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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

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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 [1]. 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 [2] decay factor exp(−(E/E₀)^δ)$\exp (-{(E/E_0)}^{\delta })$, 0<δ<1$0<\delta <1$, 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$\delta =1$) 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 2, 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 3. 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 4, 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 5, we test the averaged tachyonic Cherenkov densities by performing spectral fits to the γ  -ray pulsars PSR J1836+5925$\mathrm{J}1836+5925$, PSR J0007+7303$\mathrm{J}0007+7303$, PSR J2021+4026$\mathrm{J}2021+4026$ and the Crab pulsar [10] and [11]. In this way, the scaling exponent ρ   of the frequency-dependent tachyon mass ∝ω^ρ$\propto {\omega }^{\rho }$ can be inferred. The decay of the spectral tails is weaker than exponential because of the Weibull decay factor $\exp (-\hat{\beta }{\omega }^{1-\rho })$, 0<ρ<1$0<\rho <1$, in the asymptotic flux density, the decay exponent $\hat{\beta }$ 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$\delta =1-\rho$.) For ρ<1/2$\rho <1/2$, the radiation is superluminal [12] and [13], as happens for pulsar PSR J1836+5925$\mathrm{J}1836+5925$. 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. [14], electromagnetic radiation by acceleration of superluminal charges in Ref. [15], and Cherenkov radiation from superluminally rotating light spots in Ref. [16].

2. Quantized tachyonic Cherenkov densities of relativistic electrons

The Cherenkov radiation densities stated below are derived from the Lagrangian [9]

where ${\hat{F}}_{\mu \nu }(\mathbf{x},\omega )$ is the Fourier transform ${\int }_{-\infty }^{\infty }{F}_{\mu \nu }(\mathbf{x},t){\mathrm{e}}^{\mathrm{i}\omega t}\mathrm{d}t$ of the field tensor F_μν=A_ν,μ−A_μ,ν$F_{\mu \nu }=A_{\nu ,\mu }-A_{\mu ,\nu }$ and ${\hat{A}}_{\mu }$ and ${\hat{j}}_{\nu }(\mathbf{x},\omega )$ 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 $g_{F,A,J}^{\mu \nu }(\omega )$. $m_{\mathrm{t}}^2(\omega )$ is the frequency-dependent tachyonic mass-square. ($m_{\mathrm{t}}^2>0$ with the sign conventions in (2.1).) The first term in (2.1) containing $g_F^{\mu \nu }$ is analogous to the electrodynamic Lagrangian in a dielectric medium, in manifestly covariant notation. A second permeability tensor $g_A^{\mu \nu }$ enters in the mass term, generating different dispersion relations and group velocities for transversal and longitudinal modes. The isotropic tensors $g_{A,F}^{\mu \nu }$ are defined by positive dimensionless permeabilities ε(ω)$\epsilon (\omega )$, μ(ω)$\mu (\omega )$, μ₀(ω)${\mu }_0(\omega )$ and ε₀(ω)${\epsilon }_0(\omega )$, with $g_A^{0i}=g_F^{0i}=0$. In vacuum, ε=ε₀=1$\epsilon ={\epsilon }_0=1$ and μ=μ₀=1$\mu ={\mu }_0=1$ (Heaviside–Lorentz system). Greek indices are raised and lowered with the Minkowski metric η_μν=diag(−1,1,1,1)${\eta }_{\mu \nu }=\mathrm{diag}(-1,1,1,1)$. The 3D field strengths are ${\hat{E}}_k={\hat{F}}_{k0}$ and ${\hat{B}}^k={\epsilon }^{kij}{\hat{F}}_{ij}/2$. (ε^kij${\epsilon }^{kij}$ is the Levi-Civita tensor.) The inductive 4-potential reads ${\hat{C}}^{\mu }=g_A^{\mu \nu }{\hat{A}}_{\nu }$ and the inductive field tensor ${\hat{H}}^{\mu \nu }=g_F^{\mu \alpha }{g}_F^{\nu \beta }{\hat{F}}_{\alpha \beta }$. The 3D inductions are ${\hat{D}}^l={\hat{H}}^{0l}$, ${\hat{H}}_i={\epsilon }_{ikl}{\hat{H}}^{kl}/2$, so that the constitutive relations read $\hat{\mathbf{D}}=\epsilon \hat{\mathbf{E}}$, $\hat{\mathbf{B}}=\mu \hat{\mathbf{H}}$ and $\hat{\mathbf{A}}={\mu }_0\hat{\mathbf{C}}$, ${\hat{C}}_0={\epsilon }_0{\hat{A}}_0$. The permeability tensor $g_J^{\mu \nu }$ in Lagrangian (2.1) couples the external current to the field, The ‘dressed’ current ${\hat{j}}_{\Omega }^{\mu }=g_J^{\mu \nu }{\hat{j}}_{\nu }$ in the Lagrangian amounts to a varying coupling constant if Ω₀(ω)${\Omega }_0(\omega )$ coincides with 1/Ω(ω)$1/\Omega (\omega )$[8], so that ${\hat{j}}_{\Omega }^{\mu }={\hat{j}}^{\mu }/\Omega$, which is assumed from now on.

