1 | \documentclass{article} |
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2 | %---------------------------------------------------------------------------- |
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3 | % |
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4 | \usepackage[T1]{fontenc} |
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5 | \usepackage[latin1]{inputenc} |
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6 | \usepackage{graphicx} |
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7 | \usepackage{epsfig} |
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8 | \usepackage{amssymb} |
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9 | \usepackage{amsmath} |
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10 | \usepackage{latexsym} |
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11 | |
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12 | %---------------------------------------------------------------------------- |
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13 | \begin{document} |
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14 | When $\delta_{CP}= 0^o$ and Matter Effects are neglected then, |
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15 | \begin{eqnarray} |
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16 | P_{\nu_\mu\rightarrow\nu_e} |
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17 | & \simeq & 4 c_{13}^2 s_{13}^2 s_{23}^2 \sin^2 \Delta_{31} \\ |
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18 | & + & (8 c_{12} s_{12} c_{13}^2 s_{13} s_{23} c_{23} - 8 s_{12}^2 c_{13}^2 s_{13}^2 s_{23}^2) |
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19 | \cos \Delta_{23} \sin\Delta_{31} \sin\Delta_{21} |
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20 | \end{eqnarray} |
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21 | As $\Delta_{23} = \Delta_{21} + \Delta_{13}$ then |
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22 | $$ |
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23 | \cos \Delta_{23} = \cos \Delta_{21} \cos \Delta_{13} - \sin \Delta_{21} \sin \Delta_{13} |
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24 | $$ |
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25 | and in numerical application $\Delta_{21} = 1.27 \delta m^2_{21} L/E \approx O(10^{-2})$. So, |
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26 | \begin{eqnarray} |
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27 | P_{\nu_\mu\rightarrow\nu_e} |
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28 | & \simeq & 4 c_{13}^2 s_{13}^2 s_{23}^2 \sin^2 \Delta_{31} \\ |
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29 | & + & (8 c_{12} s_{12} c_{13}^2 s_{13} s_{23} c_{23} - 8 s_{12}^2 c_{13}^2 s_{13}^2 s_{23}^2) |
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30 | \Delta_{21} \cos \Delta_{13} \sin\Delta_{31} |
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31 | \end{eqnarray} |
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32 | If one uses $s_{12}^2 = 0.314$ and $s_{23}^2 = 0.44$ then one realizes that in the parenthesis the second term is of the order $s_{13}$ compared to the first term, so it may be neglected hereafter as we will focus on $s_{13} < 10^{-2}$. Then, it yields |
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33 | \begin{eqnarray} |
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34 | P_{\nu_\mu\rightarrow\nu_e} |
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35 | & \simeq & \alpha \cos\beta \sin^2 \Delta_{31} + \alpha \sin\beta \cos \Delta_{13} \sin\Delta_{31} |
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36 | \end{eqnarray} |
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37 | with |
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38 | \begin{eqnarray} |
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39 | \alpha \cos\beta & \equiv & 4 c_{13}^2 s_{13}^2 s_{23}^2 \\ |
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40 | \alpha \sin\beta & \equiv & 8 c_{12} s_{12} c_{13}^2 s_{13} s_{23} c_{23} |
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41 | \end{eqnarray} |
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42 | It is remarkable that now the oscillation probability may be written as |
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43 | \begin{eqnarray} |
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44 | P_{\nu_\mu\rightarrow\nu_e} |
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45 | & \simeq & \frac{\alpha}{2}\left[ |
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46 | \cos\beta - \cos\left( 2\Delta_{13} + \beta\right) \right] |
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47 | \end{eqnarray} |
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48 | The maximum of the probability is obtained at |
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49 | \begin{eqnarray} |
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50 | \Delta_{31} & = & \frac{\pi}{2} - \frac{\beta}{2} \\ |
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51 | \mathrm{with}\ \tan \beta & = & 2 \Delta_{21}\frac{c_{12} s_{12}c_{23} }{ s_{13}s_{23}} |
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52 | \end{eqnarray} |
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53 | The "usual" 2-famillies case is obtain with $\beta = 0$. In case of $E\sim 0.3$~GeV, $L\sim130$~km and $\sin^22\theta_{13} = 10^{-3}$ then |
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54 | \begin{eqnarray} |
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55 | \left(\delta m_{31}^2\right)_{max} & = & 2.9\, 10^{-3} \ \mathrm{if} \ \beta = 0 \\ |
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56 | & = & 1.8\, 10^{-3} |
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57 | \end{eqnarray} |
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58 | |
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59 | |
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60 | |
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61 | |
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62 | |
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63 | |
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64 | %---------------------------------------------------------------------------- |
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65 | \end{document} |
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