Basics

S-matrix and cross sections (simplified)

  1. S-matrix and transition amplitudes
  2. Definition of cross section
  3. Connection to S-matrix

    TRANSITION PROBABILITY, ONCE AGAIN

    In the previous disucssion, it was found that the transition probability per unit volume of space-time is given by
    Since in the notation used here the states are not normalised to one but to their energies, see above, this normalisation has to be taken into account, leading to
    for the properly normalised transition probability. In scattering experiments in particle physics, usually two particles (target and projectile) interact, producing many particles. According to the definitions above, the cross section for such a 2→ n scattering reads
    The integration over the final state particle states in the equation above just takes into account that a sum over different final states has to be performed. Of course, at this point certain constraints on this phase space may be applied, which usually depend on experimental conditions and on the process under consideration. Note here that there is no integration over the initial state particles, reflecting that they are usually well prepared, i.e. in a "clean state" (momentum eigenstate). Finally, the relative velocity of beam and target particle are cast into a Lorentz-invariant form (i.e. in a form that depends on Lorentz-scalars such as the scalar products of four momenta only). To this end, note that
    leading to the

    FINAL EXPRESSION

    Obviously, at this point, the expression for the cross section above is in units of energy/momenta rather than in units of length. The connection of energy and length, however, is given by
    indicating that cross sections have units of inverse squares of energy.

  4. Cross section, luminosity, and event rate
  5. Width and life-time
  6. Problems