Muon Inelasticity in Underground Cosmic -ray Ne Utrino Interactions*

The inelasticity of the secondary muon. in high-energy neutrino interactions, defined in a manner appropriate for the deep-mine cosmic-ray experiments, is computed. On the basis of the hypothesis of locality of the Iepton current, < E,/EV> = 0 0 62 f 0.12. (Submitted to Nuovo Cimento.) * Work supported by the U. S. Atomic Energy CGmmission. In order to interpret underground cosmic-ray neutrino experiments, 1,2 it is necessary to estimate the fraction of energy c = retained by the muon in the interaction which absorbs the neutrino. The purpose of this note is to show that, provided only that the lepton current is of local V-A form, the effective 6 for the high-energy contribution is given by CE = 0.62 k .12, where the limits are absolute. Let Ev = neutrino energy, E c1 = muon energy at production, and let v = Ev -E . P Assume that the muon ionization loss dEP/dx Z constant and range fluctuations may be ignored. Assume the incident differential neutrino spectrum can be approximated 3-5,6 by a power law7 spectrum w E; ‘. Then the rate for detecting muons underground in a given element of solid angle is 00 EV R=C dEv E-‘$ V 0 0 (1) where the nature of the constant C is not important for the argument here. Conventionally, in the cosmic-ray literature it is assumed that the muon carries off a unique fraction c of the neutrino energy; i.e., da =I dv =a = o(Ev) a(EP EEv) 1. (2) where E in principle could vary with Ev . With this artifice, the rate underground becomes R=C / ,=)dE ‘El-‘a(E )c v v V (3) 0 If one assumes that the lepton current is local the cross section for the process ~~(2) + N -+P&‘) + hadrons