A great deal of particle phenomenology, including both the properties of the observed particles and their reactions in many accelerators, deals with energy scales that are very small in comparison with the mass of the weak vector bosons, MW or MZ. A technique that has been used to good effect at various points in the previous chapters is the expansion of low-energy scattering amplitudes in inverse powers of the W- or Z-boson masses. This expansion greatly simplified the corresponding calculations and was justified in each case by the fact that the typical energies involved in the amplitudes under consideration were much smaller than MW and MZ.

Concrete examples where this type of expansion is justified are given by the weak decays of a light meson such as the muon, as was computed in Chapter 5, since the energy scales involved are much smaller than the mass of the virtual W boson that mediates these decays. A similar simplification is justified in Chapter 6 in the amplitudes for electron–positron annihilation at energies that are low compared to the Z-boson mass.

All of these examples furnish special cases of the general technique of low-energy expansions. This technique appears ubiquitously throughout physics because many physical systems have the property that they involve two (or more) degrees of freedom that each have very different masses.