Volume 378, Issues 32–33, 27 June 2014, Pages 2337–2344

# Tachyonic quantum densities of relativistic electron plasmas: Cherenkov spectra of γ-ray pulsars

## Highlights

Quantized tachyonic Cherenkov densities lead to subexponential spectral decay.

γ  -Ray spectral fits to Crab pulsar, PSR J1836+5925, PSR J0007+7303, PSR J2021+4026.

The polarization of γ-rays is analyzed in the quasiclassical regime and quantum limit.

Three degrees of polarization due to the negative mass-square of the Maxwell–Proca field.

Weibull decay of spectral tails caused by frequency scaling of the tachyon mass.

## Abstract

Tachyonic Cherenkov radiation in second quantization can explain the subexponential spectral tails of GeV γ  -ray pulsars (Crab pulsar, PSR J1836+5925, PSR J0007+7303, PSR J2021+4026) recently observed with the Fermi-LAT, VERITAS and MAGIC telescopes. The radiation is emitted by a thermal ultra-relativistic electron plasma. The Cherenkov effect is derived from a Maxwell–Proca field with negative mass-square in a dispersive spacetime. The frequency variation of the tachyon mass results in attenuation of the asymptotic Cherenkov energy flux, where is a decay constant related to the electron temperature and ρ   is the frequency scaling exponent of the tachyon mass. An exponent in the range 0<ρ<1 can reproduce the observed subexponential decay of the energy flux. For the Crab pulsar, we find ρ=0.81±0.02, inferred from the substantially weaker-than-exponential decay of its spectral tail measured by MAGIC over an extended energy range. The scaling exponent ρ determines whether the group velocity of the tachyonic γ-rays is sub- or superluminal.

## Keywords

• Quantized tachyonic Cherenkov densities;
• Frequency-dependent tachyon mass;
• Superluminal radiation in a permeable spacetime;
• Transversal/longitudinal Cherenkov energy flux;
• Tachyonic spectral fits to γ-ray pulsars;
• Subexponential Weibull decay of spectral tails

## 1. Introduction

Stunning high-precision spectra of several γ-ray pulsars which defy traditional radiation models have recently been recorded by the Fermi satellite and the terrestrial γ-ray telescopes MAGIC and VERITAS. An account of these developments can be found in the review . These spectra are difficult to explain, as the spectral tails decay more slowly than an exponential cutoff but faster than a power law, so that something in between these two extremes is needed.

Here, we will show that a Weibull decay factor exp(−(E/E0)δ), 0<δ<1, in the asymptotic energy flux can reproduce the very accurately measured subexponential decay of the spectral tails. To this end, we will perform spectral fits to four Fermi-LAT pulsars (including the Crab pulsar), covering a representative range of shape parameters δ  . First, however, we have to find a radiation model which produces an energy flux admitting asymptotic Weibull decay; this is the principal aim of this article. Time-honored electromagnetic radiation mechanisms such as inverse-Compton scattering or synchrotron and curvature radiation result in straight exponential cutoffs (δ=1) if the cross-section or radiation density is averaged over a relativistic equilibrated or non-thermal electronic source plasma.

We will invoke the tachyonic Cherenkov effect based on a Maxwell–Proca field theory with negative mass-square, and demonstrate that the Cherenkov flux density produced by an ultra-relativistic electron population decays subexponentially. This holds true in the quasiclassical regime as well as in the extreme quantum limit of the Cherenkov flux, with different shape parameters δ in the Weibull exponential. The spectral fits to the GeV pulsar spectra will be performed in the semiclassical regime, but the quantum limit is inescapable at sufficiently high frequency and can be manifested in TeV γ-rays.

Apart from subexponential Weibull decay, which is the main focus, there are also other features of tachyonic Cherenkov radiation quite different from electromagnetic radiation. For instance, three degrees of polarization, two transversal and one longitudinal due to the tachyonic mass-square of the radiation field. The Cherenkov emission angles differ for transversal and longitudinal γ-rays. Most importantly, the γ-rays can have a slightly superluminal group velocity depending on the Weibull shape parameter δ, and we will illustrate this with a pulsar spectrum.

