  Fracture functions are parton distributions of an initial hadron in the
presence of an almost collinear particle observed in the final state. They are
important ingredients in QCD factorization for processes where a particle is
produced diffractively. There are different fracture functions for a process in
different kinematic regions. We take the production of a lepton pair combined
with a diffractively produced particle in hadron collisions to discuss this.
Those fracture functions can be factorized further if there are large energy
scales involved. They can be factorized with twist-2 parton distribution
functions and fragmentation functions. We perform one-loop calculations to
illustrate the factorization in the case with the diffractively produced
particle as a real photon. Evolution equations of different fracture functions
are derived from our explicit calculations. They agree with expectations. These
equations can be used for re-summations of large log terms in perturbative
expansions.
