Consider a neutrino beam produced in a proton accelerator facility. The accelerated proton beam is driven on a target. The emerging pions are focussed in the forward direction and then drift in a vacuum pipe in which they decay. At the end of the decay volume all the charged and neutral particles are absorbed, with the exclusion of neutrinos. In the ‘long base line’ experiments studying neutrino oscillations the detector is located at several hundred kilometres from the source, at L = 730 km in the OPERA experiment at Gran Sasso on the CNGS beam from the SpS accelerator at CERN and L= 295 km in the T2K experiment with the Super-Kamiokande detector on the beam from the J-PARC 50 GeV high-intensity accelerator in Japan. Assume a typical pion beam E_π = 80 GeVin the first case and E_π = 7 GeV in the second. In both cases:

a) calculate the pion decay length,

b) calculate the maximum and minimum neutrino energy.

c) In the CM, neutrinos are emitted isotropically in θ^*. 

Hence half of them are emitted forward, namely with θ^* ≥ 0. Find the corresponding angle in the L frame and the beam ‘radius’ at the far detectors.