Within the hypothesis that neutrinos are spacelike (tachyonic) fermions quantized in a preferred cosmological frame, ultra-high-energy neutrinos can undergo the three-body decay να→νανβνˉβ\nu_{\alpha}\to\nu_{\alpha}\nu_{\beta}\bar{\nu}_{\beta} over cosmological distances, which has been proposed as an explanation for the non-observation of neutrinos above roughly 100 PeV. The published treatment computed only the total decay width and the survival probability of primaries aimed at Earth, and explicitly deferred the study of the differential (energy–angle) distributions of the decay secondaries. The open problem is to compute the fully differential decay distribution dΓ/dΦ\mathrm{d}\Gamma/\mathrm{d}\Phi for this tachyonic three-body decay and to use it to determine whether primaries emitted outside the Earth's solid angle can still deliver high-energy secondaries into the detector's solid angle with appreciable flux and with measurable arrival-time delays relative to the electromagnetic counterpart, thereby altering the expected simultaneity of high-energy neutrinos and photons.

Background

Several direct neutrino-mass measurements in tritium beta decay (Mainz, Troitsk, and more recently KATRIN) have repeatedly returned slightly negative central values for the fitted neutrino mass square mν2m_\nu^2, which motivates the speculative hypothesis that neutrinos are tachyonic, i.e. spacelike particles whose four-momentum satisfies k2=−κ2<0k^2=-\kappa^2<0 with κ\kappa a real tachyonic mass parameter. A consistent quantization of such fermions was developed by Rembieliński, Caban and Ciborowski in a preferred cosmological frame (identified with the cosmic neutrino-background rest frame), in which helicity eigenstates are well defined and the weak interactions can be formulated as in the Standard Model. In this framework a neutrino can decay into a neutrino plus a neutrino–antineutrino pair, να→να νβνˉβ\nu_{\alpha}\to\nu_{\alpha}\,\nu_{\beta}\bar{\nu}_{\beta}, a channel kinematically forbidden for ordinary massive or massless neutrinos; the related constraint that Lorentz-violating superluminal neutrinos rapidly shed energy was emphasized by Cohen and Glashow in a different (Lorentz-covariant) setting. Building on earlier work in this framework, which had computed only the energy-differential width dΓ/dE\mathrm{d}\Gamma/\mathrm{d}E and a one-dimensional energy-cascade equation for solar neutrinos, Rembieliński and Ciborowski computed the Standard Model total width for the three-body decay, folded it into an exact survival probability in an expanding Universe, and showed that the neutrino flux at 100 PeV and above would be strongly depleted over cosmological distances for a tachyonic mass κ≳0.35\kappa\gtrsim 0.35 eV, with depletion at 1 EeV setting in already at κ≈0.15\kappa\approx 0.15 eV, consistent with the non-observation of ultra-high-energy (cosmogenic) neutrinos at IceCube, the cubic-kilometer neutrino detector at the South Pole that has observed PeV-scale astrophysical neutrinos but none at hundreds of PeV and above. Their analysis treated only the disappearance of primary neutrinos emitted toward Earth. The decay is a one-to-three-body process: the primary energy is shared unequally among three secondaries emitted at non-zero angles with respect to the primary direction of flight. A primary neutrino emitted by the source outside the (extremely small) solid angle subtended by the Earth may therefore decay in flight and still produce a secondary that enters the detector's solid angle; such a secondary can retain a very high energy but travels a longer path, arriving with a time delay relative to the photons (and the non-decayed neutrinos) from the same source. Whether this off-axis repopulation and time-delay channel is a significant correction to the expected coincidence between high-energy neutrinos and their electromagnetic counterparts has never been assessed, because the required differential distributions of the decay — in the paper's notation dΓ/dΦ\mathrm{d}\Gamma/\mathrm{d}\Phi, where dΦ\mathrm{d}\Phi denotes the Lorentz-invariant tachyonic phase-space measure θ(u⋅p) θ(u⋅l) θ(u⋅r) δ4(k−p−l−r) δ(p2+κ2) δ(l2+μ2) δ(r2+μ2) d4p d4l d4r\theta(u\cdot p)\,\theta(u\cdot l)\,\theta(u\cdot r)\,\delta^4(k-p-l-r)\,\delta(p^2+\kappa^2)\,\delta(l^2+\mu^2)\,\delta(r^2+\mu^2)\,\mathrm{d}^4p\,\mathrm{d}^4l\,\mathrm{d}^4r, with uu the four-velocity of the preferred frame, κ\kappa the tachyonic mass of the primary (and of the same-flavor secondary) and μ\mu that of the emitted neutrino–antineutrino pair — were not evaluated. Related analyses in adjacent frameworks (tachyonic or Lorentz-violating superluminal dispersion relations) computed decay rates, thresholds, and energy-loss rates, including the kinematic bound that confines the fastest secondary to a cone of opening angle θ<θth\theta<\theta_{\mathrm{th}} around the primary direction, but none derived the flux-level angular distribution of the secondaries nor studied the geometry of detection and timing.

