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## Decay kinematics in the aether: Pion decay, kaon decay, and superluminal Cherenkov radiation

Roman Tomaschitz

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Roman Tomaschitz 2013 EPL 102 61002
doi:10.1209/0295-5075/102/61002
Received 5 April 2013, accepted for publication 12 June 2013
Published 12 July 2013

### Abstract

Pion and kaon decay, π → μ + νμ, K → μ + νμ, into muons and GeV neutrinos is investigated with regard to a possible superluminal neutrino speed. Photonic Cherenkov emission by superluminal high-energy charges is shown to be forbidden by causality violation. A proper causality interpretation of decay processes outside the lightcone involving superluminal particles requires an absolute spacetime conception based on a distinguished frame of reference (aether frame). The universal reference frame is physically manifested as the rest frame of the cosmic microwave background (CMB) radiation. The propagation of particles and radiation modes in the CMB rest frame is determined by dispersive wave equations coupled to isotropic permeability tensors. The decay of ~50 GeV pions and ~85 GeV kaons generating the CERN neutrino beam to Gran Sasso (CNGS) is analyzed in this context. Causality constraints on the group velocity of the ~17 GeV muon neutrinos produced in the decay are derived and compared to recent experimental bounds on the neutrino speed.

### Introduction

We investigate pion and kaon decay into muons and muon neutrinos, which is the source of the CNGS neutrino beam [1]. This is motivated by upper bounds on a superluminal neutrino velocity recently established by the OPERA [2,3], BOREXINO [4], LVD [5] and ICARUS [6] experiments. Outside the lightcone, a proper causality interpretation of decay processes requires an absolute spacetime conception, as the time order of spacelike connections established by superluminal signals can be overturned in the rest frames of the interacting subluminal constituents [7,8]. The absolute spacetime, the aether, is manifested by dispersive permeability tensors, which affect the wave propagation of particles and radiation modes, in particular their group velocity. The permeability tensors are isotropic in a distinguished frame of reference defined by vanishing temperature dipole anisotropy of the CMB [912]. The coupling of Dirac and gauge fields to frequency-dependent permeability tensors has been explained in [13,14]. Here, we focus on specific decay and emission processes outside the lightcone.

We study the decays π → μ + νμ and K → μ + νμ as well as the hypothetical Cherenkov emission of photons by superluminal charges [1517]. Susceptibility functions are introduced, which allow to treat sub- and superluminal group velocities on equal footing and to develop the decay kinematics irrespectively of whether the particles and radiation quanta are sub- or superluminal. We obtain causality constraints on the group velocities in the CMB rest frame to be satisfied in addition to energy-momentum conservation. In the high-energy limit, these causality conditions can be made explicit as linear inequalities for the frequency-dependent susceptibility functions of the respective particles. These conditions are always met if all constituents of the interaction are subluminal, but they prohibit the often invoked [16, 17] photonic Cherenkov radiation by superluminal charges.

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### Dispersive two-particle decay in a permeable spacetime: energy-momentum conservation, refractive indices and susceptibility functions

We focus on pion and kaon decay as well as on Cherenkov radiation, but the formalism developed is applicable to two-particle decay in general. We start with energy-momentum conservation in the CMB rest frame (aether frame), ω in = ω out + ω ν, kin = kout + kν, where (ω in,kin ) denote the energy and momentum variables of the incoming pion or kaon, and (ω out,kout ) the variables of the muon (or the outgoing pion in the case of Cherenkov radiation π → π + γ). The frequency and wave vector of the neutrino (or photon) are denoted by (ω ν,kν ). The outgoing momenta are split into longitudinal and transversal components,

kout=λkin,0+λk,0,kν=λLkin,0λk,0,(1)

so that kin,0 k⊥,0 = 0. Subscript zeros denote unit vectors, kin =  kin (ω in )kin,0, kout =  kout (ωout )kout,0, kν =  kν(ων)kν,0, where the wave numbers are determined by dispersion relations. We square kout and kν in (1), λ 2 + λ 2 =  kout2 (ω out ), λ L2 + λ 2 =  kν 2 (ω ν ), and substitute ω out = ω in - ω ν into kout (ω out ). Subtracting these two equations, factorizing, and using momentum conservation λ L + λ  =  kin (ω in ), we obtain

