Abstract
In this work, we study the correlation between the photon index (${\Gamma}$) of the X-ray spectrum and the 2-10 keV X-ray luminosity ($L_X$) for black hole X-ray binaries (BHBs). The BHB sample is mainly from the quiescent, hard and intermediate states, with values of $L_X$ ranging from ${\sim}10^{30.5}$ to $10^{37.5}$ erg $s^{-1}$. We find that the photon index ${\Gamma}$ is positively or negatively correlated with the X-ray luminosity $L_X$, for $L_X$ above or below a critical value, ${\sim}10^{36.5}$ erg $s^{-1}$. This result is consistent with previous works. Moreover, when $L_X{\leq}{\sim}10^{33}$ erg $s^{-1}$, we found that the photon index is roughly independent of the X-ray luminosity. We interpret the above correlations in the framework of a coupled hot accretion flow - jet model. Besides, we also find that in the moderate-luminosity region, different sources may have different anti-correlation slopes, and we argue this diversity is caused by the different value of ${\delta}$, which describes the fraction of turbulent dissipation that directly heats electrons.