Self-induced conversion in dense neutrino gases: Pendulum in flavor space

Abstract

Neutrino-neutrino interactions can lead to collective flavour conversion effects in supernovae and in the early universe. We demonstrate that the case of "bipolar" oscillations, where a dense gas of neutrinos and antineutrinos in equal numbers completely converts from one flavour to another even if the mixing angle is small, is equivalent to a pendulum in flavour space. Bipolar flavour conversion corresponds to the swinging of the pendulum, which begins in an unstable upright position (the initial flavour), and passes through momentarily the vertically downward position (the other flavour) in the course of its motion. The time scale to complete one cycle of oscillation depends logarithmically on the vacuum mixing angle. Likewise, the presence of an ordinary medium can be shown analytically to contribute to a logarithmic increase in the bipolar conversion period. We further find that a more complex (and realistic) system of unequal numbers of neutrinos and antineutrinos is analogous to a spinning top subject to a torque. This analogy easily explains how such a system can oscillate in both the bipolar and the synchronised mode, depending on the neutrino density and the size of the neutrino-antineutrino asymmetry. Our simple model applies strictly only to isotropic neutrino gasses. In more general cases, and especially for neutrinos streaming from a supernova core, different modes couple to each other with unequal strength, an effect that can lead to kinematical decoherence in flavour space rather than collective oscillations. The exact circumstances under which collective oscillations occur in non-isotropic media remain to be understood.

Self-induced decoherence in dense neutrino gases

“…In this case the equations for ̺ 0 and̺ 0 form a closed set, describing the dynamics of the "flavor pendulum" studied in Ref. [26]. In addition, the higher multipoles ̺ n with n ≥ 2, if initially nonzero, will simply oscillate under the action of the vacuum term and of the density term (̺ −̺) 0 .…”

“…When this condition is satisfied, collective effects are important, even if the ordinary matter effect is much larger than that from the neutrino-neutrino interaction [23,26]. One characteristic feature of collective oscillations is the phenomenon of "self-maintained coherence" [6], meaning that all modes oscillate with the same frequency even though the energy spectrum may be broad.…”

“…Therefore, if the energy spectrum is broad, the energy dependence of the oscillation frequency quickly leads to kinematical decoherence, i.e., along a neutrino beam the overall flavor content quickly approaches an average value. The situation changes radically when neutrinos themselves provide a significant refractive effect, leading to collective oscillation modes [1,2,3,4,5,6,7,8,9,10,11,12] that can be of practical interest in the early universe [13,14,15,16] or in core-collapse supernovae [17,18,19,20,21,22,23,24,25,26]. Defining the parameter…”

“…A symmetric νν gas oscillates in the "bipolar mode" where pairs of neutrinos of a given flavor coherently oscillate into the other flavor and back with the "bipolar frequency" [6,8,23,26] κ = 2ωµ .…”

Self interactions of Supernova neutrinos

“…Note that the previous equation has two unknowns, the energy of the mode and the radius at which the resonance occurs. When the collective oscillations begin (bipolar regime), all the mode precess with the same frequency ω bipolar ≃ √ µω ave Q z ∼ √ µω ave D z (see [4] for a definition of Q z and [3] for a definition of ω ave ). When ω ω bipolar , even after the start of the collective oscillations, the corresponding mode remains decoupled, while for ω < ω bipolar the mode participate to the collective oscillations.…”