Cosmic-Ray Neutrinos from the Decay of Long-Lived Particle and the Recent IceCube Result

Motivated by the recent IceCube result, we study high energy cosmic-ray neutrino flux from the decay of a long-lived particle. Because neutrinos are so transparent, high energy neutrinos produced in the past may also contribute to the present neutrino flux. We point out that the PeV neutrino events observed by IceCube may originate in the decay of a particle much heavier than PeV if its lifetime is shorter than the present cosmic time. It is shown that the mass of the particle responsible for the IceCube event can be as large as $\sim 10^{10}\ {\rm GeV}$. We also discuss several possibilities to acquire information about the lifetime of the long-lived particle.

Neutrino astronomy provides a new window to the early universe. This is particularly because, contrary to particles with electromagnetic interactions (i.e., e ± , p andp, γ and so on), neutrinos have very weak interaction and are very transparent. Thus, the cosmic-ray neutrino spectrum in the present universe contains various information about the production processes of energetic neutrinos.
Recently, the IceCube experiment reported the result of their analysis on high-energy neutrino events [1]. The IceCube experiment observed 28 high-energy neutrino events with E EM > 30 TeV (with E EM being the deposited electromagnetic-equivalent energy in detector), which is substantially larger than the number of events expected from the atmospheric backgrounds (which is 10.6 +5.0 −3.6 events). Notably, the IceCube experiment found two events with deposit energy of ∼ PeV (1041 +132 −144 TeV and 1141 +143 −133 TeV, nicknamed as Ernie and Bert, respectively), while no event with larger E EM has been observed.
IceCube claims the existence of a source of high energy cosmic-ray neutrinos other than the atmospheric one [1]. We may consider particle-physics or astrophysical origin of such high energy neutrinos, the former of which is the subject of our study. (Possible astrophysical origins include Active Galactic Nuclei [2,3,4], γ-ray burst [5,6,7,8], hypernova remnants [9,10], star-forming galaxies [11], Galactic cosmic-rays [12,13,14], neutron-star mergers [15] and cosmogenic neutrinos [16,17]. For other astrophysical discussion, see also [18,19].) Although it is premature to make any conclusion, the negative observation of the events with larger energy deposit may indicate that the cosmic-ray neutrino spectrum has a cutoff at the energy around ∼ PeV. In fact, we also note that no event is observed in the energy bins between 0.4 − 0.63 PeV and 0.63 − 1 PeV. It may be a consequence of a peak of the cosmic-ray electron neutrino spectrum at ∼ PeV; this is because, within experimental uncertainties, the deposited energy is equal to the energy of the initial-state neutrino for ν e charged current events, while it is below the energy of the neutrino for other types of events.
From particle-physics point of view, the structure in the neutrino spectrum mentioned above may be realized with a new physics at the energy scale higher than ∼ PeV. We consider such a case in this letter. In particular, we study whether the decay of a new particle X with its mass m X 1 PeV is responsible for the high-energy neutrino events observed by IceCube. For example, if neutrinos are produced by the decay of a particle X with m X ∼ PeV in the present universe, a neutrino spectrum with the cutoff at ∼ PeV may be obtained. The case where the dark-matter particle plays the role of X was considered in [20,21,22]. (For early discussion about related issues, see [23,24,25,26,27,28,29].) On the contrary, even if the mass of X is much larger than ∼ PeV, there still exists a possibility to produce present cosmic-ray neutrinos with E ∼ PeV. This is because, if the decay of X occurs earlier than the present epoch, the energy of the neutrino produced by the X decay is redshifted.
In this letter, we study cosmic-ray neutrinos produced by a long-lived particle X. We show that some of the high energy neutrino events observed by IceCube can be due to neutrinos produced by the decay of X. In particular, we discuss that the peak in the cosmicray neutrino spectrum may show up at E ∼ 1 PeV even with the mass of X much larger than PeV, if the lifetime of X is shorter than the present age of the universe. For such a scenario to work, the decay of X is required to occur at z O(10 3 ) (with z being the redshift), which implies that the mass of X can be as large as 10 10 GeV. We also discuss how the neutrino spectrum depends on the properties of X.
