Atmospheric neutrino oscillation analysis with subleading effects in Super-Kamiokande I, II, and III

We present a search for nonzero ${\theta }_{13}$ and deviations of ${sin}^2{\theta }_{23}$ from 0.5 in the oscillations of atmospheric neutrino data from Super-Kamiokande I, II, and III. No distortions of the neutrino flux consistent with nonzero ${\theta }_{13}$ are found and both neutrino mass hierarchy hypotheses are in agreement with the data. The data are best fit at $\Delta m^2=2.1\times {10}^{-3}{\mathrm{eV}}^2$, ${sin}^2{\theta }_{13}=0.0$, and ${sin}^2{\theta }_{23}=0.5$. In the normal (inverted) hierarchy ${\theta }_{13}$ and $\Delta m^2$ are constrained at the one-dimensional 90% C.L. to ${sin}^2{\theta }_{13}<0.04(0.09)$ and $1.9(1.7)\times {10}^{-3}<\Delta m^2<2.6(2.7)\times {10}^{-3}{\mathrm{eV}}^2$. The atmospheric mixing angle is within $0.407\le {sin}^2{\theta }_{23}\le 0.583$ at 90% C.L.

Figure 1

The left side of the figure shows the calculated ${\nu }_e$ transition probability $P_{ex}$ for atmospheric neutrinos with an energy $E_{\nu }$ and neutrino zenith angle, $\cos {\Theta }_{\nu }$, using $\Delta m_{12}^2=7.7\times {10}^{-5}{\mathrm{eV}}^2$ ${sin}^2{\theta }_{12}=0.3$ [28], ${sin}^2{\theta }_{23}=0.5$, ${sin}^2{\theta }_{13}=0.0$ and $\Delta m_{23}^2=2.1\times {10}^{-3}{\mathrm{eV}}^2$. Matter effects within the Earth are taken into account. Negative $\cos {\Theta }_{\nu }$ corresponds to upward-going neutrinos and 0 is the horizon. The electron neutrino flux ratio ${\Phi }_e/{\Phi }_e^0$ is shown in the right side of the figure. An expected excess (deficit) for atmospheric mixing in the first (second) octant is shown in the upper (lower) panel. The island shapes are regions of probability driven by the solar oscillation parameters. The center panel shows no significant region of excess or deficit when ${sin}^2{\theta }_{23}=0.5$.

Figure 2

The three flavor oscillation probability ${\nu }_{\mu }\leftrightarrow {\nu }_e$ for ${\theta }_{13}$ at the Chooz limit for neutrinos under the normal hierarchy in the one mass scale dominant framework is shown at left. In the right panel the ${\nu }_e$ flux ratio ${\Phi }_e^{\mathrm{Chooz}}/{\Phi }_e^0-1$ for oscillations with ${\theta }_{13}$ at the Chooz limit relative to those at ${\theta }_{13}=0$. Large matter-induced resonances between 2–10 GeV appear for upward-going neutrinos traversing the core ($\cos {\Theta }_{\nu }<-0.84$) and mantle regions ($-0.84<\cos {\Theta }_{\nu }<-0.45$). Atmospheric mixing is assumed at $\Delta m_{23}^2=2.1\times {10}^{-3}{\mathrm{eV}}^2$ and ${sin}^2{\theta }_{23}=0.5$.

Figure 3

The distributions used in the ${\pi }^0$ selection for five momentum regions: (a) $P_e<250\mathrm{MeV}/c$, (b) $250\mathrm{MeV}/c\le P_e<400\mathrm{MeV}/c$, (c) $400\mathrm{MeV}/c\le P_e<630\mathrm{MeV}/c$, (d) $630\mathrm{MeV}/c\le P_e<1000\mathrm{MeV}/c$, and (e) $1000\mathrm{MeV}/c\le P_e$. Solid (dashed) lines represent CCQE (NC) events in the FC sub-GeV single-ring $e$-like Monte Carlo. Events with an invariant mass above $100\mathrm{MeV}/c^2$ are selected as ${\pi }^0$-like. To separate ${\pi }^0$-like and electronlike more efficiently, an additional likelihood selection is applied for events with momentum above $250\mathrm{MeV}/c$. The distributions of the three likelihood variables are shown: the fraction of energy carried by the second fitted ring [$E_2/(E_1+E_2)$], the ${\pi }^0$ mass, and $\Delta$-likelihood (described in the text). All distributions have been normalized to unit area.

