In the study of radiative corrections a special role is played by those corrections due to ‘soft’ photons: photons whose energy and momentum are much less than the masses and energies characteristic of the process in question. Not only are these corrections often so large that they must be summed to all orders of perturbation theory; they are so simple that this summation is not difficult. The contribution of photons of infinitely long wavelength takes the form of divergent integrals, but as we shall see these ‘infrared divergences’ all cancel.

In most of this chapter we will deal with photons interacting with charged particles of arbitrary type and spin, including particles like atomic nuclei that have strong as well as electromagnetic interactions. But it is not difficult to adapt the calculations presented here to the infrared effects of other massless particles, such as the gluons of quantum chromodynamics. In Section 13.4 we shall explicitly consider very general theories of massless particles, and will show the cancellation of infrared divergences on general grounds.

After these generalities, we shall return to photons, and take up two topics of practical importance: the scattering of soft photons by charged particles with arbitrary non-electromagnetic interactions and arbitrary spin, and the treatment of heavy charged particles like atomic nuclei as a source of an external electromagnetic field.

Soft Photon Amplitudes

In this section we shall derive a universal formula that gives the amplitude for emission of any number of very-low-energy photons in a process α → β involving any number of higher-energy charged particles of any types.