**Optics Communications**

Volume 282, Issue 9, 1 May 2009, Pages 1710-1719

### Abstract

The diffraction of superluminal radiation fields in crystal lattices is studied. The negative mass-square of the tachyonic wave modes affects the modulation function of diffraction gratings and the scattering amplitude. The Bragg condition for tachyon diffraction as well as the longitudinal and transversal cross sections are derived. Scalar and vectorial Kirchhoff identities for superluminal Proca fields are obtained from Sommerfeld’s dipole functionals, in analogy to electromagnetic theory. These surface-integral representations of the tachyon potential and the tachyonic field strengths are used to calculate the asymptotic diffracted modes and the intensity ratios. The dependence of the primary and secondary intensity peaks on the tachyon mass is analyzed in the reciprocal lattice, and the conversion of transversal into longitudinal radiation by way of Bragg scattering is explained. Specifically, tachyonic spectral fits are performed to the TeV spectra of three active galactic nuclei, H2356 − 309, 1ES 1218 + 304, and 1ES 1101 − 232, obtained with the imaging atmospheric Cherenkov detectors HESS, MAGIC, and VERITAS. The curvature in the spectral maps of these blazars is shown to be intrinsic, generated by ultra-relativistic electron populations in the galactic nuclei rather than by intergalactic absorption, and is reproduced by a tachyonic cascade fit.

**Keywords: **Tachyon diffraction;
Transversal and longitudinal polarization; Green’s function for
superluminal wave propagation; Tachyonic Maxwell equations and
Fraunhofer far-field approximation; Cross sections and intensity ratios
of tachyonic Bragg scattering; Superluminal cascade spectra of TeV
blazars

**PACS classification codes: **42.25.Fx;
42.25.Ja; 61.05.cc; 95.30.Gv

### Article Outline

- 1. Introduction
- 2. Superluminal radiation fields
- 2.1. Proca equation with negative mass-square
- 2.2. Tachyonic dipole fields
- 2.3. Kirchhoff representation of the tachyon potential and the field strengths
- 3. Fraunhofer diffraction of superluminal radiation at a plane aperture
- 3.1. Tachyonic energy flux
- 3.2. Polarized superluminal modes: conversion of transversal into longitudinal tachyons by diffraction
- 3.3. Intensity ratios for the conversion of longitudinal into transversal radiation
- 4. Tachyonic Bragg scattering
- 4.1. Diffraction gratings: negative mass-square and Bragg condition
- 4.2. Tachyon diffraction in crystal lattices: transversal and longitudinal scattering cross sections
- 5. Tachyonic flare spectra of TeV blazars
- 6. Conclusion: tachyonic X-rays and Bragg spectrometers
- Acknowledgements
- References

### 1. Introduction

We investigate the diffraction of superluminal wave modes, outlining a theory of tachyon diffraction based on Kirchhoff’s surface-integral representation of Proca fields. We work out two specific examples, diffraction at plane apertures such as diffraction gratings, and tachyonic Bragg scattering in crystal lattices. The formalism is developed in close analogy to electromagnetic diffraction theory, even though there are substantial differences owing to the negative mass-square of tachyons [1], [2], [3], [4] and [5] and the occurrence of longitudinally polarized modes [6], [7] and [8].The negative mass-square refers to the radiation rather than the source. This in strong contrast to the traditional approach based on superluminal source particles emitting electromagnetic radiation [9]. The tachyonic radiation discussed here implies superluminal energy transfer, the radiation quanta moving faster than light, in contrast to the rotating superluminal light sources studied in Refs. [10], [11] and [12] and the vacuum Cherenkov radiation suggested in [13], [14], [15] and [16]. Tachyons are radiation modes, a kind of photons with negative mass-square, coupled by minimal substitution to the electron current, cf. Section 2. The tachyonic Maxwell equations admit a static potential analogous to the Coulomb potential, but oscillating because of the negative mass-square, and much weaker due to the small tachyonic fine structure constant [17]. Photons can only be radiated by accelerated charges, in contrast to tachyonic quanta, where the emission rate primarily depends on the electronic Lorentz factor rather than on acceleration [18]. Here, we investigate how the intensity peaks and scattering cross sections are affected by the tachyon mass, and study the effect of diffraction on the polarization of superluminal modes. We disentangle the transversal and longitudinal polarization components in the spectral maps of the BL Lacertae objects H2356 − 309, 1ES 1218 + 304, and 1ES 1101 − 232, and show that the TeV spectra of these blazars can be fitted with tachyonic cascades radiated by the thermal electron plasma in the active galactic nuclei. In the spectral maps, the tachyon–electron mass ratio enters in the cutoff energy of the cascades.

In Section 2, we discuss the tachyonic Maxwell equations in Fourier space, including the material equations relating tachyonic inductions and field strengths. We calculate the superluminal radiation fields generated by dipole currents, and introduce dipole functionals to derive the Kirchhoff identities for the scalar and vector potentials as well as the tachyonic field strengths. In Section 3, we study tachyon diffraction at a plane aperture, calculate the asymptotic diffracted wave modes in the far-field regime, and assemble the intensity ratios determining the conversion efficiency from transversal to longitudinal radiation and vice versa. In Section 4, we consider the specific case of a grating aperture, and calculate the modulation function, from which the intensity peaks of the diffracted superluminal wave fields can be read off. We discuss Bragg diffraction of tachyons in crystal lattices, in particular the effect of the negative mass-square on the transversal and longitudinal scattering cross sections. In Section 5, we perform tachyonic cascade fits to blazar spectra, separate the transversal and longitudinal flux components, and derive estimates of the electronic source populations in the active galactic nuclei. In Section 6, we present our conclusions with regard to tachyonic X-ray spectra obtained with Bragg gratings.

