Neutrino splitting for Lorentz-violating neutrinos: Detailed analysis

Lorentz-violating neutrino parameters have been severely constrained on the basis of astrophysical considerations. In the high-energy limit, one generally assumes a superluminal dispersion relation of an incoming neutrino of the form $E\approx |\overrightarrow{p}|v$, where $E$ is the energy, $\overrightarrow{p}$ is the momentum and $v=\sqrt{1+\delta }>1$. Lepton-pair creation due to a Cerenkov-radiation-like process ($\nu \to \nu +e^{-}+e^{+}$) becomes possible above a certain energy threshold, and bounds on the Lorentz-violating parameter $\delta$ can be derived. Here, we investigate a related process, ${\nu }_i\to {\nu }_i+{\nu }_f+{\overline{\nu }}_f$, where ${\nu }_i$ is an incoming neutrino mass eigenstate, while ${\nu }_f$ is the final neutrino mass eigenstate, with a superluminal velocity that is slightly slower than that of the initial state. This process is kinematically allowed if the Lorentz-violating parameters at high energy differ for the different neutrino mass eigenstates. Neutrino splitting is not subject to any significant energy threshold condition and could yield quite a substantial contribution to decay and energy loss processes at high energy, even if the differential Lorentz violation among neutrino flavors is severely constrained by other experiments. We also discuss the $SU(2{)}_L$-gauge invariance of the superluminal models and briefly discuss the use of a generalized vierbein formalism in the formulation of the Lorentz-violating Dirac equation.

Figure 1

The Feynman diagrams for the LPCR (a) and NPCR (b) processes illustrate the exchange of a virtual $Z$ boson. They become kinematically possible for an incoming neutrino mass eigenstate $|{\nu }_i^{(m)}⟩$, which decays into a state of the same mass, but lower energy, also labeled $|{\nu }_i^{(m)}⟩$, and an electron-positron pair (LPCR) or a neutrino-antineutrino pair (NPCR).

Figure 2

The physical region (shaded) in the $(M^2,|{\overrightarrow{p}}_3|)$ plane for “slow” superluminal neutrino pair emission from a “fast” superluminal neutrino. $|{\overrightarrow{p}}_3{|}_{\max }$ and $|{\overrightarrow{p}}_3{|}_{\min }$ are defined in Eq. (58), while $M_{\mathrm{cut}}^2$ is given in Eq. (63). The variable $M^2$ is introduced in Eq. (42) [see also Eq. (51)].

Figure 3

In genuine Cerenkov radiation $\nu \to \nu +\gamma$ (for superluminal neutrinos), the photon is emitted from a $W$ loop (a) or from a vacuum-polarization insertion into the $Z$ propagator (b); the emission takes place from a flavor eigenstate $|{\nu }_i^{(f)}⟩$. The photon emission thus involves an extra factor ${\alpha }_{\mathrm{QED}}\approx 1/137.036$ at the $W$-photon vertex as compared to LPCR and NPCR.