#### Abstract

Current experiments do not exclude the possibility that one or more neutrinos are very slightly superluminal or that they have a very small tachyonic mass. Important bounds on the size of a hypothetical tachyonic neutrino mass term are set by lepton pair Čerenkov radiation (LPCR), that is, by the decay channel , which proceeds via a virtual boson. Here, we use a Lorentz-invariant dispersion relation which leads to very tight constraints on the tachyonic mass of neutrinos; we also calculate decay and energy loss rates. A possible cutoff seen in the IceCube neutrino spectrum for PeV, due to the potential onset of LPCR, is discussed.

#### 1. Introduction

The early arrival of a neutrino burst from the 1987A supernova [1] still motivates speculations about a possible superluminal nature of neutrinos, even if it is generally assumed that the delay in the arrival of electromagnetic radiation (light) is caused by the time the shock wave from the core collapse needs in order to reach the surface of the exploding star. If neutrinos are ever so slightly superluminal, then they may emit Čerenkov radiation in the form of light lepton pairs. In this paper, we attempt to answer three questions: (i) How would the energy threshold for the decay channel (lepton pair Čerenkov radiation, LPCR) have to be calculated if we assume a strictly Lorentz-covariant, space-like dispersion relation for the relevant neutrino flavor eigenstate? (ii) How would the decay rate and the energy loss rate have to be calculated under this assumption? Can the tachyonic Dirac equation [2–5] and its bispinor solutions [6, 7] be used in that context? (iii) What implications could be derived for astrophysics under the assumption that a possible cutoff seen by IceCube for neutrinos with energies PeV is confirmed by future experiments?

Theoretical arguments can be useful in restricting the possible degree of superluminality of neutrinos and maximum attainable neutrino velocities [8–10]. In [8, 9], a Lorentz-noncovariant dispersion relation was used, where is a constant parameter. This assumption leads to an energy-dependent effective “mass” square . The effective mass then grows linearly with the neutrino energy. (Natural units with are used in this paper, yet we shall include explicit factors of when indicated by the context.) Indeed, at the time, a best fit to the available experimental neutrino mass data including the initial OPERA claim [11] suggested the conceivable existence of an “energy-dependent mass” of the neutrino, as evidenced in Figure of [12]. The choice of the relation made in [8] was consistent with the need to model the initial OPERA claim [11] and is perfectly compatible with the concept of perturbative Lorentz breaking terms in the neutrino sector [9]. A Dirac-type equation leading to the Lorentz-noncovariant dispersion relation used by Cohen and Glashow [8] can be obtained [9] from the current operator given in Eq. (2) of [13] upon a particular choice of parameters in the generalized fermionic current operator (in the notation adopted in [13]). Then, assuming a constant neutrino speed , one can effectively describe the apparent absence of energy dependence of the deviation of the neutrino speed from the speed of light (in the range GeV), according to the (falsified) initial claim made by OPERA [11], while remaining compatible with the framework of perturbative Lorentz breaking [13].

However, while there are advantages to assuming a Lorentz-noninvariant dispersion relation for superluminal neutrinos (such as the preservation of the timelike positive quantity ), there are also a number of disadvantages. For example, if the dispersion relation holds in one particular Lorentz frame, then, under a Lorentz boost, in general, one has in the moving frame [8, 9]. In order to illustrate the consequences of Lorentz noncovariance, let us consider a boost along positive -axis into a frame which moves with velocity . A particle moving along positive -axis of the lab frame with four-momentum is mapped onto and thus is “at rest” in the moving frame. However, the general dispersion relation in the moving frame,is much more complicated. (Throughout this paper, we denote the spatial components of the four-vector by and keep explicitly in order to avoid confusion between and .)

An alternative, commonly accepted dispersion relation for so-called tachyons (these are space-like, faster-than-light particles described by a Lorentz-invariant wave equation) reads as ; that is, it is the “normal” dispersion relation with the negative sign of the mass square term (see [2–7, 14–23]). Here, we calculate the threshold energy and the decay rate under the assumption of a Lorentz-invariant dispersion relation for the neutrino. We find that the alternate dispersion relation imposes tight restrictions on superluminality and has important phenomenological implications for neutrino masses.