### 2. Superluminal radiation fields

#### 2.1. Proca equation with negative mass-square

The tachyonic radiation field in vacuum is a real vector field
with negative mass-square, satisfying the Proca equation ,
subject to the Lorentz condition *A*^{μ},_{μ}=0
[6]. *m*_{t}
is the mass of the superluminal Proca field *A _{μ}*,
and

*q*the tachyonic charge carried by the subluminal electron current

*j*= (

^{μ}*ρ*,

**j**). In the Proca equation, the mass term is added with a positive sign, and the sign convention for the metric defining the d’Alembertian ∂

*∂*

^{ν}*is diag(−1, 1, 1, 1), so that is the negative mass-square of the radiation field. The 3D version of Proca’s equation is a set of Maxwell equations,*

_{ν}**E**=

*A*

_{0}− ∂

**A**/∂

*t*and

**B**= rot

**A**. The Lorentz condition, div

**A**− ∂

*A*

_{0}/∂

*t*= 0, follows from the field equations and current conservation, div

**j**+ ∂

*ρ*/∂

*t*= 0.

In a permeable medium, the potential and field strengths in
the inhomogeneous vacuum equations are replaced by inductions, (*A*_{0}, **A**) → (*C*_{0}, **C**),
(**E**, **B**) → (**D**, **H**),
defined by material equations [19], [20] and [21]. We will
mostly consider monochromatic waves, ,
and analogously for the scalar potential *A*_{0},
the current, charge density, field strengths, and inductions. Fourier
amplitudes are denoted by a hat. The tachyonic Maxwell equations (2.1) read in
Fourier space as

The inductive potentials as well as and are related to the primary fields by frequency-dependent dielectric and magnetic permeabilities. In an anisotropic medium, we have to use tensorial permeabilities, e.g. . The Fourier amplitudes of the field strengths and potentials are connected by and . Current conservation, , implies the Lorentz condition . In a dissipative medium, the permittivities (

*ε*

_{0},

*ε*) and permeabilities (

*μ*

_{0},

*μ*) are complex, resulting in exponential attenuation of the wave fields [22].

#### 2.2. Tachyonic dipole fields

We substitute the potential representation of the field
strengths into the inhomogeneous field equations in (2.2), and make
use of the Lorentz condition to find

*ε*

_{0}=

*ε*and

*μ*

_{0}=

*μ*, otherwise different dispersion relations are obtained for the scalar and vector potentials, implying different group velocities of the transversal and longitudinal modes. On the left-hand side of (2.4), we can thus identify the squared wave number as

We consider real

*ε*,

*μ*, and a positive

*k*

^{2}; the permeabilities may even be negative [23], but then we restrict to a frequency range with positive squared wave numbers. The Green function inverting the wave equations (2.4) is

Here, Re

*k*> 0, so that

*G*(

*x*,

*x*

_{0};

*ω*) gives retarded solutions,

We will exclusively use the whole-space Green function (2.7), symmetric with respect to the first and second argument.

The field strengths are found via the potential representation
stated after (2.3). We define
the bivector

*x*(although we may switch to the

*x*

^{′}gradient if convenient, by substituting ), and obtain

The gradient refers to

*x*, but we may replace

*G*→ −′

*G*, where the prime indicates

*x*

^{′}differentiation. The d

^{3}

*x*

^{′}integration extends over the whole space.

*G*satisfies the radiation condition

*r*(i

*k*−

**n**)

*G*=

*G*, and

*G*=O(1/

*r*), with

*r*=|

*x*−

*x*

^{′}| and

**n**= (

*x*−

*x*

^{′})/

*r*.

We consider a dipole current and the corresponding charge
density,

**p**denotes an arbitrary constant vector (possibly depending on

*x*

_{0}), the charge density being obtained from the continuity equation, cf. after (2.3). The gradient in (2.12) refers to

*x*, but we may substitute → −

_{0}, where

_{0}is the

*x*

_{0}gradient. The corresponding scalar and vector potentials are

The field generated by the dipole is , or

and the magnetic counterpart reads

(The dipole vector

**p**is independent of

*x*.) By making use of wave equation (2.6) for

*G*(

*x*,

*x*

_{0};

*ω*), we write the dipole field in (2.14) as

The substitutions → −

_{0}and rot → −rot

_{0}can be performed whenever convenient; the subscript zero denotes differentiation with respect to argument

*x*

_{0}in the Green function. We write for the field (2.14) generated by a dipole

**p**, and analogously in (2.15). If

**q**is another arbitrary constant dipole vector, then , and the same holds true for in (2.13). We also note the anti-symmetry . If the dipole vectors

**q**and

**p**depend on

*x*rather than

*x*

_{0}, these symmetries remain valid, provided that the gradients and rotors are replaced by the substitutions indicated after (2.16).

