Abstract
We show that particles are unstable with respect to a splitting process, which is the quantum analog of the modulational instability in anomalous dispersive media, only when their group velocity exceeds their phase velocity. In the case of a neutrino, when the concavity results from a term E(P)~Pk, the neutrino will decay to two neutrinos and an antineutrino after traveling a distance proportional to E2+3k. Unlike the Cohen-Glashow instability, the splitting instability exists even if all particles involved in the interaction have the same dispersion relations at the relevant energy scales. We show that this instability leads to strong constraints even if the energy E is a function of both the momentum P and also of the background density ρ; for example, we show that it alone would have been sufficient to eliminate any model of the MINOS/OPERA velocity anomaly which modifies the neutrino dispersion relation while leaving those of other particles intact.
Highlights
It was argued that one could escape by rendering other particles, such as the electron, superluminal [35, 36] after all the dispersion relations could well be density dependent [37,38,39,40,41,42,43] and all strong constraints on electron velocities appear to come from experiments in a vacuum [42, 43]
The orthogonal momenta, as we have described, would be constrained to vanish if the dispersion relation were linear
It may well be that the neutrino velocity is a function of energy which is reasonably constant but has very deep subluminal troughs which are so thin that no OPERA neutrinos have hit them, and so they have not been observed
Summary
The velocity anomaly reported by OPERA 1 was the result of experimental error 2 , while the weeks after OPERA’s announcement were characterized by a frenzied and often mutually inconsistent attempts at profound advances in fundamental physics. It was argued that one could escape by rendering other particles, such as the electron, superluminal [35, 36] after all the dispersion relations could well be density dependent [37,38,39,40,41,42,43] and all strong constraints on electron velocities appear to come from experiments in a vacuum 42, 43 Another escape route is if, while traveling, the neutrinos are noninteracting, they are taking a shortcut through another dimension [44,45,46,47] or converted into a sterile flavor 48–51. For example in the case of neutrinos the 4 neutrino Fermi vertex is allowed, which yields the desired process 2.1
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