Simulation of coherent nonlinear neutrino flavor transformation in the supernova environment: Correlated neutrino trajectories

We present results of large-scale numerical simulations of the evolution of neutrino and antineutrino flavors in the region above the late-time post-supernova-explosion proto-neutron star. Our calculations are the first to allow explicit flavor evolution histories on different neutrino trajectories and to self-consistently couple flavor development on these trajectories through forward scattering-induced quantum coupling. Employing the atmospheric-scale neutrino mass-squared difference ($|\delta m^2|\simeq 3\times {10}^{-3}{\mathrm{eV}}^2$) and values of ${\theta }_{13}$ allowed by current bounds, we find transformation of neutrino and antineutrino flavors over broad ranges of energy and luminosity in roughly the “bi-polar” collective mode. We find that this large-scale flavor conversion, largely driven by the flavor off-diagonal neutrino-neutrino forward scattering potential, sets in much closer to the proto-neutron star than simple estimates based on flavor-diagonal potentials and Mikheyev-Smirnov-Wolfenstein evolution would indicate. In turn, this suggests that models of $r$-process nucleosynthesis sited in the neutrino-driven wind could be affected substantially by active-active neutrino flavor mixing, even with the small measured neutrino mass-squared differences.

Figure 1

The geometric picture of the neutrino bulb model. An arbitrary neutrino beam (solid line) is shown emanating from a point on the neutrino sphere with polar angle $\Theta$. This beam intersects the $z$-axis at point $P$ with angle $\vartheta$. Because neutrinos are emitted from the neutrino sphere of radius $R_{\nu }$, point $P$ sees only neutrinos traveling within the cone delimited by the dotted lines. One of the most important geometric characteristics of a neutrino beam is its emission angle ${\vartheta }_0$, defined with respect to the normal direction at the point of emission on the neutrino sphere (${\vartheta }_0=\Theta +\vartheta$). All other geometric properties of a neutrino beam may be calculated using radius $r$ and ${\vartheta }_0$.

Figure 2

Plot of (effective) net lepton number density $n_l(r)$. The dashed and dot-dashed lines are for the net electron density $n_e=Y_e{n}_b$ using the baryon density profile in Eq. (16) with $S=140$ and 250, respectively. The dotted line is for the net electron density assuming the baryon density profile in Eq. (17) only. The solid line is for the effective net ${\nu }_{\underline{e}}$ density along the radial trajectory [Eq. (40)].

Figure 3

Average ${\nu }_e$ survival probability $⟨P_{{\nu }_{\underline{e}}{\nu }_e}(r)⟩$ along the radial trajectory ($\cos {\vartheta }_0=1$) with the normal mass hierarchy in different numerical schemes. Here the average is done over the initial energy spectra of ${\nu }_{\underline{e}}$. The dot-dashed line uses 160 angle bins and error tolerance ${10}^{-5}$ in each step without the initial baryon density profile $n_{\mathrm{b}}^{\prime }$. The dashed and solid lines both include $n_{\mathrm{b}}^{\prime }$ and employ error tolerance ${10}^{-10}$, but use 256 and 512 angle bins, respectively. Calculations with 768, 1024 and 1407 angle bins in different binning schemes produce curves which fall on the solid line.

Figure 4

Plots of average survival probability $⟨P_{{\nu }_{\underline{e}}{\nu }_e}⟩$ (left panels) and $⟨P_{{\overline{\nu }}_{\underline{e}}{\overline{\nu }}_e}⟩$ (right panels) with the normal (upper panels) and inverted (lower panels) neutrino mass hierarchies, respectively. The solid and dotted lines give average survival probabilities along trajectories with $\cos {\vartheta }_0=1$ and $\cos {\vartheta }_0=0$, respectively, as computed in the multiangle simulations. The dot-dashed lines and the dashed lines characterize the limits where neutrinos and antineutrino undergo flavor transformation individually ($L_{\nu }=0$ and $A$ potential only) and simultaneously (full synchronization), respectively. The dashed line is not distinguishable from the dot-dashed line in panel (c).

Figure 5

Plots of $P_{{\nu }_{\underline{e}}{\nu }_e}(r)$ with the normal mass hierarchy. Panel (a) is for the radial trajectory ($\cos {\vartheta }_0=1$), and (b) is for the tangential trajectory ($\cos {\vartheta }_0=0$). The dot-dashed, dotted, dashed and solid lines are for ${\nu }_{\underline{e}}$ of energies 6.95, 8.95, 10.95 and 14.95 MeV, respectively.

Figure 6

Plots of $P_{{\nu }_{\underline{e}}{\nu }_e}(\cos {\vartheta }_0)$ (left panels) and $P_{{\overline{\nu }}_{\underline{e}}{\overline{\nu }}_e}(\cos {\vartheta }_0)$ (right panels) at $r\simeq 92.85\mathrm{km}$ for the normal mass hierarchy (upper panels) and at $r\simeq 88.57\mathrm{km}$ for the inverted mass hierarchy (lower panels). The dot-dashed, dotted, dashed and solid lines are for ${\nu }_{\underline{e}}$ or ${\overline{\nu }}_{\underline{e}}$ of energies 6.95, 8.95, 10.95 and 14.95 MeV, respectively.

Figure 7

Change of energy spectra of neutrinos (left panels) and antineutrinos (right panels) with the normal (upper panels) and inverted (lower panels) neutrino mass hierarchies. The dotted and dot-dashed lines are the spectra of neutrinos (antineutrinos) in the electron and tau flavors, respectively, at $r=R_{\nu }$, and the solid and dashed lines are the corresponding spectra at $r=250\mathrm{km}$.

