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

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 [9–12]. 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 + ν_μ and K → μ + ν_μ as well as the hypothetical Cherenkov emission of photons by superluminal charges [15–17]. 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.

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, k_in = k_out + k_ν, where (ω_in,k_in ) denote the energy and momentum variables of the incoming pion or kaon, and (ω_out,k_out ) 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,k_ν ). The outgoing momenta are split into longitudinal and transversal components,

$$\begin{equation} {\bf{k}}_{{\rm{out}}} = \lambda _{\parallel} {\bf{k}}_{{\rm{in,0}}} + \lambda _{\bot} {\bf{k}}_{{\bot}{\rm{,0}}},\quad {\bf{k}}_\nu = \lambda _{\rm{L}} {\bf{k}}_{{\rm{in,0}}} - \lambda _{\bot} {\bf{k}}_{{\bot}{\rm{,0}}} , \end{equation} \tag{ 1 }$$

so that k_in,0 k_⊥,0 = 0. Subscript zeros denote unit vectors, k_in =  k_in (ω_in )k_in,0, k_out =  k_out (ω_out )k_out,0, k_ν =  k_ν(ω_ν)k_ν,0, where the wave numbers are determined by dispersion relations. We square k_out and k_ν in (1), λ_∥² + λ_⊥² =  k_out² (ω_out ), λ_L² + λ_⊥² =  k_ν ² (ω_ν ), and substitute ω_out = ω_in - ω_ν into k_out (ω_out ). Subtracting these two equations, factorizing, and using momentum conservation λ_L + λ_∥ =  k_in (ω_in ), we obtain

$$\begin{eqnarray} \lambda _{\rm{L}} &=& \frac{{k_{{\rm{in}}}^2 - k_{{\rm{out}}}^2 + k_\nu ^2 }}{{2k_{{\rm{in}}} }},\quad \lambda _{\parallel} = \frac{{k_{{\rm{in}}}^2 + k_{{\rm{out}}}^2 - k_\nu ^2 }}{{2k_{{\rm{in}}} }},\nonumber\\ \lambda _{\bot}^2 &=& (k_\nu + \lambda _{\rm{L}} )(k_\nu - \lambda _{\rm{L}} ). \end{eqnarray} \tag{ 2 }$$

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

$$\begin{eqnarray} k_{{\rm{in}}} &=& \omega _{{\rm{in}}} n_{{\rm{in}}} (\omega _{{\rm{in}}} ),\quad k_{{\rm{out}}} = \omega _{{\rm{out}}} n_{{\rm{out}}} (\omega _{{\rm{out}}} ),\nonumber\\ k_\nu &=& \omega _\nu n_\nu (\omega _\nu), \end{eqnarray} \tag{ 3 }$$

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

$$\begin{equation} n_{{\rm{in}}} = \mu _{{\rm{in}}} \varepsilon _{{\rm{in}}} \sqrt {1 - \frac{{m_{{\rm{in}}}^2 }}{{\varepsilon _{{\rm{in}}}^2 \omega _{{\rm{in}}}^2 }}},\quad n_\nu = \mu _\nu \varepsilon _\nu \sqrt {1 - \frac{{m_\nu ^2 }}{{\varepsilon _\nu ^2 \omega _\nu ^2 }}}, \end{equation} \tag{ 4 }$$

and analogously for n_out. 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 h_in,out,ν,^αβ in the CMB rest frame [14],

$$\begin{eqnarray} &&h_{{\rm{in}}}^{00} (\omega _{{\rm{in}}} ) = - \varepsilon _{{\rm{in}}}^2 (\omega _{{\rm{in}}} ),\quad h_{{\rm{in}}}^{ik} (\omega _{{\rm{in}}} ) = \frac{{\delta ^{ik} }}{{\mu _{{\rm{in}}}^2 (\omega _{{\rm{in}}} )}},\quad h_{{\rm{in}}}^{0k} = 0,\nonumber\\ \end{eqnarray} \tag{ 5 }$$

and analogously for the tensors h_out^{α β} (ω_out ) and h_ν ^{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 (h_in^{α β} ∂_α ∂_β - m_in² )ψ_in = 0, obtained by squaring the Dirac equation of the respective particle [8]. We find h_in^{α β} k_in,αk_in,β +  m_in² = 0, h_ν ^{α β} k_ν,α k_ν,β  +  m_ν ² = 0, and similarly for k_out,α, where k_in,α = ( - ω_in,k_in), k_out,α = ( - ω_out,k_out ) and k_{ν ,α}  = ( - ω_ν ,k_ν ) 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 χ_e,ν(ω_ν ) = ε_ν - 1 and χ_m,ν(ω_ν ) = μ_ν - 1. We expand the neutrino (photon) refractive index n_ν (ω_ν ) in linear order in χ_e,ν, χ_m,ν and m_ν 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

