Neutrinos Associated With Cosmic Rays of Top-Down Origin

Top-down models of cosmic rays produce more neutrinos than photons and more photons than protons. In these models, we reevaluate the fluxes of neutrinos associated with the highest energy cosmic rays in light of mounting evidence that they are protons and not gamma rays. While proton dominance at EeV energies can possibly be achieved by efficient absorption of the dominant high-energy photon flux on universal and galactic photon and magnetic background fields, we show that the associated neutrino flux is inevitably increased to a level where it should be within reach of operating experiments such as AMANDA II, RICE and AGASA. In future neutrino telescopes, tens to a hundred, rather than a few neutrinos per kilometer squared per year, may be detected above 1 PeV.

eV, leaving a dominant proton component at GZK energies [14,15]. With this in mind, we will choose to normalize the proton spectrum from top-down scenarios to the observed ultra high-energy cosmic ray flux.
Neutrinos are produced more numerously than protons and travel much greater distances.
The main point of this paper is to point out that this "renormalization" of the observed cosmic ray flux to protons generically predicts observable neutrino signals in operating experiments such as AMANDA II, RICE and AGASA. Top-down models, if not revealed, will be severely constrained by high-energy neutrino observations in the near future.

II. NUCLEONS FROM ULTRA-HIGH ENERGY JETS
The assumption that nucleons from the decay (or annihilation) of very massive X particles are the source of the highest energy cosmic rays normalizes the decay or annihilation rate of their sources, once the shape of the spectrum of the produced nucleons is known. One needs mass M X ≥ 10 21 eV in order to explain the observed UHECR events. The presence of such very massive particles strongly indicates the existence of superparticles with masses at or below the TeV scale, since otherwise it would be difficult to keep the weak energy scale ten or more orders of magnitude below M X in the presence of quantum corrections. Moreover, we know that all gauge interactions are of comparable strength at energies near M X . These two facts together imply that the evolution of a jet with energy ≥ 10 21 eV shows some new features not present in jets produced at current particle collider experiments.
First of all, primary X decays are likely to produce approximately equal numbers of particles and superparticles, since M X is much larger than the scale M SUSY ≤ 1 TeV of typical superparticle masses. Even if the primary X decay only produces ordinary particles, superparticles will be produced in the subsequent shower evolution. Note also that (at least at high energies) electroweak interactions should be included when modeling the parton shower. Both effects taken together imply that the jet will include many massive particlessuperparticles, electroweak gauge and Higgs bosons, and also top quarks. The decays of these massive particles increase the overall particle multiplicity of the jet, and also produce quite energetic neutrinos, charged leptons and lightest supersymmetric particles (LSP). Eventually the quarks and gluons in the jet will hadronize into baryons and mesons, many of which will in turn decay.
We model these jets at the point of their origin using the program described in Ref. [16].
This program allows us to calculate spectra for different X decay modes. It then follows the supersymmetric parton cascade down to virtuality (or inverse time) of the order of M SUSY , including all gauge interactions as well as third generation Yukawa interactions. At M SUSY all massive particles are decoupled from the parton shower, and decay. Supersymmetric cascade decays are fully taken into account; the results presented below have been obtained using the same spectrum of superparticles as in ref. [16]. At virtualities below M SUSY only ordinary QCD interactions contribute significantly to the development at the jet; b and c quarks are decoupled at their respective masses, hadronize, and decay. At a virtuality near 1 GeV the light quarks and gluons hadronize, with a meson to baryon ratio of roughly thirty to one (five to one) at small (large) x. All baryons will eventually decay into protons, while the mesons (mostly pions) decay into photons, electrons [17] and neutrinos (plus their antiparticles). The heavier charged leptons (muons and taus) also decay. The final output of the code is the spectra of seven types of particles which are sufficiently long-lived to reach the Earth: protons, electrons, photons, three flavors of neutrinos, and LSPs. We assume that X decays are CP-symmetric, i.e. we assume equal fluxes of particles and antiparticles of a given species.
The calculation of Ref. [16] was based on conventional one-loop evolution equations for the relevant fragmentation functions. These may not be reliable in the region of very small x. We wish to calculate neutrino fluxes at energies down to ∼ 10 15 eV (1 PeV), which corresponds to x ∼ 10 −6 (10 −10 ) for M X = 10 21 (10 25 ) eV. At these very small x values color coherence effects are expected to suppress the shower evolution [18]. We try to estimate the size of these effects by matching our spectra computed using conventional evolution equations to the socalled asymptotic MLLA spectra; details of this procedure will be described elsewhere [19].
The effect of this modification on the neutrino event rate is relatively modest for primary jet energy near 10 21 eV, but becomes significant at 10 25 eV. However, even at this higher energy the proton flux, which we only need at x ≥ 10 −5 , is not affected significantly.
This calculation gives us the shape of the spectra of the stable particles at source. The spectra on Earth might differ significantly due to propagation effects. As stated in the introduction, we will assume that (almost) all UHE photons get absorbed. This is actually expected to be true for a homogeneous source distribution. However, according to current estimates of the strengths of the magnetic fields and of the radio wave background in (the  [14]. Note that all observed super GZK events can be explained by this mechanism. halo of) our own galaxy most UHE photons produced in the halo of our galaxy are expected to reach the Earth. As stated in the Introduction, this seems to be in conflict with observation. We will therefore assume that the interaction length of UHE photons in our galaxy has been greatly over-estimated, and explore the consequences of this assumption for neutrino signals.
As well known, (anti)protons lose energy when traveling through the intergalactic medium, mostly through scattering off photons of the ubiquitous cosmic microwave background (CMB). We calculate the observed spectrum of protons taking into account scattering off the CMB at the ∆−resonance and scattering by e + e − pair production; energy losses through the Hubble expansion of the Universe are also included [14,20]. Note that the photoproduction of charged pions contributes to the observed neutrino flux on Earth. In order to solve the ultra high-energy cosmic ray problem, the (anti)proton flux must accommodate the events above the GZK cutoff. Observations indicate on the order of a few times 10 −27 events m −2 s −1 sr −1 GeV −1 in the energy range above the GZK cutoff (5 × 10 19 eV to 2 × 10 20 The formalism of a generic top-down scenario is sufficiently flexible to explain the data from either the HIRES [3] or AGASA [2] experiments. Figure 1 compares HIRES and AGASA data to the proton spectrum predicted for a galactic distribution of decaying particles with mass M X = 2 · 10 21 eV. The drop near a few times 10 19 eV is a manifestation of the GZK cutoff. Note, however, that there are sufficient semi-local events to explain all observed super GZK events. Similarly, figure 2 compares HIRES and AGASA data to the spectrum predicted for M X = 2 · 10 25 eV, rather than 2 · 10 21 eV, decaying particles for the same distribution. Although HIRES and AGASA data differ at face value, especially above the GZK cutoff, top-down scenarios can accommodate all events observed above the GZK cutoff in either experiment.
If the cosmic ray sources are not distributed with a large overdensity in the galaxy, the resulting cosmic ray and neutrino spectrum will be modified. For example, using a homogeneous distribution, the GZK cutoff will again be manifest and the observed cosmic ray spectrum will be difficult to explain. A galactic overdensity of 10 3 to 10 4 or more seems necessary to fit the data. The figure 1 shows a 10 5 overdensity, which is the overall overdensity of matter in our galaxy at the location of the Sun. Note that for less extreme overdensities, the average distance at which a proton is produced will be larger. This implies larger energy losses, and hence a reduced proton flux on Earth for a given number of sources.

