Figure 4 shows the ratio of data *"real"* events and Monte Carlo expected events in the absence of oscillations for various $L/E_\nu$. The dashed lines correspond to the predictions of the best fit considering a $\nu_\mu \leftrightarrow \nu_\tau$ oscillation. Once again we see a deficit of $\mu$-like events showing that the data collected by Super-Kamiokande is consistent with a $\nu_\mu \leftrightarrow \nu_\tau$ oscillation scenario. The conclusion states that the observed events are consistent with a two flavour $\nu_\mu \leftrightarrow \nu_\tau$ oscillation scenario a gives new limits for $\sin^2(2\theta)$ and $\Delta m^2$. Figure 1 (bottom) shows a strong asymmetry for $\mu$-like events detected by Super-Kamiokande. The hatched region shows the Monte Carlo results considering no neutrino oscillations. We see that the data (black dots) deviate from these predictions. The dashed line corresponds to $$\nu_\mu \leftrightarrow \nu_\tau$$ oscillations and seems to be fairly consistent with the observations. Zenith angle distributions for $e$-like and $\mu$-like events. The hatched region shows the Monte Carlo expectation in the case of no oscillations. The bold line corresponds to the best fit considering a $\nu_\mu \leftrightarrow \nu_\tau$ oscillation. We can see that the data (black dots) is consistent with an oscillation scenario for both types of events and various energy ranges. Curiosity: In Particle Physics, there is a convention of a **five-sigma effect (99.99994% confidence)** being required to qualify any detected event as a **"Discovery"**. Phase space *("allowed solutions")* for $\Delta m^{2}$ and $\sin^2(2\theta)$ for the results of the best fit to the data considering a two flavour oscillation scenarion from $\nu_\mu$ to $\nu_\tau$ for different confidence levels (68%, 90%, 99%). We see an overlap at the 90% confidence level between Super-Kamiokande and Kamiokande. Atmospheric neutrinos are produced in hadronic showers that result from the interaction of a cosmic ray at the top of the atmosphere. The cosmic rays start a cascade of reactions that produce pions that will decay and produce neutrinos.  There are three flavours of neutrinos (electron, muon, tau): \begin{eqnarray*} e - \nu_e\\ \mu - \nu_\mu\\ \tau - \nu_\tau\\ \end{eqnarray*} (and the corresponding antineutrinos) Picture of the first neutrino observation in a Bubble chamer (1970) via it's charged current interaction with a proton. Once can see that the neutrino detection is made indirectly by studying the paths of the particles produced when the neutrino interaction with matter occurred.  It is very hard to detect neutrinos. Neutrinos are electrically neutral, so they are not affected by Electromagnetism; they are leptons so they are not affected by the Strong force. So they are only affected by the Weak force (short range interaction) and Gravity (extremly weak at the subatomic scale). Neutrinos are detected indirectly via their weak interactions with matter where they produce the charged leptons associated with them. Curiosity: This paper is mentioned in two different Physics Nobel Prize ceremonies 2002 and 2015. See here: [Physical Review Letters.](http://journals.aps.org/prl/50years/milestones#1998) Let us compute the oscillation probability. Consider a two Neutrino oscillation hypothesis from flavour a to flavour b, ($\nu_a\,,\nu_b$): \begin{eqnarray*} \left(\begin{array}{c} \nu_{a} \\ \nu_{b} \\ \end{array}\right) &=& \left(\begin{array}{cc} cos{\theta} & sin{\theta} \\ -sin{\theta} & cos{\theta} \\ \end{array}\right) \left(\begin{array}{c} \nu_{1} \\ \nu_{2} \\ \end{array}\right) \end{eqnarray*} In bra-ket notation: \begin{eqnarray*} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle #1|} \ket{\nu_a} &=& \cos\theta\ket{\nu_1} + \sin\theta\ket{\nu_2}\\ \ket{\nu_b} &=& -\sin\theta\ket{\nu_1} + \cos\theta\ket{\nu_2} \end{eqnarray*} Using the Schrödinger equation one can write the mass eigenstates of a particle as: \begin{eqnarray*} \ket{\nu_1(t)} &=& e^{-iE_1t}\ket{\nu_1} = e^{-i\sqrt{m_1^2+p_1^2}\,t}\ket{\nu_1} \nonumber\\ &\simeq& e^{-i(p+\frac{m_1^2}{2p})t}\ket{\nu_1}\\ \ket{\nu_2(t)} &\simeq& e^{-i(p+\frac{m_2^2}{2p})t}\ket{\nu_2} \end{eqnarray*} In natural units the energy of any given mass eigenstate $i$ can be written as: $E_i=\sqrt{m_i^2+p^2}\simeq p+\frac{m_i^2}{2p}$ where we consider that $m\ll p$. One can then rewrite the wave function for a flavour a neutrino ($\nu_a$) as: \begin{eqnarray*} \ket{\nu_a(t)} &=& c\ket{\nu_1(t)} + s\ket{\nu_2(t)}\\ &\simeq& e^{-ip}\left( ce^{-i\frac{m_1^2}{2p}t}\ket{\nu_1} + se^{-i\frac{m_2^2}{2p}t}\ket{\nu_2} \right) \end{eqnarray*} where $c \equiv \cos\theta$ and $s\equiv \sin\theta$. We do the same for flavour b ($\nu_b$) and then compute the probability function for the oscillation from flavour $\ket{\nu_a}$ to flavout $\ket{\nu_b}$ as: \begin{eqnarray*} P_{\nu_a\rightarrow \nu_b}(t) &=& \left| \left< \nu_b | \nu_a(t) \right> \right|^2 \nonumber \\ &=& \left| -sce^{-i\frac{m_1^2}{2p}t} + cse^{-i\frac{m_2^2}{2p}t} \right|^2 \nonumber \\ &=& 2s^2c^2 \left( 1-\cos{\frac{m_1^2-m_2^2}{2p}t} \right) \nonumber \\ &\simeq& \sin^2(2\theta)\sin^2(\delta_{12}) \end{eqnarray*} where:\begin{eqnarray*} \delta_{ij} = \frac{c^3}{\hbar}\frac{\Delta m_{ij}^2}{4E} L \simeq \frac{\Delta m_{ij}^2}{2p} L \simeq 1.27\, \frac {\Delta m_{ij}^2}{E} L \end{eqnarray*} with $\Delta m^2_{ij}=m^2_i-m^2_j$ and again we use natural units so that $t \simeq L$, $p \simeq E$, $L$ (km) is the distance travelled by the neutrino $E$ (GeV) the neutrino energy $\theta$ mixing angle. So finally one can write: \begin{eqnarray*} P_{\nu_a \rightarrow \nu_b} \left( L/E \right) &=& sin^2\left( 2\theta \right) \, sin^2\left( 1.27\,\Delta m^2 L/E \right).\\ \end{eqnarray*}