The external current ${\hat{j}}^{\mu }$ 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,

so that $p_{\mathrm{spin}}^{\mathrm{T}}=p_{\mathrm{spin}}^{\mathrm{T}(1)}+p_{\mathrm{spin}}^{\mathrm{T}(2)}$ is the total transversal density. The various symbols in (2.4) and (2.5) are defined below. The longitudinal radiation component is where m   denotes the electron mass, γ   the electronic Lorentz factor and m_t(ω)>0$m_{\mathrm{t}}(\omega )>0$ the frequency-dependent tachyon mass. The permeabilities ε  , μ  , μ₀${\mu }_0$ and ε₀${\epsilon }_0$ are positive. The dimensionless constant α_t0=q²/(4πħc)${\alpha }_{\mathrm{t}0}=q^2/(4\pi ħc)$ is the tachyonic counterpart to the electric fine-structure constant e²/(4πħc)≈1/137$e^2/(4\pi ħc)\approx 1/137$. Ω²(ω)${\Omega }^2(\omega )$ is the scale factor of the frequency-dependent tachyonic fine-structure constant α_t(ω)=α_t0/Ω²(ω)${\alpha }_{\mathrm{t}}(\omega )={\alpha }_{\mathrm{t}0}/{\Omega }^2(\omega )$, see after (2.3). We introduce the rescaled tachyon mass and write the generalized transversal and longitudinal mass-squares $M_{\mathrm{T},\mathrm{L}}^2$ in densities (2.4), (2.5) and (2.6) as These mass-squares must be positive for radiation to occur, see after (2.14). Finally, the factor $\Theta (D_{\mathrm{t}}^{\mathrm{T},\mathrm{L}})$ in the spectral densities (2.4), (2.5) and (2.6) is the Heaviside step function with argument where k_T,L(ω)$k_{\mathrm{T},\mathrm{L}}(\omega )$ are the transversal/longitudinal wavenumbers defined by the dispersion relations so that $k_{\mathrm{T},\mathrm{L}}^2={\omega }^2+M_{\mathrm{T},\mathrm{L}}^2$. The wavenumbers (2.10) are identical only if the permeabilities satisfy ε₀μ₀=εμ${\epsilon }_0{\mu }_0=\epsilon \mu$. 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. [8]) and to use the integral representation of the asymptotic flux vectors stated in Eq. (3.7) of Ref. [8], 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 $P^{\mathrm{T}(j),\mathrm{L}}={\int }_0^{m\gamma }{p}_{\mathrm{spin}}^{\mathrm{T}(j),\mathrm{L}}(\omega )\mathrm{d}\omega$, where the upper integration boundary is the energy mγ   of the radiating charge. The total transversal power is P^T=P^T(1)+P^T(2)$P^{\mathrm{T}}=P^{\mathrm{T}(1)}+P^{\mathrm{T}(2)}$, and the total power radiated is P=P^T+P^L$P=P^{\mathrm{T}}+P^{\mathrm{L}}$. If we use a spinless Klein–Gordon current as radiation source instead of the electronic Dirac current, the quantized radiation densities read $p_{\mathrm{KG}}^{\mathrm{T}(1)}(\omega )=0$ and Here, we use the same notation as in (2.4), (2.5) and (2.6). The transversal component $p_{\mathrm{spin}}^{\mathrm{T}(1)}$ in (2.4) is thus exclusively due to the electron spin. The vacuum version of densities (2.11) and (2.12) has been derived in Ref. [18].

