a physical observable is written as a sum of products of short-distance coefficients with long-distance operator matrix elements. The short-distance coefficients may be calculated as a perturbative series in the strong coupling constant $\alpha_s$, evaluated at the short-distance scale. The operator matrix elements contain all of the sensitivity of the physical observable to low-momentum (infrared) processes. Because of this infrared (IR) sensitivity, the operator matrix elements are generally not amenable to a perturbative calculation and are usually determined by comparison of physical quantities with experimental values or through nonperturbative methods, such as lattice QCD. Some well-known examples of this approach are the light-cone expansion in deep-inelastic scattering, the factorization expressions for hard hadron-hadron cross sections, Heavy-Quark Effective Theory (HQET), which is useful in the study of heavy-light mesons, and Nonrelativistic Quantum Chromodynamics (NRQCD), which is useful in the study of heavy quarkonium.

The conventional wisdom is that the perturbation expansions for the short-distance coefficients in QCD factorization formulas are, at best, asymptotic series [@thooft]. One can see why this might be the case by examining the behavior of the perturbation series in the context of a simple model, which gives the leading large-$N_f$ behavior of the theory. This model has 