IceCube PeV-EeV Neutrinos and Secret Interactions of Neutrinos

We show that the PeV neutrinos detected by IceCube put unique constraints on"secret"interactions of neutrinos with the cosmic neutrino background (C$\nu$B). The coupling must be $g<0.03$ for the mediating boson mass $m_{X} \lesssim 2$ MeV, $g/m_{X}<5$ GeV$^{-1}$ for $m_{X} \gtrsim 20$ MeV, and $g/m_{X}<0.07$ GeV$^{-1}$ in between. We also investigate the possibility that neutrino cascades degrade high-energy neutrinos to PeV energies by upgrading C$\nu$B where the energy flux of PeV neutrinos can coincide with the Waxman-Bahcall bound or the cosmogenic neutrino flux for protons, thanks to energy conservation. However a large coupling is required, which is disfavored by laboratory decay constraints. The suppression of PeV-EeV neutrinos is a testable prediction for the Askaryan Radio Array.

We show that the PeV neutrinos detected by IceCube put unique constraints on "secret" interactions of neutrinos with the cosmic neutrino background (CνB). The coupling must be g < 0.03 for the mediating boson mass m X 2 MeV, g/m X < 5 GeV −1 for m X 20 MeV, and g/m X < 0.07 GeV −1 in between. We also investigate the possibility that neutrino cascades degrade high-energy neutrinos to PeV energies by upgrading CνB where the energy flux of PeV neutrinos can coincide with the Waxman-Bahcall bound or the cosmogenic neutrino flux for protons, thanks to energy conservation. However, a large coupling is required, which is disfavored by laboratory decay constraints. The suppression of PeV-EeV neutrinos is a testable prediction for the Askaryan Radio Array. Introduction. Recently IceCube reported the detection of two PeV neutrinos and 26 additional events, more than expected from atmospheric backgrounds [1,2]. The arrival direction is consistent with the isotropic distribution, suggesting that at least some of the events are of cosmological origin. The Hillas condition to accelerate primary cosmic-rays up to ∼ 100 PeV allows a dozen possibilities [3,4], such as gamma-ray bursts [5][6][7][8], active galactic nuclei [9,10], galaxy clusters and groups [11,12], star-forming galaxies [12][13][14], and heavy dark matter [15][16][17]. We are witnessing the birth of high-energy neutrino astrophysics. The IceCube events remind us of Supernova (SN) 1987A, which placed unique limits on the properties of neutrinos, especially "secret" interactions of neutrinos with the cosmic neutrino background (CνB) [18,19]. The neutrino-neutrino interactions [20,21], even stronger than the weak interactions of the standard model, remain largely unconstrained below the electroweak energy scale [22][23][24] because of their weakness and the difficulties in focusing the neutrino beam [25]. High-energy neutrinos are attenuated by CνB if the cross section is large enough [26]. The much longer distance and higher energy of the IceCube events than those of SN 1987A can tighten restrictions on the secret interactions [27], as well as neutrino decays [28,29], leptoquark couplings [30], and so on.
The IceCube neutrinos may even result from secret interactions of neutrinos. By energizing CνB, high-energy neutrinos may develop cascades in intergalactic space, like gamma-ray cascades [e.g., [31][32][33]. Since such self-interactions conserve the total energy, the neutrinos keep the energy flux while reducing the typical energy. Thus, this scenario can naturally account for a possible "coincidence problem": why the observed neutrino flux is comparable to the Waxman-Bahcall bound [34] or equivalently the cosmogenic neutrino flux at EeV energies produced by ultrahigh-energy cosmic-ray (UHECR) protons [e.g., [35][36][37]. The lack of > 2 PeV events indicates either a soft spectrum or a break at several PeV [2,3], implying different processes at PeV and EeV energies. It is a coincidence that two different processes separated by three orders of magnitude in energy give almost the same flux.