In Section , we introduce the Lagrangian of the underlying field theory, a Proca field with negative mass-square, coupled to dispersive permeability tensors and an external spinor current. We discuss tachyonic radiation densities in second quantization, their polarization components and the effect of electron spin. We determine the frequency intervals in which Cherenkov radiation from an inertial Dirac electron current is possible, depending on the permeabilities in the Lagrangian and the electronic Lorentz factor, and derive the Cherenkov emission angles. The quantized electromagnetic Cherenkov densities [3], [4], [5], [6] and [7] are recovered in the limit of zero tachyon mass, and the classical tachyonic radiation densities [8] and [9] for small tachyon–electron mass ratio.

To relate the tachyonic Cherenkov densities to pulsar spectra, we have to average them over a relativistic electron plasma, cf. Section . We consider thermal equilibrium distributions as well as non-thermal power-law distributions for the radiating electron populations of the pulsars. The averaged differential flux densities are then fitted to the measured GeV pulsar spectra. There is a longitudinal radiation component, which is even more intense than the linearly polarized transversal radiation in the semiclassical regime. A second transversal degree of polarization emerges in the quantum regime, where the longitudinal energy flux weakens.

In Section , we study tachyonic flux densities in the quasiclassical regime. We explain the frequency variation of the tachyon mass and the tachyonic fine-structure constant, specializing to permeability tensors which give identical dispersion relations and group velocities for transversal and longitudinal quanta. Whether the group velocity is sub- or superluminal depends on the frequency variation of the tachyon mass, which is inferred from the decaying spectral tails of the pulsars.

In Section , we test the averaged tachyonic Cherenkov densities by performing spectral fits to the γ  -ray pulsars PSR J1836+5925, PSR J0007+7303, PSR J2021+4026 and the Crab pulsar [10] and [11]. In this way, the scaling exponent ρ   of the frequency-dependent tachyon mass ∝ωρ can be inferred. The decay of the spectral tails is weaker than exponential because of the Weibull decay factor , 0<ρ<1, in the asymptotic flux density, the decay exponent being related to the temperature of the ultra-relativistic electron plasma and the tachyonic mass amplitude. (The above mentioned Weibull slope is thus δ=1−ρ.) For ρ<1/2, the radiation is superluminal [12] and [13], as happens for pulsar PSR J1836+5925. In the case of the Crab pulsar, the tachyon mass admits a scaling exponent ρ close to one, so that the Weibull exponential approximates power-law decay of the spectral tail, and the radiation is slightly subluminal despite of the tachyonic mass-square in the dispersion relations, cf. Section 6. Apart from the tachyonic Cherenkov effect discussed here, the thermodynamics of superluminal signal transfer has been studied in Ref. , electromagnetic radiation by acceleration of superluminal charges in Ref. , and Cherenkov radiation from superluminally rotating light spots in Ref. .

## 2. Quantized tachyonic Cherenkov densities of relativistic electrons

The Cherenkov radiation densities stated below are derived from the Lagrangian

equation2.1
where is the Fourier transform of the field tensor Fμν=Aν,μ−Aμ,ν and and are the transforms of the 4-potential and external current. The tachyonic radiation field is modeled after electrodynamics, a Proca field with negative mass-square, minimally coupled to an electron current in a dispersive spacetime, the Minkowski metric being replaced by permeability tensors . is the frequency-dependent tachyonic mass-square. ( with the sign conventions in .) The first term in containing is analogous to the electrodynamic Lagrangian in a dielectric medium, in manifestly covariant notation. A second permeability tensor enters in the mass term, generating different dispersion relations and group velocities for transversal and longitudinal modes. The isotropic tensors are defined by positive dimensionless permeabilities ε(ω), μ(ω), μ0(ω) and ε0(ω),
equation2.2
with . In vacuum, ε=ε0=1 and μ=μ0=1 (Heaviside–Lorentz system). Greek indices are raised and lowered with the Minkowski metric ημν=diag(−1,1,1,1). The 3D field strengths are and . (εkij is the Levi-Civita tensor.) The inductive 4-potential reads and the inductive field tensor . The 3D inductions are , , so that the constitutive relations read , and , . The permeability tensor in Lagrangian couples the external current to the field,
equation2.3
The ‘dressed’ current in the Lagrangian amounts to a varying coupling constant if Ω0(ω) coincides with 1/Ω(ω)[8], so that , which is assumed from now on.

