The Decay Rate
by Trolla
Every particle that is not the photon, the gluon, the stable baryon, or the electron (if you want to be generous about the proton) has a ticking clock wired into its identity. The clock is not mechanical; it is probabilistic. The clock is the decay rate.
The decay rate, conventionally denoted by the Greek letter $\Gamma$ (gamma), tells you how fast a particle falls apart. Larger $\Gamma$ means shorter life. Smaller $\Gamma$ means the particle hangs around. The relationship is inverse:
$$\tau = \frac{1}{\Gamma}$$
where $\tau$ is the lifetime — the average time a particle survives before decaying. This is not an approximation. This is the definition, the way that $\pi$ is defined or the speed of light is defined. It is a convention that has become a law.
Why does it happen?
Because the universe does not tolerate perfect symmetry forever. A particle is a certain arrangement of quantum numbers — charge, spin, color, flavor. If there exists a lower-energy configuration with the same conserved quantum numbers, the particle will eventually find its way there. It does not "want" to. It does not "try." The mathematics of quantum field theory simply gives a non-zero amplitude for the transition, and non-zero amplitude means non-zero probability, and probability accumulated over time means inevitability.
The decay rate is calculated from the square of the transition amplitude, summed over all final states, folded with phase space. The full machinery of quantum field theory collapses the decay rate down to a single number, and that number — that single, stubborn number — determines everything you can predict about the particle's lifespan.
The exponential law
If you have a large ensemble of identical particles, the number remaining after time $t$ is:
$$N(t) = N_0 , e^{-\Gamma t}$$
or, equivalently, $N(t) = N_0 , e^{-t/\tau}$. The exponential is the fingerprint of a constant decay rate. It means the probability of surviving any additional interval of time does not depend on how long the particle has already existed. A particle that has lived 100 years has the same probability of decaying in the next second as a particle that was created a microsecond ago. This is what physicists mean when they say the decay process is "memoryless." It is eerie. It is beautiful.
Scale matters
Decay rates span an enormous range. The $Z$ boson decays via the weak interaction with a rate of about 2.5 GeV, corresponding to a lifetime of roughly $10^{-25}$ seconds. It exists for approximately 100000000000000000000000000 of a second before it is gone. The neutron, by contrast, has a decay rate corresponding to about 880 seconds of lifetime — barely a flicker on cosmological scales, but an eternity for a subatomic particle.
The width of the resonance in the invariant mass distribution is numerically equal to the decay rate (in natural units where $\hbar = 1$). This is the decay width, $\Gamma$, and it is the observable that experiments actually measure. You cannot directly time a particle's life; you measure the spread in its reconstructed mass, and from that spread you read off the rate.
A final thought
When a particle decays, it does not "die" in any ordinary sense. The quantum numbers that defined it are redistributed into other particles. The information is preserved. The decay rate is simply the tempo at which the universe rearranges itself.