Neutrino Pair Cerenkov Radiation for Tachyonic Neutrinos
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 neutrinos 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 value 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.
pacs:31.30.jh, 31.30.J-, 31.30.jf
After early attempts at the construction of tachyonic neutrino theories Bilaniuk et al. (1962); Dhar and Sudarshan (1968); Bilaniuk and Sudarshan (1969); Feinberg (1967, 1978); Recami and Mignani (1974); Maccarone and Recami (1980), 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 Ref. Boulware (1970)). Despite the difficulties, work on tachyonic theories has continued up to this day, for both classical theories as well as spin-zero and spin- quantum theories Chang (2002); Recami (2009); Bilaniuk (2009); Bose (2009). A very interesting hypothesis was brought forward by Chodos, Hauser and Kostelecky Chodos et al. (1985), 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 Jentschura and Wundt (2012a) 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 Bender and Boettcher (1998); Bender and Dunne (1999); Bender et al. (1999); Bender and Weniger (2001); Bender et al. (2002); Mostafazadeh (2002a, b, c, 2003); Jentschura et al. (2009, 2010). Furthermore, the bispinor solution of the tachyonic equation have been determined Jentschura and Wundt (2013), 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 Jentschura and Wundt (2012a). 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 Jentschura and Wundt (2013).
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 Refs. Kostelecky and Lehnert (2001); Kostelecky and Mewes (2012)). An example is the Lorentz-breaking dispersion relation with (units with are used throughout this paper). This dispersion relation follows Cohen and Glashow (2011); Bezrukov and Lee (2012) from a Lorentz violating “metric” . A quite illuminating analysis of the model dependence of the calculation Cohen and Glashow (2011), with reference to conceivably different forms of the interaction Lagrangian, is given in Ref. Bezrukov and Lee (2012). 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 Jentschura and Wundt (2013).
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 highly energetic neutrino M. G. Aartsen et al. [IceCube Collaboration] (2013, 2014). 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 Stecker and Scully (2014); Stecker (2014). In a recent paper, lepton pair Cerenkov radiation has been analyzed as an energy loss mechanism for high-energy tachyonic neutrinos Jentschura and Ehrlich (2016). 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 Fig. 1(a)]. For the creation of an electron-positron pair, the threshold momentum for the virtual boson is where is the electron mass.
For both the Lorentz-violating as well as the tachyonic neutrino models, one has not yet considered the additional decay and energy-loss channel which proceeds via a virtual boson and has a neutrino-antineutrino pair (as opposed to an electron-positron pair) in the exit channel [see Fig. 1(b)]. This process is not parametrically suppressed in comparison to the one with electrically charged particles in the exit channel, because of the weakly rather than electromagnetically interacting virtual particle (the boson) in the middle. For the Lorentz-violating theories, the kinematics in this case becomes involved because one has to implement Lorentz-violating parameters for all four particle in the process: (i) the oncoming and exiting neutrino, and (ii) the created neutrino-antineutrino pair. Previous studies Cohen and Glashow (2011); Bezrukov and Lee (2012) have rather concentrated on the lepton pair Cerenkov radiation process as the dominant energy loss mechanism than the neutrino pair Cerenkov radiation; the kinematics in this case appears to be a lot easier to analyze than for neutrino-pair Cerenkov radiation.
For the tachyonic case, one needs to calculate the process of neutrino-antineutrino pair Cerenkov radiation in full tachyonic kinematics, for both the in and out states. In particular, it is necessary to generalize the pair production threshold to the creation of a tachyonic neutrino-antineutrino pair. We organize this paper as follows. In Sec. II, we derive the kinematic conditions for tachyon-antitachyon pair production. The calculation of the threshold conditions and the energy loss mechanism for neutrino pair Cerenkov radiation proceeds in Sec. III. Consequences for tachyonic neutrino theories are summarized in Sec. IV.
Ii Pair Production Threshold
For two tardyonic (“normal”) particles of mass , pair production threshold is reached when the pair is emitted collinearly, with two four-vectors that fulfill
The situation is completely different for the production of a tachyonic pair. Here, a well-defined lower threshold for is missing. E.g., we have for the collinear pair with tachyonic mass parameter , and ,
which is negative. For two neutrinos of different energy, emitted collinearly ( and ), one has
In the limit of a small tachyonic mass parameter , a Taylor expansion of the latter term leads to the expression
In the limits , or alternatively , , the latter expression may assume very large negative numerical values (see also Fig. 2). There is thus no lower threshold for tachyonic pair production, expressed in .