The external current is generated by a uniformly moving subluminal charge q   with mass m   and Lorentz factor γ  . When quantizing, we use an inertial Dirac current. The Cherenkov emission of the charge has two transversal and one longitudinal polarization components. The spectral densities in second quantization read, for the two linear transversal polarizations,

equation2.4
equation2.5
so that is the total transversal density. The various symbols in and are defined below. The longitudinal radiation component is
equation2.6
where m   denotes the electron mass, γ   the electronic Lorentz factor and mt(ω)>0 the frequency-dependent tachyon mass. The permeabilities ε  , μ  , μ0 and ε0 are positive. The dimensionless constant αt0=q2/(4πħc) is the tachyonic counterpart to the electric fine-structure constant e2/(4πħc)≈1/137. Ω2(ω) is the scale factor of the frequency-dependent tachyonic fine-structure constant αt(ω)=αt02(ω), see after . We introduce the rescaled tachyon mass
equation2.7
and write the generalized transversal and longitudinal mass-squares in densities (2.4), (2.5) and (2.6) as
equation2.8
These mass-squares must be positive for radiation to occur, see after . Finally, the factor in the spectral densities (2.4), (2.5) and (2.6) is the Heaviside step function with argument
equation2.9
where kT,L(ω) are the transversal/longitudinal wavenumbers defined by the dispersion relations
equation2.10
so that . The wavenumbers are identical only if the permeabilities satisfy ε0μ0=εμ. To derive the radiation densities (2.4), (2.5) and (2.6), it suffices to calculate the asymptotic transversal and longitudinal radiation fields in dipole approximation (see Eq. (3.1) of Ref. ) and to use the integral representation of the asymptotic flux vectors stated in Eq. (3.7) of Ref. , replacing the classical transversal/longitudinal current components in this integral representation by the respective matrix elements of the spinor current of the radiating inertial charge q   as explained in Refs. [17], [18], [19] and [20] for the vacuum case. The polarized power components are , where the upper integration boundary is the energy mγ   of the radiating charge. The total transversal power is PT=PT(1)+PT(2), and the total power radiated is P=PT+PL. If we use a spinless Klein–Gordon current as radiation source instead of the electronic Dirac current, the quantized radiation densities read and
equation2.11
equation2.12
Here, we use the same notation as in (2.4), (2.5) and (2.6). The transversal component in is thus exclusively due to the electron spin. The vacuum version of densities and has been derived in Ref. .

The zeros of the argument in the Heaviside function, cf. , determine the frequency intervals (at fixed γ  ) in which , i.e. the transversal and longitudinal spectrum radiated by an inertial spinning or spinless charge of mass m   with Lorentz factor γ>1. Alternatively, we may keep ω   fixed, so that the zeros of with respect to the second variable determine the γ   intervals in which a frequency ω   can be radiated. That is, the charge must have a Lorentz factor in these intervals to radiate at a given frequency ω  . In addition, the energy condition mγ≥ω has to be satisfied. We substitute the dispersion relations into so that inequality is equivalent to

equation2.13
If , the second-order polynomial has a root satisfying as well as , which is given by
equation2.14
Inequality in is then equivalent to .

Inequality cannot be satisfied for any γ≥1 if is negative. [Inequality is not satisfied for γ→∞ if . Thus it can only hold for γ   below . Inequality is equivalent to if . This polynomial has a double-root , so that the inequality cannot be satisfied. Accordingly, if .] Thus we only need to consider frequency intervals in which the mass-squares are positive, as Cherenkov radiation cannot occur for negative . If is positive and negative, only transversal emission occurs at this frequency, and vice versa.