$$\begin{equation} \delta n_{{\rm{in}}} = n_{{\rm{in}}} - 1\sim \chi _{{\rm{in}}} - \frac{{m_{{\rm{in}}}^2 }}{{2\omega _{{\rm{in}}}^2 }},\quad \delta n'_{{\rm{in}}} \sim \chi '_{{\rm{in}}} + \frac{{m_{{\rm{in}}}^2 }}{{\omega _{{\rm{in}}}^3 }}, \end{equation} \tag{ 6 }$$

and analogously δn_out and δ n_ν. 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, n_in in (6) is linearized in χ_e,in, χ_m,in and m_in² /ω_in².) The index variations δ n_in,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 n_in,out,ν. The primes on δn'_in,out,ν denote frequency derivatives; δn'_in in (6) stands for (δ n_in)' taken at ω_in.

We substitute the wave numbers (3) and ω_out = ω_in - ω_ν into the longitudinal momentum coefficients (2),

$$\begin{eqnarray} &&\lambda _{{\rm{L,\parallel}}} = \frac{{\omega _{{\rm{in}}}^2 n_{{\rm{in}}}^2 (\omega _{{\rm{in}}} ) \mp (\omega _{{\rm{in}}} - \omega _\nu )^2 n_{{\rm{out}}}^2 (\omega _{{\rm{out}}} ) \pm \omega _\nu ^2 n_\nu ^2 (\omega _\nu )}}{{2\omega _{{\rm{in}}} n_{{\rm{in}}} (\omega _{{\rm{in}}} )}},\nonumber\\ \end{eqnarray} \tag{ 7 }$$

where the upper sign refers to λ_L. The squared transversal coefficient λ_⊥² in (2) is the product of (ω_ν n_ν (ω_ν ) ±λ_L ). We expand (7) in linear order in the index variations δ n_in,out,ν, cf. (6),

$$\begin{eqnarray} &&\lambda_{\rm{L}}\!\sim \omega_\nu\!+\!\omega_{{\rm{in}}}\left(\!1\!-\!\frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}\!\right)\delta n_{{\rm{in}}}\!-\!\omega_{{\rm{in}}} \left(\!1\!-\!\frac{{\omega_\nu }}{{\omega_{{\rm{in}}}}}\!\right)^2 \delta n_{{\rm{out}}}\!+\!\frac{{\omega_\nu^2 }}{{\omega_{{\rm{in}}} }}\delta n_\nu,\nonumber\\ &&\lambda_{\parallel}\sim \omega_{{\rm{in}}} \left(\!1\!-\!\frac{{\omega_\nu }}{{\omega_{{\rm{in}}}}}\!\right)\!+\omega_\nu \delta n_{{\rm{in}}}\!+\omega_{{\rm{in}}}\left(\!1\!-\!\frac{{\omega_\nu }}{{\omega_{{\rm{in}}}}}\!\right)^2 \delta n_{{\rm{out}}}\!-\!\frac{{\omega_\nu ^2 }}{{\omega_{{\rm{in}}}}}\delta n_\nu,\nonumber\\ &&\lambda_{\bot}^2\sim 2\omega_\nu (\omega_\nu n_\nu\!-\!\lambda_{\rm{L}} )\sim 2\omega_\nu \omega_{{\rm{in}}}\left(\!1-\frac{{\omega_\nu }}{{\omega_{{\rm{in}}}}}\!\right)\Delta n, \end{eqnarray} \tag{ 8 }$$

where Δ n denotes the refractive-index increment

$$\begin{equation} \Delta n = \left( {1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\delta n_{{\rm{out}}} + \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}\delta n_\nu - \delta n_{{\rm{in}}}. \end{equation} \tag{ 9 }$$

The only approximation in (8) and (9) is linearization in the index variations δ n_in,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).