#### 2.3. Kirchhoff representation of the tachyon potential and the field strengths

We denote the singular dipole current (2.12) and the
dipole fields (2.13), (2.14), (2.15) and (2.16) by a
subscript *δ*, e.g.
and ,
and consider a closed surface *S* around the dipole
at *x*_{0}. An independent second
set of fields, ,
and ,
solves the free tachyonic Maxwell equations (2.2) inside the
cavity defined by *S*. (These fields are supposed to
be generated by a current distribution outside the enclosure *S*,
but we will only be concerned with the fields in the interior.) We
define the flux functional [24] and [25]

*div*

_{v}**F**

*d*

_{δ}^{3}

*x*= −∫

_{s}**F**

_{δ}**n**d

*S*

^{′}, where

**n**is the inward-pointing surface normal, and d

**S**=

^{′}**n**d

*S*

^{′}the surface element. The volume integration of the

*δ*current gives , cf. (2.12). As for the surface integral, we substitute the identities

Here,

**n**(

*x*

^{′}), , etc., depend on a point

*x*

^{′}on the boundary

*S*. The Green’s function

*G*(

*x*

^{′},

*x*

_{0};

*ω*) is symmetric with respect to an interchange of

*x*

^{′}and

*x*

_{0}. It is also assumed that the substitutions → −

_{0}and rot → −rot

_{0}are performed in the dipole fields. Finally, we write

*x*for

*x*

_{0}(an arbitrary point inside the cavity), and drop the scalar multiplication with the arbitrary dipole vector

**p**, to obtain the surface-integral representation

Here, d

**S**

^{′}=

**n**(

*x*

^{′})d

*S*

^{′}, where

**n**(

*x*

^{′}) is the unit normal vector pointing into the interior of the cavity, and the integration d

*S*

^{′}is over the closed boundary surface

*S*. The field strengths and potentials satisfy the free field equations (2.2) and (2.3) (with

*ε*

_{0}=

*ε*and

*μ*

_{0}=

*μ*) inside the cavity, that is, with zero current and charge density.

*G*(

*x*,

*x*

^{′};

*ω*) is the Green function (2.7), where

*x*ranges inside the cavity, and the integration is over the surface variable

*x*

^{′}. The rotor and gradient refer to

*x*. The potentials and in the surface integrals can be replaced by field strengths (via substitution of the field equations), and the operator by (

*μ*

*ε*

*ω*

^{2}+div), cf. (2.16).

We use
to obtain the Kirchhoff identity for the scalar potential,

*x*in the Green’s function. The Kirchhoff identity for the vector potential is found by means of ,

where we may substitute for (rot rot−

*μω*

^{2}). The analogous surface-integral representation of the magnetic field is obtained via ,

where rot rot = (

*k*

^{2}+ div). These identities follow from the electric field strength (2.19), by applying the field equations or potential representation, without actually solving differential equations. We also note that the Green function in these identities is the whole-space Green function (2.7), and is not required to satisfy any particular boundary conditions on the closed surface

*S*, for technical simplicity. Therefore, these identities are only valid for solutions of the field equations, so that the boundary values cannot be arbitrarily prescribed. In the next section, however, we will prescribe boundary conditions, and convince ourselves that the fields calculated by means of the above Kirchhoff identities are asymptotic solutions of the free tachyonic Maxwell equations (2.2). This is in fact the practical use of these identities: Even though they will not give exact solutions for arbitrary boundary values, the resulting fields may well be approximate solutions in the far-field regime, which has to be checked on a case-by-case basis, by substitution into the field equations.

### 3. Fraunhofer diffraction of superluminal radiation at a plane aperture

#### 3.1. Tachyonic energy flux

As in Section 2.2, we put *ε*_{0} = *ε*
and *μ*_{0} = *μ*.
When studying diffraction in the far-field limit [26] and [27], it
suffices to use the dipole approximation |*x* − *x*^{′}| *r*(1 − **nx**^{′}/*r*),
where *r* = |**x**|
and **n** = **x**/*r*,
so that (for large *r* and |**x**^{′}| = O(1))

*x*,

*y*) plane with inward-pointing normal vector

**e**

_{3}= (0, 0, 1). We consider an aperture

*A*in the (

*x*,

*y*) plane, centered at the coordinate origin, and an incoming plane-wave propagating in the lower half-space towards the aperture. The wave diffracted through the aperture into the upper half-space is obtained from the Kirchhoff identities Eq. (2.19), (2.20), (2.21) and (2.22), where we identify the (

*x*,

*y*) plane with the boundary surface

*S*. At the perfectly conducting boundary, the complement of the aperture, we assume zero potentials and field strengths, and in the aperture

*A*we use the incoming plane wave as boundary value. The mentioned hemisphere (defining a finite cavity as an intermediate step) does not give a contribution to the surface integrals when expanded to infinity, as the Green’s function satisfies the radiation condition, cf. after (2.11).

To find the energy flux carried by polarized superluminal
modes incident upon the aperture, we use a plane-wave ansatz in the
tachyonic Maxwell equations (2.2) (with
vanishing charge and current, as well as *ε*_{0} = *ε*
and *μ*_{0} = *μ*),
,
and analogously for the scalar potential and the field strengths. Here,
**k** = *k*(*ω*)**k**_{0},
where *k* is the wave number in the Green function (2.7), and **k**_{0}
a constant unit vector. The transversality condition is ,
and the set of transversal modes reads

We substitute these plane-wave solutions into the tachyonic Poynting vector [28], and perform a time average, to obtain the transversal and longitudinal components of the energy flux,

Here, real permeabilities and a positive wave number

*k*(

*ω*) are implied.

