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https://en.wikipedia.org/wiki/Particle_decay#Probability_of_survival_and_particle_lifetime lists the mean lifetime of particles. But particles may decay in different ways, so is the mean lifetime the average over all possible decay modes?

Does each decay mode have its own decay rate?

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The decay rates are additive so the total decay rate is:

$$\Gamma\equiv\sum_i\Gamma_i$$

where $\left\{\Gamma_i\right\}$ are the partial decay rates to each of the possible decays. This makes intuitive sense as for example if you have a bucket of pebbles and you have two people taking pebbles out of the bucket the total rate is the sum of each persons rate.

This means that the lifetimes follow the following relation (harmonic sum):

$$\frac{1}{\tau}=\sum_i\frac{1}{\tau_i}$$

where $\left\{\tau_i\right\}$ is the life time for each decay mode.

This is exactly the same as for example the scattering of electrons in conductors where there are multiple scattering sources such as impurities and lattice vibrations. $\tau$ would be the mean time between scattering events and $\tau_i$ would be the mean time between being two consecutive lattice vibration scattering events or the mean time between two consecutive impurity scattering events.

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    $\begingroup$ This is the answer I came here to write. Note that the "mean lifetime $\tau$" is the denominator in the decay relation $N(t)=N_0 e^{-t/\tau}$ and is related to the half-life by a factor $\ln2$. The mean lifetime is the "harmonic sum" of the partial lifetimes, as shown in this answer; it is not their average, as suggested in the question. $\endgroup$
    – rob
    Jul 3 '21 at 13:24
  • $\begingroup$ Exactly, the harmonic sum (which I will lable in the answer for clarity) is true, however, for both mean life time and half life as the factors will divide through; hence, I left this ambiguous. $\endgroup$
    – Chris Long
    Jul 3 '21 at 13:29
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    $\begingroup$ Note that the value of the particle's lifetime measured experimentally using any decay mode will be the same. In other words, probabilities of each allowed decay mode (aka branching fractions) are not functions of time. An important note is that for particles which are mixtures of two eigenstates of different lifetimes (e.g. the $B_s^0$ meson, a mixture of the heavy & light eigenstates), certain decay modes may be allowed only for one of the two eigenstates, in that case the measured "effective" lifetime might differ from one decay mode to another. $\endgroup$
    – Martino
    Jul 4 '21 at 14:42
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There is a defined quantity called "partial lifetime", which is described in Chris Long's answer, but it has no physical interpretation as a time interval. Different decays don't take different amounts of time in any measurable sense.

If you prepare a large number of identical particles, measure the time each one takes to decay, and bin them by what they decay into, then the mean lifetime of each bin will be the lifetime of the particle, not the "partial lifetime" of the decay mode.

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  • $\begingroup$ I mean, you can picture the particle as a bucket full of water with various sized holes at the bottom. The rate the i-mode can is filling up under the i-th hole is $1/\tau_i$, and after the bucket is empty, that can has $\tau/\tau_i$ of the bucket's content, the BR. $\endgroup$ Jul 6 '21 at 18:15

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