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

$$\begin{eqnarray} &&(\omega_{{\rm{in}}} - \omega_\nu )(1 + \delta n_{{\rm{out}}} ){\bf{k}}_{{\rm{out,0}}} = \lambda_{\parallel} {\bf{k}}_{{\rm{in,0}}} + \lambda_{\bot} {\bf{k}}_{{\bot}{\rm{,0}}} ,\nonumber\\ &&\omega_\nu (1 + \delta n_\nu ){\bf{k}}_{\nu ,0} = \lambda_{\rm{L}} {\bf{k}}_{{\rm{in,0}}} - \lambda_{\bot} {\bf{k}}_{{\bot}{\rm{,0}}}, \end{eqnarray} \tag{ 10 }$$

and multiply these identities by k_in,0, using k_in,0 k_⊥,0 = 0. Defining the decay angles by cos θ_out = k_in,0 k_out,0 and cos θ_ν = k_in,0 k_{ν ,0}, we obtain

$$\begin{eqnarray} &&\cos \theta_{{\rm{out}}} \sim \frac{{\lambda_{\parallel} (1 - \delta n_{{\rm{out}}} )}}{{\omega_{{\rm{in}}} (1 - \omega_\nu /\omega_{{\rm{in}}} )}}\sim 1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}\frac{{\Delta n}}{{1 - \omega_\nu /\omega_{{\rm{in}}} }},\nonumber\\ &&\cos \theta_\nu \sim \frac{{\lambda_{\rm{L}} }}{{\omega_\nu }}(1 - \delta n_\nu )\sim 1 - \frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }}\left( {1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\Delta n,\quad \end{eqnarray} \tag{ 11 }$$

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 n_in,out,ν, cf. (6) and (9). Expanding the cosines, we find the decay angles

$$\begin{equation} \theta_{{\rm{out}}} \sim \frac{{\sqrt {2\Delta n} }}{{\sqrt {\omega_{{\rm{in}}} /\omega_\nu - 1} }},\qquad \theta_\nu \sim \left(\frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }} - 1\right)\theta_{{\rm{out}}}. \end{equation} \tag{ 12 }$$

The ratio θ_ν /θ_out does not depend on the refractive indices, and the angle between the wave vectors k_out,0 and k_{ν ,0} of the outgoing particles is θ_ν + θ_out ∼θ_out ω_in /ω_ν. Thus,

$$\begin{equation} \cos (\theta_\nu + \theta_{{\rm{out}}} )\sim 1 - \frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }}\frac{{\Delta n}}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}, \end{equation} \tag{ 13 }$$

with cos (θ_ν + θ_out ) = k_out,0 k_{ν ,0}.

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

$$\begin{eqnarray} &&\upsilon_\nu \sim 1 - \delta n_\nu - \omega_\nu \delta n'_\nu,\nonumber\\\\ &&\upsilon_{{\rm{in,out}}} \sim 1 - \delta n_{{\rm{in,out}}} - \omega_{{\rm{in,out}}} \delta n'_{{\rm{in,out}}}.\nonumber \end{eqnarray} \tag{ 14 }$$

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

$$\begin{eqnarray} &&{\boldsymbol{\upsilon }}_\nu {\boldsymbol{\upsilon }}_{{\rm{in}}} = \upsilon_\nu \upsilon_{{\rm{in}}} \cos \theta_\nu \sim 1 - \Delta \upsilon_{\nu \times {\rm{in}}},\nonumber\\ &&{\boldsymbol{\upsilon }}_{{\rm{out}}} {\boldsymbol{\upsilon }}_{{\rm{in}}} = \upsilon_{{\rm{out}}} \upsilon_{{\rm{in}}} \cos \theta_{{\rm{out}}} \sim 1 - \Delta \upsilon_{{\rm{out}} \times {\rm{in}}} , \\ &&{\boldsymbol{\upsilon }}_{{\rm{out}}} {\boldsymbol{\upsilon }}_\nu = \upsilon_{{\rm{out}}} \upsilon_\nu \cos (\theta_\nu + \theta_{{\rm{out}}} )\sim 1 - \Delta \upsilon_{{\rm{out}} \times \nu },\quad\nonumber \end{eqnarray} \tag{ 15 }$$

