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.
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