that generates a νe flux. A pure neutron (νe) neutrino source is in fact unrealistic
since it is natural that neutron production is associated with pion production and therefore with some νµ emission.
We will comment more extensively on these issues in the following.
For the [0,1,0] ([1,0,0]) model the most likely value for the observable flavor ratio is Reµ ≃ 0.55 (Reµ ≃ 2.7). In both
cases Rµτ is centered at the value unity, this is a simple consequence of the large (approximately maximal) mixing
between νµ and ντ . In the case of a [1,0,0] source the Rµτ distribution is noticeably broader extending to the interval
Rµτ ∈ (0.5, 1.5).
Neutrino decay with a very long lifetime is one of the most interesting possibilities that can be investigated with
astrophysical neutrinos. Assuming [12, 16] that only the lowest mass eigenstate is stable, and that the distance of the
source is much longer than the decay lengths, the observable flavor ratios depend uniquely on the flavor composition
of the lightest eigenstate. Depending on the sign of ∆m223 this is ν1 (direct mass hierarchy) or ν3 (inverse hierarchy).
The flavor ratios for these two models are also shown in fig. 1 and 2 as dashed lines. The Reµ ratio is particularly
interesting. In the case of direct mass hierarchy, the most probable value of the eµ ratio is large (Reµ ≃ 4.2), the
distribution is also remarkably wide, allowing in principle to constrain the mixing parameters [16]. For the inverse
hierarchy, when the stable neutrino is the ν3 that has a small (or perhaps vanishing) overlap with νe, the observable
Reµ is close to zero.
It is remarkable that the predictions for the eµ flavor ratio in neutrino decay are in both (direct and inverse mass
hierarchy) models significantly different from the standard model predictions for nearly all assumptions about the
flavor composition of the emission. Only a pure νe emission has some overlap with the decay model with stable ν1.
The study of the eµ flavor ratio of astrophysical neutrinos can therefore give evidence in favor or against the existence
of ν decay.
The determination of the neutrino mixing parameters in the standard oscillation scenario is more problematic. For
the most sensitive eµ ratio, the range of possible values due to the present uncertainties on the mixing parameters is
in fact smaller than the variations that result from different flavor abundances at the source. Therefore the possibility
to obtain interesting bounds on the mixing parameters depends crucially on having a sufficiently precise knowledge
of the source.
To illustrate this problem, we can write an approximate expression for the flavor ratio Reµ in terms of the relevant
mixing matrix parameters expanding in first order around the best fit values
)[1,1.86,0]
≃ 1.026− 0.0218 δθ◦23 + 0.0087 θ◦13 cos δ − 0.0014 δθ◦12 + 0.370 δ
)[0,1,0]
≃ 0.547− 0.0296 δθ◦23 + 0.0119 θ◦13 cos δ + 0.0197 δθ◦12 + 1.152 δ
)[1,0,0]
≃ 2.65 + 0.093 δθ◦23 − 0.037 θ◦13 cos δ − 0.131 δθ◦12 − 3.846 δ
In these equations δθ◦12 and δθ
23 are the deviations in degrees of the mixing angles from their best fit value (θ12 = 34
θ23 = 45
◦), θ◦13 is the value (always in degrees) of the third angle and δ is the CP violating phase. The superscript
labels of the flavor ratio indicate the source model. The last term in equations (3), (4) and (5) gives an estimate of
the variation in the observable ratio due to the uncertainty in the flavor ratio at the source obtained expanding in first
order in δ(να/νβ)0 around the flavor abundances of the model considered. One can see that uncertainties in flavor
abundances at the source can be so large to make impossible a meaningful constraint on the mixing parameters.
In the remaining sections of this paper we will discuss how the properties of the astrophysical sources determine
the flavor abundances of their emission, and estimate the associated theoretical uncertainties.
III. PRODUCTION OF ASTROPHYSICAL NEUTRINOS
Astrophysical neutrinos are generated when a population of relativistic hadrons (protons or nuclei) interacts with
some target material (gas or a radiation field) inside, near, or outside the acceleration site. These interactions produce
weakly decaying secondary particles whose decay generate neutrinos either directly, or indirectly with the subsequent
decay of muons. The description of these processes therefore requires the following elements:
1. The description of the energy spectrum and composition of the primary particles.
2. The definition of the target material with which the primary particles interact.
3. The modeling of the properties of particle production in hadronic interactions.
4. The description of the properties of the medium where the interactions are taking place, to determine the
relevant mechanisms for energy loss.
5. The (well known) properties of weak decays.
In our discussion we will describe the source as one stationary homogeneous region. This is an important limitation,
because in general we can expect that the sources will have non trivial space structures, with the neutrinos emerging
from different regions that can produce different spectral shapes and flavor contents; most high energy radiation sources
are also expected to show important time variabilities. Because of the limited statistics and angular resolution, the
observations of astrophysical neutrinos will necessarily integrate over the entire volume of the source (that in most
cases will appear as point–like) and average over most or all time variations. Describing this situation requires the
averaging over different source regions and conditions.
Some of the most interesting postulated sources of neutrinos, in particular the jets of Active Galactic Nuclei (AGN)
and Gamma Ray Bursts (GRB), are expected to be associated with astrophysical jets with ultra–relativistic bulk
motion. These relativistically moving sources are best described in the rest frame of the jet, where the emitted
radiation is approximately isotropic, and the electromagnetic fields can be described as a purely magnetic field. The
observable fluxes can then be obtained by an appropriate Lorentz boost.
The ν emission from a given source can be described by the emissivities Qνα(Eν ,Ω) (in units (s sr GeV)
−1) that
give the number of neutrinos of type α emitted with energy Eν in the direction Ω. The ν fluxes at the Earth are
obtained from the neutrino emissivities as:
φνα(Eν) =
(1 + z)2
4πd2L
Pνβ→να(Eν) Qνβ [Eν(1 + z),Ω⊕] (6)
where dL is the source luminosity distance, z is its redshift, and Pνβ→να are the flavor transition probabilities. The
observable fluxes depend only on the emission in the direction Ω⊕ that corresponds to the line of sight from the source
to the Earth. In the following we will not consider the angular dependence of the source emission, since this is not
observable for any individual source.
The calculation of the neutrino emissivities Qνα(Eν), starts from a decription of the energy spectrum and com-
position of the primary particles. In the following we will assume that the primary particles are protons and are
described by the energy spectrum Np(Ep) (in units GeV
−1). The extension to the case where there is a significant
contribution from nuclei is straightforward, and to a very good approximation can be calculated introducing effective
proton and neutron spectra (Np(Ep) and Nn(En)) that take also into account the bound nucleons. The description
of the environment of the source must include the density and properties of the target material and of all other fields
that can be a source of energy loss. The interactions of the primary particles result in the emission of secondary
particles (π+, π−,K+, . . .) that are sources of neutrinos. These particles must be propagated in the source medium
until they decay. All muons generated by these decays must also be propagated and their decay studied.
The scheme of the calculation is therefore the following. The rate of production of secondary particle of type a can
be calculated with the convolution:
Qa(Ea) =
dEp Np(Ep) Kp(Ep)
dnp→a
(Ea;Ep) (7)
where Kp(Ep) is the interaction probability per unit time of a proton of energy Ep, and dnp→a/dEa(Ea;Ep) is the
number of particles of type a in the final state in the energy interval (Ea, Ea + dEa).
The energy distribution of particle a at decay Qdeca (E) will in general differ from the distribution at production and
can be obtained with the integration:
Qdeca (E) =
dEi Qa(Ei)
dpadec
(E;Ei). (8)
where dpadec/dE(E;Ei) is the probability density that particle a created with energy Ei will decay with energy E.
The neutrinos created in the direct decay of particle a can be calculated as:
Qa→να(Eν) =
dEa Q
a (Ea)
dna→να
(Eν ;Ea) (9)
folding the parent energy distribution (at decay) with the appropriate decay spectra dna→ν/dEν(Eν ;Ea).
The muon production rate can be calculated with an expression that is the analogous of (9) substituting the
appropriate decay distributions:
Qµ(Eµ) =
dEa Q
a (Ea)
dna→µ
(Eµ;Ea). (10)
The sum runs over all weakly decaying hadrons.
In out calculation we explicitely included charged pions, kaons and neutrons, neglecting the much smaller contri-
butions of heavier particles. An important complication is that the spectra of the particles generated in muon decay
depend on the muon helicity (see discussion in section IV). It is therefore convenient to calculate separately the pro-
duction of muons with different helicity: {µ+L , µ
L and µ
R} (where L and R indicate the helicity). It is expected
that the helicity of the muons is to a good approximation conserved even in the presence of a strong magnetic field
and of significant energy losses, and therefore the energy distributions of the muon of different helicities at decay can
be obtained with a convolution similar to (8):
(E) =
dEi Qµ±
(E;Ei). (11)
The last step is to compute the neutrinos generated by muon decay:
(Eν) =
dEµ Q
(Eν ;Eµ). (12)
The neutrino flux is obtained summing over all possible sources.
In the following sections we will give more details about the neutrino production. Our discussion starts from the
neutrinos and goes backward along the chain of processes that lead to their production. The next section contains a
discussion of the decay energy distributions; after discussing the consequences for exact power law spectra in section V,
in section VI we will discuss possible mechanisms for energy loss and the calculation of the decay energy distribution;
in section VII we will discuss the interaction probability Kp(Ep) and the properties of particle production in hadronic
interactions.
IV. ENERGY SPECTRA IN WEAK DECAYS
For an accurate prediction of the neutrino fluxes it is necessary to include a precise description of the energy spectra
produced in weak decays. The energy distribution of particle b in the decay a → b + . . . in the frames where the
parent particle a is ultrarelativistic, takes the scaling form:
dna→b
(Eb;Ea) =
. (13)
The functions Fa→b(x) for the relevant weak decays are calculable using the measured branching fractions in the
possible final states, and their matrix elements.