#### 2. Dispersion Relations and Thresholds

For tachyonic particles, starting from the pioneering work of Sudarshan et al. [14–16], continuing with the works of Feinberg [17, 18], and including the tachyonic neutrino hypothesis [2–6, 19–23], the following dispersion relation has been assumed for the tachyonic (space-like) solutions: where we use the suggestive subscript for “neutrino.” These relations imply that instead of . Here, the tachyonic Lorentz factor appears, which is . Tachyonic and tardyonic dispersion relations are unified upon assuming an imaginary value for in the tachyonic case (starting from the tardyonic case, one has , where the latter equation holds for tachyons). With the standard definitions of and , one has for both tardyons and tachyons.

In order to obtain the threshold energy for the LPCR decay , we use the following conventions (see Figure 1), inspired by Chap. 10 of [24], and define and as the oncoming and outgoing neutrino energies, with being the four-momentum of . Pair production threshold is reached for and . For collinear geometry, with all momenta pointing along -axis, we haveFurthermore, threshold obviously requires . (This is possible for tachyonic particles, when . In this limit, the tachyonic particle becomes infinitely fast and loses all of its energy, which implies that it is impossible to detect it [25]. The counterintuitive loss of energy for tachyons under acceleration is a consequence of standard tachyonic kinematics [2, 6, 7, 14–18, 26–28].) When the relations and are substituted into (3), this yieldsIdentifying with the threshold momentum, one easily findsThe threshold energy is then easily found asBecause we are using a tachyonic dispersion relation, the threshold energy can be expressed as a function of only the mass parameters. Larger tachyonic masses lead to lower threshold energies. In view of the tachyonic dispersion relation , where is the neutrino velocity* at threshold*, we may convert the threshold energy into a function of the electron mass and the neutrino threshold velocity. For given , the limit is equivalent to the limit because . In this limit, we haveSubstituting the exact dispersion relation into the threshold condition , and solving for , one obtainsThe exact expression (8) confirms (7) in the limit , which corresponds to the phenomenologically important limit of high-energy neutrinos. Smaller values of (approaching zero) correspond to smaller tachyonic neutrino masses and therefore to larger threshold energies. For given neutrino speed , neutrinos with energy (or larger), under the hypothetical assumption of the tachyonic dispersion relation, have a tachyonic neutrino mass term large enough to make the decay via LPCR kinematically possible. Expressed differently, the tachyonic mass term in this case is large enough to lead to LPCR decay at energy , according to (6).

#### 3. Decay Rate and Timelike Noncovariant Dispersion Relation

Given the complexities of calculating the decay rate due to LPCR using a tachyonic dispersion relation, it is extremely useful to first discuss the case of a Lorentz-noncovariant form , using lab frame variables. For collinear incoming and outgoing neutrinos, threshold for pair production is reached at , from which one derives (setting ) the following threshold values (in agreement with [8]):Here, is Fermi’s coupling constant and and are the standard fundamental positive-energy and negative-energy bispinor solutions of the Dirac equation [29]. The invariant matrix element isHere, , and [see Eq. (5.57) on p. 153 of [30]]. Following [9], we now make the additional assumption that the functional form of the projector sum over the spin orientations remains the same as for the ordinary Dirac equation even if the underlying dispersion relation is Lorentz-noncovariant (for a general discussion on such models, see [31, 32]). In this case, the sum over final state and the averaging over the initial spins leads to . This enters the lab frame expression for the decay rate [33]:where . The azimuthal symmetry suggests the use of cylindrical coordinates. The domain of integration contains, for given , all permissible , where . With , the momentum transfer iswhere we require . Solving (12) for , one obtains the boundary of the region of permissible vectors. An example is given in Figure 2(a) in the form of a “sharpened ellipsoid” with “sharp” top near , , and a “rounded” bottom with , and . After somewhat tedious integration over the allowed vectors, one obtainsfor the energy loss per unit length, confirming the results given in Eq. () and Eq. () of [8] and in [9]. This confirmation of the results given in [8] (under the assumptions made in the cited paper, namely, the dispersion relation ), but using a different method, namely, phase-space integration directly in the laboratory frame, encourages us to apply the same method to the calculation of the tachyonic neutrino decay rate, where the use of the laboratory frame is indispensable. The confirmation also underlines the consistency of the theoretical formalism under a change of the assumptions made in the calculation.