#### 3.2. Polarized superluminal modes: conversion of transversal into longitudinal tachyons by diffraction

We consider an incident transversal plane wave, propagating in
the lower half-space toward the aperture *A* in the (*x*, *y*)
plane. The Fourier amplitude of the vector potential is ,
cf. (3.2). By making
use of the Kirchhoff identity (2.19) and the
limit procedure outlined after (3.1), we find
the leading asymptotic order (*r* → ∞,
Fraunhofer regime [29]) of the
diffracted wave in the upper half-space as

where

**x**

^{′}= (

*x*

^{′},

*y*

^{′},

*z*

^{′}= 0) [20]. The scalar potential reads, cf. (2.20),

and the vector potential, cf. (2.21),

The magnetic field strength is found as, cf. (2.22),

When calculating the intensity of the diffracted superluminal flux, we need the transversal and longitudinal projections of the vector potential, that is and , respectively. Here,

**ε**

*and*

_{j}**n**=

**x**/

*r*constitute the orthonormal triad defining the outgoing linear polarizations,

**n**being the unit wave vector of the outgoing spherical wave. The incident transversal wave is linearly polarized, ; the transversal linear polarization vectors

**ε**

_{0,1}and

**ε**

_{0,2}of the incoming wave are real and define with its unit wave vector

**k**

_{0}an orthonormal triad, so that

**k**

_{0}=

**ε**

_{0,1}×

**ε**

_{0,2}cyclically.

As for the outgoing transversal polarization vectors **ε*** _{j}*
of the diffracted wave in the upper half-space, we choose two real
vectors

**ε**

_{1,2}orthogonal to

**n**, so that

**ε**

_{1}lies in the plane generated by

**n**and the incident unit wave vector

**k**

_{0},

**n**=

**ε**

_{1}×

**ε**

_{2}, and |

**n**×

**k**

_{0}|

^{2}= 1 − (

**nk**

_{0})

^{2}. The scalar products of the transversal polarization vectors of the incoming and outgoing waves read

The longitudinal polarization vectors of the in- and outgoing waves are the unit wave vectors

**k**

_{0}and

**n**, respectively. The angular parametrization of these products is done with polar coordinates in the coordinate frame defined by the right-handed triad

**ε**

_{0,1},

**ε**

_{0,2}, and

**k**

_{0}of the incoming wave. Thus,

**nk**

_{0}= cos

*θ*,

**nε**

_{0,1}= cos

*φ*sin

*θ*, and

**nε**

_{0,2}= sin

*φ*sin

*θ*, and d

*Ω*= sin

*θ*d

*θ*d

*φ*is the solid angle element. Accordingly, |

**n**×

**k**

_{0}| = sin

*θ*, and

The

**e**

_{3}projection of the outgoing polarization triad is related to the

**e**

_{3}projection of the incoming triad as

The angular parametrization of the polarized components of the diffracted wave (3.8) can readily be performed by substitution of these scalar products. The transversal

**ε**

*projections of (3.8) read*

_{j}The polarization vector of the incident wave, , is indicated as argument. The longitudinal projection of the outgoing wave (3.8) is

We consider the special case where the incoming wave vector is normal to the plane of incidence. As

**k**

_{0}=

**e**

_{3}, the angular parametrization of the cross sections greatly simplifies, but the above parametrization of the polarization triads can be used for any other incident wave vector as well. On performing the average over the initial transversal polarizations and a summation over the transversal outgoing polarizations, we obtain, cf. (3.14),

where we used

The angular parametrization of the longitudinal outgoing component (3.15) at normal incidence

**k**

_{0}=

**e**

_{3}reads

These averages suffice to calculate the intensity ratios determining the conversion efficiency, cf. (3.25).

In the far-field limit, the outgoing spherical waves are
structured as [28]

The transversal component is an arbitrary complex linear combination of the two linear polarization components , cf. (3.14). The plane-wave counterpart to (3.20) is stated in (3.2) and (3.3). The time-averaged superluminal flux vectors

**S**

^{T,L}can be assembled with these asymptotic spherical waves as done in (3.4) for plane waves. The transversal and longitudinal components of the diffracted energy flux are

where we substitute the amplitudes (3.14) and (3.15). The subscripts

*i*and

*j*refer to the respective incoming and outgoing transversal polarization states. These averages are obtained from the asymptotic time-averaged Poynting vectors, cf. after (3.3),

The diffracted flux components are to be compared to the flux carried by the incident transversal plane wave, cf. (3.4),

As for the outgoing flux vectors (3.21), we perform the same average/summation over the transversal polarizations as done in (3.16) and (3.18). The ratio of the outgoing flux transversally or longitudinally diffracted into the solid angle element d

*Ω*and the transversal flux incident upon the aperture area is found as

These are dimensionless intensity ratios, cross sections divided by the aperture area. On substituting the averages (3.16) and (3.18), we find the intensity ratios for the diffraction of transversal radiation,

Diffraction of transversal wave fields generates longitudinal modes, the conversion efficiency being determined by the ratio d

*σ*

_{T→L}/(d

*σ*

_{T→T}+ d

*σ*

_{T→L}).