One might ask if arbitrarily large are compatible with the relativistic tachyonic pair production kinematics. In order to answer this question, we consider the production of an anti-collinear pair,
For large and , this expression assumes arbitrarily large positive numerical values.
The conclusion is that the tachyonic kinematics do not exclude any range of from the kinematically allowed range of permissible momentum transfers, and neutrino pair Cerenkov radiation (or, more generally, tachyon–antitachyon pair production) is allowed in the entire range
The latter condition only ensures that the energy emitted into the pair is positive. For a decaying tachyonic neutrino, the condition (6) implies that there is no lower energy threshold for the production of a tachyon-antitachyon pair from an oncoming neutrino, within the process depicted in Fig. 1(b).
Iii Calculation of the Pair Production
We calculate the decay width of the incoming tachyonic neutrino, in the lab frame, employing a relativistically covariant (tachyonic) dispersion relation, with both incoming as well as outgoing neutrinos on the tachyonic mass shell ( for ), in the conventions of Fig. 1. In the lab frame, the decay rate is
Here, refers to the specific way in which the average over the oncoming helicity states, and the outgoing helicities, needs to be carried out for tachyons Jentschura and Ehrlich (2016).
We use the Lagrangian
where is the Weinberg angle, is the boson field, and is the neutrino field. The effective four-fermion interaction is
where is the Fermi coupling constant. The matrix element is
where is a tachyonic positive-energy bispinor (particle) solution, while is a tachyonic negative-energy (antiparticle) solution. The positive-energy solutions read as follows Jentschura and Wundt (2013),
|while the negative-energy solutions are given by|
where we identify the on-shell spinors with the , where and . The symbols denote the fundamental helicity spinors (see p. 87 of Ref. Itzykson and Zuber (1980)). We note that the helicity of the antineutrino solution is positive, while in the massless limit, it has negative chirality.
For the tachyonic spin sums, one has the following sum rule for the positive-energy spinors Jentschura and Wundt (2012b),
where is the unit vector in the direction. Upon promotion to a four-vector, one has . The sum rule can thus be reformulated as
where is a time-like unit vector.
In Refs. Jentschura and Wundt (2012b, 2013), it has been argued that a consistent formulation of the tachyonic propagator is achieved when we postulate that the right-handed neutrino states, and the left-handed antineutrino states, acquire a negative Fock-space norm after quantization of the tachyonic spin- field. Hence, in order to calculate the decay process of an oncoming, left-handed, positive-energy neutrino, we must first project onto negative-helicity states, according to Ref. Jentschura and Ehrlich (2016),
The squared and spin-summed matrix element for the tachyonic decay process thus is
where the latter identity provides for an implicit definition of the function . The traces and are
We have chosen the convention to denote the by the momentum of the outgoing antiparticle.
For the outgoing pair, we use the fact that the helicity projector is approximately equal to the chirality projector in the high-energy limit, which simplifies the Dirac gamma trace somewhat. On the tachyonic mass shell, one has . After the trace over the Dirac matrices, some resultant scalar products vanish, e.g., the scalar product of the time-like unit vector and the space-like unit vector ().
After the and integrations, we are left with an expression of the form
Both the expressions for as well as are too lengthy to be displayed in the context of the current paper.
For the kinematics, we assume that
The condition is naturally imposed for tachyonic kinematics. The squared four-momentum transfer then reads as
where it is convenient to define .
The integrations are done with the kinematic conditions that all , and all for the pair are allowed (see Sec. II), leading to
where and we have used the identity
The differential energy loss, for a particle traveling at velocity , undergoing a decay with energy loss , due to the energy-resolved decay rate , in time , reads as follows,
Now we set , divide both sides of the equation by and integrate over the energy of the outgoing particle. One obtains
Hence, the energy loss rate is obtained as
After a long, and somewhat tedious integration one finds the following expressions,
These formulas are valid for , which is easily fulfilled for all neutrino masses . There is no threshold energy; i.e., formulas (27a) and (27b) are, in particular, valid in the range . Parametrically, they are of the same order-of-magnitude as those given in Ref. Jentschura and Ehrlich (2016) for (charged) lepton pair Cerenkov radiation, but the threshold is zero for the neutrino pair emission. Hence, neutrino pair emission is the dominant decay channel in the medium-energy domain, for an oncoming tachyonic neutrino flavor eigenstate.
