Basics

S-matrix and cross sections (simplified)

1. S-matrix and transition amplitudes
   
2. Definition of cross section
   
3. Connection to S-matrix
   
4. Cross section, luminosity, and event rate
   
5. Width and life-time
   

   MASTER FORMULA FOR CALCULATING THE WIDTH

   In a way similar to the cross section, the decay width Γ of unstable particles can be calculated. Again, a transition amplitude for the decay is constructed, with the usual connection to a corresponding S-matrix element. Then, for a decay 1→ n of one particle into n particles, the width reads Again, in order to define the flux, a Lorentz-invariant quantity is needed. For an one-particle state, this is the mass, and in fact the flux in such a case merely is twice the mass (remember the normalisation of states !). Therefore,
   
   

   CONNECTION TO LIFETIME

   In analogy to lineshapes related to the decays of excited states into the ground state in atomic transitions, the masses of unstable particles are smeared out. In both cases this is charaterised by a width which is a measure of the lifetime of the respective state. In fact, in atomic physics, this connection is made through the energy-time uncertainty relation, In the case of unstable particles, the lifetime τ of the particle is linked to the width by
   
   
6. Problems