Figure 4

Variables used in the likelihood definition to create the SK-I multi-GeV multiring $e$-like sample. The energy bins correspond to the most energetic ring (MER) and the distributions have been scaled to the SK-I live time. Signal events (CC ${\nu }_e+{\overline{\nu }}_e$) are shown as the solid line and background events (CC ${\nu }_{\mu }+{\overline{\nu }}_{\mu }$ and NC) are shown as the dashed line. The shapes of these distributions do not differ appreciably among SK-I, SK-II, and SK-III.

Figure 5

${\chi }^2\mathrm{\text{−}}{\chi }_{\min }^2$ distribution as a function of $\sin 2{\theta }_{23}$ for oscillations without the 1–2 parameters (dotted line) and with the 1–2 parameters (solid line). For each $\sin 2{\theta }_{23}$ point, $\Delta m_{23}^2$ is chosen so that ${\chi }^2$ is minimized. The horizontal line corresponds to the 68% (90%) confidence level which is located at ${\chi }_{\min }^2+1.0(2.7)$.

Figure 6

Asymmetry $(\mathrm{\text{up}}-\mathrm{\text{down}})/(\mathrm{\text{up}}+\mathrm{\text{down}})$ for the $\mathrm{SK}\mathrm{\text{−}}\mathrm{I}+\mathrm{II}+\mathrm{III}$ single-ring $e$-like 0-decay data set in the octant analysis. Up is defined as events with $\cos \Theta <-0.2$ and down as $\cos \Theta >0.2$. The solid line represents the MC expectation at the best-fit point and the dashed (dotted) line shows the expected asymmetry for ${sin}^2{\theta }_{23}=0.4(0.6)$. The error bars are statistical.

Figure 7

Normal hierarchy allowed regions at 68% (thin line), 90% (medium line), and 99% (thick line) C.L. for the $\mathrm{SK}\mathrm{\text{−}}\mathrm{I}+\mathrm{II}+\mathrm{III}$ data. The shaded region in the first panel shows the Chooz 90% exclusion region.

Figure 8

Inverted hierarchy allowed regions at 68% (thin line), 90% (medium line), and 99% (thick line) C.L. for the $\mathrm{SK}\mathrm{\text{−}}\mathrm{I}+\mathrm{II}+\mathrm{III}$ data. The shaded region in the first panel shows the Chooz 90% exclusion region.

Figure 9

The $\Delta {\chi }^2$ distributions for the $\mathrm{SK}\mathrm{\text{−}}\mathrm{I}+\mathrm{II}+\mathrm{III}$ data set in the normal (solid) and inverted (dashed) hierarchies. The horizontal lines represent the position of the one-dimensional cut value corresponding to the (from top to bottom) 99%, 90%, and 68% CL.

Figure 10

$\mathrm{SK}\mathrm{\text{−}}\mathrm{I}+\mathrm{II}+\mathrm{III}$ zenith angle and lepton momentum distributions of the event samples used in the analyses. Black dots represent the data with statistical errors, the solid lines are the MC expectation at the best fit from the ${\theta }_{13}$ analysis, and dashed lines show the expectation at the best-fit atmospheric variables but with ${\theta }_{13}$ at the Chooz limit.

Figure 11

Asymmetry $(\mathrm{\text{up}}-\mathrm{\text{down}})/(\mathrm{\text{up}}+\mathrm{\text{down}})$ for the $\mathrm{SK}\mathrm{\text{−}}\mathrm{I}+\mathrm{II}+\mathrm{III}$ data set. Up is defined as events with $\cos \Theta <-0.2$ and down as $\cos \Theta >0.2$. Solid lines represent the MC expectation at the best fit from the ${\theta }_{13}$ analysis and the shaded regions show the expectation at the best-fit atmospheric variables but with ${\theta }_{13}$ at the Chooz limit for the single (multi) ring $e$-like events at left (right). Error bars are statistical.