We start our discussion without assuming any particular model for the heavy particle X. Instead, we parametrize the properties of X with the following three quantities: where m X and τ X are the mass and the lifetime of X, respectively. In addition, Y X is the so-called yield variable with s being the entropy density; with Y X , the number density of X is given by In the following, to make our point clearer, we concentrate on the case where the neutrino produced by the decay of X is monochromatic (with the energyĒ ν ); the energy distribution of the neutrino produced by the decay of X is expressed as where H is the expansion rate of the universe, S ν (t, E) is the source term: with n X being the number density of X, and γ ν (t, E) is the scattering rate: with f BG being the distribution function of the background (target) particle and σ tot (s) the total scattering cross section of high energy neutrinos for the processes with the center-ofmass energy √ s. By solving the Boltzmann equation, the neutrino spectrum at the present epoch t 0 is given by where a is the scale factor and a 0 ≡ a(t 0 ). In addition, (Notice that the arguments of D ν (E; z) are chosen to be the present neutrino energy E and the redshift at the time of the neutrino production z.) When the decay process produces monochromatic neutrino, we obtain In our analysis, the effects of secondary neutrinos produced by the scattering processes are not included. This can be justified as far as the damping factor D ν (E; z) is close to 1. Hereafter, we mostly consider such a case.
In the case of our interest, important scattering processes of the high energy neutrinos are with background neutrinos, and hence f BG (k) = (e k/Tν + 1) −1 , where T ν is the neutrino temperature and is related to the temperature of the background radiation as T ν = (4/11) 1/3 T . Here we neglect the neutrino masses in the distribution function, since, as we will see later, the damping factor becomes effective only for z O(10 3 ), where T ν O(0.1 eV). Then, the scattering rate is given by For the calculation of the damping rate of ν ℓ , σ tot is obtained by taking account of the effects of the following processes: • ν ℓ +ν ℓ,BG → ν ℓ +ν ℓ , • ν ℓ +ν ℓ,BG → ℓ +l, where f denotes quarks and leptons, ℓ denotes charged leptons, and the subscript "BG" is for background neutrinos. (Cross sections for these processes are given in Appendix.) As we will see, for a cosmic-ray neutrino whose present energy is E ∼ 1 PeV, the damping becomes important only for z O(10 3 ), for which the typical center-of-mass energy of the scattering process is √ s 1 GeV. Therefore, in our calculation of σ tot , we neglect the masses of leptons as well as those of first-and second-generation quarks. We have checked that the damping factor D ν (E; z) is insensitive to the behavior of σ tot (s) with √ s a few GeV.
In Fig. 2, we plot Φ (Cosmo) ν (E) given in Eq. (12) for several values of τ X . (Hereafter, we concentrate on the flux at present, so we omit t 0 from the argument.) The normalization of the spectrum in Fig. 2 is arbitrary. One can easily see a peak in the spectrum; the position of the peak depends on the initial energy of the neutrino as well as on the lifetime of X. If τ X ≪ t 0 and z * 10 3 (for which the damping factor is negligible), the position of the peak is approximately given by where This behavior can easily be understood by using the fact that most of X decays at the cosmic time of t ∼ τ X , and that the energy of the emitted neutrino is redshifted by the factor of ∼ (1+z * ) −1 . One can also see that, compared to the flux for 1+z * = 10, that for 1+z * = 500 shows a significant suppression for E 0.1 PeV. This is due to the fact that the damping factor D ν becomes extremely small for z O(10 3 ). (The flux for 1 + z * = 10 is almost unaffected by the damping.) For 1 + z * = 500, D ν ≃ 0.5 for E = 0.08 PeV. For smaller E, secondary neutrinos produced by the scattering processes may also contribute and the flux may be affected. However, for such a redshift, the damping factor becomes close to 1 for E 0.1 PeV, so we believe that Eq. (12) well describes the contribution from cosmological distance in the energy region of our interest. Contrary to the case of τ X ≪ t 0 , the shape of the spectrum for τ X ≫ t 0 is insensitive to τ X as far asĒ ν is fixed. We can also see that the shape of the spectrum for τ X ≫ t 0 is quite different from that for τ X ≪ t 0 . In particular, Φ (Cosmo) ν has a sharp edge at E =Ē ν . Next, we consider the contribution from the Milky-Way Galaxy. Because the Galactic contribution is not isotropic, we definẽ where l.o.s. stands for the line-of-sight integral, andl denotes the direction of the line-of-sight. We assume that the density of X is proportional to that of dark matter: where Ω X and Ω DM are the density parameters of X and dark matter, respectively. In addition, ρ DM ( l) is the energy density of dark matter in the Galaxy; in our numerical calculation, we adopt the NFW density profile [30,31]: where r is the distance from the galactic center, ρ ⊙ ≃ 0.4 GeV/cm 3 is the local halo density, r c ≃ 20 kpc is the core radius, and r ⊙ ≃ 8.5 kpc is the distance between the galactic center and the solar system. Then, we define Φ (Galaxy) ν as the directional average ofΦ Now we are at the position