A common situation in physics is that in investigating phenomena on a certain distance scale, one sees no hint of those phenomena that happen at much shorter distance scales. In a classical situation this observation seems evident. For example, one can treat fluid dynamics without any knowledge of the atomic physics that generates the actual properties of the fluids. However, in a quantum field theory this decoupling of short-distance phenomena from long-distance phenomena is not self-evident at all.

Consider an e+ — e- annihilation experiment at a center-of-mass energy well below 10GeV, the threshold for making hadrons containing the b-quark. There is, for practical (or experimental) purposes, no trace of the existence of this quark. However, the quark is present in Feynman graphs as a virtual particle, and can have an apparently significant effect on cross-sections. Our task in this chapter is therefore to prove what is known as the decoupling theorem. This states that a Feynman graph containing a propagator for a field whose mass is much greater than the external momenta of the graph is in fact suppressed by a power of the heavy mass. The physics at low energy is described by an effective low-energy theory that is obtained by deleting all heavy fields from the original theory.