Kinematic Parameter for Estimating the Primary Energy and for Studying Nuclear Interactions from 17 to ∼103GeV

The kinematic parameter $\eta =\mathrm{arctanh} \left(\beta cos\theta \right)$ for a secondary of a jet transforms from the laboratory system (LS) to ${\eta }^{*}$ in a frame of reference moving with a velocity ${\beta }_c$ with respect to the LS in the direction of the incident primary, according to the relation $\eta ={\eta }^{*}+\mathrm{arctanh}{\beta }_c$. Then the velocity ${\beta }_S$ of a "symmetric system," of a group of produced secondaries for which the mean value of $\eta$ statistically vanishes, is obtained from the formula arctanh ${\beta }_S=〈\eta 〉$, which usually reduces to $\ln {\gamma }_S=-〈\ln tan\theta 〉-〈\ln \left[{(1+x^2)}^{\frac{1}{2}}{x}^{-1\mathrm{}}\right]〉$, where $x=\frac{p_t}{m}$. For the usual situation where only the emission angles of a subset of charged secondaries of a jet is known, a parameter $\eta \left(\theta \right)\cong 0.46-\ln tan\theta$, which depends only on $\theta$ but which is consistent with the definition of $\eta$, is introduced, and the rather well-known distribution of the transverse momentum of pion secondaries is used in place of knowledge of values of $\beta$ to calculate results based on the use of $\eta$. The $E\left(\theta \right)$ method, in which one substitutes $〈\eta \left(\theta \right)〉$ for $〈\eta 〉$ to find the velocity of the symmetric system, ${\beta }_S$, is an improvement over the spectrum-independent formula by Castagnoli et al., $\ln {\gamma }_{\mathrm{Cast}}=-〈\ln tan\theta 〉$, 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 ${10}^3$ GeV. Also studied are the magnitude of the statistical error in using the $E\left(\theta \right)$ method and various aspects of the problem of multiple production of particles which have been determined by examining the $\eta \left(\theta \right)$ distributions of jets and their dependence on the primary energy.