to compare the predicted neutrino spectrum with the IceCube result. In our analysis, we particularly pay attention to the fact that IceCube has observed two events in the bin of 1 < E EM < 1.6 PeV while no event in 0.4 < E EM < 0.63 PeV, 0.63 < E EM < 1 PeV, and E EM > 1.6 PeV. Thus, we concentrate on the case where the spectrum of the neutrinos from X decay, which is assumed to contain a sizable fraction of ν e , becomes largest at ∼ PeV. To make a comparison with the IceCube result, we define the "averaged" neutrino spectrum for 1 < E < 1.6 PeV: and similar quantitiesΦ where we assumed that the total energy deposit is (almost) equal to the energy of the initial-state neutrino, which is the case of the charged current events of ν e . Here, we used the effective area of 15 m 2 [1] as a reference value, although this value also includes the effects of neutral current. We take this flux as a canonical value for our study. We also note here that, with the present setup, high-energy cosmic-ray neutrino events with E EM < 1 PeV may also be induced by charged current interactions of neutrinos other than ν e and by neutral current ones. For the excess of the events with E EM < 1 PeV, we may also consider non-monochromatic initial neutrino injection as another possibility [22]. As we have mentioned, the neutrino flux is proportional to the yield variable Y X . We estimate Y X which givesΦ ν,PeV = 3 × 10 −16 m −2 sec −1 str −1 GeV −1 . For τ X ≪ t 0 , for which Φ (Galaxy) ν,PeV is negligible, we choose m X so that E (Cosmo) peak = 1.1 PeV (which is close to the deposited energies of the most energetic events, i.e., Ernie and Bert). Then, the best-fit value of Y X to realize Eq. (21) is given by Notice that, for τ X ≪ t 0 , the best-fit value is insensitive to the lifetime of X. On the contrary, for τ X ≫ t 0 , we chooseĒ ν = 1.1 PeV and obtain In the case of τ X ≫ t 0 , Y X corresponds to the present yield value of X. Then, the present mass density of X is estimated as Ω X ≃ 6 × 10 14 × Y X (m X /1 PeV). Combining this relation with Eq. (23), we can see that dark matter may play the role of X if τ X ∼ O(10 29 sec) [20,21,22]. On the contrary, scenarios with longer lifetime do not work because of the over-closure of the universe. In Fig. 3, we show the neutrino spectrum for (τ X ,Ē ν ) = (5.2 × 10 14 sec, 2.2 × 10 2 PeV) and (10 29 sec, 1.1 PeV); the yield variable is determined so thatΦ ν,PeV is equal toΦ (IceCube) ν,PeV given in Eq. (21). We can see that the enhancement of the flux at E ∼ 1 PeV is possible both for τ X ≪ t 0 and τ X ≫ t 0 .
One of the important check points of the present scenario, in particular for the case of τ X ≪ t 0 , is the flux at the "Glashow resonance" [32]. For E ≃ 6.3 PeV, the event rate of IceCube for ν e is enhanced because the center-of-mass energy hits the W -boson pole [33,34]. Indeed, the effective area of the IceCube experiment for E ≃ 6.3 PeV is about 40 times larger than that for E ≃ 1 PeV. One might wonder if the present scenario is consistent with the negative observation of the events for such an energy region because, if τ X ≪ t 0 , the spectrum is non-vanishing even at E ∼ 6.3 PeV. Notably, the flux for E ≫ E ν (see Eq. (12)). For the case where X decays in radiation-and matter-dominated epochs, for example, this quantity is given by e −t E /τ X = e −(E/E * ) 2 and e −(E/E * ) 3/2 , respectively, with ) is given by 0.014, 0.012, and 0.001, (0.027, 0.024, and 0.003) for 1 + z * = 10, 100, and 1000, respectively. Using the fact that IceCube has not observed any event at around the Glashow resonance, relatively large value of z * may be preferred. However, the statistics are still poor, and it is premature to exclude the possibility of small z * . With more data, IceCube may see events at around E ≃ 6.3 PeV in particular in the case with small z * .
Here, let us comment on the possibilities to acquire information about the lifetime of X in the present scenario. Because the detailed shape of the spectrum depends on the lifetime, it may be possible to distinguish the cases with τ X ≪ t 0 and τ X ≫ t 0 . As we can see, the spectrum smoothly continues to E > E peak if τ X ≪ t 0 . On the contrary, for τ X ≫ t 0 , the neutrino flux is dominated by the one originating in the Galaxy. Then, the spectrum is sharply peaked at E =Ē ν . Thus, if the spectrum of the neutrinos is precisely determined in the future, it will provide important information about the lifetime of X. Another possibility is to use the directional information about the high energy neutrinos. In the case of τ X ≪ t 0 , high energy neutrinos originate in the decay of X at high redshift so that they are isotropic. On the contrary, for the case of τ X ≫ t 0 , a large fraction of the high energy neutrino events are Galactic origin. Consequently, the neutrino flux is enhanced for the direction of the Galactic center. We define θ as the angle between the direction of the high-energy neutrino and that of the Galactic center, with θ = 0 being the direction of the Galactic center. Then, we calculateΦ ν,PeV (θ > 90 • ) ≃ 2. Thus, significant angular dependence is expected for τ X ≫ t 0 , while the flux is isotropic for τ X ≪ t 0 . Experimental determination of the directional distribution is important to distinguish the scenarios with τ X ≪ t 0 and τ X ≫ t 0 .
