  We investigate the path integral representation of the scalar propagator in a
background gluon field, extending beyond the eikonal approximation by
considering all gauge field components and incorporating its $x^-$ dependence.
Utilizing the worldline formalism, we integrate the Schwinger proper time to
express the scalar propagator in light-cone coordinates, facilitating a direct
comparison with known results in the literature. The derived propagator
captures the change of longitudinal momentum of the projectile within the
medium. In the high-energy limit, our result simplifies to the effective gluon
propagator employed in the BDMPS-Z formalism. Hence, we propose that our
outcome serves as a foundational point for investigating corrections to the
BDMPS-Z spectrum arising from the longitudinal momentum transfer of the
radiated gluon with the medium, as well as for studying collisional energy loss
phenomena. Lastly, by employing an expansion around the classical saddle point
solution, we systematically derive an eikonal expansion in inverse powers of
the boost parameter, encompassing corrections related to longitudinal momentum
transfer and interactions of the projectile with the transverse component of
the field.
