Higher-Order Calculations: Tree and Loops

So far, we have considered the calculation of scattering amplitudes to lowest order in $\alpha$. 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

Figure 7.14: Two-photon exchange contribution to $2\rightarrow 2$ scattering of order $\alpha^2$ in amplitude.

By applying four-momentum conservation at each vertex one finds there is a free ``loop'' momentum, $k$. It is found that the correct procedure is to integrate over the loop momentum $\int\frac{d^4k}{(2\pi)^4}$. Unfortunately, most of the integrals that arise in the calculation of such loop diagrams are divergent and we have the paradoxical result that order $\alpha^2$ 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.