The zeros of the argument $D_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )$ in the Heaviside function, cf. (2.9), determine the frequency intervals (at fixed γ  ) in which $D_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )>0$, i.e. the transversal and longitudinal spectrum radiated by an inertial spinning or spinless charge of mass m   with Lorentz factor γ>1$\gamma >1$. Alternatively, we may keep ω   fixed, so that the zeros of $D_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )$ with respect to the second variable determine the γ   intervals $D_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )>0$ 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$m\gamma \ge \omega$ has to be satisfied. We substitute the dispersion relations (2.10) into $D_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )$ so that inequality $D_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )\ge 0$ is equivalent to

If $M_{\mathrm{T},\mathrm{L}}^2(\omega )>0$, the second-order polynomial ${\mathrm{\Delta }}_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )$ has a root $\gamma ={\gamma }_{\min }^{\mathrm{T},\mathrm{L}}$ satisfying ${\gamma }_{\min }^{\mathrm{T},\mathrm{L}}\ge 1$ as well as $m{\gamma }_{\min }^{\mathrm{T},\mathrm{L}}\ge \omega$, which is given by Inequality ${\mathrm{\Delta }}_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )\ge 0$ in (2.13) is then equivalent to $\gamma \ge {\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega )$.

Inequality (2.13) cannot be satisfied for any γ≥1$\gamma \ge 1$ if $M_{\mathrm{T},\mathrm{L}}^2$ is negative. [Inequality (2.13) is not satisfied for γ→∞$\gamma \to \infty$ if $M_{\mathrm{T},\mathrm{L}}^2<0$. Thus it can only hold for γ   below ${\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega )$. Inequality ${\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega ,M_{\mathrm{T},\mathrm{L}}^2)>1$ is equivalent to $M_{\mathrm{T},\mathrm{L}}^4+4m\omega M_{\mathrm{T},\mathrm{L}}^2+4m^2{\omega }^2<0$ if $M_{\mathrm{T},\mathrm{L}}^2<0$. This polynomial has a double-root $M_{\mathrm{T},\mathrm{L}}^2=-2m\omega$, so that the inequality cannot be satisfied. Accordingly, ${\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega )<1$ if $M_{\mathrm{T},\mathrm{L}}^2<0$.] Thus we only need to consider frequency intervals in which the mass-squares $M_{\mathrm{T},\mathrm{L}}^2(\omega )$ are positive, as Cherenkov radiation cannot occur for negative $M_{\mathrm{T},\mathrm{L}}^2$. If $M_{\mathrm{T}}^2(\omega )$ is positive and $M_{\mathrm{L}}^2(\omega )$ negative, only transversal emission occurs at this frequency, and vice versa.

By expanding Eq. (2.14) in the parameter M_T,L/(2m)≪1$M_{\mathrm{T},\mathrm{L}}/(2m)\ll 1$ in leading order, we find the limit ${\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega )\sim \sqrt{1+{\omega }^2/M_{\mathrm{T},\mathrm{L}}^2}$ obtained in Ref. [8] 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 [8] can be recovered by performing the limit m→∞$m\to \infty$ in the quantum densities (2.4), (2.5) and (2.6). The quantized electromagnetic Cherenkov densities [7] are recovered in the limit of vanishing tachyon mass, m_t(ω)→0$m_{\mathrm{t}}(\omega )\to 0$, if we replace the tachyonic fine-structure constant q²/(4πΩ²)$q^2/(4\pi {\Omega }^2)$ by the electric counterpart e²/(4π)$e^2/(4\pi )$, cf. after (2.6). In this electromagnetic limit, the permeabilities have to satisfy εμ>1$\epsilon \mu >1$, otherwise the generalized mass-square $M_{\mathrm{T}}^2$ is not positive, cf. (2.8), whereas tachyonic Cherenkov radiation is quite possible for 0<εμ≤1$0<\epsilon \mu \le 1$ and 0<ε₀μ₀≤1$0<{\epsilon }_0{\mu }_0\le 1$.