**(a)**

**(b)**

#### 4. Decay Rate and Space-Like Covariant Dispersion Relation

For an incoming tachyon, the particle state (space-like neutrino) may transform into an antiparticle state upon Lorentz transformation, and its trajectory may reverse the time ordering (see Figure 3). Thus, the interpretation of a tachyonic neutrino state as a particle or antiparticle may depend on the frame of reference, and we should calculate the process directly in the lab frame. The necessity to transform certain tachyonic particle field operators into antiparticle operators under Lorentz boosts has been stressed in [6, 17, 18]. Incoming and outgoing states are required to be above-threshold positive-energy states in the lab frame (causality and tachyonic trajectories are discussed in [2, 14–18] and Appendix of [34]).

We consider the matrix elementHere, and are Dirac spinor solutions of the tachyonic Dirac equation [6, 7]. The bar denotes the Dirac adjoint. In the helicity basis (see Chap. 23 of [35] and [6, 7]), these are given bywhere are the fundamental helicity spinors (see p. 87 of [29]). Following [6, 7, 19], we use the tachyonic sum rule of the fundamental tachyonic bispinor solutions [see Eq. (34a) of [6]]:where is the four-momentum and is a helicity quantum number. We refer to [6, 7] for a thorough discussion; roughly speaking, factor in (16) restores the correct sign in the calculation of the time-ordered product of tachyonic field operators (the propagator) for the contribution of all virtual degrees of freedom of the tachyonic field [see Eqs. (46)–(57) and Eq. (73)–(75) of [7]]. matrix in (16) is a part of the natural Dirac “adjoint” for the tachyonic spinor. Namely, the adjoint equation to the tachyonic Dirac equation, , reads as . As explained in Eqs. (73)–(75) of [17], right-handed particle states and left-handed antiparticle states (those with the “wrong” helicity) are excluded from the physical spectrum of the tachyonic field by a Gupta-Bleuler condition; these cannot contribute to the oncoming and outgoing neutrino states in Figure 1 [while they do contribute to the virtual states, that is, the propagator; see Eqs. (46)–(57) of [7]]. Both the incoming and the outgoing neutrinos in Figure 1 are real rather than virtual neutrinos. Hence, in order to calculate the LPCR decay rate, we use the modified sum over tachyonic spinors:where is a timelike unit vector, is the unit vector in direction, and upon promotion to a four-vector, we have , so that becomes a left-handed helicity projector.

We thus calculate with an incoming, positive-energy, left-helicity tachyonic neutrino. One obtains the modified sum over spins in the matrix element:Here, is the familiar trace from the outgoing fermion pair. The decay rate is given by (11) under the replacement . The integrals over the momenta of the outgoing fermion pair ( and ) are done using (). ConsiderIt remains to analyze the domain of allowed vectors [see the “cupola structure” in Figure 2(b)], which is defined by the requirement , for . The dispersion relation implies thatHere, is the polar angle in spherical coordinates:Pair production threshold is reached, for given and , by solving (20) for , setting . After somewhat tedious integration over the allowed vectors (no masses can be neglected), one obtains for the decay rate, andfor the energy loss rate. In the high-energy limit, one may (somewhat trivially) rewrite the expressions as follows ():These results confirm that it is possible to use the tachyonic bispinor formalism [2–7] for the calculation of decay rates of tachyonic particles.

#### 5. Constraints on the Mass of a Tachyonic Neutrino

Our threshold relation (8) is based on a Lorentz-covariant dispersion relation. Only neutrinos with survive the possibility of generalized leptonic Čerenkov radiation over a sufficiently long path length. The hypothetical observation of an absence of neutrinos above some energy could thus be interpreted as a constraint on the neutrino mass. Let us assume a neutrino mass of eV, where is generally assumed to be of order unity or less. Then, threshold is reached for eV, , and GeV.