Iv Discussion and Conclusions
In principle, tachyonic spin- theories have a number of properties which make them more attractive than their spin-zero counterparts. One distinctive feature is that the mass parameters enters only linearly in the Lagrangian Jentschura and Wundt (2013), thus preventing the vacuum from becoming manifestly unstable against tachyon-antitachyon pair production. Also, it has been possible to calculate the time-ordered product of field operators, which leads to the Feynman propagator of the tachyonic field Jentschura and Wundt (2012b, 2013). One also observes that the generalized Dirac Hamiltonian for the tachyonic spin- fields is pseudo-Hermitian, so that it becomes possible to formulate the quantum dynamics of tachyonic wave packets without having to overcome unsurmountable challenges Jentschura and Wundt (2012a). In Ref. Jentschura et al. (2014), it has been argued that in view of the small neutrino interaction cross sections, it would be difficult to transport information faster than the speed of light using a neutrino beam, if neutrino are just a bit superluminal (tachyonic). The sign of the mass square of neutrinos has not yet been determined experimentally, in contrast to differences of mass squares among neutrino flavor eigenstates.
Here, we calculate the decay rate and energy loss rate, for a hypothetically tachyonic neutrino flavor, against neutrino-pair Cerenkov radiation. It needs to be checked if the absence of a threshold would lead to a disagreement with high-energy data on neutrinos of cosmic origin. In fact, the IceCube experiment M. G. Aartsen et al. [IceCube Collaboration] (2013, 2014) has observed 37 neutrinos having energies during the first three years of data taking. Three of these events (“Ernie”, “Bert” and “Big Bird”) had energies , while “Big Bird” is famous for having an energy of . A blazar has been identified as a possible source of this highly energetic neutrino M. Kadler et al. (2016). Neutrinos registered by IceCube have to “survive” the possibility of energy loss by decay, and if they are tachyonic, then lepton and neutrino pair Cerenkov radiation processes become kinematically allowed.
The results given in Eqs. (27a) and (27b) for the decay rate and energy loss rate due to neutrino pair Cerenkov radiation are not subject to a threshold energy; parametrically are of the same order-of-magnitude as those given for lepton pair Cerenkov radiation in Ref. Jentschura and Ehrlich (2016), but the threshold energy is zero. Let us estimate the relative energy loss due to neutrino pair Cerenkov radiation over a distance
assuming a (relative large) neutrino mass parameter of . One obtains for the relative energy loss according to Eq. (27b),
This means that even at the large “Big Bird” energy of , the relative energy loss over 15 billion light years does not exceed parts in , which is negligible.
The decay rate is obtained as follows (again, assuming that ),
Even for “Big Bird”, this means that the decay rate does not exceed , which is equivalent to a lifetime of years, far exceeding the age of the Universe. The neutrino pair Cerenkov radiation process, even if threshold-less, has such a low probability due to the weak-interaction physics involved, that it cannot constrain the tachyonic models. Indeed, even for relatively large tachyonic neutrino mass parameters of the order of , and for the largest neutrino energies observed, the process leads only to a vanishingly small relative energy loss for an oncoming neutrino of cosmic origin 15 billion light years away. The lifetime of the tachyonic neutrino far exceeds the age of the Universe. Our quick estimate shows that “Big Bird” would have survived the travel from the blazar PKS B1424-418 (see Ref. M. Kadler et al. (2016)). In other words, neutrino pair Cerenkov radiation does not pressure the tachyonic neutrino hypothesis.
We thus take the opportunity here to correct claims recently made by by one of us (U.D.J.) in Ref. Jentschura and Ehrlich (2016), where a hypothetical cutoff of cosmic neutrino spectrum at the Big Bird energy was related to the threshold energy for (charged) lepton pair Cerenkov radiation, and thus, to a neutrino mass parameter. In Ref. Jentschura and Ehrlich (2016), it was overlooked that (i) a further decay process exists for tachyonic neutrinos which is not subject to a threshold condition, and (ii) that the absolute value of both (charged) lepton as well as neutrino pair Cerenkov radiation is too small (both above as well as below threshold) to lead to any appreciable energy loss of an oncoming tachyonic neutrino flavor eigenstate, over cosmic distances and time scales. Hence, it is not possible, in contrast to the conclusions of Ref. Jentschura and Ehrlich (2016), to relate the lepton pair threshold to the tachyonic mass parameter. The (more optimistic) conclusion thus is that neither lepton nor neutrino pair Cerenkov radiation processes pressure the tachyonic model.
