Mean lifetime (physics, $\tau =\hslash /\Gamma$)

For an unstable particle undergoing exponential decay in its rest frame, the decay waiting time has an exponential distribution with mean $\tau$. Its probability of surviving to time $t\ge 0$ is $e^{-t/\tau }$, and its total decay width is $\Gamma =\hslash /\tau$. In units $\hslash =1$, $\tau =1/\Gamma$.

The mean lifetime of an unstable particle measured in its rest frame, using elapsed proper time. In the exponential decay model, survival to proper time $\tau$ has probability $e^{-\tau /{\tau }_0}$. For constant Lorentz factor $\gamma$, time dilation gives laboratory mean lifetime $\gamma {\tau }_0$. A mean lifetime is a statistical mean, rather than a fixed decay time.

For a freely moving unstable particle of fixed speed $v=\beta c$ and proper mean lifetime ${\tau }_0$, the mean travelled distance before decay is ${\ell }_d=v\gamma {\tau }_0$. Survival beyond distance $L$ has probability $e^{-L/{\ell }_d}$, since elapsed proper time is $L/(\gamma v)$. For rest mass $m$ and three-momentum magnitude $p$, ${\ell }_d=p{\tau }_0/m$.