Before closing this letter, several comments are in order. First, we consider a possible scenario to produce X in the early universe. As we have seen, forΦ ν,PeV = 3 × 10 −16 m −2 sec −1 str −1 GeV −1 , very small value of Y X is required. In particular, if τ X ≪ t 0 , Y X ∼ O (10 −(26−27) ) is necessary, which is much smaller than the typical thermal relic abundance. Even if we assume that X is in the thermal bath, however, Y X can be significantly suppressed with large entropy production. A mini inflation after the first inflation (which is responsible for the density perturbation of the universe) may be an example. Another possibility is to introduce a field which has a very small branching ratio into X; if such a field is produced in the early universe, X can be produced by its decay.
We also comment on possible constraints from high energy cosmic rays, in particular, γ-ray. With the mass of X as large as ∼ 10 10 GeV, electroweak gauge bosons and charged leptons are also produced by electroweak jet cascade even if X dominantly decays into neutrinos [35]. (If X has decay modes into gauge bosons or into charged leptons, they also contribute.) If too much γ-ray is generated, the present scenario conflicts with the observations of extragalactic γ-ray. In the present scenario, however, we expect that the flux is small enough by estimating the total amout of energy injection due to the decay of X. Assuming that the productions of electroweak gauge bosons and charged leptons are subdominant compared to the neutrino production, the energy density of radiation (i.e., γ-ray) from X should be smaller than E 2 γ Φ γ O(10 −4 m −2 sec −1 str −1 GeV) (see Eq. (21)). Notice that, if the production process of γ-ray is suppressed, the flux becomes smaller; this is the case for the electroweak jet cascade processes. For the energy range for which the measurement of the extragalactic γ-ray flux is available (i.e., E γ 100 GeV), the γray flux in the present scenario is found to be smaller than the observed one (which is [36]. Thus, we believe that the present scenario is not excluded by the current measurements of high energy extragalactic γ-ray flux. With future improvements of the measurements, signals of the decay of X may be seen. The detailed understanding of the signals requires a precise calculation of the spectrum of γ-ray, which is beyond the scope of this letter. In summary, in this letter, we have studied the cosmic-ray neutrinos produced by a long-lived particle X. The PeV neutrino events observed by IceCube may be due to the neutrinos produced by a heavy particle X. We have discussed that such a scenario works even with the mass of X much higher than ∼ PeV if the lifetime is shorter than the present cosmic time. To make such a scenario viable, we have seen that the decay of X should occur at z O(10 3 ), and that the scale of the new physics responsible for the IceCube events can be as large as O(10 9−10 GeV). Detailed study of the propagation of neutrinos taking into account the effects of the secondary neutrinos is necessary to understand the case with m X O(10 9−10 GeV). Finally, we discuss particle-physics models which contain a long-lived particle decaying into neutrino. Even assuming that X is a neutral scalar particle, one may consider the case where X is embedded into SU(2) L triplet (with the hypercharge of +1) [20]. Another possibility is to introduce SU(2) L doublet boson (with the hypercharge of −1/2) other than ordinary Higgs boson, which couples to lepton doublet and right-handed neutrino ν c R . Then, identifying the neutral component of the doublet as X, the decay process X → ν L ν c R becomes possible if the neutrino mass is Dirac type or the Majorana mass of ν c R is small enough. So far, we have considered the case where X is neutral. For the case of τ X ≪ t 0 , however, X may be charged (or even colored) because the constraints on stable superheavy charged particle (in particular, those using sea water [37]) do not apply. From particle-physics point of view, there exist various beyond-standard-model physics which contain the candidate of X, in particular when the scale of the new physics is around O(10 9−10 GeV). One example is a fermion in Peccei-Quinn (PQ) sector [38,39] (or its supersymmetric partner) in hadronic axion model [40,41]. The PQ (s)fermion may be stable if it has no mixing with standardmodel fermions. With introducing a very small mixing, the PQ (s)fermion decays into standard-model particles with a very long lifetime. Similar argument holds for a messenger (s)fermion in gauge-mediation supersymmetry breaking model [42,43,44]. Some of those particles may play the role of X if they decay into neutrinos at z O(10 3 ).
In this Appendix, we summarize the cross sections for the neutrino scattering processes which are relevant for the calculation of the damping rate of high energy neutrinos. The fermion masses (in particular, lepton masses) are neglected except in Eq. (26).