By expanding Eq. in the parameter MT,L/(2m)≪1 in leading order, we find the limit obtained in Ref. for classical Cherenkov radiation. The mass m   of the subluminal radiating charge does not enter in the classical Cherenkov densities. In fact, the classical densities can be recovered by performing the limit m→∞ in the quantum densities (2.4), (2.5) and (2.6). The quantized electromagnetic Cherenkov densities are recovered in the limit of vanishing tachyon mass, mt(ω)→0, if we replace the tachyonic fine-structure constant q2/(4πΩ2) by the electric counterpart e2/(4π), cf. after . In this electromagnetic limit, the permeabilities have to satisfy εμ>1, otherwise the generalized mass-square is not positive, cf. , whereas tachyonic Cherenkov radiation is quite possible for 0<εμ≤1 and 0<ε0μ0≤1.

Applying energy–momentum conservation, we find the Cherenkov emission angle of transversal/longitudinal tachyonic quanta as

equation2.15
with electron energy E=mγ and mass-square in . is the angle between the tachyonic wave vector and the velocity of the radiating charge. Since E≥ω, is positive, so that varies in the interval , and the radiation is thus emitted into a forward cone. This holds true for positive permeabilities; emission into a backward cone can occur in the case of a negative refractive index . We also note that inequality is equivalent to , cf. . For E→m, the right-hand side of Eq. diverges; the minimal electron energy for emission is thus obtained by solving , and we find , cf. . The energy conditions and hold, and . In the opposite limit, E→∞, we find the maximal emission angle
equation2.16
so that . Finally, the emission angle for classical tachyonic Cherenkov radiation is recovered in the limit m→∞, , where is the velocity of the radiating subluminal charge and kT,L are the tachyonic wavenumbers .

## 3. Polarized tachyonic flux densities of an ultra-relativistic electron plasma

We average the tachyonic Cherenkov densities (2.4), (2.5) and (2.6) over an electronic power-law distribution [22], [23], [24] and [25],

equation3.1
parametrized with the electronic Lorentz factor γ  . Aα,β is a dimensionless normalization constant, β=m/(kBT) is the dimensionless temperature parameter and m   the electron mass. A Maxwell–Boltzmann equilibrium distribution requires the electron index α=−2. The spectral average of the radiation densities is carried out as
equation3.2
where is the minimal Lorentz factor in . We will write BT(j),L for . The unpolarized radiation density is 〈pT+L(ω)〉α,β=BT+L, with BT+L=BT+BL and BT=BT(1)+BT(2). The differential energy flux and the differential number flux dNT(j),L/dω are related to the spectral functions by
equation3.3
where d   is the distance to the source. It is customary in high-frequency bands to write E   for the energy ω   of the γ  -rays (ħ=c=1), if no confusion with the electron energy E=mγ can arise. We have rescaled with a power ωk to make steep spectral slopes better visible, using k=1 in the spectral fits. The total differential flux is obtained by adding the polarization components, substituting BT+L into .

To make the flux densities more explicit, we start by defining the combined amplitude and the rescaled frequency and mass parameters

equation3.4
with MT,L(ω) in . The constants Aα,β, β   and α   denote normalization factor, temperature parameter and electron index of the source distribution . αt0=q2/(4π), cf. after , d   is the source distance, cf. , and m   the electron mass. The minimal electronic Lorentz factor can now be written as
equation3.5
The quasiclassical regime is attained for , the classical limit being (that is m→∞, so that the electron mass drops out in the radiation densities), and the quantum regime is realized in the opposite limit, , cf. Section .

The spectral average can be expressed in terms of incomplete gamma functions. The averaged transversal polarization components and read

equation3.6
and
equation3.7
where we have to substitute , cf. . The total transversal flux density is . The longitudinal density reads
equation3.8
where we substitute in . For a thermal electron distribution with α=−2, the spectral functions simplify since Γ(1,βγ)=e−βγ. The unpolarized flux is .

In the semiclassical regime , cf. , it is convenient to factorize , where, cf. ,

equation3.9
We substitute βγ=xT,LηT,L into the averaged spectral densities (3.6), (3.7) and (3.8) and note that ηT,L≥1 and ηT,L is also bounded from above for and arbitrary . We will use xT,L as expansion parameter, which can be large or small depending on 0<β<∞ and .

## 4. Coinciding transversal and longitudinal dispersion relations in the quasiclassical regime

Identical dispersion relations require the identity εμ=ε0μ0, cf. after . We can then drop the T and L subscripts of MT,L, and , cf. and , and also in and , writing βγmin(ω)=xη and replacing βγ by in the spectral densities (3.6), (3.7) and (3.8).