Figure 8

Plots of $⟨{\mathit{\varsigma }}_{{\nu }_{\underline{e}}}(r)⟩$ (left panels) and $⟨{\mathit{\varsigma }}_{{\overline{\nu }}_{\underline{e}}}(r)⟩$ (right panels) with the normal (upper panels) and inverted (lower panels) mass hierarchies, respectively. The dashed, dotted and solid lines represent $⟨{\varsigma }_x(r)⟩$, $⟨{\varsigma }_y(r)⟩$ and $⟨{\varsigma }_z(r)⟩$, respectively. Note that $⟨P_{{\nu }_{\underline{e}}{\nu }_e}⟩=1/2+⟨{\varsigma }_{{\nu }_{\underline{e}}z}⟩$ and $⟨P_{{\overline{\nu }}_{\underline{e}}{\overline{\nu }}_e}⟩=1/2-⟨{\varsigma }_{{\overline{\nu }}_{\underline{e}}z}⟩$. The insets in panels (c) and (d) are blowups of the corresponding plots in the range $62\mathrm{km}\le r\le 66\mathrm{km}$. These are single-angle calculation results.

Figure 9

Plots of ${\varsigma }_{{\nu }_{\underline{e}}z}(E_{{\nu }_{\underline{e}}})$ (left panels) and ${\varsigma }_{{\overline{\nu }}_{\underline{e}}z}(E_{{\overline{\nu }}_{\underline{e}}})$ (right panels) for both the normal (upper panels) and inverted (lower panels) mass hierarchies at $r=400\mathrm{km}$. The dashed and solid lines are for $L_{\nu }={10}^{51}$ and $5\times {10}^{51}\mathrm{erg}/\mathrm{s}$, respectively. These are single-angle calculation results.

Figure 10

The toy scenario explaining the evolution of NFIS $\mathit{\varsigma }$. The NFIS $\mathit{\varsigma }$ can be viewed as a “magnetic spin” which is coupled with a constant field ${\mathbf{H}}_{\mathrm{V}}$ and a field $\mathbf{\Sigma }$ rotating with angular frequency $\omega$. We actually consider here two “magnetic spin” couplings, one with“magnetic moment” ${\mu }_{\mathrm{V}}$ and the other with ${\mu }_{\nu }$. The vector ${\mu }_{\nu }\mathbf{\Sigma }$ rotates in the ${\widehat{\mathbf{e}}}_x^{\mathrm{v}}-{\widehat{\mathbf{e}}}_y^{\mathrm{v}}$ plane in the clockwise sense when viewed from above, looking in the $-{\widehat{\mathbf{e}}}_z^{\mathrm{v}}$ direction. The “magnetic spin” $\mathit{\varsigma }$ is aligned initially with the dominant field $\mathbf{\Sigma }$ at time $t=0$ [panels (a) and (d)]. The problem is easily solved in the corotating frame where $\mathbf{\Sigma }$ is not rotating. In this corotating frame, the “magnetic spin” $\mathit{\varsigma }$ rotates around the effective field ${\tilde{\mathbf{H}}}^{\mathrm{eff}}=({\mu }_{\mathrm{V}}-\omega ){\mathbf{H}}_{\mathrm{V}}+{\mu }_{\nu }\mathbf{\Sigma }$ [panels (b) and (e)]. If $\mathbf{\Sigma }$ slowly reduces its length to zero, the angle between the spin $\mathit{\varsigma }$ and the effective field ${\tilde{\mathbf{H}}}^{\mathrm{eff}}$ is constant (adiabatic process), and $\mathit{\varsigma }$ ends up almost aligned with ${\mathbf{H}}_{\mathrm{V}}$ at $t=\infty$ if ${\mu }_{\mathrm{V}}-\omega >0$ [panel (c)] or almost antialigned with ${\mathbf{H}}_{\mathrm{V}}$ if ${\mu }_{\mathrm{V}}-\omega <0$ [panel (f)].

Figure 11

Plots of $r_{\mathrm{X}}(L_{\nu })$ for both the normal [panel (a)] and inverted [panel (b)] neutrino mass hierarchies. The dashed and solid lines are based on single-angle simulations with $S=140$ and 250, respectively. The cross and star symbols are based on the average of $r_{\mathrm{X}}$ on the radial and tangential trajectories in our multiangle simulations with $S=140$ and 250, respectively. The error bars associated with the cross and star symbols indicate the range of values of $r_{\mathrm{X}}$ on different trajectories. These are too small to be visible in most of the cases. For the normal mass hierarchy case [panel (a)], $r_{\mathrm{X}}$ asymptotically approaches $r_{\mathrm{X}/\mathrm{MSW}}(E_{\mathrm{sync}})$ with large $L_{\nu }$, which is 73.9 (dot-dashed lines) and 34.4 km (dotted lines) for $S=140$ and 250, respectively. The dot-dashed line in panel (b) represents a crude estimate for where the collective neutrino flavor transformation exits the synchronization mode and enters the bi-polar mode [Eq. (57)].

Figure 12

The difference between two almost identical systems grows exponentially in the region where collective neutrino flavor transformation changes from the synchronized mode to the bi-polar mode in the inverted neutrino mass hierarchy case. The solid line shows the exponential growth of $|⟨\delta {\mathit{\varsigma }}_{{\nu }_{\underline{e}}}⟩|$ (difference between the energy-averaged ${\nu }_{\underline{e}}$ NFIS’s of the two systems) as a function of radius in the transition region. The dot-dashed line is a linear fit to $\ln |⟨\delta {\mathit{\varsigma }}_{{\nu }_{\underline{e}}}(r)⟩|$. This line has slope $\sim 2.5/\mathrm{km}$.