  The factorization theorems of quantum chromodynamics (QCD) apply equally well
to most simple quantum field theories that require renormalization but where
direct calculations are much more straightforward. Working with these simpler
theories is convenient for stress-testing the limits of the factorization
program and for examining general properties of the parton density functions
(pdfs) or other correlation functions that might be necessary for a factorized
description of a process. With this view in mind, we review the steps of
factorization in a real scalar Yukawa field theory for both deep inelastic
scattering (DIS) and semi-inclusive deep inelastic scattering (SIDIS) cross
sections. In the case of SIDIS, we illustrate how to separate the small
transverse momentum region, where transverse momentum dependent (TMD) pdfs are
needed, from a purely collinear large transverse momentum region, and we
examine the influence of subleading power corrections. We also review the steps
for formulating TMD factorization in transverse coordinate space, and we study
the effect of transforming to the well-known $b_*$-scheme. Within the Yukawa
theory, we investigate the consequences of switching to a generalized parton
model (GPM) approach, and compare with a fully factorized approach. Our results
highlight the need to address similar or analogous issues in QCD.