λLλ2==k2ink2out+k2ν2kin,λ=k2in+k2outk2ν2kin,(kν+λL)(kνλL).(2)

We factorize the wave numbers of the in- and outgoing particles and the neutrino (or photon),

kinkν==ωinnin(ωin),kout=ωoutnout(ωout),ωνnν(ων),(3)

where the refractive indices nin,out,ν read [8]

nin=μinεin1m2inε2inω2in,nν=μνεν1m2νε2νω2ν,(4)

and analogously for nout. The permeabilities (ε in (ω in), μ in (ω in )), (ε out (ω out ),μ out (ω out )) and (ε ν (ω ν ),μ ν (ω ν )) are positive functions of the indicated variables, defining isotropic permeability tensors hin,out,ν,αβ in the CMB rest frame [14],

h00in(ωin)=ε2in(ωin),hikin(ωin)=δikμ2in(ωin),h0kin=0,(5)

and analogously for the tensors houtα β (ω out ) and hν α β (ω ν ). The subscript ν labels the neutrino or photon variables, and is not to be confused with a tensor index. The dispersion relations are derived from Klein-Gordon equations such as (hinα β αβ - min2 )ψ in = 0, obtained by squaring the Dirac equation of the respective particle [8]. We find hinα β kin,αkin,β +  min2 = 0, hν α β kν,α kν,β  +  mν 2 = 0, and similarly for kout,α, where kin,α = ( - ωin,kin), kout,α = ( - ω out,kout ) and kν ,α  = ( - ω ν ,kν ) are the 4-momenta of the in- and outgoing particles and the neutrino.

In high-energy interactions, where the speed of the sub- and superluminal particles is close to the speed of light, it is efficient to define susceptibility functions for each particle species, in analogy to dielectrics, which serve as expansion parameters. The electric and magnetic susceptibilities of the incoming particle are denoted by χ e,in = ε in - 1 and χ m,in = μ in - 1, and analogously for the out-state. The neutrino (or photonic) susceptibilities are χe,ν(ω ν ) = ε ν - 1 and χ m,ν(ω ν ) = μ ν - 1. We expand the neutrino (photon) refractive index nν (ω ν ) in linear order in χe,ν, χ m,ν and mν 2 /ω ν 2, cf. (4), and analogously the refractive indices of the in- and out-states. This triple Taylor expansion is possible if the velocities of all particles involved are close to the speed of light, so that the permeability tensors (5) are close to the Minkowski metric η α β  = diag( - 1,1,1,1), with electric and magnetic susceptibilities close to zero. In the case of pion and kaon decay, the squared mass/energy ratios in the GeV region are small as well, ensuring refractive indices close to 1 and thus small index variations

δnin=nin1χinm2in2ω2in,δninχin+m2inω3in,(6)

and analogously δnout and δ nν. Here, we expanded in linear order in the enumerated parameters, and defined the shortcut χ in = χ e,in + χ m,in, and analogously for χ out and χ ν. (That is, nin in (6) is linearized in χ e,in, χ m,in and min2 /ω in2.) The index variations δ nin,out,ν and susceptibilities χ in,out,ν can have either sign, being small parameters close to zero like the squared mass/energy ratios. If not indicated otherwise, the energy dependence of the susceptibility functions is χ in (ω in), χ ν (ω ν ), and χ out (ω out ), and similarly for δ nin,out,ν. The primes on δn'in,out,ν denote frequency derivatives; δn'in in (6) stands for (δ nin)' taken at ω in.

