PX156 - J1 - decays

fermi's golden rule

the transition rate from one state to another, ie: number of transitions per unit time, is proportional to the physics as drawn in the feynmann diagrams:

P (i \to f) \propto (a m p l i t u d e)^{2} (p h a s e s p a c e)

measured experiment $\leftrightarrow$ calculate theory
decays: $1 \to n$
collisions: $2 \to n$

decays

every particle decays, except the ones that have no lighter particles to decay to: electron, proton(?), and neutrino(?)
usually looked at in a particle's rest frames
otherwise, special relativity needs to be used
controlled by a decay law
given an ensemble of $N$ particles at time, $t = 0$ , the number of particles left at time, $t$ is:

N (t) = N (0) \exp (- \frac{t}{τ})

where, the parameter, $τ \equiv$ average lifetime of a particle

in particle physics, the decay rate, $Γ = \frac{1}{τ}$ , is used:

N (t) = N (0) \exp (- Γ t)

note: $Γ$ is a transition rate $\to L H S$ of fermi's golden rule

derivation

\begin{aligned} N (t + Δ t) & = N (t) - N (t) Γ Δ t \\ \frac{N (t + Δ t) - N (t)}{Δ t} & = - N (t) Γ \\ lim_{t \to 0} : \frac{d N (t)}{d t} & = - N (t) Γ \\ \ln (\frac{N (t)}{N (0)}) & = - Γ t \\ ∴ N (t) & = N (0) e^{- Γ t} \end{aligned}