III. NEUTRINOS FROM ULTRA HIGH-ENERGY JETS
As discussed earlier, the program computing the proton flux at source also gives the neutrino flux at source. Neutrinos, not being limited by scattering, travel up to the age of the universe at the speed of light (∼ 3000 Mpc in an Euclidean approximation). The only nontrivial effect of neutrino propagation is due to oscillations. In our case the propagation distance of neutrinos amounts to many oscillation lengths, if oscillation parameters are fixed by the currently most plausible solutions of the atmospheric and solar neutrino deficits [21].
As a result, the UHE neutrino flux on Earth is the same for all three flavors, and amounts to the average of the fluxes of the three neutrinos flavors at source. The predicted neutrino flux is shown in figures 3 and 4. At E ν ≪ E jet the main contribution comes from π ± → µ ± ν µ → e ± ν e ν µ decays, but at larger E ν there can be significant contributions from the decays of heavy (s)particles. The peak in the dotted curves at E ν = E jet results from our assumption that in this scenario X decays directly into first or second generation SU(2) doublet (s)leptons, which implies that 50% of all X decays give rise to a primary neutrino; in this case the ratio of neutrino and proton fluxes has a maximum at high energy. On the other hand, if primary X decays are purely hadronic, the neutrino flux at the largest energy is only slightly above the proton flux at that energy. The reason is that neutrinos from meson decays only carry a fraction of the energy of the meson, so a five to one meson to proton ratio at large x leads to a nearly one to one neutrino to proton ratio. We see that the neutrino flux at the highest energy depends quite strongly on how the X particles decay; there is also some dependence on the parameters of the SUSY model [16,19]. For given proton flux the neutrino flux at smaller x is much less model dependent.
At very small x a new uncertainty appears due to coherence effects. These have so far only been studied in a pure QCD parton shower; our treatment of these effects is therefore of necessity rather crude.

