![chart](152,94,497,272) <table>
  <thead>
    <tr>
      <th>r (km)</th>
      <th>Gνₑ</th>
      <th>|Gνₑ⊥|</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>0</td>
      <td>3.5</td>
      <td>2.5</td>
    </tr>
    <tr>
      <td>50</td>
      <td>2.0</td>
      <td>1.5</td>
    </tr>
    <tr>
      <td>100</td>
      <td>0.8</td>
      <td>0.7</td>
    </tr>
    <tr>
      <td>150</td>
      <td>0.3</td>
      <td>0.3</td>
    </tr>
    <tr>
      <td>200</td>
      <td>0.1</td>
      <td>0.1</td>
    </tr>
    <tr>
      <td>250</td>
      <td>0.05</td>
      <td>0.05</td>
    </tr>
    <tr>
      <td>300</td>
      <td>0.0</td>
      <td>0.0</td>
    </tr>
  </tbody>
</table>

![chart](500,94,842,272) <table>
  <thead>
    <tr>
      <th>r (km)</th>
      <th>Gν̄ₑ</th>
      <th>|Gν̄ₑ⊥|</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>0</td>
      <td>3.0</td>
      <td>2.5</td>
    </tr>
    <tr>
      <td>50</td>
      <td>1.5</td>
      <td>1.0</td>
    </tr>
    <tr>
      <td>100</td>
      <td>0.5</td>
      <td>0.3</td>
    </tr>
    <tr>
      <td>150</td>
      <td>0.2</td>
      <td>0.1</td>
    </tr>
    <tr>
      <td>200</td>
      <td>0.1</td>
      <td>0.05</td>
    </tr>
    <tr>
      <td>250</td>
      <td>0.05</td>
      <td>0.02</td>
    </tr>
    <tr>
      <td>300</td>
      <td>0.02</td>
      <td>0.01</td>
    </tr>
  </tbody>
</table>

![caption](113,287,880,325) 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).

![table](111,338,934,577) <table>
  <thead>
    <tr>
      <th>Symbol</th>
      <th>Name</th>
      <th>Expression</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td> $h_0$ </td>
      <td>Vacuum contribution to the Hamiltonian</td>
      <td> $h_0 = \omega \begin{pmatrix} -c_{2\theta} & s_{2\theta} \\ s_{2\theta} & c_{2\theta} \end{pmatrix}$ </td>
    </tr>
    <tr>
      <td> $h_{\text{mat}}$ </td>
      <td>Diagonal matter Hamiltonian</td>
      <td> $h_{\text{mat},\alpha\beta} = \frac{\sqrt{2}}{2} G_F n_B \delta_{\alpha\beta} \left[ 2\delta_{\alpha e} Y_e - (1 - Y_e) \right]$ </td>
    </tr>
    <tr>
      <td> $\hat{\epsilon}_q^* \cdot \vec{V}_{\text{mat}}$ </td>
      <td>Off-diagonal matter Hamiltonian</td>
      <td> $\hat{\epsilon}_q^* \cdot \vec{V}_{\text{mat},\alpha\beta} = \frac{\sqrt{2}}{2} G_F n_B \beta \delta_{\alpha\beta} \left[ 2\delta_{\alpha e} Y_e - (1 - Y_e) \right]$ </td>
    </tr>
    <tr>
      <td> $G_{\nu_\alpha}$ </td>
      <td>Geometrical factor</td>
      <td> $G_{\nu_\alpha} = \int_{\Omega_{\nu_\alpha}} \mathrm{d}\Omega (1 - \hat{q} \cdot \hat{p})$ </td>
    </tr>
    <tr>
      <td> $G_{\nu_\alpha}^{\perp}$ </td>
      <td>Perpendicular geometrical factor</td>
      <td> $G_{\nu_\alpha}^{\perp} = \int_{\Omega_{\nu_\alpha}} \mathrm{d}\Omega (\hat{\epsilon}^*(\hat{q}) \cdot \hat{p})$ </td>
    </tr>
    <tr>
      <td> $h_{\nu\nu}$ </td>
      <td>Diagonal self-interaction Hamiltonian</td>
      <td> $h_{\nu\nu} = \sqrt{2} G_F \sum_\alpha \int \mathrm{d}p \left[ G_{\nu_\alpha} \rho_{\nu_\alpha} \frac{L_{\nu_\alpha} f_{\nu_\alpha}}{\pi^2 R_{\nu_\alpha}^2 \langle E_{\nu_\alpha} \rangle} - \bar{\rho}_{\bar{\nu}_\alpha} G_{\bar{\nu}_\alpha} \frac{L_{\bar{\nu}_\alpha} f_{\bar{\nu}_\alpha}}{\pi^2 R_{\bar{\nu}_\alpha}^2 \langle E_{\bar{\nu}_\alpha} \rangle} \right]$ </td>
    </tr>
    <tr>
      <td> $h_{\nu\nu}^{\perp}$ </td>
      <td>Off-diagonal self-interaction Hamiltonian</td>
      <td> $h_{\nu\nu}^{\perp} = \sqrt{2} G_F \sum_\alpha \int \mathrm{d}p \left[ G_{\nu_\alpha}^{\perp} \rho_{\nu_\alpha} \frac{L_{\nu_\alpha} f_{\nu_\alpha}}{\pi^2 R_{\nu_\alpha}^2 \langle E_{\nu_\alpha} \rangle} - \bar{\rho}_{\bar{\nu}_\alpha} G_{\bar{\nu}_\alpha}^{\perp} \frac{L_{\bar{\nu}_\alpha} f_{\bar{\nu}_\alpha}}{\pi^2 R_{\bar{\nu}_\alpha}^2 \langle E_{\bar{\nu}_\alpha} \rangle} \right]$ </td>
    </tr>
  </tbody>
</table>

![caption](113,601,803,612) Table 4.1: Symbols, names and expressions of the relevant quantities involved in the neutrino evolution equations (see text).

![text](112,638,888,875) 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.

![text](133,884,887,900) The neutrino total luminosities and spectra at the neutrinospheres are an essential ingredient of the self-

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