Case Study on the γ-algorithm: Possible Tests from the Luminosity versus Displacement Correlation of High Mass X-Ray Binaries

Using the apparent correlation of luminosity (L (X)) versus displacement (R) of high mass X-ray binaries (HMXBs), we aim to constrain the common envelope (CE) mechanism, which is vital in the formation and evolution of compact binaries. We find that under the assumption of the gamma-algorithm, the apparent correlation can also be reconstructed generally within a reasonable range of key parameters adopted, though the population of HMXBs is distinct with that in the canonical alpha (CE)-formalism. We compare the spatial distribution of HMXBs under the two CE mechanisms, and suggest the difference in L (X) versus R distribution may provide an additional clue for the study of the CE phase and to discriminate between CE models.