  We first revisit impact-parameter dependent collisions of ultra-relativistic
particles in quantum field theory. Two conditions sufficient for defining an
impact-parameter dependent cross section are given, which could be violated in
proton-proton collisions. By imposing these conditions, a general formula for
the impact-parameter dependent cross section is derived. Then, using
soft-collinear effective theory, we derive a factorization formula for the
impact-parameter dependent cross section for inclusive hard processes with only
colorless final-state products in hadron and nuclear collisions. It entails
defining thickness beam functions, which are Fourier transforms of transverse
phase-space parton distribution functions. By modelling non-perturbative modes
in thickness beam functions of large nuclei in heavy-ion collisions, the
factorization formula confirms the cross section in the Glauber model for hard
processes. Besides, the factorization formula is verified up to one loop in
perturbative QCD for the inclusive Drell-Yan process in quark-antiquark
collisions at a finite impact parameter.
