Three flavor neutrino oscillation analysis of atmospheric neutrinos in Super-Kamiokande

We report on the results of a three-flavor oscillation analysis using Super-Kamiokande I atmospheric neutrino data, with the assumption of one mass scale dominance ($\Delta m_{12}^2=0$). No significant flux change due to matter effect, which occurs when neutrinos propagate inside the Earth for ${\theta }_{13}\ne 0$, has been seen either in a multi-GeV ${\nu }_e$-rich sample or in a ${\nu }_{\mu }$-rich sample. Both normal and inverted mass hierarchy hypotheses are tested and both are consistent with observation. Using Super-Kamiokande data only, 2-dimensional 90% confidence allowed regions are obtained: mixing angles are constrained to ${sin}^2{\theta }_{13}<0.14$ and $0.37<{sin}^2{\theta }_{23}<0.65$ for the normal mass hierarchy. Weaker constraints, ${sin}^2{\theta }_{13}<0.27$ and $0.37<{sin}^2{\theta }_{23}<0.69$, are obtained for the inverted mass hierarchy case.

Figure 1

Oscillation probability of ${\nu }_e\to {\nu }_{\mu }$ (or ${\nu }_{\mu }\to {\nu }_e$) transition. For both figures, the horizontal axis shows neutrino energy and the vertical axis shows the zenith angle of neutrino direction; $\cos {\Theta }_{\nu }=-1$ and $\cos {\Theta }_{\nu }=0$ correspond to vertically upward and horizontal directions, respectively. Angles with $\cos {\Theta }_{\nu }<-0.84$ correspond to neutrinos passing through the earth core layers. Assumed oscillation parameters are ($\Delta m^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$, ${sin}^2{\theta }_{23}=0.5$, ${sin}^2{\theta }_{13}=0.04$). The top figure assumes neutrino oscillation in vacuum and the bottom figure takes into account Earth’s matter effect. In the bottom figure, three high probability ($>40\%$) regions are shown which correspond to the MSW resonance at 3 GeV in the core layer, the MSW resonance at 7 GeV in the mantle layer, and the enhancement due to the core-mantle transition interference [20] at the energy between the two MSW regions.

Figure 2

Livetime normalized MC distributions used in the ${\nu }_e$-enriched likelihood selection for multi-GeV multi-ring $e$-like events. Plots correspond to $1.33<E_{\mathrm{vis}}<5\mathrm{GeV}$, $5<E_{\mathrm{vis}}<10\mathrm{GeV}$, and $E_{\mathrm{vis}}>10\mathrm{GeV}$ from top to bottom. From left to right are shown PID likelihood for the most energetic ring (more negative means more electron-like), momentum fraction of the most energetic ring, number of Michel decay electrons, and square root of distance between decay electron and neutrino interaction vertex divided by visible energy of most energetic ring in $(\mathrm{cm}/\mathrm{MeV}{)}^{1/2}$. Solid histograms are ${\nu }_e+{\overline{\nu }}_e$ CC, dashed are backgrounds ($\mathrm{NC}+{\nu }_{\mu }\mathrm{CC}+{\overline{\nu }}_{\mu }\mathrm{CC}$). The ${\nu }_e$ or ${\overline{\nu }}_e$ CC signals tend to give more electron-like PID and higher momentum fraction. In contrast, backgrounds tend to give more muon decay electrons and the longer decay electron distance due to energetic muons produced by ${\nu }_{\mu }$ or ${\overline{\nu }}_{\mu }\mathrm{CC}$ interactions.

Figure 3

Zenith angle distributions of FC $e$-like, $\mu$-like, PC, and $\mathrm{UP}\mu$ are shown for data (filled circles with statistical error bars), MC distributions without oscillation (boxes) and best-fit distributions (dashed). The nonoscillated MC shows the distribution without fitting and the box height shows the statistical error. In the case of nonzero ${\theta }_{13}$, matter-enhanced excess of electron-like events is expected in the zenith angle of $-1<\cos \theta <-0.2$ regions in the multi-GeV 1-ring and multi-ring electron-like samples. The ${\nu }_{\mu }$ in the resonance regions populate mainly in the multi-GeV single-ring muon, multi-ring muon, two PC, and UP stopping $\mu$ samples.

Figure 4

Asymmetry ($\mathrm{\text{UP}}-\mathrm{\text{DOWN}}$)/($\mathrm{\text{UP}}+\mathrm{\text{DOWN}}$) as a function of particle momentum for FC single-ring $e$-like events (top) and as a function of total energy for FC multi-ring $e$-like events (bottom), where UP (DOWN) refers to the number of events in $-1.0<\cos \Theta <-0.2$ $(0.2<\cos \Theta <1.0)$. Filled circles represent data (error bars are statistical), the line represents best-fit distributions under a normal hierarchy assumption, and the filled area on the best-fit line represents the expected excess due to matter effect for $(\Delta m^2,{sin}^2{\theta }_{23},{sin}^2{\theta }_{13})=(2.5\times {10}^{-3}{\mathrm{eV}}^2,0.5,0.04)$.

Figure 5

90% (thick line) and 99% (thin line) confidence level allowed regions are shown in ${sin}^2{\theta }_{13}$ vs ${sin}^2{\theta }_{23}$ (top), $\Delta m^2$ vs ${sin}^2{\theta }_{23}$ (middle), and $\Delta m^2$ vs ${sin}^2{\theta }_{13}$ (bottom). Normal mass hierarchy ($\Delta m^2>0$) is assumed. The shaded area in the bottom figure shows the region excluded by the CHOOZ reactor neutrino experiment.

Figure 6

90% (thick line) and 99% (thin line) confidence level allowed regions assuming inverted mass hierarchy ($\Delta m^2<0$), shown in the same orientation as in Fig. 5.

Figure 7

For normal (top) and inverted (bottom) mass hierarchy assumption, ${\chi }^2-{\chi }_{\min }^2$ are shown as a function of ${sin}^2{\theta }_{13}$. Minimum ${\chi }^2$ values for each $\Delta m^2$ and ${sin}^2{\theta }_{23}$ are selected and plotted at each ${sin}^2{\theta }_{13}$. Solid (dashed) lines show 2-dimensional 90% (99%) confidence level allowed regions.