Zhenwei Li, Dandan Wei, Shi Jia, Hailiang Chen, Hongwei Ge, Zhuo Chen, Yangyang Zhang, Xuefei Chen, Zhanwen Han

The massive binary common envelope (CE) phase plays a pivotal role in the formation of close black hole/neutron star (BH/NS) binaries, yet significant uncertainties remain in our understanding of this process. In this study, we aim to constrain the massive binary CE phase by systematically reconstructing three observed BH X-ray binaries (BHXBs): GRO J1655-40, SAX J1819.3-2525, and 4U 1543-47. Through comprehensive binary evolution simulations and parametric supernova (SN) modeling, we establish lower limits for the CE efficiency parameters under different energy considerations within the standard energy formalism. Specifically, we derive minimum values for three cases: $α_{\rm 0.5U}$ and $α_{\rm U}$ representing CE efficiencies with half and all of the internal energy contributing to the envelope ejection, respectively, and $α_{\rm H}$ accounting for the envelope's enthalpy. Our analysis reveals that the self-consistent formation of these three BHXBs requires CE efficiency parameters satisfying: $α_{\rm 0.5U}\gtrsim 6.7$, $α_{\rm U}\gtrsim 4.2$ and $α_{\rm H}\gtrsim 1.7$. Notably, we find no viable solutions with CE efficiency values below unity, even when considering the most extreme scenarios in which the envelope binding energy is significantly reduced through enthalpy inclusion. {Our results strongly imply that either additional energy sources are required, or the formalism itself must be revised.} Furthermore, we quantitatively assess the impact of BH natal kicks on our results. A key finding is that 4U 1543-47's formation requires substantial natal kicks ($\gtrsim 50 \;\rm km/s$), as lower kick velocities are incompatible with isolated binary evolution.