On the $D^0$ -- $D_s$ lifetime difference and $\tau \to 7\pi +{\nu }_{\tau }$ decays

In this paper we discuss some aspects of inclusive decays of charmed mesons and also decays of the $\tau$ lepton into ${\nu }_{\tau }+7\pi$. We find that phase space effects are likely to explain the observed lifetime ratio $\tau(D_s^+) / \tau(D^0)$ = 1.17. In particular one need not appeal to a large annihilation contribution in the inclusive $D^0$ decay which, being absent in $D_s^{+}$ decays could also contribute to the enhanced $D^0$ decay rate relative to that of the $D_s^{+}$. Examining a separate problem, we find that the rate for $\tau\to \nu_\tau + 7\pi$ is almost completely dominated by the tiny phase space for the final eight particle state. Using an effective chiral Lagrangian to estimate the matrix element yields a branching ratio into the channel of interest far smaller than the present upper bound.