In this chapter, I treat the complications caused by renormalizability of the underlying field theory when one analyzes the asymptotics of processes like DIS. There are four inter-related issues:

• The leading regions include hard-scattering subgraphs that can be of arbitrarily high order in the coupling.

• There are logarithmic unsuppressed contributions from momenta that interpolate between the different regions for a graph.

• The definitions of the parton densities are modified to remove their UV divergences. This we do by renormalization.

• The parton densities acquire a scale argument µ, the dependence on which is governed by renormalization-group (RG) equations, the famous Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations. In applications, we set µ of order Q, the large scale in the hard scattering.

I will give a derivation of factorization that in the absence of gauge fields is complete and satisfactory, and is also reasonably elementary. In QCD, the same factorization theorem is also valid for simple processes, like DIS, but its derivation needs enhancement, to be given in later chapters.

Factorization: overall view

To motivate the factorization idea, we still use the ideas about the space-time structure of DIS that motivated the parton model. As illustrated in the spatial diagram of Fig. 8.1, an electron undergoes a wide-angle hard scattering off a single parton in a high-energy target hadron. In the center-of-mass frame, the target is time-dilated and Lorentz contracted.