Central assertions in setting up the parton model for DIS (Sec. 2.4) were that hard scattering occurs off a single parton constituent of the target, and that the hard scattering is just the Born approximation for electron-quark scattering. In fact, both assertions fail if taken literally. So in this chapter I show how to derive correct statements about the dominant configurations in DIS and the many other cases of interest. I will interleave a general treatment with a detailed discussion of specific examples.

Key insights were found by Sterman (1978) and Libby and Sterman (1978b), who systematized a correspondence between divergences in massless perturbative calculations and important configurations for high-energy processes. For any suitable process (like DIS) with an energy scale Q much larger than relevant particle masses, the main results are:

1. A one-to-one correspondence between mass divergences in massless perturbation theory and non-UV regions in loop-momentum space that give the large Q asymptote.

2. That mass divergences are at surfaces where the integral over loop momenta cannot be deformed away from singularities of propagators. These surfaces are called pinch singular surfaces (PSSs).

3. Simple and very general geometrical arguments in four-dimensional momentum space to locate the PSSs for a massless theory. The PSSs are in the typically higher-dimension space of all loop momenta.

4. Simple power-counting results for the strengths of the possible PSSs, and for the power dependence on Q of the contribution of the region associated with each PSS.

5. […]