3 - Cross sections and lifetimes

Summary

Most of the applications of the standard model to experimental situations are concerned with processes in which almost free particles interact briefly and over short distances. These processes could be the collisions of various elementary particles within an accelerator (Chapter 6 and Chapter 9) or they could be the decay of an unstable elementary particle in flight (Chapter 4 and Chapter 5). Scattering (S-matrix) theory is the formalism that has been devised to study these systems.

This chapter presents a whirlwind review of the quantum theory of scattering. The purpose is to gather into one place all of the results that are required in order to use the Lagrangian of Chapter 2 to predict the outcomes of experiments. The first section sets up the notion of scattering states, which are meant to represent in a precise way the idea that the particles involved do not interact except for a short time interval. This is followed by a review of the calculation of scattering amplitudes using time-dependent perturbation theory.

In later chapters this formalism is finally used to compute the Feynman rules that describe the interactions contained within the standard model Lagrangian.

Readers in a hurry, or who find themselves bogged down in this section, should try to understand Section 3.2 and will need to learn the results at the end of Section 3.3, particularly Eq. (3.40) and Eq. (3.43).