So far, we have considered the calculation of scattering amplitudes to
lowest order in .
By putting together the vertices and propagators that we have found in
these lowest-order calculations, we can find expressions for
higher-order perturbation theory corrections and their corresponding
Feynman diagrams.
One such diagram is
By applying four-momentum conservation at each vertex one finds there
is a free ``loop'' momentum, .
It is found that the correct procedure is to integrate over the loop
momentum
.
Unfortunately, most of the integrals that arise in the calculation of
such loop diagrams are divergent and we have the paradoxical result
that order
corrections to the lowest-order scattering
amplitude are infinite.
The precise programme for manipulating and ``taming'' these infinities is known as ``renormalization'' of the theory. ``Renomalization'' because all the infinities are miraculously swept up into formal expressions for the quantities like physical mass and charge of the particle.
We note however the useful distinction between ``tree'' diagrams - one with no loops - and ``loop'' diagrams - ones with one or more loops. For theories in which the physical coupling constant is small, such as, electromagnetic interactions, the tree diagram calculations are usually a good approximation to compare with experiment. Moreover, at the tree level, all the subtleties of renormailization never enter and so for many purposes it is sufficient to know how to calculate just these tree diagrams.