The IceCube experiment [36, 37] has observed 37 neutrinos having energies TeV during 3 years of data taking. Three of these events had energies PeV, and one (often referred to as “Big Bird”) had PeV. According to the IceCube collaboration [37], the spectrum of the 37 neutrinos is well fitted by a slope ~, which includes astrophysical as well as background atmospheric neutrinos, the latter being exclusively below PeV. However, their best fit to the spectrum predicts 3.1 additional events for PeV, and yet none were seen. Preliminary data for the fourth year includes 17 additional events, with none seen for PeV [38]. These facts suggest to the IceCube authors [36, 37] the possibility that there may be a cutoff for the spectrum for neutrinos above PeV. The hypothesis is given further support by models which show that the Glashow resonance [39] (resonant ) should add between zero and three times the number of events that appear in the interval as part of a broad peak centered around PeV [40]. While evidence for the cutoff is disputed and alternative explanations have been proposed [41], the significance of such a cutoff has been analyzed in the light of superluminal neutrinos [42, 43].

Let us add a few clarifying remarks here. First, we note that the plots in the paper [37] refer to the neutrino flux as a function of neutrino energy; the events were apparently sufficiently well reconstructed so that no excess neutrino energy in addition to the energy deposited inside the detector is expected. Our Figure 4 is based on Figure of [37]. Meanwhile, members of the IceCube collaboration have presented preliminary evidence for a throughgoing muon of energy PeV which could be interpreted as a decay product of a neutrino of even higher energy [44, 45]. If the throughgoing muon could indeed be assigned to an ultra-high-energy neutrino of nonatmospheric origin, then it would push the conceivable cutoff seen by IceCube to even higher energies, further constraining the tachyonic mass term of the relevant neutrino flavor. So far, the authors of [37] (see the right column on page 4 of [37]) observe that “this [the lack of high-energy events] may indicate, along with the slight excess in lower energy bins, either a softer spectrum or a cutoff at high energies.”

Assuming PeV we would find using (8) that , and we would, furthermore, find that eV (i.e., eV^{2}) for one or more of the three neutrino flavors (conceivably, the one with the smallest absolute value of ). A shifted cutoff [44, 45] of PeV would be consistent with a tachyonic neutrino mass of eV. One might object that it is not possible to have one (or more) tachyonic flavor masses () and satisfy both neutrino oscillation data and the recent findings from cosmology for the sum of the flavor masses; that is, eV [46, 47]. However, such consistency can be achieved using 3 active-sterile (tardyon-tachyon) neutrino pairs [48]. The curves in Figure 4 were generated using an assumed pure power law for flux beyond the assumed threshold, . We then use our formula (22b) for to find the modified spectrum. Good agreement is found with the IceCube data at a threshold PeV, although much more statistics will be needed to determine if the cutoff is real.

#### 6. Conclusions

Three main conclusions of the current investigation can be drawn. (i) As described in Section 2, the assumption of a Lorentz-covariant, tachyonic dispersion relation leads to tight bounds on conceivable tachyonic neutrino mass terms, for whatever neutrino flavor is causing the possible 2 PeV cutoff. The tachyonic decay rate due to LPCR is most conveniently calculated in the laboratory frame because of the space-like kinematics involved in the process, which leads to a nonunique time ordering of the trajectories, as discussed in Section 4. (ii) We may apply the formalism of the tachyonic bispinor solutions of the tachyonic Dirac equation [2–5] recently developed in [6, 7, 19] to the calculation of the tachyonic neutrino decay, as outlined in Section 4. (iii) A comparison of recent IceCube data with the results for the calculated tachyonic decay rates reveals that a tachyonic neutrino could possibly explain a possible sharp cutoff in IceCube data but only if the neutrino flavor involved has a very specific tachyonic mass. In a more general context, the calculation of tachyonic thresholds and decay rates based on Lorentz-covariant dispersion relations could be of phenomenological significance for string theories, some of which predict the existence of tachyons [49, 50]. The same is true for the precise calculation of the tail of the beta decay spectrum, which is influenced by a conceivably tachyonic neutrino mass term [51].

#### Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

#### Acknowledgments

This research has been supported by the National Science Foundation (Grant no. PHY-1403973).