We scale the magnetic permeability μ   into the fine-structure constant αt(ω), cf. after ,

equation4.1
and specify the frequency dependence of and the rescaled tachyon mass in as power laws [32] and [33],
equation4.2
where and are constant positive amplitudes, and σ   and ρ   are real exponents to be determined from spectral fits. Substituting into the wave numbers , we find the tachyonic group velocity as
equation4.3
so that υgr is superluminal for tachyonic mass exponents ρ<1/2. It is also evident from and that the frequency dependence of the tachyon mass as well as the frequency-dependent factor Ω2 of the tachyonic fine-structure constant can be absorbed into the permeabilities. That is, we can use a constant tachyon mass and fine-structure constant, employing dispersive permeabilities [9], [34], [35] and [36].

The amplitude factor atμ/Ω2 in flux densities (3.6), (3.7) and (3.8) reads, cf. , and ,

equation4.4
We further specialize the permeabilities as εμ=ε0μ0=1, so that the variables in and simplify,
equation4.5
The unpolarized flux is obtained by adding the averaged spectral densities (3.6), (3.7) and (3.8),
equation4.6
where we substitute the variables, cf. and ,
equation4.7
equation4.8
The independent fitting parameters are , , , σ   and ρ  . We also note η→1 for . In a finite frequency interval defined by the available data sets, the semiclassical regime is realized for sufficiently small tachyon–electron mass ratio , cf. . If holds in addition to (which means , cf. and ), then is not any more a fitting parameter, as it drops out of the flux density . We introduce the parameters
equation4.9
η0=σ+2ρ−1+k,
and replace the first two ratios in the flux density by an identical expression,
equation4.10
so that we can use A and η in as fitting parameters instead of and σ  . Finally, in the case of a thermal source plasma with electron index α=−2, the unpolarized flux simplifies to
equation4.11
with and η0, cf. .

## 5. Tachyonic Cherenkov fits to γ-ray pulsars in the GeV band

The quasiclassical regime is realized if the tachyon mass is small, , and the electron gas ultra-relativistic, β≪1, so that the rescaled temperature variable in stays moderate. We can then approximate , cf. , η∼1, cf. , and , , cf. . Performing these approximations in the total flux density, we obtain, cf. and ,

equation5.1
The transversal polarization component in vanishes in this limit, so that the transversal radiation is linearly polarized; is given by with the two factors of 2 dropped, and , cf. . For a thermal source plasma with electron index α=−2,
equation5.2
The high-frequency limit of flux density reads
equation5.3
The attenuation factor with tachyonic mass exponent 0<ρ<1 causes the subexponential decay of the spectral tails determined by the Weibull slope δ=1−ρ. The leading order in is entirely due to the longitudinally polarized flux component; the transversal radiation contributes the ratio to the next-to-leading order.

The low-frequency limit of the unpolarized flux is

equation5.4
which remains valid for integer α≥−1 if epsilon expansion is applied [12] and [37]. For electron indices α<1, the leading order of reads
equation5.5
with scaling exponent η0=σ+2ρ−1+k, cf. . The amplitude ratio of the low- and high-frequency limits is , cf. . In the low-frequency regime, the transversal and longitudinal flux components have equal strength, each contributing A0ωη0/2 to the flux .

In this section, we have studied the limit , , cf. . In particular, the low-frequency limit in and was derived under the condition . This is the relevant limit for γ  -ray spectra. In the radio band, a different realization of the low-frequency limit of flux density applies, namely and , see Ref. . In this case, we can approximate x∼β, cf. , and η∼1, cf. , to find the low-frequency limit of density as , with amplitude

equation5.6
This replaces the flux limit in a frequency regime where . In the GeV γ-ray band, we employ flux density and its high- and low-frequency limits and .

In the spectral fits of the γ  -ray pulsars depicted in Fig. 1, Fig. 2, Fig. 3 and Fig. 4, we use a thermal electron index α=−2. First, we perform a low-frequency power-law fit to estimate A0 and η0. In this way, we also find initial estimates of A and η, see and after . Estimates of and ρ   are then obtained by fitting the spectral tail . Finally, using these estimates as initial guess for the fitting parameters A, η, and ρ  , we perform a least-squares χ2 fit with flux density , cf. .