For the two body decay π+ → µ+νµ, that accounts for approximately 100% of charged pion decay, the scaling
distributions are fully determined by elementary kinematics:
Fπ+→νµ(x) =
1− rπ
Θ(1− rπ − x) (14)
Fπ+→µ+(x) =
1− rπ
Θ(x− rπ) (15)
where Θ(x) is the step function and rπ = (mµ/mπ)
2. It is also necessary to take into account the spin state of the
final state muons, that can be described by the helicity h = PR − PL, where PR,L is the probability that the muon
has spin parallel (anti–parallel) to its momentum. The helicity, in a frame where the parent pion is ultrarelativistic,
is a function of the fractional energy x [27]:
hπ+→µ+(x) =
(1− rπ) x
− 1 + rπ
1− rπ
. (16)
Because of CP invariance:
hπ−→µ−(x) = −hπ+→µ+(x) (17)
The helicity hπ+→µ+(x) takes the value −1 for x → 1 (forward muon emission) and +1 for x → rπ (backward
muon emission) reflecting the fact that the µ+ is created as a left–handed particle in the pion rest frame in order to
compensate the angular monentum of a left–handed neutrino.
A useful method to take into account the effect of muon polarization in the presence of energy loss for the muons is
to consider separately the production of left–handed and right–handed muons, that have two well determined decay
spectra. Combining equations (15) and (16) one obtains the distributions:
Fπ+→µ+
(x) = Fπ−→µ−
(x) =
rπ(1 − x)
(1 − rπ)2 x
Θ(x− rπ) (18)
Fπ+→µ+
(x) = Fπ−→µ+
(x) =
x− rπ
(1− rπ)2 x
Θ(x− rπ) (19)
Because of CP invariance one has: Fπ+→µ+
(x) = Fπ−→µ−
The above discussion is also valid for the decay mode K+ → µ+νµ (and charge conjugate), with the simple
replacement mπ → mK .
The scaling functions that describe the decay of a muon of helicity h are:
Fµ+→νµ(x;h) =
− 3 x2 + 4 x
+ 3 x2 − 8 x
Fµ+→νe(x;h) =
2− 6 x2 + 4 x3
2− 12 x+ 18 x2 − 8 x3
Because of CP invariance the spectra for the charged conjugate decays are obtained with the replacement h → −h.
The spectra for unpolarized muons can be obtained setting h = 0.
If the muon energy loss before decay is negligible, it is straightforward to convolute the previous expressions to
obtain the ν spectra after a chain decay (π → µ → ν) or (K → µ → ν). In the more general case the energy loss of
the muon must be taken into account.
The energy spectra of neutrinos and muons emitted in the three body decays of kaons (K
e3 and K
µ3 ) can be
written in terms of vector form factors that have been experimentally determined [28].
Neutrons are also a source of νe. The energy spectrum of these neutrinos is also well known. The νe carry a very
small fraction of the parent neutron energy and are negligible in most (but not all) cases.
Several works on the fluxes of astrophysical neutrinos make simplifying assumptions about the energy distributions
of the neutrinos produced in weak decays. For example the calculation of Kashti and Waxman [29] approximates the
decay spectra of charged pions and muons into muons and neutrinos as:
Fπ+→νµ(x) = δ
Fπ±→µ±(x) = δ
Fµ+→νµ(x) = Fµ+→νe(x) = δ
These approximations, together with the assumption that muon energy loss before decay is negligible, imply that a ν
source dominated by pions has a flavor ratio (νµ + νµ)/(νe + νe) = 2. With the use of the correct expressions (18),
(19), (20) and (21) for the decay spectra one obtains the result that even for a pure pion source the flavor ratio is not
exactly two, and is in general a function of the ν energy determined by the shape of the neutrino spectrum.
V. POWER LAW SPECTRA
The situation where the neutrino energy spectrum is a power law of form φν ∝ E−αν is phenomenologically very
important, and is simple to discuss. A power law ν spectrum implies that the spectrum of the parent particles (pions
and kaons) is also a power law with the same slope (απ,K = αν). Such a spectrum arises when the interacting primary
particles have a power law spectrum and the target is either composed of normal matter at rest, or is a photon field
with an energy distribution that has again a power law form. In the first case the slope of the secondary particles
spectrum is equal to the one for the primary particles (απ,K ≃ αp); in the second case the slope of the secondaries
is απ,K ≃ αp − β + 1 where αp the slope of the primary particles and β the slope of the target photon distribution
(nγ ∝ ε−β).
In general the spectrum of neutrinos produced in the decay of particle type a can be obtained convoluting the
energy spectrum of the parent particles with the appropriate weak decay spectrum (see equation (9)). If the parent
particle spectrum has a power law form, because of the scaling form (13) of the decay spectra the resulting neutrino
energy distribution can be written as:
Qν(Eν) =
dEa Qa(Ea)
dna→ν
(Eν ;Ea)
= Ca E
dx xα−1 Fa→ν(x) = Ca Za→ν(α) E
ν . (25)
In other words the neutrino spectrum is a power law with the same slope of the parent particles, with a proportionality
factor that is commonly called the “Z–factor”, and is the momentum of order α− 1 of the decay spectrum:
Za→νj (α) =
dx xα−1 Fa→νj (x). (26)
Za→νj (1) is the average multiplicity of νj in the final state, and Za→νj (2) is the fraction of the parent particle energy
carried away by neutrinos of type j.
In the case of π± chain decay the three relevant Z–factors are:
Zπ+→νµ(α) =
(1− rπ)α−1
Zπ+→µ+→νµ(α) =
3− 2 rπ − (3 + α) rαπ + (2 + α) r1+απ
α2 (2 + α) (3 + α) (1− rπ)2
Zπ+→µ+→νe(α) =
24 (α (1− rπ)− rπ (1− rαπ ))
α2 (1 + α) (2 + α) (3 + α) (1− rπ)2
with rπ = (mµ/mπ)
2. The Z–factors for charged conjugate modes are identical because of CP invariance. The
Z–factors given above are calculated neglecting the energy loss of muons before decay, and assuming that the muon
helicity is exactly conserved. This assumption remains valid, to a very good approximation, also in the presence of a
magnetic field, because the bending of the momentum and the spin precession exactly cancel for a particle of electric
charge ±1 and magnetic moment of one Bohr magneton. Neglecting the effects of the muon polarization leads to
an overestimate (underestimate) of the Z–factor for the π+ → µ+ → νµ (π+ → µ+ → νe) channel. For α = 2 the
Z–factors for pion decay take the values:
{Zπ+→νµ , Zπ+→µ+→νµ , Zπ+→µ+→νe}α=2 =
1− rπ
3 + 4 rπ
2 + rπ
≃ {0.2135, 0.2646, 0.2573} ,
i.e. the three neutrinos carry approximately one quarter of the charged pion energy. This happens because in the
first decay the muon carries away a large fraction ((1 + rπ)/2 ≃ 0.787) of the pion energy.
Because of the different shapes of the energy distributions of the three neutrinos emitted in a pion decay, the flavor
ratio
Rµe =
(νµ + νµ)
(νe + νe)
Zπ+→νµ + Zπ+→µ+→νµ
Zπ+→µ+→νe
for a power law spectrum of parent pions is a function of its slope. The ratio is shown in fig. 3. For α = 1 one has
Rµe = 2, since in this case the Z factors correspond to the neutrino multiplicities. With increasing α the flavor ratio
decreases monotonically, this reflects the fact that the muon neutrinos produced in the direct pion decay are softer
than the neutrinos of the muon decay. For the value α = 2 one has Rµe ≃ 1.858. Figure 3 also shows the predictions
obtained neglecting the effects of muon polarization to illustrate the importance of their inclusion.
The second most important source of neutrinos is the decay of kaons. Charged kaons can produce neutrinos with
three different decay channels: the two body mode K+ → µ+νµ (with branching ratio 0.6343), and the three body
modes K±e3 (K
+ → π◦e+νe) and K±µ3 (K+ → π◦µ+νµ) that have branching ratios B
e3 ≃ 0.0487 and B
µ3 ≃ 0.0327
[28]. The KL can also produce neutrinos in the K
e3 decay mode (KL → π∓e±νe(νe)) with combined branching ratio
B0e3 ≃ 0.3881, and K◦µ3 mode (KL → π∓µ±νµ(νµ)) with combined branching ratio B0µ3 ≃ 0.2719. The Rµe ratios for
neutrinos produced by the chain decay of charged or neutral kaons is shown in fig. 4. In the case of charged kaons
the 2–body decay mode is the dominant one, the Z–factors for this mode have the same form as in pion decay, with
the replacement rπ → rK , however for charged kaons the flavor ratio grows when α increases, reflecting the fact that
the muon neutrinos produced in the direct decay carry nearly half ((1 − rK)/2 ≃ 0.48) of the parent particle energy
while the neutrinos produced in the second stage (from muon decay) take on average only 18% (νµ) and 16% (νe) of
the kaon energy. The inclusion of the three body decay modes (K±e3 and K
µ3) that have smaller branching fractions
reduces the flavor ratio, but it is a small correction.
In the case of the decay of KL, the three body decays are the only source of neutrinos. The K
e3 mode has a
larger branching fraction than the K◦µ3 channel, because of the larger phase space available. When α = 1, when the
flavor ratio reflects the neutrino multiplicities in the final state, the flavor ratio is Rµe = 2B
µ3/(B
e3 + B
µ3) ≃ 0.82.
Increasing α the ratio decreases, because the νe from the K
e3 decay have the hardest spectrum, and their contribution
is enhanced.
Note that for power laws with slope ≃ 2 the net effect of including kaon decay as a ν source is a only a small
(positive) correction, because of a cancellation between the contributions of charged kaons (that increase the relative
importance of muon neutrinos) and neutral kaons (that do the opposite).
VI. ENERGY LOSS MECHANISMS
In the most general case, the unstable particles that create neutrinos can lose a significant amount of energy during
the time that elapses from the moment of their creation to the moment of their decay. This effect can have very
important consequences for the spectrum and flavor composition of the neutrinos.