The cross section for scattering νν → νν is generally written as [e.g., 18, 44] where √ s is the center-of-mass energy and Γ X ≃ g 2 m X /4π is the decay width. In the lowenergy limit the interaction is described by the Fermi's four-fermion theory, while in the high-energy limit the boson mass is negligible. At a resonance s ≈ m 2 X , we obtain σ νν ∼ π/m 2 X . For cosmological sources at z, a δ-function approximation for the resonance gives [44]. In addition, the annihilation νν → XX → νννν contributes σ νν ∝ (g 4 /s) ln(s/m 2 X ) for s ≫ m 2 X . We do not distinguish the types of bosons nor neutrino-antineutrino. Our discussion is basically applicable if a single flavor or a single pair of flavors exchange energy, e.g., ν e ν τ → ν e ν τ , because of flavor mixing.
For high-energy neutrinos interacting with CνB, the cross section (1) may be regarded as a function of the energy ε ν of the high-energy neutrinos by using the relation s ≃ 2m ν ε ν , where we take a neutrino mass m ν ∼ 0.05 eV as a fiducial value. From neutrino oscillations, at least one flavor has mass m ν 0.05 eV and the other has m ν 0.009 eV, while the cosmological observations limit the sum of the masses as m ν 0.3 eV [e.g., 45]. Note that we should use T ν instead of m ν if m ν is less than the CνB temperature T ν = (4/11) 1/3 T γ ≃ 1.95 K = 1.68 × 10 −4 eV. For different masses m ′ ν (or T ν ), our results can be scaled by The high-energy neutrinos are attenuated if the mean free path λ ν = 1/n ν σ νν is smaller than the distance to the source d, where n ν = 1 2 × 3 11 n γ ≃ 56 cm −3 is the current number density of CνB for each type (ν orν), neglecting neutrino asymmetry [46]. For extragalactic sources at a cosmological distance d ∼ cH −1 0 , the attenuation condition is With the PeV events in 3 × 10 4 GeV ε ν 2 × 10 6 GeV, we can constrain the coupling g in Eq.
(1) for each m X in Fig. 1 (shaded). It scales as g ≈ const. for m X 2 MeV, g ∝ m (2). We illustrate spiky constraints for a given neutrino energy, although we use σ eff νν for the resonance constraints. Also shown are the constraints by the Z-decay width for the light vector mediator [23] and by the τ -decay rate. The left regions of the vertical lines are relevant to BBN and supernovae.
The IceCube limits are independent of the previous constraints [22][23][24]: • Decay measurements: The decay width of the Z gauge boson strongly constrains the hidden neutrino interactions. Emission of bosons or neutrinos from a final state neutrino or their loop increases the decay width, which is precisely measured by experiments. We extrapolate the result for the light vector mediator [23] as in Fig. 1. This extrapolation would become worse as m X gets closer to the Z mass. In the heavy mass limit, the contact four-fermion neutrino interactions cannot be larger than the standard neutral current interactions [22], i.e., g ∼ 0.6 at m X ∼ 90 GeV in Fig. 1. The decay rates of mesons such as π, K, D, and B also provide g ej < 10 −2 , g µj < 0.5, and g τ j < 3 [24,47]. The decay rates of µ and τ also give g µj < 10 −2 and g τ j < 0.2 (see Fig. 1). • Neutrinoless double beta decay (0νββ): The 0νββ bounds g ee < 10 −5 [48].
• Supernova observations: Neutrino detections from SN 1987A gave g/m X < 12 MeV −1 [18,19]. New bosons could modify the supernova dynamics if m X 10 MeV. In the majoron models, the lepton number is not conserved, leading to entropy production and a thermal bounce of the core at a subnuclear density [49]. For a large cross section σ νν , the neutrino-sphere becomes large, but this does not cause too long a delay of the neutrino arrival time nor different neutrino heating for explosions, because the evolution of the relativistic neutrino gas is largely independent of σ νν [50], similar to gamma-ray burst fireballs. Nevertheless the flavor evolution could be different [51]. • Big bang nucleosynthesis (BBN): If m X < 1 MeV, the bosons increase the effective relativistic degrees of freedom, while they may help the small-scale structure problems in the ΛCDM cosmology [52]. BBN limits are strong if the right-handed neutrinos participate in the interaction [53].

