Figure 4.2: Geometrical factors for neutrinos (left) and antineutrinos (right figure), as a function of distance, in our schematic model based on binary star mergers. The two curves correspond to $G_{\nu}$ (4.38) (solid line) and $G_{\nu}^{\perp}$ (4.41) (dashed line) in the self-interaction Hamiltonian Eq. (4.37) and Eq. (4.40) respectively. Results correspond to Model C (Table 4.3).
| Symbol | Name | Expression |
|---|---|---|
| $h_0$ | Vacuum contribution to the Hamiltonian | $h_0 = \omega \begin{pmatrix} -c_{2\theta} & s_{2\theta} \\ s_{2\theta} & c_{2\theta} \end{pmatrix}$ |
| $h_{\text{mat}}$ | Diagonal matter Hamiltonian | $h_{\text{mat},\alpha\beta} = \frac{\sqrt{2}}{2} G_F n_B \delta_{\alpha\beta} \left[ 2\delta_{\alpha e} Y_e - (1 - Y_e) \right]$ |
| $\hat{\epsilon}_q^* \cdot \vec{V}_{\text{mat}}$ | Off-diagonal matter Hamiltonian | $\hat{\epsilon}_q^* \cdot \vec{V}_{\text{mat},\alpha\beta} = \frac{\sqrt{2}}{2} G_F n_B \beta \delta_{\alpha\beta} \left[ 2\delta_{\alpha e} Y_e - (1 - Y_e) \right]$ |
| $G_{\nu_\alpha}$ | Geometrical factor | $G_{\nu_\alpha} = \int_{\Omega_{\nu_\alpha}} \mathrm{d}\Omega (1 - \hat{q} \cdot \hat{p})$ |
| $G_{\nu_\alpha}^{\perp}$ | Perpendicular geometrical factor | $G_{\nu_\alpha}^{\perp} = \int_{\Omega_{\nu_\alpha}} \mathrm{d}\Omega (\hat{\epsilon}^*(\hat{q}) \cdot \hat{p})$ |
| $h_{\nu\nu}$ | Diagonal self-interaction Hamiltonian | $h_{\nu\nu} = \sqrt{2} G_F \sum_\alpha \int \mathrm{d}p \left[ G_{\nu_\alpha} \rho_{\nu_\alpha} \frac{L_{\nu_\alpha} f_{\nu_\alpha}}{\pi^2 R_{\nu_\alpha}^2 \langle E_{\nu_\alpha} \rangle} - \bar{\rho}_{\bar{\nu}_\alpha} G_{\bar{\nu}_\alpha} \frac{L_{\bar{\nu}_\alpha} f_{\bar{\nu}_\alpha}}{\pi^2 R_{\bar{\nu}_\alpha}^2 \langle E_{\bar{\nu}_\alpha} \rangle} \right]$ |
| $h_{\nu\nu}^{\perp}$ | Off-diagonal self-interaction Hamiltonian | $h_{\nu\nu}^{\perp} = \sqrt{2} G_F \sum_\alpha \int \mathrm{d}p \left[ G_{\nu_\alpha}^{\perp} \rho_{\nu_\alpha} \frac{L_{\nu_\alpha} f_{\nu_\alpha}}{\pi^2 R_{\nu_\alpha}^2 \langle E_{\nu_\alpha} \rangle} - \bar{\rho}_{\bar{\nu}_\alpha} G_{\bar{\nu}_\alpha}^{\perp} \frac{L_{\bar{\nu}_\alpha} f_{\bar{\nu}_\alpha}}{\pi^2 R_{\bar{\nu}_\alpha}^2 \langle E_{\bar{\nu}_\alpha} \rangle} \right]$ |
Table 4.1: Symbols, names and expressions of the relevant quantities involved in the neutrino evolution equations (see text).
where the dependence on the emission variables is shown. The explicit expression for $\Gamma^{\perp}$ is given by Eqs. (C.9)-(C.10) C. Figures 4.2 and 4.3 show the geometrical factor (4.41) as a function of the distance travelled by the neutrinos from the neutrinospheres. The results correspond to the cases A and C (Table 4.3) which can be considered as representative of the the typical behaviors of $G_{\nu}^{\perp}$ , as we have been observing in our runs. One can see that $G_{\nu}^{\perp}$ have a similar $r$ dependence as $G_{\nu}$ (4.38), as expected. Their absolute values turn out to be suppressed by a few percents up to several factors, with respect to the $G_{\nu}$ value. As we will discuss, the $r$ dependence of $G_{\nu}$ plays a crucial role on the possibility to have multiple crossings and a nonlinear feedback mechanism in presence of helicity coherence (see Section 4.4). By including Eq.(4.41) into (4.40) one gets the same expression Eq. (4.37) for $h_{\nu\nu}^{\perp}$ with $G_{\nu_\alpha}^{\perp}$ Eq.(4.41) replacing $G_{\nu_\alpha}$ Eq.(4.36). Table 4.1 sums up the relevant quantities involved in the generalized Hamiltonian.
The neutrino total luminosities and spectra at the neutrinospheres are an essential ingredient of the self-
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