Applying energy–momentum conservation, we find the Cherenkov emission angle ${\theta }_{\mathrm{T},\mathrm{L}}$ of transversal/longitudinal tachyonic quanta as

with electron energy E=mγ$E=m\gamma$ and mass-square $M_{\mathrm{T},\mathrm{L}}^2$ in (2.8). ${\theta }_{\mathrm{T},\mathrm{L}}$ is the angle between the tachyonic wave vector and the velocity of the radiating charge. Since E≥ω$E\ge \omega$, $\cos {\theta }_{\mathrm{T},\mathrm{L}}$ is positive, so that ${\theta }_{\mathrm{T},\mathrm{L}}$ varies in the interval $0\le {\theta }_{\mathrm{T},\mathrm{L}}<\pi /2$, 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 [21]. We also note that inequality $\cos {\theta }_{\mathrm{T},\mathrm{L}}\le 1$ is equivalent to $2E{D}_{\mathrm{t}}^{\mathrm{T},\mathrm{L}}(\omega ,\gamma )\ge 0$, cf. (2.9). For E→m$E\to m$, the right-hand side of Eq. (2.15) diverges; the minimal electron energy $E_{\min }^{\mathrm{T},\mathrm{L}}$ for emission is thus obtained by solving $\cos {\theta }_{\mathrm{T},\mathrm{L}}=1$, and we find $E_{\min }^{\mathrm{T},\mathrm{L}}=m{\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega )$, cf. (2.14). The energy conditions $E_{\min }^{\mathrm{T},\mathrm{L}}\ge m$ and $E_{\min }^{\mathrm{T},\mathrm{L}}\ge \omega$ hold, and ${\theta }_{\mathrm{T},\mathrm{L}}(E_{\min }^{\mathrm{T},\mathrm{L}})=0$. In the opposite limit, E→∞$E\to \infty$, we find the maximal emission angle so that $0\le {\theta }_{\mathrm{T},\mathrm{L}}\le {\theta }_{\max }^{\mathrm{T},\mathrm{L}}<\pi /2$. Finally, the emission angle for classical tachyonic Cherenkov radiation is recovered in the limit m→∞$m\to \infty$, $\cos {\theta }_{\mathrm{T},\mathrm{L}}^{\mathrm{cl}}=\omega /(k_{\mathrm{T},\mathrm{L}}\upsilon )$, where $\upsilon =\sqrt{1-1/{\gamma }^2}$ is the velocity of the radiating subluminal charge and k_T,L$k_{\mathrm{T},\mathrm{L}}$ are the tachyonic wavenumbers (2.10).

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],

parametrized with the electronic Lorentz factor γ  . A_α,β$A_{\alpha ,\beta }$ is a dimensionless normalization constant, β=m/(k_BT)$\beta =m/(k_{\mathrm{B}}T)$ is the dimensionless temperature parameter and m   the electron mass. A Maxwell–Boltzmann equilibrium distribution requires the electron index α=−2$\alpha =-2$. The spectral average of the radiation densities is carried out as where ${\gamma }_{\min }^{\mathrm{T},\mathrm{L}}$ is the minimal Lorentz factor in (2.14). We will write B^T(j),L$B^{\mathrm{T}(j),\mathrm{L}}$ for $B^{\mathrm{T}(j),\mathrm{L}}(\omega ,{\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega ))$. The unpolarized radiation density is 〈p^T+L(ω)〉_α,β=B^T+L${〈p^{\mathrm{T}+\mathrm{L}}(\omega )〉}_{\alpha ,\beta }=B^{\mathrm{T}+\mathrm{L}}$, with B^T+L=B^T+B^L$B^{\mathrm{T}+\mathrm{L}}=B^{\mathrm{T}}+B^{\mathrm{L}}$ and B^T=B^T(1)+B^T(2)$B^{\mathrm{T}}=B^{\mathrm{T}(1)}+B^{\mathrm{T}(2)}$. The differential energy flux $F_{\omega }^{\mathrm{T}(j),\mathrm{L}}$ and the differential number flux dN^T(j),L/dω$\mathrm{d}{N}^{\mathrm{T}(j),\mathrm{L}}/\mathrm{d}\omega$ are related to the spectral functions $B^{\mathrm{T}(j),\mathrm{L}}(\omega ,{\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega ))$ by 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$ħ=c=1$), if no confusion with the electron energy E=mγ$E=m\gamma$ can arise. We have rescaled $F_{\omega }^{\mathrm{T}(j),\mathrm{L}}$ with a power ω^k${\omega }^k$ to make steep spectral slopes better visible, using k=1$k=1$ in the spectral fits. The total differential flux ${\omega }^k{F}_{\omega }^{\mathrm{T}+\mathrm{L}}[{(\text{GeV})}^k{\text{cm}}^{-2}{\text{s}}^{-1}]$ is obtained by adding the polarization components, substituting B^T+L$B^{\mathrm{T}+\mathrm{L}}$ into (3.3).