#### 3.3. Intensity ratios for the conversion of longitudinal into transversal radiation

We consider an incident longitudinal plane wave, ,
cf. (3.3), and
proceed analogously to the transversal case in Section 3.2. Employing
the surface integrals (2.19), (2.20), (2.21) and (2.22), we find
the diffracted electric field strength in the Fraunhofer regime,

the vector potential,

and the magnetic field strength

The transversal polarization components of the diffracted vector potential in (3.28) read, cf. (3.14),

where we explicitly indicate the incoming longitudinal polarization

**k**

_{0}as argument. The outgoing longitudinal component is

The angular parametrization is explained in (3.11), (3.12) and (3.13). At normal incidence

**k**

_{0}=

**e**

_{3}, we find and, cf. (3.16),

The transversally diffracted radiation is thus linearly polarized. The squared longitudinal component (3.31) is parametrized as, cf. (3.18),

Regarding the energy flux, we note the polarized outgoing Poynting vectors, cf. (3.21),

the incoming longitudinal flux vector, cf. (3.4),

and the intensity ratios for the conversion of longitudinal radiation, cf. (3.24),

The conversion efficiency of longitudinal radiation into linearly polarized transversal tachyons is thus determined by the ratios, cf. (3.25),

with the squared amplitude projections (3.32) and (3.33) substituted.

### 4. Tachyonic Bragg scattering

#### 4.1. Diffraction gratings: negative mass-square and Bragg condition

We start with a grating defined as an array of 2*N* + 1
slits parallel to the **e**_{2}
axis (*y* coordinate). The slits are rectangles of
(large) height 2*b* and width 2*a*.
The *x* coordinate along **e**_{1}
ranges in equidistantly spaced intervals [*nd* − *a*, *nd* + *a*],
where *n* = −*N*, …, 0, …, *N*,
and *d* > 2*a*.
The *y* coordinate of this grating aperture ranges in
[−*b*, *b*], and the
modulation function (3.6) factorizes
accordingly as

*q*

_{1,2}:= (

**k**

_{0}−

**n**)

**e**

_{1,2}. We note the identity

as well as two limit definitions of the delta function,

*δ*

_{(1),(2)}(

*x*,

*b*→ ∞) =

*δ*(

*x*), where

The modulation factor of the grating can thus be written as

where

*η*:=

*kq*

_{1}

*d*/2. In the squared modulation factor, we substitute . The area of the aperture is 4(2

*N*+ 1)

*ab*. In the intensity ratios (3.25) and (3.37), the

*b*factors in area(

*A*) and cancel, so that the height of the slits only enters in

*δ*

_{(2)}(

*kq*

_{2},

*b*). Thus the

*b*→ ∞ limit is well defined and gives

*δ*(

*kq*

_{2}). The principal maxima of

*M*

^{2}are determined by the interference factor, the square of the first ratio in (4.4). They are located at

*η*=

*nπ*for integer

*n*. In between the principal intensity maxima, there are secondary ones, separated by zeros located at

*η*=

*nπ*/(2

*N*+ 1), where

*n*is integer but not a multiple of 2

*N*+ 1. The square of the second ratio in (4.4) attenuates the maxima at large

*η*. However, since 2

*a*/

*d*< 1, it does not significantly affect the location of the maxima. The principal maxima occur at

*η*=

*nπ*, which we may write, with

*k*= 2

*π*/

*λ*, as a Bragg condition

**k**

_{0}

**e**

_{1}−

**ne**

_{1}=

*nλ*/

*d*. We consider normal incidence

**k**

_{0}=

**e**

_{3}, and choose the transversal polarization vectors along the coordinate axes,

**ε**

_{0,}

*=*

_{i}**e**

*. In the (*

_{i}**e**

_{1},

**k**

_{0}) plane orthogonal to the slits, we then have

*φ*= 0 and

**ne**

_{1}= sin

*θ*, so that the principal maxima are recovered at scattering angles defined by , with integer

*n*.

#### 4.2. Tachyon diffraction in crystal lattices: transversal and longitudinal scattering cross sections

We consider a monochromatic superluminal radiation mode, ,
hitting a crystal lattice, and apply this field to the electron density
of the crystal. This generates a tachyonic current **j**(**x**, *t*) = *qn*_{e}(**x**)**v**(*t*),
where *n*_{e}(**x**)
is the periodic electron density in the crystal lattice. The velocity
of the electrons carrying tachyonic charge *q* is
determined by .
(In the Heaviside–Lorentz system, *q*^{2}/(4*π**c*)≈1.0×10^{-13},
estimated from Lamb shifts in hydrogenic ions [17].) In dipole
approximation, we may neglect the spatial dependence of the amplitude ,
so that
with .
We thus find the Fourier amplitude of the current
as [19]

where we have put

*ε*

_{0}=

*ε*= 1 and

*μ*

_{0}=

*μ*= 1 (vacuum permeabilities in the Heaviside–Lorentz system). The charge density follows from current conservation, . The tachyonic radiation fields generated by current (4.5) in the crystal lattice can be split into transversally and longitudinally polarized components , like the diffracted waves (3.19). We are interested in the asymptotic radiation fields outside the crystal, determined by the current transform [28]

where is the tachyonic wave number (2.5). The projections of onto a right-handed triad of polarization vectors

**ε**

_{1,2}and

**n**of the radiation field are

Here,

**n**=

**x**/

*r*is the coordinate unit vector used as longitudinal polarization vector, and

**ε**

_{i}_{=}

_{1,2}(

**x**) are real transversal polarization vectors defining two degrees of linear polarization, so that

**ε**

*and*

_{i}**n**constitute an orthonormal triad, which we choose as in (3.10). The outgoing transversal and longitudinal field components are stated in (3.19) and (3.20), where we put

*ε*=

*μ*= 1, replace the superscript T by T(

*i*), and substitute as defined in (4.7) and (4.8).