Table 1.

Parameters of the tachyonic energy flux of the pulsars in Fig. 1, Fig. 2, Fig. 3 and Fig. 4. The γ  -rays are radiated by a thermal plasma with electron index α   = −2, cf. . The fine-structure scaling exponent σ=η−2ρ is inferred from the scaling exponent ρ   of the tachyon mass, cf. , and from the exponent η in determining the slope of the initial power-law ascent of the flux density. The fitting parameters are η, ρ  , the decay exponent , cf. , and the flux amplitude A, cf. . The spectral fits depicted in the figures are based on the differential flux density , cf. and . The Weibull shape parameter determining the subexponential decay of the spectral tails is 1 − ρ  , cf. . The indicated standard deviations are extracted from the covariance matrix of the χ2 functional.

σ ρ η
PSR J1836 + 5925 −0.013 0.419 ± 0.029 0.826 ± 0.045 2.10 ± 0.19 6.04±0.60×10−8
PSR J0007 + 7303 −0.259 0.504 ± 0.042 0.749 ± 0.062 1.73 ± 0.29 2.57±0.36×10−8
PSR J2021 + 4026 −0.436 0.630 ± 0.051 0.824 ± 0.183 3.93 ± 0.10 2.55±1.69×10−7
Crab pulsar −0.824 0.809 ± 0.019 0.795 ± 0.159 6.83 ± 1.45
Full-size table

## 6. Conclusion: spectral decay and the extreme quantum regime

The basic results have been summarized in the introduction. Here, we conclude with some remarks on Weibull decay of spectral tails, polarization and the quantum limit. We have not discussed the extreme quantum limit of the flux densities (3.6), (3.7) and (3.8) (realized by large tachyon–electron mass ratios , cf. ) in any detail, as the spectral fits in Fig. 1, Fig. 2, Fig. 3 and Fig. 4 are performed in the quasiclassical regime.

### 6.1. Sub- and superexponential decay of spectral tails

The second-quantized flux densities (3.6), (3.7) and (3.8) decay for large electronic Lorentz factors, since (βγ)α+2Γ(−α−1,βγ)∼e−βγ. The minimal transversal/longitudinal Lorentz factor in increases at least linearly with ω  , so that the ultimate decay of the flux densities is exponential or superexponential for ω→∞. In contrast, if ω   varies in a finite interval (defined by data sets) and the tachyonic mass amplitude is sufficiently small and the electron temperature high, then the approximation used in and is applicable in this interval, and the decay factor in the flux density is subexponential for 0<ρ<1 and superexponential for negative ρ  . Subexponential decay within a finite interval can be realized in the quasiclassical regime, where the mass ratios defined in and are small, but also in the opposite limit, the extreme quantum regime where , see the following remark.

### 6.2. Spectral decay and polarization in the quantum limit

Instead of using the factorization designed for the quasiclassical regime, we split the minimal Lorentz factor as , where, cf. ,

equation6.1
and substitute βγ=yT,LκT,L into the flux densities (3.6), (3.7) and (3.8). The factor κT,L is larger than one and also bounded from above for (quantum regime) and arbitrary , cf. . The expansion parameter in the flux densities is now yT,L, which can be large or small, depending on β  , and . As in Section , we assume permeabilities constrained by εμ=ε0μ0=1, which give identical mass-squares for transversal and longitudinal quanta, cf. , and specify the frequency scaling of the tachyon mass and the tachyonic fine-structure constant as stated in . We can then drop the T and L subscripts in and consider a finite frequency interval in which holds, cf. , so that with , uniformly within this interval. For yκ≫1, the decay factor exp(−yκ) of the spectral tails is thus subexponential if 0<ρ<1 and superexponential for ρ>1. (The quasiclassical counterpart is the Weibull exponential in , albeit in the opposite limit , .) For and , we find strictly exponential decay with linear yκ∼βω/m, cf. and . The polarization changes in the quantum regime; there are two transversal degrees, cf. and , which are of comparable magnitude for yκ≫1, whereas the longitudinal flux is weaker by a factor 1/(yκ). Polarization in the quasiclassical regime has been discussed in Section .