We substitute the wave numbers (3) and ω out = ω in - ω ν into the longitudinal momentum coefficients (2),

λL,=ω2inn2in(ωin)(ωinων)2n2out(ωout)±ω2νn2ν(ων)2ωinnin(ωin),(7)

where the upper sign refers to λ L. The squared transversal coefficient λ 2 in (2) is the product of (ω ν nν (ω ν ) ±λ L ). We expand (7) in linear order in the index variations δ nin,out,ν, cf. (6),

λLων+ωin(1ωνωin)δninωin(1ωνωin)2δnout+ω2νωinδnν,λωin(1ωνωin)+ωνδnin+ωin(1ωνωin)2δnoutω2νωinδnν,λ22ων(ωνnνλL)2ωνωin(1ωνωin)Δn,(8)

where Δ n denotes the refractive-index increment

Δn=(1ωνωin)δnout+ωνωinδnνδnin.(9)

The only approximation in (8) and (9) is linearization in the index variations δ nin,out,ν. The condition λ 2 > 0 thus means Δ n  > 0. All three frequencies in ω out = ω in - ω ν are positive, and positivity of the index increment Δ n in (9) is a necessary condition for momentum conservation, cf. (8).

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### Decay angles and group velocities in the CMB rest frame

To find the decay (or emission) angles, we write the wave vectors (1) as, cf. (3) and (6),

(ωinων)(1+δnout)kout,0=λkin,0+λk,0,ων(1+δnν)kν,0=λLkin,0λk,0,(10)

and multiply these identities by kin,0, using kin,0 k⊥,0 = 0. Defining the decay angles by cos θ out = kin,0 kout,0 and cos θ ν = kin,0 kν ,0, we obtain

cosθoutλ(1δnout)ωin(1ων/ωin)1ωνωinΔn1ων/ωin,cosθνλLων(1δnν)1ωinων(1ωνωin)Δn,(11)

with the longitudinal momentum coefficients λ and λ L in (8) and the refractive-index increment Δ n in (9). The only approximation in (11) is systematic linearization in the index variations δ nin,out,ν, cf. (6) and (9). Expanding the cosines, we find the decay angles

θout2Δnωin/ων1,θν(ωinων1)θout.(12)

The ratio θ ν /θ out does not depend on the refractive indices, and the angle between the wave vectors kout,0 and kν ,0 of the outgoing particles is θ ν + θ out ~θ out ω in /ω ν. Thus,

cos(θν+θout)1ωinωνΔn1ων/ωin,(13)

with cos (θ ν + θ out ) = kout,0 kν ,0.

The particle velocities are group velocities obtained as reciprocal frequency derivative of the wave number (3): 1/υ ν =  nν + ω ν n'ν, where υν=υνkν,0$\boldsymbol{\upsilon}_{\nu}=\upsilon_{\nu}{\bf k}_{\nu,0}$, cf. after (1), and analogously for υ in,out. We parametrize the absolute values υ in,out,ν with nin,out,ν = 1 + δ nin,out,ν, cf. (6), and expand in linear order,

υν1δnνωνδnν,υin,out1δnin,outωin,outδnin,out.(14)

The scalar products of the group velocities linearized in the index variations read

υνυin=υνυincosθν1Δυν×in,υoutυin=υoutυincosθout1Δυout×in,υoutυν=υoutυνcos(θν+θout)1Δυout×ν,(15)

where, cf. (9),

Δυν×in=δnν+ωνδnν+δnin+ωinδnin+(ωinων1)Δn,(16)

Δυout×in=δnin+ωinδnin+δnout+ωoutδnout+ωνωinΔn1ων/ωin,(17)

Δυout×ν=δnν+ωνδnν+δnout+ωoutδnout+ωinωνΔn1ων/ωin.(18)