We further specify the wave field incident upon the crystal as
a plane wave ,
cf. before (3.2),
generating the current
in (4.5). The
polarized components
of the current transform (4.7) are thus
found as

*F*(

**k**,

**n**) denotes the scattering amplitude

By making use of (3.19) and (3.20), we obtain the asymptotic outgoing field strengths,

As for the cross sections, we start with the transversal and longitudinal flux vectors

**S**

^{T(i),L}in (3.22), and substitute the scattered fields (3.19) and (3.20),

The squared current amplitudes (4.9) read

The total transversal flux

**S**

^{T}is obtained by adding the transversal polarization components

**S**

^{T(i)}, which amounts to replacing the linearly polarized current transforms in (4.12) by in (4.13), according to (4.8).

We also need the flux density of the incident plane wave, ,
decomposed into transversal and longitudinal components, cf. (3.2) and (3.3). The flux
carried by the transversal component, ,
reads, cf. (3.4),

The transversal cross section for superluminal Bragg diffraction is thus found as

where the solid angle element d

*Ω*is centered at the outgoing wave vector

**k**

^{′}=

*k*(

*ω*)

**n**. We may replace the factor by sin

^{2}

*θ*, where

*θ*is the angle between and

**k**

^{′}. If the incident transversal radiation is unpolarized, we have to replace sin

^{2}

*θ*by the average (1 + cos

^{2})/2, where is the scattering angle between the in- and outgoing wave vectors

**k**and

**k**

^{′}. In the longitudinal cross section,

we may replace the ratio by cos

^{2 }.

In the scattering amplitude *F*(**k**, **n**),
cf. (4.10), we write
**k**^{′} for *k*(*ω*)**n**,
and substitute the Fourier series *n*_{e}(**x**)=∑*n*_{G}e^{iGx}
over the reciprocal lattice **G**,

**G**by −

**G**in the individual terms. The integrals in (4.18) are taken over the crystal volume, and give a sizeable contribution only if

**k**−

**k**

^{′}very nearly coincides with a reciprocal lattice vector. We thus arrive at the Laue condition

**k**−

**k**

^{′}=

**G**, so that the respective integral just gives the crystal volume. We square

**k**

^{′}=

**k**−

**G**to arrive at 2

**kG**= |

**G**|

^{2}, which is the diffraction condition for the incident wave vector. Alternatively, |

**G**| = |

**k**

^{′}−

**k**| = 2

*k*sin (/2), with scattering angle as above. The parallel lattice planes of the direct lattice orthogonal to a fixed

**G**are equidistantly spaced, at distance

*d*= 2

*π*/|

**G**

_{0}|, where

**G**

_{0}is the shortest reciprocal lattice vector parallel to

**G**, the latter being an integer multiple of

**G**

_{0}, |

**G**| =

*n*|

**G**

_{0}|, cf., e.g., Ref. [30]. Replacing the wave number by frequency via the dispersion relation , we find the Bragg condition for tachyon diffraction,

The angle /2 is the glancing angle between lattice plane and wave vector; incidence and reflection angle coincide as in the electromagnetic case, irrespectively of the polarization. Owing to the negative mass-square, tachyonic Bragg diffraction can only occur at wavelengths

*λ*= 2

*π*/

*k*< 2

*d*, where

*d*is usually a few Å. This matches well with the tachyonic Compton wavelength of 2

*π*/

*m*

_{t}≈ 5.7Å [17], which is the maximal wavelength attainable by tachyonic vacuum modes.

### 5. Tachyonic flare spectra of TeV blazars

Fig. 1, Fig. 2 and Fig. 3 depict
tachyonic cascade fits to the TeV spectra of the γ-ray blazars
H2356 − 309, 1ES 1218 + 304, and
1ES 1101 − 232, obtained with imaging air Cherenkov
telescopes [31]. The
cascades are plots of the *E*^{2}-rescaled
flux densities

*d*is the distance to the source, and

*p*

^{T,L}(

*ω*=

*E*/) the tachyonic spectral density of a uniformly moving charge [8],

averaged over a thermal electron distribution. The superscripts T and L indicate the transversal and longitudinal polarization components defined by and

*Δ*

^{L}= 0.

