The $\sigma_{\rm H\beta}$-based dimensionless accretion rate and its connection with the corona for AGN

With respect to the $\rm H\beta$ full width at half-maximum ($\rm FWHM_{H\beta}$), the broad $\rm H\beta$ line dispersion ($\sigma_{\rm H\beta}$) was preferred as a velocity tracer to calculate the single-epoch supermassive black hole mass ($M_{\rm BH}$) suggested by \cite{Yu2020b}. For a compiled sample of 311 broad-line active galactic nuclei (AGN) with measured hard X-ray photon index ($z<0.7$), $\sigma_{\rm H\beta}$ and the optical Fe II relative strength ($R_{\rm Fe}$) are measured from their optical spectra, which are used to calculate $\sigma_{\rm H\beta}$-based virial $M_{\rm BH}$ and dimensionless accretion rate ($\dot{\mathscr{M}}$). With respect to $\rm FWHM_{\rm H\beta}$, it is found that the mean value of $\sigma_{\rm H\beta}$-based $M_{\rm BH}$ is on average larger by 0.26 dex, and the mean value of $\sigma_{\rm H\beta}$-based $\dot{\mathscr{M}}$ is on average smaller by 0.51 dex. It is found that there exists a non-linear relationship between the Eddington ratio ($L_{\rm Bol}/L_{\rm Edd}$) and $\dot{\mathscr{M}}$, i.e., $L_{\rm Bol}/L_{\rm Edd} \propto \dot{\mathscr{M}}^{0.56\pm 0.01}$. This non-linear relationship comes from the accretion efficiency $\eta$, which is smaller for AGN with higher $\dot{\mathscr{M}}$. We find a strong bivariate correlation of the fraction of energy released in the corona $F_{\rm X}$ with $\dot{\mathscr{M}}$ and \mbh, $F_{\rm X} \propto \dot{\mathscr{M}}^{-0.57\pm 0.05} M_{\rm BH}^{-0.54\pm 0.06}$. The flat slope of $-0.57\pm 0.05$ favours the shear stress tensor of the accretion disk being proportional to the geometric mean of gas pressure and total pressure. We find a strong bivariate relation of $\Gamma$ with $\dot{\mathscr{M}}$ and $F_{\rm X}$, $\Gamma \propto \dot{\mathscr{M}}^{-0.21\pm 0.02}F_{\rm X}^{0.02\pm 0.04}$. The hard X-ray spectrum becomes softer with increasing of $F_{\rm X}$, although the scatter is large.