  We study a transverse momentum dependent (TMD) factorization framework for
the processes of di-jet and heavy meson pair production in
deep-inelastic-scattering in an electron-proton collider, considering the
measurement of the transverse momentum imbalance of the two hard probes in the
Breit frame. For the factorization theorem we employ soft-collinear and
boosted-heavy-quark effective field theories. The factorized cross-section for
both processes is sensitive to gluon unpolarized and linearly polarized TMD
distributions and requires the introduction of a new soft function. We
calculate the new soft function here at one loop, regulating rapidity
divergences with the $\delta$-regulator. In addition, using a factorization
consistency relation and a universality argument regarding the heavy-quark jet
function, we obtain the anomalous dimension of the new soft function at two
loops.