where, cf. (9),

$$\begin{eqnarray} &&\Delta \upsilon_{\nu \times {\rm{in}}} = \delta n_\nu + \omega_\nu \delta n'_\nu + \delta n_{{\rm{in}}} + \omega_{{\rm{in}}} \delta n'_{{\rm{in}}} + \left( {\frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }} - 1} \right)\Delta n,\nonumber\\ \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \Delta \upsilon_{{\rm{out}} \times {\rm{in}}} &=& \delta n_{{\rm{in}}} + \omega_{{\rm{in}}} \delta n'_{{\rm{in}}} + \delta n_{{\rm{out}}} + \omega_{{\rm{out}}} \delta n'_{{\rm{out}}}\nonumber\\ &&+ \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}\frac{{\Delta n}}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}, \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} \Delta \upsilon_{{\rm{out}} \times \nu } &=& \delta n_\nu + \omega_\nu \delta n'_\nu + \delta n_{{\rm{out}}} + \omega_{{\rm{out}}} \delta n'_{{\rm{out}}}\nonumber\\ &&+ \frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }}\frac{{\Delta n}}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}. \end{eqnarray} \tag{ 18 }$$

The causality constraints are $\boldsymbol{\upsilon}_{i}\boldsymbol{\upsilon}_{j}< 1$, 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 $\boldsymbol{\upsilon}_{\rm out}\boldsymbol{\upsilon}_{\rm in} > 1$. 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 $\boldsymbol{\upsilon}_{i}\boldsymbol{\upsilon}_{j}< 1$ 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].

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

$$\begin{eqnarray} \Delta n&\sim& \left( {1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\chi_{{\rm{out}}} + \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}\chi_\nu - \chi_{{\rm{in}}}\nonumber\\ &&+ \frac{1}{2}\left( {\frac{{m_{{\rm{in}}}^2 }}{{\omega_{{\rm{in}}}^2 }} - \frac{{m_{{\rm{out}}}^2 /\omega_{{\rm{in}}}^2 }}{{1 - \omega_\nu /\omega_{{\rm{in}}} }} - \frac{{m_\nu ^2 }}{{\omega_{{\rm{in}}} \omega_\nu }}} \right), \end{eqnarray} \tag{ 19 }$$

where we invoked energy conservation ω_out = ω_in (1 - ω_ν /ω_in ). We find

$$\begin{eqnarray} \Delta \upsilon_{\nu \times {\rm{in}}} &\sim& \frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }}\left( {1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)^2 \chi_{{\rm{out}}} + \left( {2 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\chi_\nu \nonumber\\ &&+ \left( {2 - \frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }}} \right)\chi_{{\rm{in}}}+ \omega_\nu \chi '_\nu + \omega_{{\rm{in}}} \chi '_{{\rm{in}}} \nonumber\\ &&+ \frac{{m_{{\rm{in}}}^2 - m_{{\rm{out}}}^2 + m_\nu ^2 }}{{2\omega_{{\rm{in}}} \omega_\nu }}, \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \Delta \upsilon_{{\rm{out}} \times {\rm{in}}} &\sim& \left( {1 + \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\chi_{{\rm{out}}}\nonumber\\ &&+ \frac{{\omega_\nu ^2 /\omega_{{\rm{in}}}^2 }}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}\chi_\nu + \frac{{1 - 2\omega_\nu /\omega_{{\rm{in}}} }}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}\chi_{{\rm{in}}} \nonumber\\ &&+ \omega_{{\rm{in}}} \chi '_{{\rm{in}}} + \omega_{{\rm{out}}} \chi '_{{\rm{out}}} + \frac{{m_{{\rm{in}}}^2 + m_{{\rm{out}}}^2 - m_\nu ^2 }}{{2\omega_{{\rm{in}}}^2 (1 - \omega_\nu /\omega_{{\rm{in}}} )}}.\nonumber\\ \end{eqnarray} \tag{ 21 }$$

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),

$$\begin{eqnarray} \Delta \upsilon_{{\rm{out}} \times \nu } &\!\sim\!& \left(\!1 + \frac{{\omega_{{\rm{in}}} }}{{\omega_\nu }}\!\right)\chi_{{\rm{out}}}\!+\!\frac{{2 - \omega_\nu /\omega_{{\rm{in}}} }}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}\chi_\nu - \frac{{\omega_{{\rm{in}}} /\omega_\nu }}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}\chi_{{\rm{in}}} \nonumber\\ &&+ \omega_\nu \chi '_\nu + \omega_{{\rm{out}}} \chi '_{{\rm{out}}} + \frac{{m_{{\rm{in}}}^2 - m_{{\rm{out}}}^2 - m_\nu ^2 }}{{2\omega_{{\rm{in}}} \omega_\nu (1 - \omega_\nu /\omega_{{\rm{in}}} )}}.\nonumber\\ \end{eqnarray} \tag{ 22 }$$

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

For instance, the decay of a superluminal pion, π → μ + ν_μ, into a subluminal muon and a superluminal neutrino requires two kinematic causality conditions, Δυ_{out× in} > 0 and Δυ_out×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.

