  For any hard QCD amplitude with massless partons, infrared (IR) singularities
arise from pinches in the complex planes of loop momenta, called pinch
surfaces. To organize and study their leading behaviors in the neighborhoods of
these surfaces, we can construct approximation operators for collinear and soft
singularities. A BPHZ-like forest formula can be developed to subtract them
systematically. In this paper, we utilize the position-space analysis of
Erdogan and Sterman for Green functions, and develop the formalism for momentum
space. A related analysis has been carried out by Collins for the Sudakov form
factors, and is generalized here to any wide-angle kinematics with an arbitrary
number of external momenta. We will first see that the approximations yield
much richer IR structures than those of an original amplitude, then construct
the forest formula and prove that all the singularities appearing in its
subtraction terms cancel pairwise. With the help of the forest formula, the
full amplitude can also be reorganized into a factorized expression, which
helps to generalize the Sudakov form factor result to arbitrary numbers of
external momenta. All our analysis will be on the amplitude level.
