## Massive Neutrinos

In June 1998 an experimental group working at SuperKamiokande in Japan released evidence for neutrino oscillations and hence that at least one species of neutrino has mass. Here is a very basic description of the effect and its significance.

What was detected?

They detected an asymmetry in the neutrino signal in the detector arriving from above the detector (downward going) and from below (upward going). The counts arise from neutrinos produced in Earth's atmosphere from cosmic ray events and are not related to solar neutrinos produced by thermonuclear reactions in the Sun. The origins of the neutrinos are thus in the Earth's atmosphere above the detector and on the far side of the Earth respectively; thus they have different path lengths to travel.

How is this interpreted?

While the electron neutrinos did not show the asymmetry, the muon neutrinos did. The detector is not sensitive to tau neutrinos or (by definition) sterile neutrinos. The interpretation is that some muon neutrinos oscillated or changed species to an undetected sort (tau or sterile) neutrino. These oscillations are expected to be dependent on path length and on neutrino mass.

The oscillations are called vacuum oscillations because they occur due to the internal properties of the neutrinos and not the environment through which they propagate. The cross section for neutrinos to interact with matter is so low that to them passing through the Earth is like passing through vacuum. On the other hand, the solution to the solar neutrino puzzle is thought to be matter dependent MSW oscillations (see below).

Why oscillations?

When we detect particles what we are doing in a quantum mechanical sense is hitting them with some operator, e.g. some Hamiltonian applied to their wave function. The detected properties or normal modes depend on what sort of operator, i.e. what we are measuring. When we talk about neutrino species, we are talking about their flavor eigenstates, those of the weak interaction Hamiltonian. Here the normal modes (eigenstates) are electron neutrino, muon neutrino, tau neutrino, etc. However, it is possible that when we ask their mass (hit the wave functions with the free Hamiltonian) that we obtain different eigenstates - the flavor states are superpositions of the mass eigenstates. This is only possible if the masses of the species are different, i.e. they cannot all have zero mass. It then opens up the possibility of oscillation between flavor states - a given mass state is partly one species and partly another and so the species can change with time or oscillate.

Mass and path length dependence of oscillations

We know how this change with time proceeds: Heisenberg taught us that wave functions evolve as \psi \sim \exp {(i/\hbar)\Delta E t}. Even if they have masses these are small compared to their energies, so the neutrinos move nearly at the speed of light and the time t and path length L are related by t = L/c. To find the wavelength of the oscillation we ask when the phase difference adds up to 2\pi: \Delta\phi = L \Delta E = L (\Delta m2 / 2E) where we have used that E2 = m2 + p2. One finds that the wavelength is Losc = 4\pi E/\Delta m2 = 800 (E/GeV) (\Delta m2 / 10-3eV2)-1 km.

One can measure the energy of the neutrinos received and one knows the path length difference of those arising from the near and the far side of the Earth, so one can relate the different proportions of muon neutrinos received upward going and downward going to the mass difference \Delta m. The SuperKamiokande results point to \Delta m = 0.07 eV. But they cannot detect what the muon neutrinos oscillate into, so they don't know if these are tau neutrinos or a new species of sterile neutrinos (they do know they are not electron neutrinos).

How is this different from the solar neutrino puzzle?

Just about completely. These are neutrinos produced in the Earth's atmosphere with energies of order GeV and undergo vacuum oscillations. The solar neutrinos are produced in the Sun's core with energies of order MeV and are thought to undergo MSW oscillations. These oscillations, also called matter resonant oscillations, involve interactions of electron neutrinos (not the muon neutrinos of SuperKamiokande) with the electrons in the solar matter. These interactions give the neutrinos an effective mass and can cause species oscillations to be resonantly enhanced for certain values of the electron density. MSW solutions tend to give \Delta m = 0.002 eV between electron neutrinos and another species, possibly muon neutrinos. It is possible that MSW oscillations turn electron neutrinos into muon neutrinos and vacuum oscillations are turning muon neutrinos into tau neutrinos. Many experiments are underway to understand exactly which oscillations are occuring, and what they imply for all the neutrino masses.

What impact does this have cosmologically?

The Cowsik-McClelland bound implies that the cosmological energy density of an equilibrium species of massive neutrino is \Omega = (M / 92 eV). All the oscillations results give is the mass difference between species so we do not know from these if the two masses are 1 eV and 1.07 eV or 30 eV and 30.07 eV, say. There are bounds on the masses, however, from other laboratory experiments, particle physics theory, and astrophysical observations, making it unlikely that neutrinos are large enough to give critical energy density though possibly enough to be an appreciable (\Omega = 0.3) component, in the form of hot dark matter.