The causality constraints are υiυj<1$\boldsymbol{\upsilon}_{i}\boldsymbol{\upsilon}_{j}< 1$, where the subscript i labels the subluminal velocities and j the superluminal ones in the aether frame [7]. These kinematic constraints are thus tantamount to positivity of the respective increments Δυ i × j of the velocity products in (16)–(18). For instance, if the pion velocity υ in is superluminal and the muon velocity υ out subluminal, the pion trajectory appears time inverted in the proper time of the muon if υoutυin>1$\boldsymbol{\upsilon}_{\rm out}\boldsymbol{\upsilon}_{\rm in} > 1$. In this case, the pion reemerges during the muon's proper lifetime, which is causality violating, as the pion was annihilated by decay at the time the muon was created. The velocity constraints υiυj<1$\boldsymbol{\upsilon}_{i}\boldsymbol{\upsilon}_{j}< 1$ in the aether frame are necessary and sufficient to exclude causality violating time inversions in the rest frames of the subluminal particles. The velocities refer to asymptotic in- and out-states of the interacting particles and radiation modes. The inertial frames and proper times of subluminal in- and out-states are linked by Lorentz boosts to the aether frame [8], which is the universal frame of reference, manifested as the homogeneous and isotropic CMB rest frame [18].

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### Causality conditions on the susceptibility functions of the aether

We parametrize the velocity increments Δυ i × j in (16)–(18) with the susceptibility functions (6), starting with

Δn(1ωνωin)χout+ωνωinχνχin+12(m2inω2inm2out/ω2in1ων/ωinm2νωinων),(19)

where we invoked energy conservation ω out = ω in (1 - ω ν /ω in ). We find

Δυν×inωinων(1ωνωin)2χout+(2ωνωin)χν+(2ωinων)χin+ωνχν+ωinχin+m2inm2out+m2ν2ωinων,(20)

Δυout×in(1+ωνωin)χout+ω2ν/ω2in1ων/ωinχν+12ων/ωin1ων/ωinχin+ωinχin+ωoutχout+m2in+m2outm2ν2ω2in(1ων/ωin).(21)

As a consistency check, we may interchange the indices ν ↔out in (20), and use ω out = ω in - ω ν to recover (21). Similarly, cf. (18),

Δυout×ν(1+ωinων)χout+2ων/ωin1ων/ωinχνωin/ων1ων/ωinχin+ωνχν+ωoutχout+m2inm2outm2ν2ωinων(1ων/ωin).(22)

By interchanging ν↔in, we recover (21).

For instance, the decay of a superluminal pion, π → μ + νμ, into a subluminal muon and a superluminal neutrino requires two kinematic causality conditions, Δυ out× in > 0 and Δυout×ν > 0. The third condition, Δυ ν × in > 0, need not be satisfied, as both the outgoing neutrino and the incoming pion are superluminal, so that neither of them admits a rest frame where a time inversion could occur. All causality conditions refer to group velocities in the CMB rest frame. The third condition for this decay is Δ n  > 0 in (19), required by momentum conservation, cf. (8). In brief, the causality condition Δυ i × j > 0 has to be satisfied in the aether frame if one of the indices labels a superluminal particle or radiation mode and the other a subluminal one. If this condition is violated, the trajectory of the superluminal particle is time inverted in the rest frame of the subluminal particle, so that absorption happens prior to emission in the proper time of the subluminal particle.

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### Pion decay π → μ + νμ generating the CNGS beam: constraints on the neutrino velocity

