  Familiar factorized descriptions of classic QCD processes such as
deeply-inelastic scattering (DIS) apply in the limit of very large hard scales,
much larger than nonperturbative mass scales and other nonperturbative physical
properties like intrinsic transverse momentum. Since many interesting DIS
studies occur at kinematic regions where the hard scale, $Q \sim$ 1-2 GeV, is
not very much greater than the hadron masses involved, and the Bjorken scaling
variable $x_{bj}$ is large, $x_{bj} \gtrsim 0.5$, it is important to examine
the boundaries of the most basic factorization assumptions and assess whether
improved starting points are needed. Using an idealized field-theoretic model
that contains most of the essential elements that a factorization derivation
must confront, we retrace the steps of factorization approximations and compare
with calculations that keep all kinematics exact. We examine the relative
importance of such quantities as the target mass, light quark masses, and
intrinsic parton transverse momentum, and argue that a careful accounting of
parton virtuality is essential for treating power corrections to collinear
factorization. We use our observations to motivate searches for new or enhanced
factorization theorems specifically designed to deal with moderately low-$Q$
and large-$x_{bj}$ physics.
