Common Envelope Mechanisms: Constraints from the X-ray Luminosity Function of High Mass X-ray Binaries

We use the measured X-ray luminosity function (XLF) of high-mass X-ray binaries (HMXBs) in nearby star-forming galaxies to constrain the common envelope (CE) mechanisms, which play a key role in governing the binary evolution. We find that the XLF can be reproduced quite closely under both CE mechanisms usually adopted, i.e., the $\alpha_{\rm CE}$ formalism and the $\gamma$ algorithm, with a reasonable range of parameters considered. Provided that the parameter combination is the same, the $\gamma$ algorithm is likely to produce more HMXBs than the $\alpha_{\rm CE}$ formalism, by a factor of up to $\sim$ 10. In the framework of the $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ is required to fit the observed XLF, though it does not significantly affect the global number of the HMXB populations. We present the detailed components of the HMXB populations under the $\gamma$ algorithm and compare them with those in Zuo et al. and observations. We suggest the distinct observational properties, as well as period distributions of HMXBs, may provide further clues to discriminate between these two types of CE mechanisms.