The mass squares in the refractive indices (6) of pion and muon read min2 =  mπ 2 ≈ 0.0195 GeV2 and mout2 =  mμ 2 ≈ 0.0112 GeV2 [19]. A neutrino mass of below 2 eV [20] in the neutrino refractive index is negligible in the GeV range, mν 2 ≈ 0, cf. (4) and (6). As for the CNGS neutrino beam, the energy of the incoming pions is about ω in ≈ 50 GeV [1], and the average energy of the neutrinos ω ν ≈ 17 GeV [26]. The neutrino refractive index (6) is nν (ω ν ) = 1 + δ nν, with δ nν =  nν - 1 ~ χ ν. The frequency derivatives of the susceptibilities are put to zero, χ 'ν ,in,out ≈ 0. The pion refractive index is parametrized by δ nin ~χ in - min2 /(2ω in2 ), with derivative δ n'in ~ min2 /ω in3, cf. (6), and analogously for the outgoing muon, δ nout ~χ out - mout2 /(2ω out2 ), and δ n'out ~ mout2 /ω out3, taken at ω out = ω in - ω ν ≈ 33 GeV. The refractive-index increment (19) gives the constraint

Δn0.66χout+0.34χνχin+5.06×107>0,(23)

required by energy-momentum conservation. The causality conditions for this decay are obtained from the velocity increments (20)–(22), with χ 'ν ,in,out ≈ 0 and mν 2 ≈ 0:

Δυν×in1.28χout+1.66χν0.94χin+4.88×106>0,(24)

Δυout×in(1ωνωin)1.34χout+0.175χν+0.485χin+9.30×106>0,(25)

Δυout×ν(1ωνωin)2.60χout+1.66χν2.94χin+4.88×106>0.(26)

The susceptibilities refer to different frequencies, χ out (ω out ), χ ν (ω ν ) and χ in (ω in ), and the frequency variation is neglected, assuming vanishing derivatives at the respective energies. The neutrino group velocity and refractive index read, cf. (6) and (14),

υν11nνm2ν2ω2νχνωνχν,υν1+m2νω2ν+ωνχν,(27)

and analogously for the pionic and muonic group velocities υ in,out and their refractive indices nin,out. The susceptibilities χ ν ,in,out are close to zero and can have either sign, and the same holds true for υ ν ,in,out - 1 and 1 - nν ,in,out in (27). In deriving (27), we used (ω nν )' = 1/υ ν. We also put χ 'ν ,in,out ≈ 0 and mν 2 ≈ 0 as in (23)–(26), so that 1 - nν ~υ ν - 1 ~ - χ ν. A neutrino index nν < 1 or a negative susceptibility is thus tantamount to a superluminal neutrino speed υ ν > 1. The pion and muon velocities are related to their susceptibilities by υ in - 1 ~ - 3.90 × 10 - 6 - χ in and υ out - 1 ~ - 5.14 × 10 - 6 - χ out.

The OPERA Collaboration derived the bound υ ν - 1 = (2.7 ± 6.5) × 10 - 6 on the neutrino velocity, based on ~15200 events collected in 2009–2011 [2]. BOREXINO obtained υ ν - 1 = 2.7 ± 5.4 × 10 - 6 in the Oct./Nov. 2011 run, and |υν1|<2.1×106$|{\upsilon_\nu - 1}| < 2.1 \times 10^{ - 6}$ in May 2012 [4]. The OPERA upper bounds inferred from the May 2012 data are υ ν - 1 < 2.3 × 10 - 6 and υνˉ1<3.0×106$\upsilon_{\bar \nu } - 1 < 3.0 \times 10^{ - 6}$ for antineutrinos [3]. The LVD experiment obtained |υν1|<3.5×106$\left| {\upsilon_\nu - 1} \right| < 3.5 \times 10^{ - 6}$ [5], and the ICARUS bound from the May 2012 run is |υν1|<1.6×106$\left| {\upsilon_\nu - 1} \right| < 1.6 \times 10^{ - 6}$ [6]. All bounds refer to an averaged neutrino energy of 17 GeV in the CNGS beam. The OPERA bound from the 2009–2011 data [2] is based on a much larger sample than the other experiments (<100 events). The current MINOS estimate, υν − 1 = 6 ±13 ×10−6 [21], is in line with the quoted CNGS results; all these experiments report a positive neutrino excess velocity with an error bound safely consistent with the speed of light.