*γ*is the electronic Lorentz factor, and

*α*

_{q}the tachyonic fine structure constant. We use the Heaviside–Lorentz system, so that

*α*

_{q}=

*q*

^{2}/(4

*π*

*c*)≈1.0×10

^{-13}and

*m*

_{t}≈ 2.15 keV/

*c*

^{2}, as inferred from Lamb-shift estimates [17]. The tachyon–electron mass ratio is

*m*

_{t}/

*m*≈ 1/238, and a spectral cutoff occurs at

Only frequencies in the range 0

*ω*

*ω*

_{max}(

*γ*) can be radiated by a uniformly moving charge, the tachyonic spectral densities

*p*

^{T,L}(

*ω*) being cut off at the break frequency

*ω*

_{max}. A positive

*ω*

_{max}(

*γ*) requires Lorentz factors exceeding the threshold

*μ*

_{t}.

Fig. 1. Spectral map of the BL Lac object
H2356 − 309. HESS data points from Ref. [31]. The solid
line T + L depicts the *E*^{2}-scaled
differential tachyon flux d*N*^{T+L}/d*E*,
obtained by adding the flux densities *ρ*_{1,2}
of two electron populations, cf. (5.1). The
transversal (T) and longitudinal (L) flux densities d*N*^{T,L}/d*E*
add up to the total unpolarized flux T + L. The
exponential decay of the cascades *ρ*_{1,2}
sets in at about *E*_{cut} ≈ (*m*_{t}/*m*)*kT*,
cf. after (5.2), implying
cutoffs at 0.84 TeV for the *ρ*_{1}
cascade and 92 GeV for *ρ*_{2}.
The *χ*^{2} fit is done with the
unpolarized tachyon flux T + L, and subsequently
split into transversal and longitudinal components. Temperature and
number count of the electron populations are recorded in Table 1.

Fig. 2. Spectral map of the blazar 1ES
1218 + 304. MAGIC data points from Ref. [35], VERITAS
points from Ref. [36]. The upper
flux limit in the 0.1–0.2 TeV interval is based on STACEE
observations in 2006 and 2007 [37]. The
spectral fit T + L = *ρ*_{1} + *ρ*_{2}
is performed with the electron distributions quoted in Table 1; the
polarized flux components are labeled T and L. The *ρ*_{1}
cascade is cut at *E*_{cut} ≈ 0.28 TeV,
and *ρ*_{2} at 46 GeV.
Comparing to the BL Lac in Fig. 1, located
at a lower redshift, there is no indication of absorption in the
spectral slope. The electron densities generating the cascades *ρ*_{1,2}
are thermal in either case. The spectral curvature is intrinsic, caused
by the Boltzmann factor of the electron populations in the galactic
nucleus.

Fig. 3. Spectral map of the BL Lac object 1ES
1101 − 232. HESS flux points from Ref. [38]. The plots
are labeled as in Fig. 1 and Fig. 2. The *ρ*_{1}
cascade is cut at 1.6 TeV and *ρ*_{2}
at 0.21 TeV. The parameters of the electron populations are
listed in Table 1. The
spectral slope is steeper than that of 1ES 1218 + 304
in Fig. 2, even
though these blazars have almost identical redshifts, which suggests
that the shape of the plotted density *E*^{2}d*N*^{T+L}/d*E*
is intrinsic rather than affected by intergalactic absorption;
tachyonic γ-rays do not interact with background photons.

The average *p*^{T,L}(*ω*)
defining the differential flux (5.1) is taken
over thermal ultra-relativistic electron distributions .
The least-squares fit is performed with the total unpolarized flux
density d*N*^{T+L} = d*N*^{T} + d*N*^{L}.
The cascades are labeled *ρ*_{1,2}
in the figures, and the parameters of the electron populations
generating them are listed in Table 1. The
details of the spectral fitting have been explained in Ref. [32]. The
electron count is calculated as ,
where
defines the tachyonic flux amplitude extracted from the fit. The cutoff
parameter of the thermal cascades is related to the electron
temperature by *kT*[TeV] ≈ 5.11 × 10^{−7}/*β*,
and the internal energy estimates of the source populations in Table 1 are
obtained from *U*[erg] 2.46 × 10^{−6}*n*^{e}/*β*.
The distance estimates of the active galactic nuclei are based on *d* *cz*/*H*_{0},
with *c*/*H*_{0} ≈ 4.4 × 10^{3} Mpc.
Hence, *d*[Mpc] ≈ 4.4 × 10^{3}*z*,
and ,
cf. Table 1.

Electronic source distributions *ρ _{i}*
generating the tachyonic cascade spectra of the active galactic nuclei
in Fig. 1, Fig. 2 and Fig. 3. Each

*ρ*stands for a thermal ultra-relativistic Maxwell–Boltzmann density with cutoff parameter

_{i}*β*in the Boltzmann factor, cf. after (5.3). determines the amplitude of the tachyon flux generated by the electron density

*ρ*, from which the electron count

_{i}*n*

^{e}∝

*d*

^{2}is inferred at the indicated distance.

*d*is the distance to the blazar, estimated from the redshift

*z*.

*kT*is the temperature and

*U*the internal energy of the electron populations

*ρ*, cf. Ref. [42]. The distance estimates do not affect the spectral maps in Fig. 1, Fig. 2 and Fig. 3, but the electronic source count

_{i}*n*

^{e}. Each cascade depends on two fitting parameters

*β*and , extracted from the

*χ*

^{2}fit T + L in the figures.