As an illustration, muon energy losses are very important for the prediction of the atmospheric neutrino fluxes. The
largest effect is due to the fact that the muons that reach the ground lose rapidly energy because of ionization and
electromagnetic radiation processes, and decay practically at rest (or are captured by heavy nuclei [30]) producing
only very low energy neutrinos. Pion energy losses are also very important for the calculation of the atmospheric ν
fluxes. In this case the dominant mechanism for energy loss are hadronic interactions in air. The pion decay length
grows linearly with Eπ, and (for Eπ >∼ 100 GeV) it becomes (for vertical particles) comparable to the interaction
length. Most pions of higher energy interact before decaying. These hadronic interactions result in a multiplication
of the pions, but the secondary particles have much lower energy and the net effect is a strong suppression of the
neutrino flux at high energy.
There are several mechanisms for energy loss that could be present in the astrophysical environment where neutrinos
are produced. In several circumstances synchrotron radiation could be the dominant source of energy loss [29, 31]. In
the presence of a magnetic field of value B, the energy loss of a particle of electric charge e, mass m and energy E,
after averaging over all possible orientation of the magnetic field direction is given by the well known expression:
e4 B2
E2 (32)
The energy loss is significant only for E >∼ Esyn the critical energy where the the synchrotron loss time tsyn =
−[(dE/dt)/E]−1 is equal to the decay time tdec = τ E/m:
Esyn =
5.8× 1018 eV
BGauss
The critical energy for synchrotron losses scales ∝ m5/2 τ−1/2. It is smallest for muons, it becomes 18.4 times larger
for charged pions, and 628 times larger for charged kaons. There is therefore an energy range where the losses are
only significant for muons, and a second energy range where the synchroton losses are significant for both pions and
muons but not for kaons.
In the presence of ordinary matter in gas form the particles can also lose energy because of ionization and radiation
processes (with radiation dominating at large energies). These energy losses take the form −dE/dt ≃ ρ (a + bE)
where ρ is the medium density and a and b are slowly varying coefficients that depend on the gas composition. For a
medium mostly composed of protons, as it is likely in astrophysical environments, the critical energy for the energy
losses with ordinary matter in gaseous form (where the loss time equals the decay time) is for muons:
matter ≃ 6.5× 1018 (ρ−10)−1 eV (34)
where ρ−10 is the matter density in units of 10
−10 g cm−3.
In the presence of a gas of ordinary matter the dominant source of energy loss for charged pions and kaons is due
to hadronic interactions. The critical energy at which the hadronic interaction time tint = 〈A〉/(NA ρ σint) equals the
decay time τ E/m can be estimated for pions as:
Eπmatter ≃ 7.4× 1016 (ρ−10)−1 eV . (35)
This critical energy scales ∝ m/τ and for charged kaons (assuming the same hadronic cross section) is approximately
7.4 times larger.
The effects of ordinary matter on the secondary particles can be safely neglected for all energies (including those
above Ematter) and all particle types if the density is below the critical value:
ρcrit ≃ 2.0× 10−15 BGauss g cm−3 . (36)
For ρ < ρcrit the energy losses on matter are either negligibly small, or dominated by the synchrotron losses.
Figures 5 and 6 illustrate the above discussion showing the characteristic times tdec (decay), tsyn (synchrotron
losses) and tmatter for muons and pions.
Gamma Ray Bursts have been proposed by Waxman and Bahcall [25] as a neutrino source having a potentially
very strong magnetic field. In the WB model, the strength of the magnetic field is estimated by energy equipartition,
assuming that it corresponds to an energy density of the same order of magnitude as the energy density carried by
the fireball photons. It is natural to ask the question if this radiation field can be an important source of energy
loss for secondary particles produced in it. The calculation of the time tIC for Inverse Compton losses of charged
particles traveling in the fireball radiation field requires to take into account the energy spectrum of the target photons
(assumed to be isotropic in the GRB jet frame). In fact the Compton scatterings with sufficiently high energy photons
ε >∼ m2/E happen in the Klein–Nishina regime and are inefficient as a source of energy loss. For the predicted GRB
photon spectrum [25], (given also in equation (63)) that falls at high energy∝ ε−2, the energy loss for Inverse Compton
grows at high energy as −(dE/dt)IC ∝ E ργ/m2, (where ργ is the target photon energy density), and the loss time
tIC becomes constant. The results of a detailed integration are also shown in fig. 5 and 6, where we have assumed
that the densities ρB = B
2/(8π) and ργ are equal. In most circumstances (unless equipartition is very badly violated)
Inverse Compton losses can be neglected.
Charged pions (and kaons) traveling in a radiation field can also lose energy for photo–hadronic interactions. For
completeness we have calculated the interaction time of pions in the GRB radiation field. The result is also shown in
fig. 6. For this calculation the πγ hadronic cross section has been estimated as the sum of contributions for production
of the relevant resonances [28] : ρ(770), a1(1260), b1(1235) and a2(1320) and a non resonant background. At high
energy (s >∼ 30 GeV2) the cross section is described by a formula obtained using Pomeron universality and Regge poles
factorization from the fits to the γ-proton and π-proton cross sections reported in [28]. The lower energy non–resonant
background has been normalized to obtain a smooth energy dependence.
At very high energy pion–photon interactions are more important that Inverse Compton losses, however if the
energy density in photons and magnetic field are comparable, the synchrotron losses dominate.
A. Decay Energy Distribution
To describe the effect of energy losses, it is useful to consider the function dpdec/dE(E,Ei) that gives the probability
density for an unstable particle of mass m and lifetime τ , created with initial energy Ei to decay with energy E.
Assuming that the energy loss is a continuous process, and can be described by the equation:
(E) = f(E) , (37)
(with f(E) is the average loss per unit time), the distribution dpdec/dE(E,Ei) can be calculated as:
dpdec
(E;Ei) =
f(E) E
f(E′)E′
. (38)
For an energy loss of form f(E) = aEn (with n > 0) the decay probability becomes [29]:
dpdec
(E;Ei) =
with:
. (40)
The quantity ǫ has the physical meaning of the energy for which the loss-time: tloss = E/f(E) = 1/(aE
n−1) is equal
to the decay time: tdecay = τ E/m.
For Ei ≪ ǫ, the energy loss is negligible and the decay energy distribution becomes a simple delta function:
dpdec
(E;Ei)
≃ δ[E − Ei], (41)
while for Ei ≫ ǫ the decay energy distribution takes a universal form
dpdec
(E;Ei)
. (42)
An illustration of the decay energy distribution for the case f(E) = a E2, relevant for synchrotron emission is shown
in fig. 7. The important qualitative feature in fig. 7 is that all particles with sufficiently high initial energy (Ei >∼ 2 ǫ)
have nearly identical final energy distributions, with a well defined maximum at E ∼ 0.6 ǫ. In many cases this can
result in “pile–up” effects in the final state particle spectra.
VII. HADRONIC INTERACTIONS
The target for the primary particles interactions can be either normal matter in gaseous form, or a radiation field.
In the following we will consider both cases separately.
A. Gas Target
For a target material made of normal matter at rest the interaction rate of a primary particle of energy Ep is:
Kpp(Ep) = ngas σpp(s). (43)
where ngas is the number density of the target material. Equation (43) is valid for a situation where the primary
particles are protons and the target material is also dominated by protons. This is likely to be a good approximation
in most circumstances. It is straightforward to consider the more general case.
Since the target is at rest, the interaction rate is proportional to the cross section at a well defined c.m. energy
s = m2p + 2mp Ep. Since hadronic cross sections grow only logarithmically with c.m. energy (or as power law ∝ sǫ
with a small exponent ǫ), the interaction rate of the primary particles changes only very slowly with energy.
It is well known that the multiplicity and energy distributions of the particles produced in hadronic interactions
cannot be calculated from first principles, however it is believed that particle production, to a a reasonably good
approximation, satisfies Feynman scaling, defined by the condition:
(p∗‖,
s) = Fj(xF ) (44)
where p∗
is the longitudinal momentum in the c.m. frame, dσj is the inclusive differential cross section for the
production of particle type j, and xF = 2 p
s is the Feynman variable.
For large c.m. energy (s ≫ m2p) the target rest frame energy of secondary particles in the forward hemisphere
(xF > 0) is well approximated by the expression E ≃ Ep xF (where E (Ep) is the energy of the final state (projectile)
particle in this frame). It follows that the validity of Feynman–scaling in the fragmentation region implies also the
approximate validity of scaling of the inclusive cross sections in the target rest frame:
dnpp→a
(E;Ep) ≃
Fpp→a
(for E ≫ m) (45)
The approximate validity of the scaling law (45) together with the slow variation of the hadronic cross sections
with c.m. energy have the important consequence (based on essentially the same argument previously discussed in
connection with decay (25)) that a power law spectrum of primary particles generates spectra of secondaries also
having a power law form. For example for final state particles of type π:
Qπ(Eπ) = Cp ngas σpp E
dx xα−1 Fpp→π(x) = Cp ngas σpp Zpp→π(α) E
π . (46)
In general, one expects some deviations from a perfect power law. First of all, this can reflect a shape of the
primary spectrum different from a power law. In addition the energy dependence of the cross sections, and the
existence of violations of Feynman scaling (as measured in hadron colliders) introduce some distortions. Another
situation that can result in large deviations from a power law emission arises when particles of different rigidities
have different confinement volumes having different target densities (while our derivation of (46) implicitely assumed
a homogeneous source volume).
