An Off-Axis Neutrino Beam

1 Problem A typical high-energy neutrino beam is made from the decay of π mesons that have been produced in proton interactions on a target, as sketched in the figure below. Suppose that only positively charged particles are collected by the " horn ". The main source of neutrinos is then the decay π + → µ + ν µ. 1. Give a simple estimate of the relative number of other types of neutrinos than ν µ in the beam. 2. If the decay pions have energy E π m π , what is the characteristic angle θ C of the decay neutrinos with respect to the direction of the π + ? 3. If a neutrino is produced with energy E ν m π , what is the maximum angle θ max (E ν) between it and the direction of its parent pion (which can have any energy)? What is the maximum energy E ν at which a neutrino can be produced in the decay of a pion if it appears at a given angle θ with respect to the pion's direction? Parts 4 and 6 explore consequences of the existence of these maxima. 4. Deduce an analytic expression for the energy-angle spectrum d 2 N/dE ν dΩ for neutrinos produced at angle θ ≤ θ C to the proton beam. You may suppose that E ν m π , that the pions are produced with an energy spectrum dN/dE π ∝ (E p − E π) 5 , where E p is the energy of the proton beam, and that the " horn " makes all pion momenta parallel to that of the proton beam. 5. At what energy E ν,peak does the neutrino spectrum peak for θ = 0? 6. Compare the characteristics of a neutrino beam at θ = 0 with an off-axis beam at angle θ such that E ν,max (θ) is less than E ν,peak (θ = 0). Facts: m π = 139.6 MeV/c 2 , τ π = 26 ns, m µ = 105.7 MeV/c 2 , τ µ = 2.2 µs. In this problem, neutrinos can be taken as massless.