This week we talked about decay and scattering – applications of relativistic kinematics. And a brief introduction on what information we obtain / derive from decays etc.
I shall demonstrate the simplest case of these: rest frame decay
A->B+C, if they are spin-less, we only have to worry about the linear momentum and energy – thus the final momentum of the two out going particles are deterministic. In
A's rest frame, we have 4-momentum conservation:
We also have the total 4-momentum invariant conserved as a whole:
Another fun derivation is the Compton scattering, which will give a interesting relation between the wavelength change & scatter angel, the outline of the derivation goes like this:
First write down before and after 4 momentums for photon and electron:
From hyperphysics Then, you compare total 4-momentum before and after term by term, from:
you can relate to and then go to the component, use the fact:
to re-express in terms of and . Finally, use the 0-th component to obtain:
For a particle, in its rest-frame, we say that the decay rate is , for a particle can have multiple modes, under the hood: . Since we assume particles don't have 'memory' – so to speak, the decay probability between is constant. Obviously, we get a exponential decay law out of this just like radio active decay we're familiar with. We can associate a mean life time for each mode, and because the way math works:
We can also define a relative ration to indicate the branching probability of a particular mode over all modes:
An interesting trick comes in as we always want to express things in unit of energy. Notice . So we can define a decay width with unit of energy:
Of course, branching ratio can also be expressed using this width:
Later we will see that this total decay width, for particles with tiny lifetime, corresponds to Full-Width-Half-Maximum in their invariant mass distribution. This is a consequence of the said distribution follows Breit-Wigner distribution.