B. Photoproduction
In several of the proposed neutrino sources, the target of the primary particles is a radiation field. In this case
the interaction rate and the energy distribution of the particles produced in an interaction depend not only on the
density, but also on the energy and angular distribution of the target photons. The interaction probability per unit
time of a proton of energy Ep traveling in the radiation field described by nγ(ε,Ωγ) can be calculated as:
Kpγ (Ep) =
λpγ(Ep)
d cos θpγ
(1− cos θpγ) nγ(ε, cos θpγ) σpγ(ǫr) (47)
where θpγ is the angle between the photon and the proton momenta in the interaction, σpγ is the photoproduction
cross section, and ǫr is the photon energy in the proton rest frame:
(1− cos θpγ) (48)
The quantity ǫr is in one to one correspondence with the c.m. energy of the reaction. It is convenient to change the
integration variable from cos θpγ to ǫr. Restricting ourselves to a situation (and a frame) where the photon distribution
is isotropic one can then rewrite (47) as:
Kpγ(Ep) =
dǫr ǫr σpγ(ǫr)
(mpǫr)/(2Ep)
nγ(ε)
where ǫth is the threshold photon energy for pion production in the proton rest frame:
ǫth = mπ +
Equation (49) can be recast in the form
Kpγ(Ep) =
dǫr fpγ (ǫr; Ep) . (51)
This expression shows explicitely the fact that the pγ interactions of a proton of energy Ep do not correspond to a
single value of the c.m. energy but have a distribution, that in general is a function of Ep, and is determined by the
energy (and angular) distribution of the target photons. The probability distribution for ǫr is:
p(ǫr; Ep) =
f(ǫr, Ep)
Kpγ(Ep)
. (52)
A phenomenologically important case for the target radiation field is the form: nγ(ε) = Cγ ε
−β, that is an (isotropic)
power law spectrum of slope β. For this form the last integration in (49) can be performed analytically with the result:
Kpγ(Ep) = Cγ
β + 1
)β−1 ∫ ∞
dǫr ǫ
r σpγ(ǫr) = K0(β) E
p (53)
One can see that the interaction rate has the energy dependence Eβ−1p . The growth with energy (for β > 1) of the
interaction rate can be understood observing that a proton of energy Ep can interact inelastically only with photons
above a minimum energy mp ǫth/(2Ep). This minimum target energy decreases proportionally to E
p , and therefore
protons of higher energy can interact with a softer and more abundant photon population. A result essentially
equivalent to equation (53) was originally shown by Waxman and Bahcall in [25]. An important feature of equation
(53) is that the probability distribution for ǫr takes a form that is independent from Ep:
p(ǫr) = ǫ
r σpγ(ǫr)
dǫr ǫ
r σpγ(ǫr)
This fact has interesting consequences discussed below. It should be noted however that equations (53) and (54) are
calculated assuming a power law that extends to very large energy without any cutoff. This is not only unrealistic,
but in some important cases also untenable. In fact the integral over ǫr in (53) diverges at its upper limit (since
σpγ is slowly increasing with energy) for β ≤ 1, and the expression for Kpγ becomes meaningless. The divergence
corresponds to the divergence of the photon number density in the absence of an upper limit cutoff (note that in fact
the energy density of the photon population diverges at the upper end already for β ≤ 2). The introduction of a high
energy cutoff for the energy distribution of the target photons is therefore mandatory. This will be discussed in the
next section.
The properties of particle production in pγ interactions is clearly intimately related to the distribution of c.m. energy
(or equivalently ǫr) of the interactions. It is necessary to consider the energy distributions of secondary particles in
the “source frame” (where the primary proton has energy Ep). In this frame the energy of a secondary particle can
obviously be expressed as function of quantities in the c.m. frame with an appropriate Lorentz boost:
E = γ (E∗ + v p∗z) (55)
where v and γ are the velocity and gamma–factor of the c.m. of the reaction in the source frame, E∗ is the c.m.
energy of the secondary particle and p∗z the momentum component parallel to β. The Lorentz γ that connects the
source and c.m. frames is:
Ep + ε√
≃ Ep√
. (56)
In the second approximated equality we have neglected the photon energy ε with respect to Ep, this is expected to
be an excellent approximation. Similarly one can safely make the approximation v ≃ 1. With these approximations
we can rewrite the Lorentz boost from the c.m. to the source frame as:
E = Ep
E∗ + p∗z√
= Ep ξ . (57)
This equations indicates that to a good approximation all secondary particles of source frame energy E are created in
the c.m. frame of the interaction with (to a very good approximation) the same value of the quantity ξ = (E∗+p∗z)/
The energy spectrum of (for example) pions created in the interaction of a proton of energy Ep can then be written
dnpγ→π
(Eπ ; Ep) ≃
dǫr p(ǫr;Ep)
dnpγ→π
(ξ; ǫr)
ξ=Eπ/Ep
where dnpγ→π/dξ(ξ, ǫr) is the ξ distributions of the secondary particles of type π in pγ interactions with c.m. energy
that correspond to ǫr; this distribution is convoluted for a fixed value ξ = Eπ/Ep over all possible value of the ǫr with
the appropriate distribution. If the energy distribution of the target photon field has a power law form, the function
p(ǫr; Ep) is independent from Ep, and equation (58) becomes the expression of a scaling law of form:
dnpγ→π
(Eπ ; Ep) ≃
Fpγ→π
. (59)
The scaling function Fpγ→π is not truly universal, but depends on the slope of the target photon spectrum. Integrating
over all primary particle energies one can obtain the production rate of pions as:
Qπ(Eπ) =
dEp Np(Ep) Kpγ(Ep)
dnpγ→π
(Eπ;Ep)
K0(β) E
Fpγ→π
= Cp K0(β)E
−(α−β+1)
dξ ξα−β Fpγ→π(ξ; β)
= Cp K0(β) Zpγ→π[α− β + 1, β] E−(α−β+1)π . (60)
This equation shows that under the assumptions made, that is: (i) a power law spectrum of protons, and (ii) an
isotropic power law spectrum of target photons, the energy distribution of produced secondaries is again a power law
with slope α− β + 1. Note that this result is based on purely kinematical considerations, while the result (46) about
the interactions on an ordinary matter target was based on a dynamical assumption about the approximate validity
of Feynman scaling in hadronic interactions, and on the weak energy dependence of hadronic cross sections.
A result roughly equivalent to (60) was obtained by Waxman and Bahcall in [25], who concluded that in the case
β = 1 the neutrino emission has a power law spectrum with the same slope of the parent protons. However, the
validity of equation (60), as the validity of equation (53), relies on the untenable assumption that the power law
spectrum of target photons has no high energy cutoff. The important effects of the existence of an high energy cutoff
for equations (53) and (60) will be discussed in section VIII.
C. Nuclear photodisintegration
The beta decay of neutrons produces νe. In many circumstances the contribution of this anti–neutrino source is of
negligible importance. This is because the average fraction of the neutron energy En carried away by the νe (in a frame
where the neutron is ultrarelativistic) is 〈Eνe〉/En ≃ 5.1× 10−4, with end–point Emaxνe /En ≃ 2(mn−mp −me)/mn ≃
1.66 × 10−3. In most cases the flux of the softer n–decay neutrinos is neglible with respect to the contribution of
neutrinos from π/K of decay that carry a much larger fraction (of order ∼ 0.25) of their parent energy. There are
two circumstances where the n contribution can become important. The first one is when the neutrino spectrum has
a low energy cutoff. This happens naturally when the target of the primary particles is a radiation field and there is
an interaction energy threshold. In this case the νe produced in neutron decay can become the dominant component
of the neutrino flux at low energy, because most of the neutrinos from pion and kaon decay have higher energy. An
example of this situation will be shown in the following section. A second, more interesting case is when the neutrons
are produced in the photodisintegration of high energy nuclei [32]. In this case it is in principle possible to have the
emission of a pure νe flux. In fact the threshold for photodisintegration of a nucleus of mass number A expressed in
terms of energy per nucleon (E0 = Etot/A) is of order:
mpǫbind
where εγ is the energy of the target photons and ǫbind ≃ 8 MeV is the binding energy of a nucleon in the nucleus.
The threshold for pion photoproduction is:
mp mπ
. (62)
Since the binding energy ǫbind is approximately fifteen times smaller than a pion mass, it is in principle possible to have
circumstances where the primary particles are below the threshold for pion production, but above the threshold for
photodisintegration. This would clearly result in a pure νe flux. It should be noted that the photodisintegration and
pion photoproduction thresholds differ by only one order of magnitude, and therefore a pure νe flux can only extend
for a small interval of energy. In general one expects important contributions from neutrinos from meson decays.
In fact, even a very small number of pion production interactions created by the high energy tails of the photon
and/or primary particles spectra can result in an important “contamination” of νe and muon neutrinos. Most of these
neutrinos have an energy two orders of magnitude larger that the νe produced in n decay and have a correspondingly
larger cross section.
VIII. NEUTRINO PRODUCTION IN GAMMA RAY BURSTS
As an illustration of the spectra and flavor composition of neutrinos from astrophysical sources, in this section we
will consider the model of neutrino production in Gamma Ray Bursts (GRB) developed by Waxman and Bahcall
(WB) [25]. Our goal is not to analyze critically the assumptions that underline the model, but to recalculate with a
more detailed modeling of particle production the neutrino fluxes starting from the same general assumptions.
In the (WB) model [25] the GRB prompt emission is generated by the synchrotron radiation of high energy electrons
accelerated by internal shocks in a relativistic expanding wind (for more discussion see [33, 34]; alternative explanations
of the GRB mechanism exist in the literature, see for example [35]).
The internal shocks also accelerate protons, that can interact with the radiation field of the wind and produce
secondary particles that generate neutrinos. The accelerated protons have a power law spectrum of slope α ≃ 2, while
the target radiation field is isotropic in the wind frame, and approximated as a broken power law of form:
nγ(ε) =
−1 for ε ≤ εb,
Cγ εb ε
−2 for ε > εb
where εb is a break energy. The photon energy distribution in the wind frame is related to the obervable spectrum of
the GRB prompt emission by a Lorentz transformation with parameter Γ ≃ 300. Since the observed break energies
are distributed around an average value 〈εobsb 〉 ∼ 300 KeV [36], the typical value of εb in the wind frame is of order
1 KeV.
The energy spectra of individual GRB are indeed well fitted by two power laws smoothly joined at a break energy,
however the fitted values of the exponents of the spectrum have rather broad distributions centered at β ≃ 1 below the
break energy, and β ≃ 2.2 above the break energy [36]. In our calculation we have therefore considered a generalization
of (63) that leaves the two exponents as free parameters. We have also introduced a high energy cutoff εmax, because
in its absence the energy density of the radiation field would diverge for a high energy slope ≤ 2. Our description of
the energy spectrum of the target radiation field then becomes:
nγ(ε) =
(Cγ/εb) (ε/εb)
−β1 for ε ≤ εb,
(Cγ/εb) (ε/εb)
−β2 for εb < ε < εmax,
0 for ε ≥ εmax.
