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. Problems
   
   1. Calculate the cross section for (classical) scattering of point-like particles off
      • a Yukawa potential of the form exp[-br] (b is a constant)
      • a Coulomb potential of the form a/r (a is a constant)
   2. Consider 2→ 2 scattering experiments with no other preferred axis than the beam axis. Convince yourself that the (classical) cross section can be written as a function of cos θ, where θ is the angle w.r.t. the beam axis.
   3. In the following, assume all particles to have an energy of their mass plus 10 GeV. How far will the following particles on average travel in the laboratory system, before they decay:
      • electrons,
      • neutral pions, π⁰,
      • charged pions, π^±,
      • muons, μ^-,
      • charged kaons, K^±,
      • neutral kaons, K_L and K_S,
      • taus, τ^-,
      • protons, p⁺,
      • neutrons,
      • W bosons, W^±,
      • and top quarks.
      Compare this with the typical length scale of a LHC experiment, roughly 10 meters.
      Hint: From the energies and the masses, calculate the velocities and the boost factor. Use this to obtain the decay time in the lab system from the proper lifetimes given through the measured widths/lifetimes. (On the PDG homepage, browse the summary tables to find the particles and their properties).