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"text": "<table>\n <thead>\n <tr>\n <th>r (km)</th>\n <th>Gνₑ</th>\n <th>|Gνₑ⊥|</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <td>0</td>\n <td>3.5</td>\n <td>2.5</td>\n </tr>\n <tr>\n <td>50</td>\n <td>2.0</td>\n <td>1.5</td>\n </tr>\n <tr>\n <td>100</td>\n <td>0.8</td>\n <td>0.7</td>\n </tr>\n <tr>\n <td>150</td>\n <td>0.3</td>\n <td>0.3</td>\n </tr>\n <tr>\n <td>200</td>\n <td>0.1</td>\n <td>0.1</td>\n </tr>\n <tr>\n <td>250</td>\n <td>0.05</td>\n <td>0.05</td>\n </tr>\n <tr>\n <td>300</td>\n <td>0.0</td>\n <td>0.0</td>\n </tr>\n </tbody>\n</table>",
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"text": "<table>\n <thead>\n <tr>\n <th>r (km)</th>\n <th>Gν̄ₑ</th>\n <th>|Gν̄ₑ⊥|</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <td>0</td>\n <td>3.0</td>\n <td>2.5</td>\n </tr>\n <tr>\n <td>50</td>\n <td>1.5</td>\n <td>1.0</td>\n </tr>\n <tr>\n <td>100</td>\n <td>0.5</td>\n <td>0.3</td>\n </tr>\n <tr>\n <td>150</td>\n <td>0.2</td>\n <td>0.1</td>\n </tr>\n <tr>\n <td>200</td>\n <td>0.1</td>\n <td>0.05</td>\n </tr>\n <tr>\n <td>250</td>\n <td>0.05</td>\n <td>0.02</td>\n </tr>\n <tr>\n <td>300</td>\n <td>0.02</td>\n <td>0.01</td>\n </tr>\n </tbody>\n</table>",
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"text": "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).",
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"text": "<table>\n <thead>\n <tr>\n <th>Symbol</th>\n <th>Name</th>\n <th>Expression</th>\n </tr>\n </thead>\n <tbody>\n <tr>\n <td> $h_0$ </td>\n <td>Vacuum contribution to the Hamiltonian</td>\n <td> $h_0 = \\omega \\begin{pmatrix} -c_{2\\theta} & s_{2\\theta} \\\\ s_{2\\theta} & c_{2\\theta} \\end{pmatrix}$ </td>\n </tr>\n <tr>\n <td> $h_{\\text{mat}}$ </td>\n <td>Diagonal matter Hamiltonian</td>\n <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>\n </tr>\n <tr>\n <td> $\\hat{\\epsilon}_q^* \\cdot \\vec{V}_{\\text{mat}}$ </td>\n <td>Off-diagonal matter Hamiltonian</td>\n <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>\n </tr>\n <tr>\n <td> $G_{\\nu_\\alpha}$ </td>\n <td>Geometrical factor</td>\n <td> $G_{\\nu_\\alpha} = \\int_{\\Omega_{\\nu_\\alpha}} \\mathrm{d}\\Omega (1 - \\hat{q} \\cdot \\hat{p})$ </td>\n </tr>\n <tr>\n <td> $G_{\\nu_\\alpha}^{\\perp}$ </td>\n <td>Perpendicular geometrical factor</td>\n <td> $G_{\\nu_\\alpha}^{\\perp} = \\int_{\\Omega_{\\nu_\\alpha}} \\mathrm{d}\\Omega (\\hat{\\epsilon}^*(\\hat{q}) \\cdot \\hat{p})$ </td>\n </tr>\n <tr>\n <td> $h_{\\nu\\nu}$ </td>\n <td>Diagonal self-interaction Hamiltonian</td>\n <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>\n </tr>\n <tr>\n <td> $h_{\\nu\\nu}^{\\perp}$ </td>\n <td>Off-diagonal self-interaction Hamiltonian</td>\n <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>\n </tr>\n </tbody>\n</table>",
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"text": "Table 4.1: Symbols, names and expressions of the relevant quantities involved in the neutrino evolution equations (see text).",
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"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.",
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"text": "The neutrino total luminosities and spectra at the neutrinospheres are an essential ingredient of the self-",
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"text": "83",
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