### 6.3. Subexponential decay approximating power-law decay in the quasiclassical regime

If the scaling exponent ρ   of the tachyon mass in is close to one, 0<1−ρ=ε≪1, then the Weibull exponential in the flux asymptotics admits the epsilon expansion

equation6.2
If the decay exponent is large, so that is moderate, this expansion approximates power-law decay , leading to slightly curved spectral tails in double-logarithmic plots, see the spectral fit of the Crab pulsar in and . Alternatively, one may consider a logarithmic frequency dependence of the tachyon mass from the outset, replacing the power law in by . The semiclassical expansion parameter in the flux densities is then replaced by , cf. the beginning of Section , so that the Weibull exponential in becomes a power law . In contrast, the spectral fits of the pulsars in Fig. 1, Fig. 2, Fig. 3 and Fig. 4 are based on the tachyonic mass scaling relation , resulting in subexponential Weibull decay of the GeV spectral tails because of the attenuation factor in the high-frequency asymptotics of the Cherenkov flux.

## References

• [1]
• Annu. Rev. Astron. Astrophys., 52 (2014)

• [2]
• J. Appl. Mech., 18 (1951), p. 293

•  |
• [3]
• Transition Radiation and Transition Scattering

• Hilger, Bristol (1990)

• [4]
• Phys. Usp., 39 (1996), p. 973

• |  |
• [5]
• Phys. Usp., 45 (2002), p. 341

• |  |
• [6]

• Kluwer, Dordrecht (2004)

• [7]
• Proc. R. Soc. A, Math. Phys. Eng. Sci., 462 (2006), p. 689

•  |
• [8]
• Phys. Lett. A, 377 (2013), p. 3247

• | |  |
• [9]
• Europhys. Lett., 106 (2014), p. 39001

• |  |
• [10]
• Astrophys. J. Suppl. Ser., 208 (2013), p. 17

• [11]
• Astron. Astrophys., 540 (2012), p. A69

• |  |
• [12]
• Ann. Phys., 322 (2007), p. 677

• | |  |
• [13]
• Europhys. Lett., 102 (2013), p. 61002

• |  |
• [14]
• Sov. Phys. Dokl., 5 (1961), p. 782

• [15]
• Europhys. Lett., 16 (1991), p. 121

•  |
• [16]
• Phys. Scr. T, 2A (1982), p. 182

• [17]
• Eur. Phys. J. C, 49 (2007), p. 815

• |  |
• [18]
• Physica A, 320 (2003), p. 329

• | |  |
• [19]
• J. Phys. A, Math. Gen., 38 (2005), p. 2201

• |  |
• [20]
• Eur. Phys. J. D, 32 (2005), p. 241

• |  |
• [21]
• Mater. Today, 14 (2011), p. 34

• | | |  |
• [22]
• Physica A, 387 (2008), p. 3480

• | |  |
• [23]
• Physica B, 405 (2010), p. 1022

• | |  |
• [24]
• Europhys. Lett., 104 (2013), p. 19001

• |  |
• [25]
• Physica A, 394 (2014), p. 110

• | |  |
• [26]
• Astrophys. J., 712 (2010), p. 1209

• |  |
• [27]
• Astrophys. J., 744 (2012), p. 146

• |  |
• [28]
• Astrophys. J., 700 (2009), p. 1059

• |  |
• [29]
• Astrophys. J., 708 (2010), p. 1254

• |  |
• [30]
• Astrophys. J., 742 (2011), p. 43

• |  |
• [31]
• Science, 334 (2011), p. 69

•  |
• [32]
• Eur. Phys. J. C, 69 (2010), p. 241

• |  |
• [33]
• Europhys. Lett., 89 (2010), p. 39002

• |  |
• [34]
• Physica A, 307 (2002), p. 375

• | |  |
• [35]
• Opt. Commun., 282 (2009), p. 1710

• | |  |
• [36]
• Phys. Lett. A, 377 (2013), p. 945

• | |  |
• [37]
• Appl. Math. Comput., 225 (2013), p. 228

• | |  |