To make the flux densities (3.3) more explicit, we start by defining the combined amplitude and the rescaled frequency and mass parameters

with M_T,L(ω)$M_{\mathrm{T},\mathrm{L}}(\omega )$ in (2.8). The constants A_α,β$A_{\alpha ,\beta }$, β   and α   denote normalization factor, temperature parameter and electron index of the source distribution (3.1). α_t0=q²/(4π)${\alpha }_{\mathrm{t}0}=q^2/(4\pi )$, cf. after (2.6), d   is the source distance, cf. (3.3), and m   the electron mass. The minimal electronic Lorentz factor (2.14) can now be written as The quasiclassical regime is attained for ${\hat{M}}_{\mathrm{T},\mathrm{L}}(\omega )\ll 1$, the classical limit being ${\hat{M}}_{\mathrm{T},\mathrm{L}}=0$ (that is m→∞$m\to \infty$, so that the electron mass drops out in the radiation densities), and the quantum regime is realized in the opposite limit, ${\hat{M}}_{\mathrm{T},\mathrm{L}}\gg 1$, cf. Section 6.

The spectral average (3.2) can be expressed in terms of incomplete gamma functions. The averaged transversal polarization components (2.4) and (2.5) read

and where we have to substitute $\gamma ={\gamma }_{\min }^{\mathrm{T}}(\omega )$, cf. (3.5). The total transversal flux density is $F_{\omega }^{\mathrm{T}}=F_{\omega }^{\mathrm{T}(1)}+F_{\omega }^{\mathrm{T}(2)}$. The longitudinal density reads where we substitute $\gamma ={\gamma }_{\min }^{\mathrm{L}}(\omega )$ in (3.5). For a thermal electron distribution with α=−2$\alpha =-2$, the spectral functions simplify since Γ(1,βγ)=e^−βγ$\Gamma (1,\beta \gamma )={\mathrm{e}}^{-\beta \gamma }$. The unpolarized flux is $F_{\omega }^{\mathrm{T}+\mathrm{L}}=F_{\omega }^{\mathrm{T}}+F_{\omega }^{\mathrm{L}}$.

In the semiclassical regime ${\hat{M}}_{\mathrm{T},\mathrm{L}}\ll 1$, cf. (3.4), it is convenient to factorize $\beta {\gamma }_{\min }^{\mathrm{T},\mathrm{L}}(\omega )=x_{\mathrm{T},\mathrm{L}}{\eta }_{\mathrm{T},\mathrm{L}}$, where, cf. (3.5),

We substitute βγ=x_T,Lη_T,L$\beta \gamma =x_{\mathrm{T},\mathrm{L}}{\eta }_{\mathrm{T},\mathrm{L}}$ into the averaged spectral densities (3.6), (3.7) and (3.8) and note that η_T,L≥1${\eta }_{\mathrm{T},\mathrm{L}}\ge 1$ and η_T,L${\eta }_{\mathrm{T},\mathrm{L}}$ is also bounded from above for ${\hat{M}}_{\mathrm{T},\mathrm{L}}\ll 1$ and arbitrary ${\hat{\omega }}_{\mathrm{T},\mathrm{L}}$. We will use x_T,L$x_{\mathrm{T},\mathrm{L}}$ as expansion parameter, which can be large or small depending on 0<β<∞$0<\beta <\infty$ and ${\hat{\omega }}_{\mathrm{T},\mathrm{L}}$.

