Realistic Earth matter effects and a method to acquire information about small ${\theta }_{13}$ in the detection of supernova neutrinos

In this paper, we first calculate the realistic Earth matter effects in the detection of type II supernova neutrinos at the Daya Bay reactor neutrino experiment which is currently under construction. It is found that the Earth matter effects depend on the neutrino incident angle $\theta$, the neutrino mass hierarchy $\Delta m_{31}^2$, the crossing probability at the high resonance region inside the supernova, $P_H$, the neutrino temperature, $T_{\alpha }$, and the pinching parameter in the neutrino spectrum, ${\eta }_{\alpha }$. We also take into account the collective effects due to neutrino-neutrino interactions inside the supernova. With the expression for the dependence of $P_H$ on the neutrino mixing-angle ${\theta }_{13}$, we obtain the relations between ${\theta }_{13}$ and the event numbers for various reaction channels of supernova neutrinos. Using these relations, we propose a possible method to acquire information about ${\theta }_{13}$ smaller than 1.5°. Such a sensitivity cannot be achieved by the reactor neutrino deta at the Daya Bay experiment which has a sensitivity of the order of ${\theta }_{13}\sim 3°$. Furthermore, we apply this method to other neutrino experiments, i.e. Super-K, SNO, KamLAND, LVD, MinBooNE, Borexino, and Double-Chooz. We also study the energy spectra of the differential event numbers, $dN/dE$.

Figure 1

Illustration of the path of the SN neutrino reaching the detector in the Earth. $D$ is the location of the detector, $\theta$ is the incident angle of the neutrino, O is the center of the Earth, $L$ is the distance the neutrino travels through the Earth, and $\tilde{x}$ is the distance of the neutrino to the center of the Earth.

Figure 2

The event numbers observed at the Daya Bay experiment as a function of the incident angle $\theta$ when $T_{{\nu }_e}=3.5\mathrm{MeV}$, $T_{{\overline{\nu }}_e}=5\mathrm{MeV}$, $T_{{\nu }_x}=8\mathrm{MeV}$, and ${\eta }_{{\nu }_e}={\eta }_{{\overline{\nu }}_e}={\eta }_{{\nu }_x}=0$. (a) the channel ${\overline{\nu }}_e+p\to e^{+}+n$; (b) the reactions $\nu +e^{-}\to \nu +e^{-}$; (c) the neutrino-carbon reactions. The solid curves correspond to $P_H=1$ (normal hierarchy), the dashed curves correspond to $P_H=1$ (inverted hierarchy), the dotted curves correspond to $P_H=0$ (normal hierarchy), and the dot-dashed curves correspond to $P_H=0$ (inverted hierarchy). For (a), the solid curve also corresponds to $P_H=1$ (inverted hierarchies) and $P_H=0$ (normal hierarchy).

Figure 3

Similar to Fig. 2 but $T_{\alpha }$ and ${\eta }_{\alpha }$ take their maximum values: $T_{{\nu }_e}=4\mathrm{MeV}$, $T_{{\overline{\nu }}_e}=6\mathrm{MeV}$, $T_{{\nu }_x}=9\mathrm{MeV}$, ${\eta }_{{\nu }_e}=5$, ${\eta }_{{\overline{\nu }}_e}=2.5$, ${\eta }_{{\nu }_x}=2$.

Figure 4

Similar to Fig. 2 but $T_{\alpha }$ and ${\eta }_{\alpha }$ take their minimum values: $T_{{\nu }_e}=3\mathrm{MeV}$, $T_{{\overline{\nu }}_e}=5\mathrm{MeV}$, $T_{{\nu }_x}=7\mathrm{MeV}$, ${\eta }_{{\nu }_e}=3$, ${\eta }_{{\overline{\nu }}_e}=2$, ${\eta }_{{\nu }_x}=0$.

Figure 5

The crossing probability at the high resonance region inside the SN: (a) as a function of the neutrino mixing-angle ${\theta }_{13}$ and the solid curves, dotted curves, dashed curves correspond to $E=11\mathrm{MeV}$, 16 MeV, 25 MeV, respectively; (b) as a function of the neutrinos energy $E$ and the solid curves, dotted curves, dashed curves correspond to ${\theta }_{13}=0.1°$, 1°, 2°, respectively.

Figure 6

The event number observed in the channel ${\overline{\nu }}_e+p\to e^{+}+n$ at the Daya Bay experiment as a function of the neutrino mixing-angle ${\theta }_{13}$ when the incident angle $\theta =30°$. The solid curves correspond to the normal hierarchy, and the dashed curves correspond to the inverted hierarchy, where “max” (”min”) corresponds to the maximum (minimum) values of $T_{\alpha }$ and ${\eta }_{\alpha }$.

Figure 7

The ratio of the event number of ${\nu }_e$ to that of ${\overline{\nu }}_e$, $R_1$, as a function of the mixing-angle ${\theta }_{13}$ in the channel of neutrino-carbon reactions at the Daya Bay experiment. (a) the incident angle $\theta =30°$; (b) $\theta =90°$; (c) $\theta =93°$; (d) $\theta =150°$. The solid curves correspond to the normal hierarchy (max), the dashed curves correspond to the inverted hierarchy (max), the dot-dashed curves correspond to the normal hierarchy (min), the dotted curves correspond to the inverted hierarchy (min), where “max” (”min”) corresponds to the maximum (minimum) values of $T_{\alpha }$ and ${\eta }_{\alpha }$.

Figure 8

Similar to Fig. 6 but for the Super-K experiment.

Figure 9

Similar to Fig. 7 but for the ratio of the event number of ${\nu }_e$ to that of ${\overline{\nu }}_e$, $R_2$, as a function of the mixing-angle ${\theta }_{13}$ in the channel of neutrino-electron scattering at the Super-K experiment.

Figure 10

Similar to Fig. 7 for the ratio of the event number of ${\nu }_e$ to that of ${\overline{\nu }}_e$, $R_3$, as a function of the mixing-angle ${\theta }_{13}$ in the channel of neutrino-deuterium reactions at the SNO experiment.

Figure 11

In the case of inverted hierarchy, the differential event number $dN/dE$ observed in the inverse beta-decay channel at the Daya Bay experiment as a function of the neutrino energy $E$. (a) for different incident angle $\theta$ (${\theta }_{13}=0$); (b) for different ${\theta }_{13}$ ($\theta =30°$, $C=3$) when $T_{{\nu }_e}=3.5\mathrm{MeV}$, $T_{{\overline{\nu }}_e}=5\mathrm{MeV}$, $T_{{\nu }_x}=8\mathrm{MeV}$, ${\eta }_{{\nu }_e}={\eta }_{{\overline{\nu }}_e}={\eta }_{{\nu }_x}=0$.