We investigate the DC conductivity of two-dimensional tilted Dirac systems with short- and long-range impurity scattering, across the subcritical (Type I), critical, and overcritical (Type II) tilt regimes. Using the Kubo formalism, we uncover transport signatures that depend sensitively on both the tilt regime and the impurity type. For short-range impurities, the conductivity as a function of the tilt parameter $t$ exhibits a minimum at the critical point $t=1$, increasing away from it, with pronounced anisotropy emerging in the Type-II regime. Long-range impurities instead produce an enhanced conductivity near the critical point. The energy dependence of the conductivity also differs sharply between the two cases: it is negligible to moderate for short-range impurities, but strong for long-range impurities, scaling nearly quadratically for $t<1$ and linearly for $t>1$. We further show that vertex corrections are negligible at the Lifshitz transition ($t=1$). Finally, we interpret these results within a unified geometric framework, establishing the tilt parameter as a powerful knob for engineering the transport properties of disordered Dirac materials.

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• Dirac fermions
• Transport phenomena
• Two-dimensional electron system
• Diagrammatic methods