The mass squares in the refractive indices (6) of pion and muon read m_in² =  m_π ² ≈ 0.0195 GeV² and m_out² =  m_μ ² ≈ 0.0112 GeV² [19]. A neutrino mass of below 2 eV [20] in the neutrino refractive index is negligible in the GeV range, m_ν 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 [2–6]. The neutrino refractive index (6) is n_ν (ω_ν ) = 1 + δ n_ν, with δ n_ν =  n_ν - 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 n_in ∼χ_in - m_in² /(2ω_in² ), with derivative δ n'_in ∼ m_in² /ω_in³, cf. (6), and analogously for the outgoing muon, δ n_out ∼χ_out - m_out² /(2ω_out² ), and δ n'_out ∼ m_out² /ω_out³, taken at ω_out = ω_in - ω_ν ≈ 33 GeV. The refractive-index increment (19) gives the constraint

$$\begin{equation} \Delta n\sim 0.66\chi_{{\rm{out}}} + 0.34\chi_\nu - \chi_{{\rm{in}}} + 5.06 \times 10^{ - 7} > 0, \end{equation} \tag{ 23 }$$

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 m_ν ² ≈ 0:

$$\begin{eqnarray} &&\Delta \upsilon_{\nu \times {\rm{in}}} \sim 1.28\chi_{{\rm{out}}} + 1.66\chi_\nu - 0.94\chi_{{\rm{in}}} + 4.88 \times 10^{ - 6} > 0,\nonumber\\ \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray} \Delta \upsilon_{{\rm{out}} \times {\rm{in}}} \left( {1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)&\sim &1.34\chi_{{\rm{out}}} + 0.175\chi_\nu + 0.485\chi_{{\rm{in}}}\nonumber\\ &&+ 9.30 \times 10^{ - 6} > 0,\qquad \end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray} \Delta \upsilon_{{\rm{out}} \times \nu } \left({1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)&\sim & 2.60\chi_{{\rm{out}}} + 1.66\chi_\nu - 2.94\chi_{{\rm{in}}} \nonumber\\ &&+\,4.88\times 10^{ - 6} > 0. \end{eqnarray} \tag{ 26 }$$

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),

$$\begin{eqnarray} \upsilon_\nu - 1&\sim& - \frac{{m_\nu ^2 }}{{2\omega_\nu ^2 }} - \chi_\nu - \omega_\nu \chi '_\nu ,\nonumber\\\\ 1 - n_\nu &\sim& \upsilon_\nu - 1 + \frac{{m_\nu ^2 }}{{\omega_\nu ^2 }} + \omega_\nu \chi '_\nu ,\nonumber \end{eqnarray} \tag{ 27 }$$

and analogously for the pionic and muonic group velocities υ_in,out and their refractive indices n_in,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 - n_{ν ,in,out} in (27). In deriving (27), we used (ω n_ν )' = 1/υ_ν. We also put χ '_{ν ,in,out} ≈ 0 and m_ν ² ≈ 0 as in (23)–(26), so that 1 - n_ν ∼υ_ν - 1 ∼ - χ_ν. A neutrino index n_ν < 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 $|{\upsilon_\nu - 1}| < 2.1 \times 10^{ - 6}$ in May 2012 [4]. The OPERA upper bounds inferred from the May 2012 data are υ_ν - 1 < 2.3 × 10 ^{- 6} and $\upsilon_{\bar \nu } - 1 < 3.0 \times 10^{ - 6}$ for antineutrinos [3]. The LVD experiment obtained $\left| {\upsilon_\nu - 1} \right| < 3.5 \times 10^{ - 6}$ [5], and the ICARUS bound from the May 2012 run is $\left| {\upsilon_\nu - 1} \right| < 1.6 \times 10^{ - 6}$ [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, υ_ν − 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 - n_ν, 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_in ≈χ_out ≈χ 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

$$\begin{equation} - 3.90 \times 10^{ - 6} < \chi < \chi_\nu + 1.49 \times 10^{ - 6}. \end{equation} \tag{ 28 }$$