The calculation of the interaction rate of a proton of energy Ep in such a radiation field is straightforward. The target
photon field (64) can be seen as the sum of two power law spectra with sharp low and high–energy cutoffs εmin and
εmax.
We have already obtained the expression for the proton interaction rate for a power law that extends to all energies.
The expression (53) must however be modified in the presence of low and high energy cutoffs. In the presence of
a high energy cutoff εmax the interaction rate vanishes below a threshold energy Eth. Above the threshold, in the
interval:
Eth =
mp ǫth
2 εmax
≤ Ep ≤
mp ǫth
2 εmin
. (65)
the interaction rate is
Kpγ(Ep) =
2β Cγ
β + 1
)β−1 ∫ 2Epεmax/mp
dǫr σpγ(ǫr)
ǫ−βr − ǫr
2Ep εmax
coinciding with (53) in the limit εmax → ∞. The effect of a lower energy cutoff εmin is to stop the Eβ−1p growth of
the interaction rate. For Ep > mpǫth/(2εmin) equation (66) must be replaced by:
Kpγ(Ep) =
2E2p (1 + β)
−(β+1)
min − ε
−(β+1)
max )
∫ 2Ep εmin/mp
dǫr ǫr σpγ(ǫr)
2β Cγ
β + 1
)β−1 ∫ 2Epεmax/mp
2Epεmin/mp
dǫr σpγ(ǫr)
ǫ−βr − ǫr
2Ep εmax
For a qualitative understanding, it can be useful to consider the pγ cross section as approximately constant above
threshold. The integrations over ǫr in (66) and (67) are then trivial. For β 6= 1 one finds:
[Kpγ(Ep)]
β 6=1 ≃
0 for x < 1
Cγ σpγ
2 ε1−βmax
xβ−1 − 1
1 + β + (1− β)x−2
for 1 ≤ x ≤ r
Cγ σpγ ε
1−r1−β
x2 (1+β)
r2 − r1−β
for x > r
where x = Ep/Eth is the proton energy expressed in units of the threshold energy, and r = εmax/εmin. For β = 1, one
finds:
[Kpγ(Ep)]
β=1 ≃
0 for x < 1
Cγ σpγ
log[x]− 1
for 1 ≤ x ≤ r
Cγ σpγ
log[r] − 1
r2 − 1
for x > r
These expressions show that the interaction rate for a power law spectrum of target photons in a restricted range
do maintain the approximate behaviour Kpγ ∝ Eβ−1p but only in a limited energy region (in the limit β → 1 the
interaction rate grows logarithmically: Kpγ ∝ log[Ep εmax]). The interaction probability vanishes below threshold
and goes asymptotically to a constant for very large Ep. It can also be useful to consider the high energy limit
(Ep/Eth → ∞) of expressions (68) and (69):
[Kpγ(Ep)]
β 6=1
= σpγ
β − 1
min − ε
, (70)
[Kpγ(Ep)]
= σpγ Cγ log
. (71)
These expressions have the form Kpγ = σpγ Nγ with Nγ the integrated number density of the target photons, that
can be immediately recognized as the correct high energy limit.
The interaction probability for the radiation field (64) can be obtained combining two expressions corresponding
to the parts of the photon spectrum below and above the break energy εb that plays in the two cases the role of the
maximum or the minimum target photon energy. An illustration of the energy dependence of the proton interaction
rate is shown in figure 8. The curves in this and the following figures are made independent from the value of εb
measuring all energies in units of:
mp ǫth
mp mπ
≃ 6.9× 1013
eV. (72)
The energy E∗ has the physical meaning of the threshold proton energy for inelastic interactions with photons having
the break energy εb. In figure 8 we have chosen for the target photons energy spectrum the original form (63) with
exponents β1 = 1, β2 = 2 and no high energy cutoff. For low Ep the interaction rate grows approximately linearly with
energy according to the scaling law (53) because low energy protons interact inelastically only with the high energy
part of the radiation field (with slope β2 = 2). For larger energy (Ep >∼ E∗) the growth becomes logarithmic following
the qualitative behaviour of equation (69). It should be noted that the assumption made in [25] of considering the
interaction rate as approximately constant for Ep > E
∗ is not a good approximation.
In fig. 9 we show the distribution f(ǫr; Ep) of the c.m. energy of the interaction for different values of the proton
energy (Ep/E
∗ = 1, 10, 100). The distribution is proportional to the pγ cross section, and the shape of the curves in the
figure reflects the energy dependence of σpγ that has prominent resonances. The most important one (at ǫr ∼ 0.3 GeV)
corresponds to the production of the ∆ resonance. The important feature in the figure is that with increasing energy,
higher and higher c.m. energies become possible. This has significant phenomenological consequences. For example,
for ǫr sufficiently small, only production of a single pion, in the two channels pπ
◦ and nπ+, is kinematically allowed,
and therefore there is no π− production, and consequently no νe production. For larger c.m. energy multiple pion
production becomes possible, νe can be produced, and the ratio νe/νe decreases. Above the KΛ energy threshold,
also the production of kaons becomes possible introducing an additional neutrino source.
In the following we will show some examples of the neutrino fluxes that are obtained varying the parameters of the
model. Our calculations have been performed by Montecarlo methods using a detailed model for the pγ cross section
that includes the production of all relevant resonances and a non–resonant component [38]. In order to perform a
calculation one needs to specify: (i) the primary proton spectrum, (ii) the target photon spectrum, and finally the
value of the magnetic field B (all other sources of energy losses for secondary particles are considered as negligible).
The target photon energy distribution is taken with the form (64) that depends on 4 parameters: {Cγ , εb, εmax, β1,
β2}. Different values of εb correspond to different values for E∗ ∝ ε−1b . The proton spectrum is taken with the form:
Np(Ep) = Cp E
p exp
that is a power law with a smooth high energy cutoff: defined by the three parameters: {Cp, α, Emax}. In all of the
following we have kept fixed two of the parameters, the proton maximum energy: Emax/E
∗ = 104, and the photon
high energy cutoff εmax/εb = 300. The parameters Cp and Cγ determine the absolute value of the neutrino emission,
but not its shape, and will be left unspecified in the following.
The absolute value of the neutrino flux from an individual source can be obtained specifying the values of Cp (that
fixes the amount of energy in relativistic protons in the wind frame), Cγ that specifies the density of the photon
target field, the exact Lorentz Boost and obviously also the source redshift (distance). The integration over the
ensemble of all sources requires additional assumptions about the distributions of the relevant parameters used for
their description and of their cosmological evolution. In this work we will not discuss this integration.
The value B of the magnetic field is in a one to one correspondence with the muon synchrotron energy, defined in
equation (33), that can also be expressed as:
Eµsyn
= 8.4× 104
Gauss
As an intermediate step toward the calculation of the neutrino fluxes we show in fig. 10 the yields of different
mesons. The meson yields are calculated for a proton spectrum with exponent α = 2, and a radiation field with slopes
(below and above the break energy) β1 = 1 and β2 = 2. The solid lines in fig. 10 show the energy distributions of the
different mesons at the moment of their creation. In the absence of significant energy losses the curves also describe
the energy distributions at decay. One can see that all possible mesons are produced, but with different abundances.
The most abundant particles are charged pions, reflecting the fact that a large fraction of the pγ interactions happens
close to threshold where only single (neutral or positive) pion production is possible. The production of π− becomes
possible only above the threshold for two pion production, and is therefore suppressed. The production of kaons
is significantly smaller, because it is suppressed both dynamically (since it requires the creation of ss pairs) and
kinematically (because of the larger strange hadron masses). The threshold for the production of mesons containing
a strange antiquark s (K+ and K◦) is lower, since it corresponds to final states containing a strange baryon (such as
ΛK+). The production of mesons containing a strange quark s (K− and K
) has a higher threshold since the final
state must contain a minimum of two kaons (as for example: pK+K−). Accordingly one finds that the productions
of different kaon types is ordered as follows: K+ > KL > K
−. The shapes of energy spectra of the mesons are never
well approximated by a simple power law, but are always “curving” in a log–log representation. The shapes of these
spectra reflect the energy dependence of the proton interaction rate, and the opening up of the different kinematical
channels. At the highest energy the rapid drop in the meson yields is connected to the cutoff in the primary proton
energy at Emax/E
∗ ∼ 104. Note that the ratio π−/π+ and K/π are not constant but increase with energy reflecting
the increasing importance of π− and kaon production with growing c.m. energy.
The dashed lines in fig. 10 describe the decay energy distributions of the produced mesons assuming the presence
of a magnetic field that corresponds to ǫµ = 3 (or B = 2.8 × 103 εKeVb Gauss). Pions created with energy above
their critical energy for synchrotron losses (in units of E∗: ǫπ ≃ 18.4ǫµ ≃ 55) lose most of their energy before decay,
and therefore the number of pions decaying above this critical energy is strongly suppressed (∝ (Eπ/ǫπ)−2). One
can also notice that synchrotron losses result in an enhancement of the number of pions decaying with energy just
below ǫπ. This is a pile–up effect due to the fact that all high energy pions (with Eπ ≫ ǫπ) decay with similar energy
distributions. The same effects are present also for charged kaons at higher energy, because the synchrotron energy
for charged kaons (in unit of E∗) is ǫK ≃ 628ǫµ ≃ 34ǫπ ≃ 1900.
Secondary neutrons are also produced in pγ interactions. Assuming that they can freely exit from the source, the
νe spectra from their decay have been calculated and included in the following figures.
Some examples of the resulting neutrino fluxes obtained summing over all possible parent particles and all ν types
are shown in fig. 11 and fig. 12. The lines in fig. 11 are calculated assuming that energy losses for secondary particles
are negligible. The different curves correspond to different assumptions about the shape of the energy spectra for the
primary protons and target photons. The thick solid curve is calculated with the choice of slopes (for the proton flux
and for the radiation field below and above the break energy): {α, β1, β2} = {2, 1, 2}. The other three curves show the
effect of changing the slopes one by one. The largest effect is related to the slope of the primary protons, increasing
α from 2 to 2.4 results in an important softening of the neutrino spectrum. The modification of the shape of the
energy distribution of the target photons also distorts the neutrino spectrum. Changing the slope below (above) the
break energy, modifies the high (low) energy part of the neutrino spectrum. This is easily understood, since high
(low) energy neutrinos are produced by higher (lower) energy protons that mostly interact with lower (higher) energy
photons. Note that (reflecting the spectra of the parent pions and kaons) the neutrino energy distribution changes
gradually its slope and cannot be well fitted by a power law.
