Abstract-Material gain of GeSn/SiGeSn quantum wells, which can be grown on Si substrate by using a buffer layer, is analyzed based on microscopic many-body theory (MBT) for mid-infrared light sources based on Si photonics. MBT can take into account a gain spectrum broadening associated with scattering phenomena, such as Coulomb scattering, based on quantum field theory, and does not need any artificial fitting parameters, such as a relaxation time, used in conventional analysis. Not only Γ-but also carrier distributions in L-points are considered for the gain analysis. By using MBT, the quantum well structures maximizing the material gain and the differential gain at the threshold are investigated in terms of the well thickness, the strain, and the energy difference of quantum states between Γ-and L-points.Index Terms -Mid-infrared photonics, Si photonics, GeSn quantum wells, and many-body theory.