Is radio jet power linearly proportional to the product of central black hole mass and Eddington ratio in AGN?

A model for the relation between radio jet power and the product of central black hole (BH) mass and Eddington ratio of AGN is proposed, and the model is examined with data from the literature. We find that radio jet power positively correlates but not linearly with the product of BH mass ($m$ in solar mass) and Eddington ratio ($\lambda$), and the power law indices ($\mu$) are significantly less than unity for relatively low accretion ($\lambda<0.1$) AGN, $P_{j}\propto (\lambda m)^{\mu}$, in the radio galaxies and the Seyfert galaxies. This leads to a negative correlation between radio loudness and $\lambda m$ for the low luminosity AGN, i.e. $R\propto (\lambda m)^{\rho}$ with $\rho=(7/6)\mu-1<0$, which may be attributed to a contribution of BH spin to total jet power assuming that the spin induced jet is gradually suppressed as the accretion rate increases. Whereas, for the high-z quasars which often show the slope $\mu\geq1$, a positive correlation between the radio loudness and disc luminosity is predicted. We discuss that the jet powers of the high-z FRII quasars are likely dominated by the accretion disc rather than by the BH spin.