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c2a9ec8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 |  <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>
 <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>
 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>
<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>
 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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