There are thought to be three types of the particles called neutrinos: electrontype (left(u_{e} ight)), muon type

There are thought to be three types of the particles called neutrinos: electrontype \(\left(u_{e}\right)\), muon type \(\left(u_{\mu}\right)\), and tau-type \(\left(u_{\tau}\right)\). If they were all massless they could not spontaneously convert from one type into a different type. But if there is a mass difference between two types, call them types \(u_{1}\) and \(u_{2}\), the probability that a neutrino starting out as a \(u_{1}\) becomes a \(u_{2}\) is given by the oscillating probability \(P=S_{12} \sin ^{2}(L / \lambda)\), where \(S_{12}\) is called the mixing strength parameter, which we take to be constant, \(L\) is the distance traveled by the neutrino, and \(\lambda\) is a characteristic length, given in kilometers by

where \(E\) is the energy of the neutrino in units of \(\mathrm{GeV}\left(1 \mathrm{GeV}=10^{9} \mathrm{eV}\right)\) and \(\Delta(m)^{2}\) is the difference in the squares of the two masses in units \(\left.(\mathrm{eV})^{2}\right)\). Neutrinos are formed in earth's atmosphere by the collision of cosmic-ray protons from outer space with atomic nuclei in the atmosphere. The giant detector Super Kamiokande, located deep underground in a mine west of Tokyo, saw equal numbers of electron-type neutrinos coming (1) from the atmosphere above the detector (2) from the atmosphere on the other side of the earth, which pass through our planet on their way to the detector. However, Super K saw more muon-type neutrinos coming down from above than those coming up from above. This was strong evidence that muon-type neutrinos oscillated into tau-type neutrinos (which Super \(\mathrm{K}\) could not detect) as they penetrated the earth, since it requires more time to go 13,000 km through the earth than \(20 \mathrm{~km}\) through the atmosphere above the mine.

(a) Suppose \((\Delta m)^{2}=0.01 \mathrm{eV}^{2}\) between \(u_{\mu}\) and \(u_{\tau}\) type neutrinos, and that the neutrino energy is \(E=5 \mathrm{GeV}\). What is \(\lambda\) ? How would this explain the fewer number of muon neutrinos seen from below than from above?

(b) The best experimental fit is \((\Delta m)^{2}=0.0022 \mathrm{eV}^{2}\). Again assuming \(E=5 \mathrm{GeV}\), what is \(\lambda\) ? Make a crude estimate of the ratio one might expect for the number of muon neutrinos from below and from above.

Transcribed Image Text:

= E 1.27A(m)2