Given the low relative speed υ r ≈ 1.2 × 10 - 3 of the Solar system barycenter in the CMB rest frame [11,12], we can use the linearized addition law for velocities, υ CMB - 1 = (υ ν - 1)(1 + O(υ r )), where υ CMB is the neutrino speed in the CMB rest frame and υ ν the speed measured in the baseline frame (rest frame of source and detector) [8]. Hence, υ CMB - 1 ~ υ ν - 1 ~ 1 - nν, cf. after (27).

In the following, we assume that the incoming pion and the outgoing muon are subluminal, and the neutrino is superluminal. Furthermore, we assume that pion and muon have similar susceptibilities, so that we can equate χ inχ outχ in the energy-momentum and causality conditions (23)–(26). (We will later drop this assumption, cf. after (30).) The velocity condition on pion and muon is 1 - υ in,out > 0, which gives the bound - 3.90 × 10 - 6 <χ, cf. the estimates stated after (27). Since the neutrino is superluminal, we have χ ν < 0. The energy-momentum constraint (23) combined with the velocity condition gives

3.90×106<χ<χν+1.49×106.(28)

The causality conditions (24) and (26) read

(4.88χν+1.435×105)<χ<4.88χν+1.435×105,(29)

which can only be satisfied if χ ν > - 2.94 × 10 - 6. In fact, for (28) to be consistent with (29), an even stronger lower bound on χ ν is required, χ ν > - 2.694 × 10 - 6. The causality condition (25) does not apply, as the pion as well as the muon are subluminal, cf. after (22). If χ ν ≈ - 2.694 × 10 - 6, we find the unique solution χ ≈ - 1.204 × 10 - 6. (At the opposite edge χ ν ≈ 0 of the allowed χ ν interval, the admissible χ range is given by (28).) As we have put χ inχ outχ, we find the pion and muon velocities 1 - υ in ~ 2.7 × 10 - 6 and 1 - υ out ~ 3.9 × 10 - 6, respectively. The neutrino excess velocity is υ ν - 1 ~ - χ ν ~ 2.7 × 10 - 6, which coincides with the quoted OPERA and BOREXINO 2011 upper bounds [2,4].

We return to the basic energy-momentum and causality conditions (23), (24) and (26), substitute χ ν ≈ - 2.7 × 10 - 6, and drop the assumption of equal pion and muon susceptibilities χ inχ out to obtain

Δn0.66χoutχin4.1×107>0,(30)

Δυν×in1.28χout0.94χin+4.1×107>0,(31)

Δυout×ν2.60χout2.94χin+4.1×107>0.(32)

Adding the first to the second and third of these inequalities, we find χ in <χ out and χ in < 0.83χ out, the latter is weaker and can be ignored if we consider negative susceptibilities χ in,out < 0. The constraints (30)–(32) thus reduce to χ in < 0.66χ out - 4.1 × 10 - 7 and χ in <χ out for χ in,out in the range - 3.90 × 10 - 6 <χ in < 0 and - 5.14 × 10 - 6 <χ out < 0. The latter two lower bounds on the pion and muon susceptibilities are required by a subluminal particle velocity, cf. after (27). These constraints are based on the neutrino susceptibility χ ν ≈ - 2.7 × 10 - 6; a possible solution is χ inχ out ≈ - 1.2 × 10 - 6 as discussed after (29).