The causality conditions (24) and (26) read

$$\begin{eqnarray} &&- (4.88\chi_\nu + 1.435 \times 10^{ - 5} ) < \chi < 4.88\chi_\nu + 1.435 \times 10^{ - 5} ,\nonumber\\ \end{eqnarray} \tag{ 29 }$$

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_in ≈χ_out ≈χ, 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_in ≈χ_out to obtain

$$\begin{equation} \Delta n\sim 0.66\chi_{{\rm{out}}} - \chi_{{\rm{in}}} - 4.1 \times 10^{ - 7} > 0, \end{equation} \tag{ 30 }$$

$$\begin{equation} \Delta \upsilon_{\nu \times {\rm{in}}} \propto 1.28\chi_{{\rm{out}}} - 0.94\chi_{{\rm{in}}} + 4.1 \times 10^{ - 7} > 0, \end{equation} \tag{ 31 }$$

$$\begin{equation} \Delta \upsilon_{{\rm{out}} \times \nu } \propto 2.60\chi_{{\rm{out}}} - 2.94\chi_{{\rm{in}}} + 4.1 \times 10^{ - 7} > 0. \end{equation} \tag{ 32 }$$

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_in ≈χ_out ≈ - 1.2 × 10 ^{- 6} as discussed after (29).

The reasoning is analogous to that of pion decay, cf. (30)–(32). The pionic mass square is replaced by the kaon mass, m_in² =  m_K² ≈ 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

$$\begin{equation} \Delta n\sim 0.8\chi_{{\rm{out}}} + 0.2\chi_\nu - \chi_{{\rm{in}}} + 1.6 \times 10^{ - 5} > 0, \end{equation} \tag{ 33 }$$

$$\begin{equation} \Delta \upsilon_{\nu \times {\rm{in}}} \propto 1.1\chi_{{\rm{out}}} + 0.6\chi_\nu - \chi_{{\rm{in}}} + 2.7 \times 10^{ - 5} > 0, \end{equation} \tag{ 34 }$$

$$\begin{equation} \Delta \upsilon_{{\rm{out}} \times \nu } \propto 1.0\chi_{{\rm{out}}} + 0.36\chi_\nu - \chi_{{\rm{in}}} + 1.6 \times 10^{ - 5} > 0. \end{equation} \tag{ 35 }$$

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_in ≈χ_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.

In this section, the subscript index ν labels photon variables. The outgoing photon with frequency ω_ν has zero rest mass m_ν ² = 0 and a refractive index n_ν - 1 = δ n_ν ∼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 χ_ν → 0 (photonic permeability tensor coinciding with Minkowski metric) is performed in the subsequent inequalities by putting δ n_ν ≈ 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 n_in,out ∼χ_in,out - m_in² /(2ω_in,out² ), cf. (6), with derivatives δ n'_in,out ∼χ '_in,out +  m_in² /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

$$\begin{eqnarray} \Delta n&\sim& - \frac{{\omega_\nu /\omega_{{\rm{in}}} }}{{1 - \omega_\nu /\omega_{{\rm{in}}} }}\left[\left({1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)^2 \omega_{{\rm{in}}} \chi '_{{\rm{in}}}\right.\nonumber\\ &&\left.+ \left( {1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\chi_{{\rm{in}}} + \frac{{m_{{\rm{in}}}^2 }}{{2\omega_{{\rm{in}}}^2 }}\vphantom{\left({1 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)^2}\right]. \end{eqnarray} \tag{ 36 }$$

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

$$\begin{equation} \Delta \upsilon_{\nu \times {\rm{in}}} \sim \chi_{{\rm{in}}} \left( {\frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }} + \left( {2 - \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\omega_\nu \frac{{\chi '_{{\rm{in}}} }}{{\chi_{{\rm{in}}} }}} \right), \end{equation} \tag{ 37 }$$

so that Δυ_{ν × in} < 0 if both χ_in and χ '_in are negative. Increment (22) can be written as

$$\begin{eqnarray} &&\Delta \upsilon_{{\rm{out}} \times \nu } \sim \frac{{\chi_{{\rm{in}}} }}{{1 + \omega_\nu /\omega_{{\rm{in}}} }}\left( {2 + \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }} - 2\left( {1 + \frac{{\omega_\nu }}{{\omega_{{\rm{in}}} }}} \right)\omega_\nu \frac{{\chi '_{{\rm{in}}} }}{{\chi_{{\rm{in}}} }}} \right),\nonumber\\ \end{eqnarray} \tag{ 38 }$$

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.

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.