However, for the Lorentz-violating models, important limits on the available parameter space have been set in Refs. Stecker and Scully (2014); Stecker (2014), based on (charged) lepton pair Cerenkov radiation alone. Roughly speaking, the reason for the pressure on the Lorentz-violating models is that even small Lorentz violations at PeV energies correspond to high “virtualities” of the superluminal particles and hence, relatively large (energy-dependent) mass parameters. It is quite imperative that the additional decay process studied here should also be calculated for the different kinematic conditions in Lorentz-violating models, where it will further limit the available parameter space for the Lorentz-violating parameters. Note that, e.g., employing a Lorentz-violating dispersion relation Cohen and Glashow (2011); Bezrukov and Lee (2012), with , a quick calculation shows the absence of a neutrino pair production threshold in the Lorentz-violating model; the reason being simple: namely, one has for , and it thus becomes possible to generate Lorentz-violating neutrino pairs with near-zero four-momenta. The additional decay process uncovered here thus has the potential of fundamentally changing the bounds to be inferred for the Lorentz-violating parameters, from the cosmic high-energy neutrinos, within the Lorentz-violating models.
To conclude, the Lorentz-violating model is pressured at high energies, where even numerically tiny values of the Lorentz-violating parameters induce large deviations from the light-like dispersion relation, corresponding to a numerically large value of the “effective mass” with (where we assume the dispersion relation given in Refs. Cohen and Glashow (2011); Bezrukov and Lee (2012)). By contrast, the tachyonic model is fully compatible with astrophysical data collected at high energies, while the tachyonic dispersion relation predicts noticeable deviations from the speed of light only for comparatively low-energy neutrinos. A proposal to test the tachyonic hypothesis, in the low-energy domain, has recently been published in Ref. Jentschura et al. (2014). Finally, we also refer to Ref. Jentschura et al. (2014) for clarifying remarks on general aspects of the tachyonic model.
Acknowledgements.The authors acknowledge helpful conversations with R. Ehrlich. This research has been supported by the NSF (grant PHY–1403937) and by a János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The authors declare that there is no conflict of interest regarding the publication of this paper.â
- Bilaniuk et al. (1962) O. M. P. Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, ““Meta” Relativity,” Am. J. Phys. 30, 718–723 (1962).
- Dhar and Sudarshan (1968) J. Dhar and E. C. G. Sudarshan, “Quantum Field Theory of Interacting Tachyons,” Phys. Rev. 174, 1808–1815 (1968).
- Bilaniuk and Sudarshan (1969) O.-M. Bilaniuk and E. C. G. Sudarshan, “Causality and Space-like Signals,” Nature (London) 223, 386–387 (1969).
- Feinberg (1967) G. Feinberg, “Possibility of Faster-Than-Light Particles,” Phys. Rev. 159, 1089–1105 (1967).
- Feinberg (1978) G. Feinberg, “Lorentz invariance of tachyon theories,” Phys. Rev. D 17, 1651–1660 (1978).
- Recami and Mignani (1974) E. Recami and R. Mignani, “Classical Theory of Tachyons (Special Relativity Extended to Superluminal Frames and Objects),” Rivista Nuovo Cimento 4, 209–398 (1974).
- Maccarone and Recami (1980) G. D. Maccarone and E. Recami, “Two-Body Interactions through Tachyon Exchange,” Nuovo Cimento 57, 85–101 (1980).
- Boulware (1970) D. G. Boulware, “Unitarity and Interacting Tachyons,” Phys. Rev. D 1, 2426–2427 (1970).
- Chang (2002) T. Chang, “Parity Violation and Neutrino Mass,” Nucl. Sci. Technol. 13, 129–133 (2002).
- Recami (2009) E. Recami, “Superluminal waves and objects: an overview of the relevant experiments,” J. Phys. Conf. Ser. 196, 012020 (2009).
- Bilaniuk (2009) O. M. Bilaniuk, “Tachyons,” J. Phys. Conf. Ser. 196, 012021 (2009).
- Bose (2009) S. K. Bose, “Aspects of Tachyon theory,” J. Phys. Conf. Ser. 196, 012022 (2009).
- Chodos et al. (1985) A. Chodos, A. I. Hauser, and V. A. Kostelecky, “The Neutrino as a Tachyon,” Phys. Lett. B 150, 431–435 (1985).
- Jentschura and Wundt (2012a) U. D. Jentschura and B. J. Wundt, “Pseudo–Hermitian Quantum Dynamics of Tachyonic Spin– Particles,” J. Phys. A 45, 444017 (2012a).
- Bender and Boettcher (1998) C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having -Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
- Bender and Dunne (1999) C. M. Bender and G. V. Dunne, “Large-order perturbation theory for a non-Hermitian -symmetric Hamiltonian,” J. Math. Phys. 40, 4616–4621 (1999).
- Bender et al. (1999) C. M. Bender, S. Boettcher, and P. N. Meisinger, “-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
- Bender and Weniger (2001) C. M. Bender and E. J. Weniger, “Numerical evidence that the perturbation expansion for a non-Hermitian -symmetric Hamiltonian is Stieltjes,” J. Math. Phys. 42, 2167–2183 (2001).
