A Review of Neutrino Oscillations and Beyond — Epoche C2
The measurement that settled it In 1998 the Super-Kamiokande collaboration reported that muon neutrinos arriving at its detector from below, after crossing the Earth, were about half as numerous as those arriving from above after falling a few tens of kilometres (Fukuda et al., 1998). That asymmetry, rather than any direct weighing of a neutrino, is why the Standard Model's prediction of a massless neutrino is now known to be false. What follows sets out how the measurement was made, how a mass splitting produces exactly that directional dependence and no other, what the resulting number implies about the scale of whatever lies beyond the Standard Model, and which questions the method is constitutionally unable to answer. The neutrinos in question are made in the atmosphere. Primary cosmic rays, mostly protons, strike nuclei some fifteen to twenty kilometres up and produce pions; a charged pion decays as $\pi^+ \to \mu^+ \nu_\mu$, and the muon then decays as $\mu^+ \to e^+ \nu_e \bar{\nu}_\mu$, so each chain yields two muon-type neutrinos for every electron-type one, and the charge-conjugate chain does the same. This is the key to the experimental design. The absolute flux is known only to about twenty per cent, because it depends on the primary cosmic-ray spectrum and on hadronic production cross-sections; the flavour composition, being fixed by the decay chain itself, is known to a few per cent. Super-Kamiokande therefore reports a double ratio, the observed ratio of muon-like to electron-like events divided by the ratio a simulation predicts in the absence of oscillation. For events below about one GeV of visible energy the collaboration found that double ratio to be $0.63 \pm 0.03$ (statistical) $\pm\, 0.05$ (systematic), and for the multi-GeV sample $0.65 \pm 0.05 \pm 0.08$. A value of one is expected if nothing happens in flight. The detector is fifty kilotonnes of ultrapure water viewed by photomultiplier tubes, of which 22.5 kilotonnes were used as the fiducial volume for this analysis, sited under about a kilometre of rock in the Kamioka mine so that the flux of cosmic-ray muons is suppressed by some five orders of magnitude. A charged-current interaction converts the neutrino into its charged partner, and the Cherenkov ring of that lepton is sharp-edged for a muon and diffuse for an electron, which showers; that difference is what identifies the flavour. At these energies the charged lepton is emitted close to the neutrino's direction, so the zenith angle of the ring measures the direction of the incoming neutrino, and the direction fixes the path length. A neutrino from directly overhead has travelled about 15 km; one from directly below has crossed a full Earth diameter, 12,742 km. The same detector, the same beam and the same reconstruction thus deliver path lengths spanning nearly three orders of magnitude, with everything else held fixed. The decisive number is the up-down asymmetry of the multi-GeV muon-like sample, $A = (U - D)/(U + D)$, where $U$ and $D$ count upward-going and downward-going events. Super-Kamiokande measured $A = -0.296 \pm 0.048 \pm 0.010$, which corresponds to an upward rate that is $0.704/1.296 = 0.54$ of the downward rate. Two features make this hard to explain away. Above a few GeV the primary cosmic-ray flux is isotropic to good accuracy, so the up-down comparison uses the detector as its own control and the twenty per cent flux normalisation cancels entirely. And the electron-like events show no comparable asymmetry, which excludes both an instrumental up-down bias and the possibility that the missing muon neutrinos are arriving as electron neutrinos. Why a mass splitting makes the deficit depend on direction The states produced and detected by the weak interaction are the flavour states, and there is no reason of principle for these to coincide with the states of definite mass that propagate freely. Writing the relation as $|\nu_\alpha\rangle = \sum_i U_{\alpha i}^{*}|\nu_i\rangle$, with $\alpha$ running over flavours and $i$ over mass eigenvalues $m_i$, each mass component acquires its own phase in flight. For a neutrino of momentum $p$ the energies are $E_i = \sqrt{p^2 + m_i^2} \simeq p + m_i^2/(2p)$, the expansion being excellent because $m_i$ is at most of order an electronvolt while $p$ is of order a GeV. Taking the propagation time equal to the distance and dropping the phase common to all components, the amplitude to start as $\nu_\alpha$ and be detected as $\nu_\beta$ is a sum over the mass states of $U_{\alpha i}^{*}U_{\beta i}\exp(-i m_i^2 L / 2E)$. With two flavours the mixing matrix is a single rotation by an angle $\theta$, and the amplitude for the flavour to change is $\sin\theta\cos\theta\,(e^{-i\phi_2} - e^{-i\phi_1})$ with $\phi_i = m_i^2 L/2E$. Squaring, $|e^{-i\phi_2} - e^{-i\phi_1}|^2 = 2 - 2\cos(\phi_2 - \phi_1) = 4\sin^2[(\phi_2-\phi_1)/2]$, and $\sin^2\theta\cos^2\theta = \tfrac{1}{4}\sin^2 2\theta$, so the two factors of four cancel and $$P(\nu_{\alpha} \to \nu_{\beta}, L) = \sin^2(2\theta)\, \sin^2\left(\frac{\Delta m^2 L}{4E}\right)$$ where $\Delta m^2 = m_2^2 - m_1^2$. The amplitude of the oscillation is set by the mixing angle alone and its wavelength by the mass splitting alone; the two are measured by different features of the data, the depth of the deficit and its position in $L/E$. The formula is written in natural units, and the conversion is worth doing explicitly because every subsequent number depends on it. Restoring $\hbar$ and $c$, the phase is $\Delta m^2 c^4 L/(4\hbar c E)$. Using $\hbar c = 1.97327 \times 10^{-7}$ eV m, one electronvolt squared of splitting over one kilometre at one GeV gives a phase of $10^{3}/(4 \times 10^{9} \times 1.97327 \times 10^{-7}) = 1.267$ radians, so in practical units the phase is $1.267\,\Delta m^2[\mathrm{eV}^2]\,L[\mathrm{km}]/E[\mathrm{GeV}]$. Now put Super-Kamiokande's best-fit splitting, $\Delta m^2 = 2.2 \times 10^{-3}\ \mathrm{eV}^2$, into that expression. The first