Figure 12 illustrates the effect of including synchrotron losses on the neutrino fluxes. The different lines in fig. 12
show the neutrino flux (summed over all ν types) calculated with the slopes {α, β1, β2} = {2, 1, 2}, and three different
assumptions for the magnetic field that correspond to muon synchrotron energy ǫµ = ∞, 30 and 3 (the first case
corresponds to negligible losses). Increasing the magnetic field (reducing ǫµ) suppresses the neutrino flux at high
energy. In general, in the presence of a strong magnetic field in the source, one can identify four interesting neutrino
energy ranges, that are related to the ordering of the synchrotron critical energies for different particles: ǫµ < ǫπ < ǫK .
At sufficiently low energies the synchrotron losses are completely negligible; at higher energy the muons energy losses
have to be taken into account; at still higher energy also the losses of charged pions must be considered; at the
highest energy the emission of synchrotron radiation is important also for charged kaons. In general, in the presence
of important synchrotron emission, one can have some pile–up effects, due to the fact that all high energy particles
decay just below their critical synchrotron energy.
The inclusive neutrino spectra shown in the previous figures contain the contribution of the νe from n decay. The
structure of this contribution is illustrated in fig. 13. The anti–neutrinos from neutron decay have significantly lower
energies that the neutrinos from meson decay. They are the main component of the neutrino flux at the lowest
energies, where however they are difficult to observe because the flux is suppressed and the cross section is small.
The flavor ratios Reµ = (νe+νe)/(νµ+νµ) that correspond to the neutrino spectra of the previous figures are shown
in figures 14 and 15. The flavor ratio is energy and model dependent. The Reµ ratios shown in figure 14 correspond
the energy spectra of fig. 11 and are calculated assuming that all energy losses are negligible. At the lowest energy
the ratio Reµ increases rapidly due to the contribution of νe from n decay. At higher energy the ratio Reµ stays close
to the value of 1/2, however the expanded scale allows to see that the “naive” result is not exact. The value of Reµ,
for the reasons illustrated in section V, are correlated with the shape of the energy spectrum. The gradual softening
of the neutrino energy distribution with growing Eν is reflected in a slow increase of Reµ. Comparing the different
curves, one can also see that the curve calculated for α = 2.4 that gives the softest neutrino spectrum correspond (at
sufficiently high energy) to the highest Reµ.
The effects of the existence of significant synchrotron losses are very important for the flavor ratio. The main effect
is the effective loss of the neutrinos from high energy muon decay. This results in a “pile–up” effects that produced
an increase in the ratio for neutrino energies below the synchrotron muon energy ǫµ. Above this energy the ratio
falls steeply reflecting the effective absence of the main source (muon decay) of electron neutrinos. The flavor ratio
however does not vanish because kaons have decay modes into electron neutrinos, and reaches a level of a few percent.
At very large energy the role of neutral kaons (that do not suffer synchrotron losses) become enhanced and the flavor
ratio grows because of the importance of the K◦e3 decay mode.
Figure 16 and figure 17 show the neutrino/anti–neutrino ratios νe/νe and νµ/νµ calculated for the same three
situations present in fig. 12 and 15 (that is slopes {α, β1, β2} = {2, 1, 2} and three values of the magnetic field that
corresponds to ǫµ = ∞, 30 and 3).
The ν/ν ratios are difficult to measure because the detectors have no capability to measure the charge of the final
state charged lepton in charged current neutrino interactions. The most attractive idea [37] to estimate the νe/νe ratio
is the comparison of the event rate at and near the “Glashow resonance” [39] at Eν ∼ M2W /(2me). The resonance is
present only for νe in reactions such as νe + e
− → hadrons that can proceed via the formation of a W− boson in the
s channel, while non–resonant events proceed via the normal charged current interaction on nucleons.
The study of νe/νe ratio is particularly interesting because it has been proposed [37] as a method to distinguish pp
and pγ interactions as the source of the neutrinos. The idea is that the νe/νe ratio at the source is close to unity for
a pp source, when the interactions produce approximately equal numbers of positive and negative pions (that decay
into νe and νe), and is very large (or actually diverges) for pγ interactions, when most interactions happen close to
threshold, and the cross section is dominated by single pion production (via the ∆ resonance), and therefore only π+
production is present. This argument is qualitatively right, but quantitatively incorrect, in fact as discussed above, in
general increasing Ep a broader range of c.m. energies becomes possible, and this results in a growing contribution of
negative pions to neutrino production. In certain circumstances also the contribution of kaons can become important.
This is illustrated in fig. 16 that shows the νe/νe ratio. At the lowest energies the flux is dominated by the νe from n
decay, and the νe/νe ratio is therefore small. When the contribution of neutrinos from pion decay becomes dominant
the ratio νe/νe is significantly larger than one, because positive pions are more abundantly produced than negative
ones. In fact below the threshold for two pion production the ratio π+/π− diverges. With growing energy the ratio
π+/π− decreases because of the growing importance of multiple pion production. This is reflected in a decrease of the
νe/νe ratio that at large energies becomes ∼ 2. If a large magnetic field is present, at large energy the synchrotron
energy losses for muons become important and neutrinos from muon decay are suppressed. The decay of neutral
kaons (KL) becomes the dominant source of νe and νe, and the ratio νe/νe becomes unity.
The ratio νµ/νµ (shown in fig. 17) is approximately unity at low energy, reflecting the fact that each charged pion,
after chain decay, contributes a νµ and a νµ. The existence of important synchrotron losses can modify this quite
robust prediction. At high energy, when muons energy losses become significant, only the neutrinos from direct pion
decay are important, and the ratio νµ/νµ is approximately equal to the π
+/π−. At larger energy, charged kaons
become the dominant νµ and νµ source and the νµ/νµ ratio increases reflecting the higher K
+/K− ratio.
In fig. 18 we show the (νe/νµ) ratio for an Earth observer. To propagate neutrino we have used standard oscillations
with mixing parameters θ23 = 45
◦, θ12 = 34
◦ and θ13 = 0. In the figure we have also shown the same observable ratio
for a source with the “naive” composition [νe, νµ, ντ ] = [1, 2, 0]. For the “best fit” values of the mixing parameters the
observable (νe/νµ) has the well known value of unity. Changing the value of the θ23 angle by ±5◦ the ratio changes
by −13% (+7%). Similarly for θ13 = 5◦ the ratio take values in the interval (0.95,1.04) depending on the value of
the phase δ. This figure illustrates the interplay between astrophysical and particle physics uncertainties (also shown
before in equations (3), (4) and (5)). In order to perform a measurement of θ23 one clearly needs an accurate control
of the initial flux composition.
The description of the emission of neutrinos from GRB that we have used in this section following reference [25]
is very simplified, and could be improved in many aspects. The results of our calculation show that one obtains non
trivial and intriguing energy dependences of the flavor ratios, that could reveal important informations about the
source. This qualitative observation is of a much more general validity and one expects that in most cases, and also
using more realistic descriptions of the sources, one should find non trivial structures in the energy spectra and flavor
composition of the astrophysical neutrino signals.
IX. EXPERIMENTAL FLAVOR IDENTIFICATION
In this work we will not consider in detail the crucial problem of the experimental determination of the flavor ratios.
Several experimental approaches for the detection are at present being developed. The concept that is in the most
advanced stage is a large volume (cubic Kilometer) ice or water Cherenkov detector, with photon detectors distributed
inside the volume, as originally proposed by the Dumand group [40] and now in different stages of development at
the South Pole [41, 42], in the Baikal lake [43] and in the mediterranean sea [44, 45, 46]. Several other alternative
techiques are also being developed [47], that includes radio [48, 49], acoustic and air shower [50, 51] detection methods.
A very imporant problem is of course the amount of data that is required in order to reduce the statistical errors
to the level needed for a meaningful measurement. Today we only have upper limits for the astrophysical neutrino
fluxes, optimistically one can expect that these fluxes will be soon discovered just below the current upper limits,
however even in this case the event rates will remain small, and one will need very long data taking periods, or much
larger mass detectors to collect sufficient data [52, 53].
In this section we want to stress the point that in all the experimental methods that are envisaged, the measurement
of the flavor ratio is connected with a determination of (or a theoretical assumption on) the shape of the energy
distribution of the neutrino fluxes. This happens because the detection methods of different neutrino flavors have
efficiencies with different dependences on Eν . If the statistical errors are sufficiently small (with very large mass
detectors), and having achieved perfect control of the detector acceptances and efficiencies, the uncertainties in the
determination of the neutrino spectral shapes will remain as the dominant source of error in the measurement of the
flavor ratios.
To illustrate the interplay between the spectral shapes and the determination of the flavor ratio one can consider
the case of the Km3 type Cherenkov telescopes. In these detectors one expects to find different classes of neutrino
induced events [13].
(i) In “Track Events” a single up–going muon is seen entering the detector. The energy of the muons can be
estimated from the amount of Cherenkov light produced by the photons and e∓ pairs radiated from the muon
track.
(ii) In “Shower Events” one observes the release of a large amount of Cherenkov light generated by the shower
induced by a neutrino interaction in the detector volume. The amount of light can be translated in the estimate
of a visible energy release Evis.
(iii) In “Double Bang Events” [8, 54] one detects two distinct energy releases inside the detector volume, with space
and time separations that are consistent with the propagation and decay of an energetic tau lepton.
Events in class (i) are associated with the charged current (CC) interactions of muon neutrinos below the detector.
Events in class (ii) can be produced by several types of neutrino interactions. Electron (anti)neutrino CC interactions
are the main source of shower events, there are also contributions from the neutral current interactions of all neutrino
types, and in general also from the CC interactions of νµ and ντ . Finally events in class (iii) are generated by ντ
CC interactions. Clearly, the ratio of the frequencies of these three classes of events does give information about the
flavor composition of the neutrino fluxes. In fact the ratio of the event rates for “Shower” and “Tracks” events has
been proposed as the best method for a measurement of the νe/νµ flavor ratio.