SHOWER EXPERIMENTS
We will discuss two classes of experiments capable of observing high energy cosmic neutrinos: neutrino telescopes and air shower experiments.
Optical Cerenkov neutrino telescopes such as the operating AMANDA II and next generation IceCube are designed to observe muon tracks from charged current interactions as well as showers which occur in the detector. The probability of detecting a neutrino passing through the detector from its muon track is given by where n H 2 O is the number density of nucleons in the detector medium (water or ice), and the muon range R µ (E µ , θ zenith ) is the average distance traveled by a muon of energy E µ before falling below some threshold energy (we have used 100 TeV). This quantity depends on the zenith angle of the incoming neutrino because for a detector depth of ∼ 2 km, only quasi-horizontal or upgoing events can benefit from longer muon ranges. At the energies we are most concerned with, the majority of muon events will be quasi-horizontal. The number of muon events observed is then given by where T is the time observed and A eff is the effective area of the detector: one twentieth square kilometers for AMANDA II and one square kilometer for IceCube.
AMANDA II and IceCube can also observe showers generated in charged or neutral current interactions within the detector volume. The event rate from showers is not enhanced by long muon ranges, but can be generated by all three flavors of neutrinos and with greater cross section (neutral + charged current). We use a shower energy threshold of 100 TeV.
The energy threshold imposed effectively removes any background events from atmospheric neutrino events. For a review of Optical Cerenkov neutrino telescopes see Ref. [22].
The operating radio Cerenkov experiment, RICE, is capable of observing showers generated in charged current electron neutrino events. RICE's effective volume increases with energy. At 100 TeV, RICE has an effective volume less than one hundredth of a cubic kilometer. By 10 PeV, however, it increases to about ten cubic kilometers [23]. Again, we use a hard 100 TeV shower threshold.
Air shower experiments can also observe very high energy cosmic neutrinos. We consider AGASA, the largest ground array currently in operation [24], and the next generation AUGER array [25].
To determine that an air shower was initiated by a neutrino, rather than a proton or other cosmic ray, we require a slant depth greater than 4000 g/cm 2 . This corresponds to a zenith angle very near 75 degrees. Therefore, only quasi-horizontal air shower events can be identified as neutrinos. Additionally, unlike showers generated in the upper atmosphere, deeply penetrating showers provide both muon and electromagnetic shower components which help them be differentiated from showers with hadronic primaries. The probability of detecting and identifying a neutrino initiated air shower is described in terms of the array's acceptance, A, in units of volume times water equivalent steradians (we sr). The detector's acceptance increases with energy. For AGASA, the acceptance is about 0.01 km 3 we st at 10 7 GeV but increases to 1.0 km 3 we st at 10 10 GeV and above. For AUGER, the acceptance is about 0.1 km 3 we st at 10 7 GeV, 10.0 km 3 we st at 10 9 GeV and 50.0 km 3 we st at 10 12 GeV.
The number of events observed is then where T is again the time observed, n H 2 O is the number density of nucleons in water and A(E ν ) is the detector's acceptance. AGASA presently has about five years of effective running time between 1995 and 2000 analyzed. A useful treatment of air shower event rates from neutrinos can be found in Ref. [26].   assumed that the ultra high-energy photons are degraded by the universal and/or galactic radio background, leaving protons to dominate the highest energy cosmic ray flux. The neutrino flux must then be normalized to the proton flux resulting in significantly improved prospects for its detection.
A word about the uncertainties in our calculation might be in order. First of all, the uncertainty of the measured UHECR flux, and in particular the discrepancy between the HIRES and AGASA results, leads to an overall uncertainty of a factor of 2 − 3. On the theoretical side, the main uncertainty probably comes from the calculation of the particle spectra at "small" energies, where currently not very well understood coherence effects can play a role. This effect is bigger for higher primary jet energy, and can change the event rate by up to a factor of about 7 (see table). Relaxing our assumption that all UHE photons are absorbed would lead to a corresponding reduction of the fitted source density, and hence of the neutrino flux. In this context it is worth mentioning that in the scenario which seems to fit the data best, with primary jet energy near 10 21 eV and a galactic source overdensity of about 10 5 (see Fig. 1 and ref. [30]), including the photon flux fully would only reduce the predicted event rate by a factor of two to three, since in this case the flux of 10 20 eV photons at source is only slightly larger than the corresponding proton flux. This would still give a neutrino flux in easy striking range of km 2 scale detectors.
This paper shows that the neutrino flux accompanying the highest energy cosmic rays in top-down scenarios is of order of the limits placed by operating experiments such as AMANDA II, RICE and AGASA. Further data from these experiments, or next generation experiments IceCube and AUGER, can test the viability of top-down scenarios which generate the highest energy cosmic rays. If a signal is found soon, future high statistics experiments should be able to map out the neutrino spectrum, thereby allowing us direct experimental access to physics at energy scales many orders of magnitude beyond the scope of any conceivable particle collider on Earth.