  In the target fragmentation region of Semi-Inclusive Deep Inelastic
Scattering, the diffractively produced hadron has small transverse momentum. If
it is at order of $\Lambda_{QCD}$, it prevents to make predictions with the
standard collinear factorization. However, in this case, differential
cross-sections can be predicted by the factorization with fracture functions,
diffractive parton distributions. If the transverse momentum is much larger
than $\Lambda_{QCD}$ but much smaller than $Q$ which is the virtuality of the
virtual photon, both factorizations apply. In this case, fracture functions can
be factorized with collinear parton distributions and fragmentation functions.
We study the factorization up to twist-3 level and obtain gauge invariant
results. They will be helpful for modeling fracture functions and useful for
resummation of large logarithm of the transverse momentum appearing in
collinear factorization.