- Bender et al. (2002) C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
- Mostafazadeh (2002a) A. Mostafazadeh, “Pseudo–Hermiticity versus -symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205–214 (2002a).
- Mostafazadeh (2002b) A. Mostafazadeh, ‘‘Pseudo-Hermiticity versus -Symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum,” J. Math. Phys. 43, 2814–2816 (2002b).
- Mostafazadeh (2002c) A. Mostafazadeh, “Pseudo-Hermiticity versus -Symmetry III: Equivalence of pseudo-hermiticity and the presence of antilinear symmetries,” J. Math. Phys. 43, 3944–3951 (2002c).
- Mostafazadeh (2003) A. Mostafazadeh, “Pseudo-Hermiticity and Generalized PT- and CPT-Symmetries,” J. Math. Phys. 44, 974–989 (2003).
- Jentschura et al. (2009) U. D. Jentschura, A. Surzhykov, and J. Zinn-Justin, “Unified Treatment of Even and Odd Anharmonic Oscillators of Arbitrary Degree,” Phys. Rev. Lett. 102, 011601 (2009).
- Jentschura et al. (2010) U. D. Jentschura, A. Surzhykov, and J. Zinn-Justin, “Multi–Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators,” Ann. Phys. (N.Y.) 325, 1135–1172 (2010).
- Jentschura and Wundt (2013) U. D. Jentschura and B. J. Wundt, “From Generalized Dirac Equations to a Candidate for Dark Energy,” ISRN High Energy Physics 2013, 374612 (2013).
- Kostelecky and Lehnert (2001) V. A. Kostelecky and R. Lehnert, “Stability, causality, and Lorentz and CPT violation,” Phys. Rev. D 63, 065008 (2001).
- Kostelecky and Mewes (2012) V. A. Kostelecky and M. Mewes, “Neutrinos with Lorentz-violating operators of arbitrary dimension,” Phys. Rev. D 85, 096005 (2012).
- Cohen and Glashow (2011) A. G. Cohen and S. L. Glashow, “Pair Creation Constrains Superluminal Neutrino Propagation,” Phys. Rev. Lett. 107, 181803 (2011).
- Bezrukov and Lee (2012) F. Bezrukov and H. M. Lee, “Model dependence of the bremsstrahlung effects from the superluminal neutrino at OPERA,” Phys. Rev. D 85, 031901(R) (2012).
- M. G. Aartsen et al. [IceCube Collaboration] (2013) M. G. Aartsen et al. [IceCube Collaboration], “First Observation of PeV-Energy Neutrinos with IceCube,” Phys. Rev. Lett. 111, 021103 (2013).
- M. G. Aartsen et al. [IceCube Collaboration] (2014) M. G. Aartsen et al. [IceCube Collaboration], “Observation of High-Energy Astrophysical Neutrinos in Three Years of IceCube Data,” Phys. Rev. Lett. 113, 101101 (2014).
- Stecker and Scully (2014) F. W. Stecker and S. T. Scully, “Propagation of superluminal PeV IceCube neutrinos: A high energy spectral cutoff of new constraints on Lorentz invariance violation,” Phys. Rev. D 90, 043012 (2014).
- Stecker (2014) F. W. Stecker, “Limiting superluminal electron and neutrino velocities using the 2010 Crab Nebula flare and the IceCube PeV neutrino events,” Astropart. Phys. 56, 16–18 (2014).
- Jentschura and Ehrlich (2016) U. D. Jentschura and R. Ehrlich, “Lepton-pair Cerenkov radiation emitted by tachyonic neutrinos: Lorentz-covariant approach and Ice Cube data,” Adv. High Energy Phys. 2016, 4764981 (2016).
- Itzykson and Zuber (1980) C. Itzykson and J. B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
- Jentschura and Wundt (2012b) U. D. Jentschura and B. J. Wundt, “Localizability of Tachyonic Particles and Neutrinoless Double Beta Decay,” Eur. Phys. J. C 72, 1894 (2012b).
- Jentschura et al. (2014) U. D. Jentschura, D. Horváth, S. Nagy, I. Nándori, Z. Trócsányi, and B. Ujvári, ‘‘Weighing the Neutrino,” Int. J. Mod. Phys. E 23, 1450004 (2014).
- M. Kadler et al. (2016) M. Kadler et al., “Coincidence of a high-gluence blazar outburst with a PeV-energy neutrino event,” Nat. Phys. 12, 807–814 (2016).