The frequency of “Track” events can be studied as a function of the visible muon energy Eµ. The differential event
rate can be calculated as:
dNTracks
dEµ dΩ⊕ dt
(Eµ,Ω⊕) = A
Tracks
det (Eµ,Ω⊕)
νµ,νµ
dEν φν(Eν ,Ω⊕)
dYν→µ
(Eµ; Eν) (75)
where ATracksdet (Eµ,Ω⊕) is the detector effective area for the detection of muons with energy Eµ from the direction
Ω⊕, dYν→µ/dEµ(Eµ; Eν) is the “muon yield”, that is the probability that a νµ (νµ) of energy Eν produces a µ
energy Eµ at the detector. In this equation we have assumed that the muon and the neutrino are collinear. Even
for an individual source, the energy spectrum will depend on the zenith angle θ⊕, because of the effects of neutrino
absorption in the Earth. This procedure hower implies a good understanding of the neutrino energy spectrum.
The rate of “Shower” events with a visible energy release Evis can be estimated as the convolution
dNShowers
dEvis dΩ⊕ dt
(Evis,Ω⊕) ≃
Mdet(Evis)
νe,νe
[φν(Eν ,Ω⊕) σ
ν (Eν)]Eν≃Evis
All ν
dEν φν(Eν ,Ω⊕)
dσncν
(y, Eν) δ[Evis − Eν y] (76)
+ [(νµ, νµ) CC] + [(ντ , ντ ) CC]
In this equation Mdet(Evis) is the effective detector mass; the first line is the contribution from the charged current
interactions of electron (anti)–neutrinos, in this case the visible energy, in first approximation (neglecting differences
in the light to energy conversion for e.m. and hadronic showers) equals the initial ν energy; the second line is the
contribution from neutral current interactions, in this case the visible energy is only a fraction of the initial energy
because the neutrino in the final state carries away an invisible energy Eν (1− y). The contributions of νµ and ντ CC
interactions to the shower event rate in (76) has been left implicit. A fraction of these interactions can probably be
identified and eliminated selecting events with a muon in the final state, or a double bang structure, but the rest will
contribute to the “shower” class of events.
From (75) and (76) it is clear that the energy response for the two classes of events is different, with the “Track”
events selecting larger Eν . Therefore the ratio for the integral rates for “Shower” and “Tracks” events is strongly
dependent on the spectral shape of the fluxes. Use of the differential distributions in Eµ and Evis allows in principle
to unfold the shape of the neutrino energy spectra, however an important difficulty is that a narrow interval in the
muon energy Eµ (that can however be measured only with modest resolution) corresponds to a very broad interval in
Eν . Also Evis in shower events is not in a one to one correspondence with Eν . The unfolding procedure has therefore
some significant limitations.
As an illustration of these problems, one can look back at the original discovery of neutrino oscillations [6, 7, 55]
with atmospheric neutrinos. The evidence collected by the SK [55] experiment for subGeV fully contained events
(Evis ≤ 1.33 GeV) implied comparing the ratio of the observed rates of e–like and µ–like events with a MC prediction
(µ/e)MC ≃ 1.50. This prediction is significantly smaller than the no–oscillation ratio (νµ + νµ)/(νe + νe) ≃ 2 for the
combination of several reasons, related to the difference in detector acceptances for e–like and µ–like events. The
experimental cuts select for the e–like events a broader energy range, both at the low and high energy ends (at low
energy because electrons have a larger yield of Cherenkov photons, and at high energy because electrons have a shorter
range and are more easily contained). For this reason the MC prediction does depend on the shape of the neutrino
fluxes. For these sub–GeV events, the intervals of Eν for the two classes of e–like and µ–like events are very similar,
and therefore the associated systematic uncertainty in the (µ/e) prediction is relatively small, but it remains as a
relevant source of error.
The comparison of the rates of classes of events that correspond to very different atmospheric neutrino energy, such
as samples of fully contained e–like events and upgoing muons (that are produced by νµ charged current interaction
with a median energy of order 100 GeV) also allows in principle to verify the existence of νµ ↔ ντ transitions. However
in this case the systematic uncertainty in the prediction of the shape of the energy distribution is much larger, and
in practice this method gives only marginal evidence for the existence of a suppression of the upgoing muon rate.
In conclusion, the extraction of the e/µ flavor ratio from the comparison of the rates of “Track” and “Shower”
events in high energy neutrino telescopes requires excellent control of the detector performances, and a sufficiently
precise knowledge of the shapes of the neutrino spectra. The determination of these shapes could be obtained from
the data themselves (the “unfolding method”) or with the help of reliable theoretical models for the source. In both
cases the task appears as remarkably difficult.
X. CONCLUSIONS
The study of astrophysical neutrinos can be used for two distinct goals: the understanding of their sources and
the investigations of the neutrino properties. These two goals are in a sense contradictory. The investigation of the
neutrino fundamental properties requires a comparison of the observations with expectations that must necessarily be
based on a sufficiently accurate understanding of the source. On the other hand the flavor composition of a neutrino
signal encodes information about the source properties, but in order to extract this information one needs to know
the properties of flavor transitions over galactic or cosmological distances with sufficient accuracy.
Several recent works [11, 12, 13] have speculated that it is possible to discovery new neutrino properties, or even
measure precisely the parameters of the already established standard oscillations [56] from the observation of future
astrophysical neutrino sources assuming that the flavor ratios at the source are robustly predictable. The often
repeated argument is that a “standard neutrino source” is dominated by charged pion decay, and therefore emits
the different flavors (summing over ν and ν) with relative intensities: [νe, νµ, ντ ] ≃ [1, 2, 0]. This statement however
must be understood as a first approximation of only limited validity. First of all it requires that the energy losses of
the muons created in charged pion decay are negligible. This crucial assumption is satisfied for several but not all
of the proposed neutrino sources. Even for “thin environment sources” (where the density of ordinary matter and
magnetic field is sufficiently low so that the energy losses for all secondary unstable particles can be safely neglected)
the flavor ratios are (slowly varying) functions of the neutrino energy, with values that are determined by the shape of
the neutrino spectra and by the nature of target material. This dependence arises from two effects: the first is that in
general the contributions of other weakly decaying particles (kaons and in some cases also neutrons) is not negligible,
the second is that, even if charged pions are the only significant neutrino source, the three neutrinos that are emitted
in a π± chain decay have different energy spectra. Folding these decay spectra with the energy distribution of the
parent particles, one obtains slightly different shapes, and therefore energy dependent flavor ratios. In the special case
where the parent pion spectrum is a power law, the neutrino energy distributions have also a power law form with
the same slope, and the ν flavor ratios are constants. As a (phenomenologically important) example, the neutrino
fluxes with spectrum ∝ E−2ν emerging from a “thin”, charged pion source, have relative intensity (at the source)
[νe, νµ, ντ ] ≃ [1, 1.86, 0], a 7% deviation from the naive expectation.
In general, the combined effects of different shapes for the ν energy spectra, and different contributions of the relevant
weakly decaying particles correspond to variations of order 10% of the νe/νµ ratio (in some special circumstances
even larger). The size of these corrections is however comparable to the effects on the observable flavor ratios due to
the present uncertainties in the values of the standard oscillation parameters (for a quantitative illustration of this
point see equations (3), (4) and (5)). The very ambitious program [56] to attempt a precision measurement of θ23
or a determination of θ13 with astrophysical neutrinos cannot therefore ignore these problems, and requires a very
precise control of the structure and properties of the neutrino sources.
Much larger effects on the neutrino flavor ratio arise from situations where the environment of the source contains
large energy densities in magnetic field, or in ordinary matter. In the presence of sufficiently large magnetic fields,
such as those that could exist in a GRB fireball, high energy muons can lose most of their energy to synchrotron
radiation [29, 31] and their contribution to the neutrino fluxes becomes strongly suppressed. This results in a very
small νe/νµ ratio, with a value determined by the size of the kaon contribution (that generate electron neutrinos in
direct decay modes). At higher energy also pions suffer important synchrotron losses, and kaon decay become the
dominant ν source for all flavors. In some extreme circumstances neutral kaon decay will remain as the only significant
neutrino source with a νe/νµ ratio at the source of order unity.
In some circumstances it is possible that interactions with ordinary matter play a major rôle. If the density is
sufficiently large high energy pions reinteract before decaying. This supresses the neutrino fluxes and enhances the
contribution of kaon decay. Summing over charged and neutral kaons, this also results in a significantly reduced νe/νµ
ratio.
The prediction of the neutrino flavor ratios in these circumstances (strong magnetic field or large densities of
ordinary matter) is going to be very difficult, because the assumption of a “single volume” homogeneous source is
very likely a poor (and in any case not demonstrable) assumption. The neutrinos will likely emerge with different
spectra and flavor ratios from different regions of the source, and for example fitting the data including a value of the
field B, and/or a matter density ngas is likely to be only a poor approximation.
The existence of neutrino sources dominated by neutron decay are also a theoretical possibility, in situations where
photodisintegration of relativistic nuclei interacting on a radiation field is possible while pion photoproduction is
kinematically forbidden. If such sources exist, it will however be very difficult to use these fluxes for studies of
the neutrino flavor transitions because in general one expect them to have an important “contamination” of muon
neutrinos in their initial flavor composition. In fact it is unnatural to have the pion photoproduction to be absolutely
forbidden, because the energy thresholds for nuclear photodisintegration and pion production differ only by one order
of magnitude. The size of the contribution of pion decay to the neutrino fluxes in these circumstances is difficult to
control.
