Fast Pairwise Conversion of Supernova Neutrinos: A Dispersion Relation Approach

Collective pair conversion ${\nu }_e{\overline{\nu }}_e\leftrightarrow {\nu }_x{\overline{\nu }}_x$ by forward scattering, where $x=\mu$ or $\tau$, may be generic for supernova neutrino transport. Depending on the local angular intensity of the electron lepton number carried by neutrinos, the conversion rate can be “fast,” i.e., of the order of $\sqrt{2}{G}_F(n_{{\nu }_e}-n_{{\overline{\nu }}_e})\gg \mathrm{\Delta }{m}_{\mathrm{atm}}^2/2E$. We present a novel approach to understand these phenomena: a dispersion relation for the frequency and wave number $(\mathrm{\Omega },\mathbf{K})$ of disturbances in the mean field of ${\nu }_e{\nu }_x$ flavor coherence. Runaway solutions occur in “dispersion gaps,” i.e., in “forbidden” intervals of $\mathrm{\Omega }$ and/or $\mathbf{K}$ where propagating plane waves do not exist. We stress that the actual solutions also depend on the initial and/or boundary conditions, which need to be further investigated.

Figure 1

Dispersion relations (black lines) for two $\theta$ modes. The thick red line is $\mathrm{Re}(\omega )$ for real $k_z$ or $\mathrm{Re}(k_z)$ for real $\omega$. The width of the blob is $\pm \mathrm{Im}(\omega )$ or $\pm \mathrm{Im}(k_z)$. Left: Only outward modes. Right: One outward and one backward mode. Top: Both ${\nu }_e$ excess. Bottom: Forward mode ${\overline{\nu }}_e$ excess.

Figure 2

Electron lepton number (ELN) angle distribution $G_{\theta }$ of a $15M_{\odot }$ SN simulation at 280 ms post bounce and a radius 37 km. We plot an ELN number density, to be converted to a weak potential by Eq. (1). We show a mildly smoothed approximation suitable for analytic post processing.

Figure 3

Dispersion relations for $\mathbf{k}=(0,0,k_z)$, with $G_{\theta }$ shown in Fig. 2. Blue: Axially symmetric polarizations. Orange: Two degenerate axially breaking polarizations. Dots: End of a branch. Filled regions: Complex solutions in analogy to the red blobs in Fig. 1, where the semithick solid line is $\mathrm{Re}(k_z)$, and the edge of the blob indicates $\pm \mathrm{Im}(k_z)$. Gray region: Zone of avoidance for real $(\omega ,k_z)$.