Fig. 1 shows the
tachyonic spectral map of the blazar H2356 − 309 at
redshift *z* ≈ 0.165 [31]. TeV γ-ray
spectra of blazars are usually assumed to be generated by inverse
Compton scattering, which results in a flux of TeV photons thought to
be partially absorbed by interaction with infrared background photons,
so that the intrinsic spectrum has to be reconstructed on the basis of
intergalactic absorption models. By contrast, the extragalactic tachyon
flux is not attenuated by interaction with the background light, there
is no absorption of tachyonic γ-rays. Apparently, the curvature present
in the TeV spectra of blazars is not correlated with distance, at least
there is no evidence to that effect if we compare the spectral slopes
in Fig. 1, Fig. 2 and Fig. 3 to the
spectral maps of other flaring active galactic nuclei such as the BL
Lacertae objects (BL Lacs) H1426 + 428 (*z* ≈ 0.129,
570 Mpc) and 1ES 1959 + 650 (*z* ≈ 0.047,
210 Mpc) in Ref. [33], the
blazars 1ES 0229 + 200 (*z* ≈ 0.140,
620 Mpc) and
(*z* ≈ 0.188, 830 Mpc) in
Ref. [32], and the
quasar 3C 279 (*z* ≈ 0.538,
2.4 Gpc) in Ref. [34]. Fig. 2 shows the
tachyonic spectral fit of the BL Lac 1ES 1218 + 304
at *z* ≈ 0.182 [35], [36] and [37], and Fig. 3 the
spectral map of the blazar 1ES 1101 − 232 at *z* ≈ 0.186
[38]. There is
no correlation between redshift and spectral curvature visible. The
curvature in the spectral maps of BL Lacs is intrinsic, generated by
the superluminal spectral densities of the thermal electron plasma in
the active galactic nuclei [39].

### 6. Conclusion: tachyonic X-rays and Bragg spectrometers

We have outlined a diffraction theory of superluminal wave fields based on Kirchhoff identities, cf. Section 2, and discussed the specific case of tachyonic Bragg diffraction, first with regard to a grating aperture and then in crystal lattices, cf. Section 4. We analyzed the effect of diffraction on the polarization of tachyons, cf. Section 3, and separated the transversal and longitudinal flux components in the spectral maps of γ-ray blazars, cf. Section 5. A more detailed summary is given in the Introduction. Here, we briefly sketch how the negative mass-square shows in tachyonic X-ray spectra obtained with Bragg spectrometers.

Tachyonic spectral fits are based on the *E ^{n}*-scaled
flux density, cf. (5.1),

*p*(

*ω*) is the unpolarized tachyonic spectral density

*p*

^{T}+

*p*

^{L}in (5.2), averaged over thermal [40] and [41] or nonthermal [42] and [43] electronic source populations. The exponent

*n*is a conveniently chosen real power: Observational spectra are usually plotted as differential count rate d

*N*/d

*E*(counts per unit time, unit area, and unit energy), or differential energy flux

*E*d

*N*/d

*E*(which gives the power radiated if integrated over the respective energy band), or as

*E*

^{2}-rescaled differential flux

*E*

^{2}d

*N*/d

*E*(energy per unit time and unit area, adopted in Section 5). If a Bragg spectrometer is used, the primary quantity measured is the flux depending on wavelength rather than energy [44]. The energy parametrization in experimental plots is done with the assumed photonic relation

*λ*= 2

*π*/

*ω*, which substantially differs from the tachyonic dispersion relation in the X-ray bands, due to the tachyon mass of 2.15 keV. Therefore, we have to reparametrize the experimental spectra with the tachyonic dispersion relation before comparing to

*E*d

^{n}*N*/d

*E*in (6.1).

To this end, we start with an experimental plot of the
(assumed photonic) spectral density ,
parametrized by wavelength as inferred from the Bragg condition 2*d*sin(/2)=*n**λ*.
The power radiated over a finite range of wavelengths is .
We reparametrize with energy via the photonic dispersion relation *λ* = 2*π*/*ω*_{ph},

*ω*

_{ph,min}= 2

*π*/

*λ*

_{max}, and analogously for

*ω*

_{ph,max}. (Photon frequencies are denoted by a subscript ph, to distinguish them from their tachyonic counterpart.) By contrast, if the tachyonic dispersion relation is used for the energy parametrization, we find , where

This observationally determined tachyonic spectral density

*p*(

*ω*) is to be fitted with the tachyonic spectral average

*p*(

*ω*) in (6.1). More generally, if the photonic spectral density is plotted, the tachyonic density

*f*(

_{n}*ω*) :=

*ω*

^{n}^{−1}

*p*(

*ω*) is recovered as

Tachyonic and photonic frequencies are related by . This apparently requires

*ω*

_{ph}

*m*

_{t}, which is not a severe restriction, as the condition

*λ*< 2

*d*for Bragg scattering in a crystal amounts to roughly the same, cf. the end of Section 4.2.

We consider a set of photonic data points
labeled by index *i*. These flux points are inferred
from Bragg diffraction, that is, from the wavelength of the incident
quanta, and subsequently parametrized by frequency via the photonic
dispersion relation. If a tachyonic spectral fit based on density (6.1) is
performed, we have to use instead the tachyonic dispersion relation for
the energy parametrization. In effect, the photonic data points
are mapped into tachyonic points (*ω _{i}*,

*f*(

_{n}*ω*)) by

_{i}### Acknowledgements

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.