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### Kaon decay K → μ + νμ and superluminal muon neutrinos

The reasoning is analogous to that of pion decay, cf. (30)–(32). The pionic mass square is replaced by the kaon mass, min2 =  mK2 ≈ 0.244 GeV2 [19], and the energy of the incoming kaon is ω in ≈ 85 GeV [1], so that ω out = ω in - ω ν ≈ 68 GeV for the outgoing muon. The group velocities of kaon and muon are calculated as in (27), 1 - υ in ~ 1.7 × 10 - 5 + χ in, and 1 - υ out ~ 1.2 × 10 - 6 + χ out. The energy-momentum and causality constraints (19), (20) and (22) read

Δn0.8χout+0.2χνχin+1.6×105>0,(33)

Δυν×in1.1χout+0.6χνχin+2.7×105>0,(34)

Δυout×ν1.0χout+0.36χνχin+1.6×105>0.(35)

These conditions can readily be satisfied with a neutrino susceptibility in the interval 0 >χ ν > - 2.7 × 10 - 6, since in this case the χ ν terms are negligible; χ ν ≈ - 2.7 × 10 - 6 is the neutrino susceptibility defined by the quoted OPERA [2] and BOREXINO [4] upper bounds on the neutrino excess velocity. Subluminal kaon and muon velocities require the constraints - 1.7 × 10 - 5 <χ in and - 1.2 × 10 - 6 <χ out. Conditions (33)–(35) are satisfied by negative susceptibilities χ in,out subject to these velocity bounds. We do not assume χ inχ out, as the kaon and muon mass squares substantially differ. χ out (ω out ) refers to a muon energy of ω out ≈ 68 GeV, as compared to 33 GeV in the case of pion decay. The neutrino susceptibility χ ν (ω ν ) is taken at a neutrino energy of 17 GeV in both cases.

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### Causality violation prohibiting photon emission π→π+γ${\pi}\rightarrow{\pi}+{\gamma}$ by superluminal high-energy charges

In this section, the subscript index ν labels photon variables. The outgoing photon with frequency ω ν has zero rest mass mν 2 = 0 and a refractive index nν - 1 = δ nν ~χ ν, cf. (6). We assume a nearly constant photon susceptibility χ ν ≥ 0, χ 'ν ≈ 0, so that the photonic group velocity (14) is υ ν ≈ 1 - χ ν ≤ 1. We use a refractive photon index that is slightly larger than one, so that a rest frame exists for the photon. The causality conditions are Δυ ν × in > 0 and Δυ out×ν > 0, cf. (16) and (18). The vacuum limit χν → 0 (photonic permeability tensor coinciding with Minkowski metric) is performed in the subsequent inequalities by putting δ nν ≈ 0 and δ n'ν ≈ 0 in the causality constraints and the refractive-index increment (9). Energy conservation means ω out = ω in - ω ν, with positive frequencies. The refractive indices of the in- and outgoing charges are δ nin,out ~χ in,out - min2 /(2ω in,out2 ), cf. (6), with derivatives δ n'in,out ~χ 'in,out +  min2 /ω in,out3 at ω in,out. The susceptibility functions χ in,out (ω ) are identical, but taken at different energies ω in,out. Expanding χ in (ω ) ≈χ in + (ω - ω in )χ 'in in linear order at ω in, and making use of energy conservation, we can approximate χ out (ω out ) ≈χ in (ω in ) - ω ν χ 'in (ω in ) and χ 'out (ω out ) ≈χ 'in (ω in ). The refractive-index increment (19) reads in this case

Δnων/ωin1ων/ωin[(1ωνωin)2ωinχin+(1ωνωin)χin+m2in2ω2in].(36)

First, we show that one of the causality conditions Δυ ν × in > 0 and Δυ out×ν > 0 is violated for negative χ in. In fact, the velocity increment (20) reads

Δυν×inχin(ωνωin+(2ωνωin)ωνχinχin),(37)

so that Δυ ν × in < 0 if both χ in and χ 'in are negative. Increment (22) can be written as

Δυout×νχin1+ων/ωin(2+ωνωin2(1+ωνωin)ωνχinχin),(38)

so that Δυ out×ν < 0 if χ in is negative and χ 'in positive. Thus the emission π → π γ is causality violating if the susceptibility χ in (ω in ) is negative.

