Inclusive hadron spectrum ine+e−annihilation as a test for the production and decay of heavy leptons

The production and subsequent decay into a neutrino and hadrons of a heavy lepton in $e^{+}{e}^{-}$ colliding beams gives a contribution to the inclusive hadron spectrum $\frac{\mathrm{sd}\sigma }{\mathrm{dx}}$. We calculate this contribution assuming the usual $V-A$ structure for the weak hadronic current and assuming that the "hadronic" decays of the heavy lepton are dominated by the $\pi$, $\rho$, $A_1$, and ${\rho }^{\prime }$ mesons. Owing to the decay into a single pion and a neutrino the inclusive energy spectrum of hadrons has a distinct maximum. The position of this maximum is directly related to the heavy-lepton velocity. Experimental determination of the position of this maximum will therefore determine the heavy-lepton mass. Our calculation also shows that the inclusive hadron spectrum decreases with increasing $x$ less rapidly than the total measured inclusive hadron spectrum. Measuring the spectrum of a hadron in $e^{+}{e}^{-}\to \mathrm{hadron}+e\left(\mu \right)+\mathrm{anything}$ and determining the position of the maximum offers a test for the existence of heavy leptons and a way of determining their mass.