Advances in High Energy Physics

Volume 2017, Article ID 9850312, 8 pages

https://doi.org/10.1155/2017/9850312

## Neutrino Pair Cerenkov Radiation for Tachyonic Neutrinos

^{1}Department of Physics, Missouri University of Science and Technology, Rolla, MO 65409, USA^{2}MTA-DE Particle Physics Research Group, P.O. Box 51, Debrecen 4001, Hungary

Correspondence should be addressed to Ulrich D. Jentschura; ude.tsm@jlu

Received 25 May 2017; Accepted 17 October 2017; Published 12 November 2017

Academic Editor: Ming Liu

Copyright © 2017 Ulrich D. Jentschura and István Nándori. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The emission of a charged light lepton pair by a superluminal neutrino has been identified as a major factor in the energy loss of highly energetic neutrinos. The observation of PeV neutrinos by IceCube implies their stability against lepton pair Cerenkov radiation. Under the assumption of a Lorentz-violating dispersion relation for highly energetic superluminal neutrinos, one may thus constrain the Lorentz-violating parameters. A kinematically different situation arises when one assumes a Lorentz-covariant, space-like dispersion relation for hypothetical tachyonic neutrinos, as an alternative to Lorentz-violating theories. We here discuss a hitherto neglected decay process, where a highly energetic tachyonic neutrino may emit other (space-like, tachyonic) neutrino pairs. We find that the space-like dispersion relation implies the* absence* of a threshold for the production of a tachyonic neutrino-antineutrino pair, thus leading to the dominant additional energy loss mechanism for an oncoming tachyonic neutrino in the medium-energy domain. Surprisingly, the small absolute values of the decay rate and energy loss rate in the tachyonic model imply that these models, in contrast to the Lorentz-violating theories, are not pressured by the cosmic PeV neutrinos registered by the IceCube collaboration.

#### 1. Introduction

After early attempts at the construction of tachyonic neutrino theories [1–7], progress in the theoretical development was hindered by difficulties in the construction of a viable field theory involving tachyons (a particularly interesting argument was presented in [8]). Despite the difficulties, work on tachyonic theories has continued up to this day, for both classical theories and spin-zero and spin- quantum theories [9–12]. A very interesting hypothesis was brought forward by Chodos et al. [13], who developed a tachyonic neutrino model based on the so-called tachyonic Dirac equation. They recognized that a simple modification of the mass term in the Dirac equation, according to the replacement , induces a dispersion relation of the form (with the “tachyonic” sign in front of the mass term), while preserving the spin- character of the equation. Recently, it has been recognized [14] that the modified Dirac Hamiltonian corresponding to the tachyonic solutions has a property known as pseudo-Hermiticity, which has been recognized as a viable generalization of the concept of Hermiticity, for quantum mechanical systems [15–25]. Furthermore, the bispinor solutions of the tachyonic equation have been determined [26], and they have been shown to fulfill sum rules which enter the calculation of the time-ordered product of tachyonic field operators. The tachyonic pseudo-Hermitian quantum dynamics of wave packets composed of the bispinor solutions has been discussed in [14]. A surprising feature of the tachyonic Dirac equation is the natural appearance of the fifth current in the equation. In particular, the appearance of elevates the helicity basis to the most natural* ansatz* for the solution of the equation and induces parity-breaking in a natural way. States with the “wrong helicity” are eliminated from the theory by a Gupta-Bleuler type condition [26].

Just to fix ideas, we should point out here that the tachyonic neutrino differs from other faster-than-light neutrino models in that the dispersion relation is Lorentz-covariant. Explicit breaking of the Lorentz symmetry may induce faster-than-light dynamics for neutrino wave packets, with a time-like four-vector product (see [27, 28]). An example is the Lorentz-breaking dispersion relation with (units with are used throughout this paper). This dispersion relation follows [29, 30] from a Lorentz-violating “metric” . A quite illuminating analysis of the model dependence of the calculation [29], with reference to conceivably different forms of the interaction Lagrangian, is given in [30]. By contrast, the tachyonic theory implies a space-like four-vector product , thus leaving Lorentz symmetry intact and enabling the construction of bispinor solutions in the helicity basis [26].

Despite some “seductive” observations regarding the tachyonic neutrino model (most of all, pseudo-Hermiticity and natural emergence of the helicity eigenstates, as well as the suppression of states with the “wrong” helicity), any alternative neutrino model must also pass various other tests concerning the stability of highly energetic neutrinos against the emission of particle-antiparticle pairs. The IceCube collaboration has registered “big bird,” an (2.004 ± 0.236) PeV highly energetic neutrino [31, 32]. If neutrinos in this energy range are stable against lepton pair Cerenkov radiation, then this sets rather strict bounds on the values of the Lorentz-violating parameters [33, 34]. In a recent paper, lepton pair Cerenkov radiation has been analyzed as an energy loss mechanism for high-energy tachyonic neutrinos [35]. The kinematics in this case implies that the oncoming, decaying neutrino decays into a tachyonic state of lower energy, emitting an electron-positron pair (see Figure 1(a)). For the creation of an electron-positron pair, the threshold momentum for the virtual boson is , where is the electron mass.