This emission process is also forbidden in the case of a positive susceptibility χ in (ω in ). If both χ in and χ 'in are positive, this implies a negative refractive-index increment Δ n , cf. (36), so that momentum cannot be conserved. (In this case, the group velocity (27) of the incoming charge is subluminal.) If χ in is positive and χ 'in negative, the conditions Δ n  > 0 and Δυ ν × in > 0 cannot simultaneously be satisfied, cf. (36) and (37). In fact, we may drop the mass term in (36) and require (1 - ω ν /ω in )ω in χ 'in + χ in < 0, which is necessary (but not sufficient) for Δ n  > 0. The causality condition Δυ ν × in > 0 implies (ω ν /ω in - 2)ω in χ 'in - χ in < 0, cf. (37). Adding these inequalities, we obtain - ω in χ 'in < 0, in contradiction to the assumed negative derivative χ 'in. We have thus demonstrated that photon emission by superluminal high-energy charges is forbidden since it is causality violating.

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### Conclusion

We have studied two-particle decay in a dispersive spacetime, deriving bounds on a superluminal group velocity of the decay products. The nonlinear causality and energy-momentum constraints can be linearized in the high-energy regime by introducing frequency-dependent susceptibility functions for the in- and outgoing particles which serve as expansion parameters. In this way, analytically tractable causality conditions are obtained even in multi-channel interactions. These constraints on the susceptibility functions in the isotropic aether frame (identified as CMB rest frame [22]) prevent time inversions in the rest frames of the subluminal particles and radiation modes (inertial in- and out-states) of the decay process [7].

Specifically, we discussed the dispersive kinematics of pion and kaon decay in the aether, and calculated the susceptibility functions with input parameters of the CNGS neutrino beam. The causality conditions are linear inequalities to be satisfied by the susceptibilities of the respective particles. We employed these constraints to obtain velocity estimates for the muon and muon neutrino generated by the decay. Finally we used causality conditions on susceptibility functions to demonstrate, without the use of specific input parameters, that photonic Cherenkov radiation by superluminal high-energy charges is causality violating.

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### References

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Bandyopadhyay A. et al 2009 Rep. Prog. Phys. 72 106201 IOPscience
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Bi X.-J. et al 2011 Phys. Rev. Lett. 107 241802 CrossRef
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Tomaschitz R. 2012 Mon. Not. R. Astron. Soc. 427 1363 CrossRef
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↑ Close

References
Close
[1]
Elsener K. et al 1998 CERN-98-02 doi: 10.5170/CERN-1998-002 CrossRef
[2]
Adam T. et al 2012 JHEP 10 093 CrossRef
[3]
Adam T. et al 2013 JHEP 01 153 CrossRef
[4]
Alvarez Sanchez P. et al 2012 Phys. Lett. B 716 401 CrossRef
[5]
Agafonova N.Yu. et al 2012 Phys. Rev. Lett. 109 070801 CrossRef
[6]
Antonello M. et al 2012 JHEP 11 049 CrossRef
[7]
Tomaschitz R. 2012 EPL 98 19001 IOPscience
[8]
Tomaschitz R. 2013 Phys. Lett. A 377 945 CrossRef
[9]
Kogut A. et al 1993 Astrophys. J. 419 1 CrossRef
[10]
Fixsen D.J. et al 1996 Astrophys. J. 473 576 IOPscience
[11]
Hinshaw G. et al 2009 Astrophys. J. Suppl. Ser. 180 225 IOPscience
[12]
Planck Collaboration, Aghanim N. et al 2013 arXiv:1303.5087 Preprint
[13]
Tomaschitz R. 2010 Eur. Phys. J. C 69 241 CrossRef
[14]
Tomaschitz R. 2012 EPL 97 39003 IOPscience
[15]
Bandyopadhyay A. et al 2009 Rep. Prog. Phys. 72 106201 IOPscience
[16]
Bi X.-J. et al 2011 Phys. Rev. Lett. 107 241802 CrossRef