Theory of neutrino slow flavor evolution. Part I. Homogeneous medium

A bstract Dense neutrino gases can exhibit collective flavor instabilities, triggering large flavor conversions that are driven primarily by neutrino-neutrino refraction. One broadly distinguishes between fast instabilities that exist in the limit of vanishing neutrino masses, and slow ones, that require neutrino mass splittings. In a related series of papers, we have shown that fast instabilities result from the resonant growth of flavor waves, in the same way as turbulent electric fields in an unstable plasma. Here we extend this framework to slow instabilities, focusing on the simplest case of an infinitely homogeneous medium with axisymmetric neutrino distribution. The relevant length and time scales are defined by three parameters: the vacuum oscillation frequency ω E = δm 2 /2 E , the scale of neutrino-neutrino refraction energy $$ \mu =\sqrt{2}{G}_{\textrm{F}}\left({n}_{\nu }+{n}_{\overline{\nu}}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> , and the ratio between lepton and particle number $$ \epsilon =\left({n}_{\nu }-{n}_{\overline{\nu}}\right)/\left({n}_{\nu }+{n}_{\overline{\nu}}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mo>/</mml:mo> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:mrow> </mml:mfenced> </mml:math> . We distinguish between two very different regimes: (i) For ω E ≪ μϵ 2 , instabilities occur at small spatial scales of order ( μϵ ) − 1 with a time scale of order $$ \epsilon {\omega}_E^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> <mml:msubsup> <mml:mi>ω</mml:mi> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> . This novel branch of slow instability arises from resonant interactions with neutrinos moving along the axis of symmetry. (ii) For μϵ 2 ≪ ω E ≪ μ , the instability is strongly non-resonant, with typical time and length scales of order $$ 1/\sqrt{\omega_E\mu } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>E</mml:mi> </mml:msub> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msqrt> </mml:math> . Unstable modes interact with all neutrino directions at once, recovering the characteristic scaling of the traditional studies of slow instabilities. In the inner regions of supernovae and neutron-star mergers, the first regime may be more likely to appear, meaning that slow instabilities in this region may have an entirely different character than usually envisaged.