Kinematic Parameter for Estimating the Primary Energy and for Studying Nuclear Interactions from 17 to ~103 GeV
The kinematic parameter η=arctanh (βθ) for a secondary of a jet transforms from the laboratory system (LS) to η* in a frame of reference moving with a velocity βc with respect to the LS in the direction of the incident primary, according to the relation η=η*+arctanhβc. Then the velocity βS of a "symmetric system," of a group of produced secondaries for which the mean value of η statistically vanishes, is obtained from the formula arctanh βS=<η>, which usually reduces to lnγS=-<ln θ>-<ln[(1+x2)12x-1]>, where x=ptm. For the usual situation where only the emission angles of a subset of charged secondaries of a jet is known, a parameter η(θ)≅0.46-ln θ, which depends only on θ but which is consistent with the definition of η, is introduced, and the rather well-known distribution of the transverse momentum of pion secondaries is used in place of knowledge of values of β to calculate results based on the use of η. The E(θ) method, in which one substitutes <η(θ)> for <η> to find the velocity of the symmetric system, βS, is an improvement over the spectrum-independent formula by Castagnoli et al., ln γCast=-<ln θ>, for estimating the primary energies of jets. This is shown with accelerator-produced jets with energies ranging from 17 to 30.9 GeV and cosmic-ray jets with energies around 103 GeV. Also studied are the magnitude of the statistical error in using the E(θ) method and various aspects of the problem of multiple production of particles which have been determined by examining the η(θ) distributions of jets and their dependence on the primary energy.