How do infrared divergences arise in QED? What is an example case of such a divergence and how do we usually deal with such divergences? Are they absorbed like ultraviolet divergences?
Infrared divergences can arise in various ways. Usually we refer to two sources as most important:
- External particles with vanishingly small momenta (so called "soft particles"). These are particles, usually photons, that are not observable by detectors in the sense that their momenta are so small that instruments are not sensitive to them. It turns out that in the limit of vanishingly small momenta an IR divergence can arise.
- IR divergences in virtual loop integrals or, in 2 dimensions, built into the particle propagator itself. The IR divergences can arise due to the integral over internal loop momenta not determined by momentum conservation, that begins at one-loop order; part of this region contains soft internal particles. In 2 dimensional space it turns out that even the propagator of physical particles can be IR divergent.
Interestingly there is a great soft particle theorem due to Weinberg and discussed in Peskin and Schroeder Chap. 6. It has been shown that the IR divergences in virtual loop integrals can be summed exactly and cancel the IR divergences from soft particles! In this way these two main forms of IR divergences are removed from the game!
 S. Weinberg, Phys. Rev. 140, B516 (1965).