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

Identical dispersion relations require the identity εμ=ε₀μ₀$\epsilon \mu ={\epsilon }_0{\mu }_0$, cf. after (2.10). We can then drop the T and L subscripts of M_T,L$M_{\mathrm{T},\mathrm{L}}$, ${\hat{M}}_{\mathrm{T},\mathrm{L}}$ and ${\hat{\omega }}_{\mathrm{T},\mathrm{L}}$, cf. (2.8) and (3.4), and also in (3.5) and (3.9), writing βγ_min(ω)=xη$\beta {\gamma }_{\min }(\omega )=x\eta$ and replacing βγ by xη in the spectral densities (3.6), (3.7) and (3.8).

We scale the magnetic permeability μ   into the fine-structure constant α_t(ω)${\alpha }_{\mathrm{t}}(\omega )$, cf. after (2.6),

and specify the frequency dependence of ${\hat{\alpha }}_{\mathrm{t}}(\omega )$ and the rescaled tachyon mass ${\hat{m}}_{\mathrm{t}}(\omega )$ in (2.7) as power laws [32] and [33], where ${\hat{\alpha }}_{\mathrm{t}0}$ and ${\hat{m}}_{\mathrm{t}0}$ are constant positive amplitudes, and σ   and ρ   are real exponents to be determined from spectral fits. Substituting ${\hat{m}}_{\mathrm{t}}(\omega )$ into the wave numbers (2.10), we find the tachyonic group velocity ${\upsilon }_{\mathrm{gr}}=1/k_{\mathrm{T},\mathrm{L}}^{\prime }(\omega )$ as so that υ_gr${\upsilon }_{\mathrm{gr}}$ is superluminal for tachyonic mass exponents ρ<1/2$\rho <1/2$. It is also evident from (2.7) and (4.1) that the frequency dependence of the tachyon mass as well as the frequency-dependent factor Ω²${\Omega }^2$ 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 a_tμ/Ω²$a_{\mathrm{t}}\mu /{\Omega }^2$ in flux densities (3.6), (3.7) and (3.8) reads, cf. (3.4), (4.1) and (4.2),

We further specialize the permeabilities as εμ=ε₀μ₀=1$\epsilon \mu ={\epsilon }_0{\mu }_0=1$, so that the variables in (2.8) and (3.4) simplify, The unpolarized flux is obtained by adding the averaged spectral densities (3.6), (3.7) and (3.8), where we substitute the variables, cf. (3.9) and (4.5), The independent fitting parameters are ${\hat{a}}_{\mathrm{t}}$, $\hat{\beta }$, ${\hat{m}}_{\mathrm{t}0}$, σ   and ρ  . We also note η→1$\eta \to 1$ for $\hat{M}\to 0$. In a finite frequency interval defined by the available data sets, the semiclassical regime $\hat{M}\ll 1$ is realized for sufficiently small tachyon–electron mass ratio ${\hat{m}}_{\mathrm{t}0}/m$, cf. (4.5). If $\hat{\omega }\gg 1$ holds in addition to $\hat{M}\ll 1$ (which means $x\sim \hat{\beta }{\omega }^{1-\rho }$, cf. (4.5) and (4.7)), then ${\hat{m}}_{\mathrm{t}0}$ is not any more a fitting parameter, as it drops out of the flux density (4.6). We introduce the parameters and replace the first two ratios in the flux density (4.6) by an identical expression, so that we can use A_∞$A_{\infty }$ and η_∞${\eta }_{\infty }$ in (4.9) as fitting parameters instead of ${\hat{a}}_{\mathrm{t}}$ and σ  . Finally, in the case of a thermal source plasma with electron index α=−2$\alpha =-2$, the unpolarized flux (4.6) simplifies to with $A_{\infty }={\hat{a}}_{\mathrm{t}}{\beta }^2/{\hat{\beta }}^2$ and η_∞=η₀${\eta }_{\infty }={\eta }_0$, cf. (4.9).