Even considering all the caveats of our discussion, there are some circumstances where observations of astrophysical
neutrinos could lead to the discovery that new physics is involved in the propagation of neutrinos over astrophysical
distances. In particular the investigation of the existence of neutrino decay [16] is a very attractive possibility. The
observation of the absence (or very strong suppression) of a νe component in the measured fluxes cannot be explained
with even extreme variations of the source astrophysical properties, and is on the contrary predicted (for any type
of source) in a neutrino decay model with inverse mass hierarchy. For the normal mass hierarchy the neutrino decay
model predicts a large (νe/νµ)obs ratio between 3 and 6. If the neutrinos have standard properties, ratios (νe/νµ)obs
much larger than unity are only possible (and limited to <∼ 4) for sources dominated by νe emission. Therefore from
the observation of a large (νe/νµ)obs ratio it could be possible to infer the existence of ν decay in the normal mass
hierarchy case. This could require some understanding of the properties of the ν source.
In the general case the observation of flavor ratios very different from the “standard source” values (relative inten-
sities [νe, νµ, ντ ]obs ∼ [1, 1, 1]) could be interpreted as evidence for some form of new physics in neutrino propagation,
but also, and probably more economically, as an indication of the existence of some unexpected properties in the ν
source. Resolving these ambiguities is in principle possible, but certainly not easy, and requires a careful interpretation
of the multi–messenger (photons, neutrinos and possibly cosmic rays) and multi–wavelength observations.
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FIG. 1: Distribution of the observable flavor ratio Reµ = (νe/νµ)obs for different models of the ν source and different neutrino
properties. A source model is defined by the relative intensity (summing over ν and ν) of the emission for the three flavors:
[νe, νµ, ντ ]source. The distribution of Reµ is entirely due to the present uncertainties in the determination of the ν mixing
parameters. The source models [0, 1, 0], and [1, 0, 0] correspond to pure νµ or νe emission; the model [1, 1.86, 0] corresponds
to the pion dominated emission from a thin source with a power law spectrum of slope 2. The two dashed curves show the
distributions of Reµ for a ν–decay model [16] where only the lowest ν mass eigenstate (in the direct and inverse hierarchy) is
stable.
FIG. 2: Distribution of the observable flavor ratio Rµτ = (νµ/ντ )obs (see caption of fig. 1). Some of the curves have been
renormalized for an easier observation of their shape.
FIG. 3: Neutrino flavor ratio (νµ+νµ)/(νe+νe) at the source obtained from the chain decay of charged pions with a power law
energy spectrum ∝ E−απ . Energy losses are assumed to be negligible. The dashed line neglects the effect of muon polarization
in pion decay.
FIG. 4: Neutrino flavor ratio (νµ + νµ)/(νe + νe) at the source for the chain decay of different mesons having a power law
spectrum ∝ E−α. The different lines refer to the decay of π±, K± and KL. The dashed (dotdashed) line show the effect of
neglecting muon polarization in the chain decay of charged pions (kaons).
FIG. 5: Characteristic times for a muon plotted as a function of its energy. The different lines correspond to (i) the decay
time tdec; (ii) the time tmatter for energy loss in ordinary matter, assuming a gaseous medium with the average composition
of the interstellar medium (ISM) and the density ρ = 10−10 g cm−3 (this time scales as tmatter ∝ ρ
−1); (iii) the time tsyn for
synchrotron losses in a magnetic field B = 104 Gauss (tsyn ∝ B
−1), and (iv) the time tIC for Inverse Compton losses. The
target radiation field is assumed isotropic, with energy spectrum of GRB form (63) with εb = 1 KeV, and energy density
ργ = ρB (with B = 10
4 Gauss).
FIG. 6: Characteristic times for charged pions. The different lines correspond to: (i) the decay time tdec; (ii) the time tmatter
for hadronic interactions with a gaseous medium with the ISM composition and density ρ = 10−10 g cm−2; (iii) the time tsyn
for synchrotron losses (in a randomly oriented magnetic field B = 104 Gauss); (iv) the time tIC for Inverse Compton losses;
(v) the time tπγ for photo–hadronic interactions. The last two times are calculated for a radiation field of GRB form (63) with
εb = 1 KeV, and energy density ργ = ρB (with B = 10
4 Gauss). Point a (b) is the intersection of the tdec and tsyn (tmatter)
curves. The energy that correspond to the point is the critical energy above which synchrotron losses (the hadronic interaction
probability) are (is) important before decay. For the values of B and ρ chosen in this illustration the presence of a gas of
ordinary matter is negligible.
FIG. 7: Decay energy distribution dpdec/dE(E,Ei) for particles losing energy as −dE/dt = aE
2. The critical energy
m/(aτ ) is the energy where the decay time and the energy loss time are equal.
FIG. 8: Proton interaction rate in a radiation field of the form (63). The energy E∗ (defined in equation (72)) corresponds
to the threshold proton energy for interaction with photons with energy εb. The dashed line neglects the interactions with all
photons with energy ε > εb.
FIG. 9: Distribution of ǫr (the photon energy in the proton rest frame) for the pγ interactions of protons of different energy
Ep. The isotropic photon field has the energy spectrum of GRB form (63). The dashed line neglect the contribution of photons
above the break energy εb.
FIG. 10: Energy distributions (in the form Qj(E)E
2 versus E) for different secondary particles (π±, K±, KL) produced by
the pγ interactions of relativistic protons in a GRB radiation field. The shape of the primary proton spectrum is also shown
(in the form Np(E)E
2 versus E) as a thin solid line. The energy spectrum of the isotropic target photons has the form (63).
All energies are in units of the characteristic energy E∗ (defined in equation (72)). The dashed lines are the distributions of
the meson energy at decay assuming the presence of magnetic field with value B = 2.78× 103 (εb/KeV) Gauss.
FIG. 11: Neutrino energy spectrum (summed over all flavors) from a GRB source, according to the model [25]. The calculation
is performed assuming a proton spectrum Np(E) ∝ E
−α, traveling in an isotropic radiation field of form (63). The different
curves are obtained varying the exponent α of the proton spectrum, or the exponents β1 and β2 of the radiation field. All
energies are measured as fraction of E∗ = mp ǫth/(2εb). The energy losses of all secondary particles are assumed to be negligible.
FIG. 12: Neutrino energy spectrum (summed over all flavors) from a GRB source (see text). The primary proton spectrum
has slope α = 2, while the target photons have a broken power law energy distribution (see equation (63)) with slopes β1 = 1
and β2 = 2 below and above the break energy εb. The curves are calculated assuming different values for the magnetic field,
corresponding to values of the critical energy for muons ǫµ = ∞, 30 and 3 (see eq. (74)).
FIG. 13: Energy spectra for νe and νe from a GRB source (proton slope α = 2, target photon slopes β1 = 1 and β2 = 2,
no magnetic field) For the νe flux also the contributions from meson and neutron decay are separately shown. Neutron decay
dominates at low energy.
FIG. 14: Ratio (νµ + νµ)/(νe + νe) for neutrinos from a GRB source. The ratio correspond to the spectra shown in fig. 11.
The rapid increase at low energy is due to the effect of νe from n decay.
FIG. 15: Ratio (νµ + νµ)/(νe + νe) for neutrinos from a GRB source. The ratio correspond to the spectra shown in fig. 12.
The three curves are calculated assuming different values of the magnetic field in the source (and proton slope α = 2, target
photon slopes β1 = 1 and β2 = 2). The decrease of the flavor ratio at high energy for large B is associated to the important
synchrotron losses of high energy muons.
FIG. 16: Ratio (νe/νe) at the source for neutrinos from a GRB source. The ratio correspond to the spectra shown in fig. 12.
At low energy the ratio is low because of the importance of νe from n decay. At higher energy the ratio decreases because of
the increased importance of π− production. For high magnetic field, at high energy the dominant source of νe and νe becomes
KL decay, and the (νe/νe) ratio becomes unity.
FIG. 17: Ratio (νµ/νµ) at the source for neutrinos from a GRB source. The ratio correspond to the spectra shown in fig. 12.
The three curves correspond to three different values of the magnetic field in the source. At low energy (and at all energies for
small B) the ratio is unity, reflecting the fact that the chain decay of a charged pion produces a νµ and a νµ. In the presence of
a large magnetic field, the neutrinos from muon decay are suppressed, and the flavor ratio is determined by the π+/π− ratio.
At the highest energies charged kaon decay becomes the dominant source of νµ and νµ and the flavor ratio reflects the K
ratio.
FIG. 18: Observable ratio (νµ + νµ)/(νe + νe) after propagation for the neutrinos from a GRB source. The solid lines
corresponds to the source models shown in fig. 12, and 15. The three curves differ for the value of the magnetic field that
corresponds to ǫµ = 3, 30 and ∞ (no field). The neutrino oscillation parameters have been chosen as θ23 = 45
◦, θ12 = 34
◦ and
θ13 = 0. The thin dotted are calculate for the “naive” assumption of a source composition [νe, νµ, ντ ] = [1, 2, 0]. The dotted
line assumes the same mixing parameters as above. The Dashes (dot–dashed) line has θ23 = 50
◦ ( 40◦).
	Introduction
	Flavor Ratios of Astrophysical Neutrinos
	Production of Astrophysical Neutrinos
	Energy Spectra in weak decays 
	Power law Spectra
	Energy Loss Mechanisms
	Decay Energy Distribution
	Hadronic interactions
	Gas Target
	Photoproduction
	Nuclear photodisintegration
	Neutrino Production in Gamma Ray Bursts
	Experimental Flavor identification
	Conclusions
	References

  The measurement of the flavor composition of the neutrino fluxes from
astrophysical sources has been proposed as a method to study not only the
nature of their emission mechanisms, but also the neutrino fundamental
properties. It is however problematic to reconcile these two goals, since a
sufficiently accurate understanding of the neutrino fluxes at the source is
needed to extract information about the physics of neutrino propagation. In
this work we discuss critically the expectations for the flavor composition and
energy spectrum from different types of astrophysical sources, and comment on
the theoretical uncertainties connected to our limited knowledge of their
structure.

Introduction
	Flavor Ratios of Astrophysical Neutrinos
	Production of Astrophysical Neutrinos
	Energy Spectra in weak decays 
	Power law Spectra
	Energy Loss Mechanisms
	Decay Energy Distribution
	Hadronic interactions
	Gas Target
	Photoproduction
	Nuclear photodisintegration
	Neutrino Production in Gamma Ray Bursts
	Experimental Flavor identification
	Conclusions
	References