Problem Statement

Within the preferred-frame QFT of spacelike (tachyonic) neutrinos, compute the fully differential decay distribution dΓ/dΦ\mathrm{d}\Gamma/\mathrm{d}\Phi — equivalently the joint energy and emission-angle distributions of the secondary neutrinos — for the three-body decay να→νανβνˉβ\nu_{\alpha}\to\nu_{\alpha}\nu_{\beta}\bar{\nu}_{\beta} of an ultra-high-energy tachyonic neutrino, and use it to answer the question posed by Rembieliński and Ciborowski: can a primary neutrino emitted by a cosmological source outside the solid angle subtended by the Earth still produce, via in-flight decay, a secondary neutrino that (i) enters the detector's solid angle on Earth and (ii) retains an ultra-high energy, and with what probability and angular scale (as a function of the tachyonic mass κ\kappa, the source redshift, and the emission energy, including cosmological propagation effects and the possibility of further decays)? Determine the resulting arrival-time distribution of such off-axis secondaries relative to the electromagnetic counterpart of the source, and assess whether this effect is significant enough to alter the expected simultaneity between high-energy neutrinos and photons. The problem requires (i) the differential energy–angle distribution dΓ/dΦ\mathrm{d}\Gamma/\mathrm{d}\Phi of the secondaries, (ii) a quantitative estimate — with the normalization made explicit — of the probability and angular scale with which an off-axis primary delivers an ultra-high-energy secondary into the detector's solid angle, together with the resulting arrival-time delay distribution, and (iii) a definite verdict on significance reached by comparing the off-axis secondary flux and timing with the on-axis flux and the expected simultaneity; the cosmological propagation effects and the possibility of further decays named above are part of this estimate.

References

  1. Jakub Rembieliński and Jacek Ciborowski, "Reduction of ultra-high energy neutrino flux for spacelike neutrinos from extragalactic sources" (arXiv e-print titled "Possible explanation of not observing ultra-high energy cosmic neutrinos"), Journal of High Energy Astrophysics 41 (2024) 81–88, DOI 10.1016/j.jheap.2024.02.001, arXiv:2310.02763, https://arxiv.org/abs/2310.02763
  2. Paweł Caban, Jakub Rembieliński, and Kordian A. Smoliński, "Decays of spacelike neutrinos", arXiv:hep-ph/9707391, https://arxiv.org/abs/hep-ph/9707391
  3. Jakub Rembieliński, Paweł Caban, and Jacek Ciborowski, "Quantum field theory of space-like neutrino", European Physical Journal C 81 (2021) 716, DOI 10.1140/epjc/s10052-021-09494-x, arXiv:2103.13982, https://arxiv.org/abs/2103.13982
  4. Ulrich D. Jentschura, István Nándori, and Robert Ehrlich, "Calculation of the decay rate of tachyonic neutrinos against charged-lepton-pair and neutrino-pair Cerenkov radiation", Journal of Physics G 44 (2017) 105201, DOI 10.1088/1361-6471/aa84da, arXiv:1709.07711, https://arxiv.org/abs/1709.07711
  5. G. Somogyi, I. Nándori, and U. D. Jentschura, "Neutrino splitting for Lorentz-violating neutrinos: Detailed analysis", Physical Review D 100 (2019) 035036, DOI 10.1103/PhysRevD.100.035036, arXiv:1904.10505, https://arxiv.org/abs/1904.10505
  6. Robert Ehrlich, "Tachyonic Neutrinos: From the Cosmic Rays to Extragalactic Supernovae", Symmetry 15 (2023) 1624, DOI 10.3390/sym15091624, https://www.mdpi.com/2073-8994/15/9/1624
  7. Andrew G. Cohen and Sheldon L. Glashow, "Pair Creation Constrains Superluminal Neutrino Propagation", Physical Review Letters 107 (2011) 181803, DOI 10.1103/PhysRevLett.107.181803, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.181803