Two-dimensional (2D) semi-Dirac bands hybridize Schrödinger and Dirac fermions, exhibiting quadratic dispersion along the $k_x$ direction and linear dispersion along the $k_y$ direction. Tilting along either direction enhances band anisotropy and can drive the corresponding Lifshitz transition when the tilt is tuned appropriately. To characterize the two fundamentally distinct Lifshitz transitions in 2D semi-Dirac bands, we propose for the first time a chemical-potential-dependent tilt parameter ($t_x=v_{tx}/\sqrt{4a\mu }$) for tilting solely along the quadratic dispersion direction ($k_x$ direction), in stark contrast to the conventional tilt parameter ($t_y=v_{ty}/v_F$) for tilting purely along the linear dispersion direction ($k_y$ direction). Within linear response theory, we rigorously prove the particle-hole symmetry of optical conductivities in 2D tilted semi-Dirac bands, and analytically compute, at zero temperature, the corresponding longitudinal optical conductivities (LOCs) and the joint density of states. The interband LOCs exhibit characteristic piecewise line shapes, critical frequencies, and high-frequency backgrounds for different tilting directions, especially in the type-II phase. Remarkably, we find a robust fixed point at $\omega =2\mu$ for both tilting directions, where the interband LOC and the joint density of states become exactly half of their untilted values. However, the range of tilt parameter for this fixed point differs markedly: $0<t_x\le 1$ for tilting along the $k_x$ direction versus $0<t_y\le 2$ for tilting along the $k_y$ direction. Furthermore, for tilting along the $k_x$ direction, the Drude weight is always cutoff independent (for $t_x>0$); whereas for tilting along the $k_y$ direction, the Drude weight is cutoff-free for $0\le t_y<1$ but cutoff dependent for $t_y\ge 1$. All of the piecewise line shapes, high-frequency asymptotic background, and range of tilt parameter for the fixed point in the interband LOCs serve as distinctive fingerprints for identifying the tilting direction and Lifshitz transitions in future optical experiments on semi-Dirac materials.

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• Optical conductivity
• Dirac semimetal
• Two-dimensional electron system
• Linear response theory