Neutrino cascades via secret interactions. The total energy of the neutrinos is conserved
in the neutrinophilic interactions because the boson immediately decays back to neutrinos X → νν with a rate Γ X ∼ g 2 m X /4π as long as m X > 2m ν . Then high-energy neutrinos, in particular EeV cosmogenic neutrinos, can cascade down to PeV neutrinos by upscattering CνB. Interestingly, this scenario could explain why the PeV neutrino flux is comparable to the Waxman-Bahcall bound or the cosmogenic neutrino flux for pure protons.
To calculate the neutrino spectra, we consider the simplified Boltzmann equations, is the homogeneous and isotropic distribution function (the number of particles per comoving volume per energy) for UHECR protons (i = p) and neutrinos (i = ν), ε i and t are the energy and the proper time measured by the comoving observer, and we have changed the independent variables as (ε i , t) → (ε obs i , z) ≡ (ε i /(1 + z), z) to take the cosmological redshift into account. Note |dt/dz| We consider neutrinos from UHECR protons via pγ interactions with CMB, assuming the typical neutrino energy ε ν ≈ 0.05ε p and the mean inelasticity K π ∼ 0.2. The mean free paths for protons and neutrinos are given by where γ p = ε p /m p c 2 and we evaluate Eq. (5) with the ∆-resonance approximation σ ∆ pγ ∼ 4 × 10 −28 cm −2 , ε ′ ∆ ∼ 0.3 GeV, and ∆ε ′ ∼ 0.2 GeV [5,55]. For simplicity, we neglect the Bethe-Heitler process. Although the resulting neutrino flux is overestimated by a factor of 2-3, it is enough for our purpose due to uncertainty of redshift evolution models. Although an approximation ε ν ≈ 0.05ε p also affects the low-energy side of the peak of cosmogenic neutrinos by a factor and multi-pion production is relevant at high energies [37], we may neglect these effects as long as details of the original spectrum are smeared by cascades.
For simplicity, we assume that a high-energy neutrino imparts half of its energy to CνB via new interactions. The influence on details of spectra is less than a factor of two.
The energy generation rate of UHECRs is normalized by the observations at 10 19.5 eV [14] where we use p = 2 and the source evolution R(z) = (1 + z) 4 for z < 1.2 and R(z) ∝ (1 + z) −1.2 for 1.2 < z for demonstrative purposes [cf. 37,56]. with the CνB and reappear at ε ν ∼ 1 PeV where the universe becomes transparent since the opacity peaks at ε ν ∼ m 2 X /2m ν ∼ 10 17 eV. The spectral shape reflects the low-energy cross section in Eq. (1) and could be compatible with the IceCube data given uncertainties of the atmospheric prompt neutrino background.
If the UHECR energy generation rate is high at high redshifts, e.g., from Population III star activities [56][57][58], we expect an enhancement at low energies, as in Fig. 2 (Pop III model). If there are two or more mediators, e.g., mu-and tau-types, the spectrum may have dips (2X model). A dip could lead to the possible deficit in the IceCube data.
Discussions. Requiring that the observed neutrinos are not affected, we obtained astrophysical constraints on the secret interactions. Although our work is greatly simplified, detailed studies are possible, including various interaction types, scattering angles of particles, and the neutrino-antineutrino difference. Although a large coupling is already disfavored by the laboratory decay constraints, an appealing point of the cascade scenario is that the PeV flux can coincide with the Waxman-Bahcall bound. We must tune the mass to bring a peak to PeV energies, but the neutrino flux is the same for different parameters. The cascade scenario predicts the suppression of >PeV neutrinos, whatever the source is. Future neutrino detectors such as the Askaryan Radio Array [59] can test this possibility.