A Scaling Relation of LRDs between Broad H$\alpha$ and Bolometric Luminosities: Enhanced Broad H$\alpha$ Emission Relative to Low-$z$ Type 1 AGN

We investigate the demography of little red dots (LRDs) using 37 objects at $z\sim3$-$7$ with JWST/NIRSpec PRISM and grating spectra compiled from various JWST programs. We focus on spectroscopic quantities of the broad H$\alpha$ luminosity $L_\mathrm{H\alpha,broad}$ (and the broad H$\beta$ luminosity $L_\mathrm{H\beta,broad}$ where available) and the bolometric luminosity $L_\mathrm{bol}$ represented by modified blackbody emission, avoiding quantities contaminated by host-galaxy emission (e.g., total H$\alpha$ luminosity). We identifiy a tight scaling relation between $L_\mathrm{H\alpha,broad}$ and $L_\mathrm{bol}$, supporting the interpretation that these emissions are primarily powered by the central engine. Interestingly, the $L_\mathrm{H\alpha,broad}$-$L_\mathrm{bol}$ scaling relation of LRDs is enhanced by a factor of $\sim40$ in $L_\mathrm{H\alpha,broad}$ relative to that of low-$z$ Type 1 AGN. A similar trend is found in the $L_\mathrm{H\beta,broad}$-$L_\mathrm{bol}$ relation, although the enhancement in $L_\mathrm{H\beta,broad}$ is smaller, only by a factor of $\sim10$. We explore the physical origin of these enhancements and find that \textsc{Cloudy} photoionization modeling within the classic locally optimally-emitting cloud (LOC) framework can explain them through an increase in the covering factor from $\sim20$\% (Type 1 AGN) to $\sim100$\% (LRDs), together with an increase in the hydrogen column density from $N_\mathrm{H}\sim10^{23}\,\mathrm{cm}^{-2}$ to $\gtrsim10^{24}\,\mathrm{cm}^{-2}$, with a preferred gas density of $\sim10^{10}\,\mathrm{cm}^{-3}$, successfully reproducing the modified blackbody emission. Such a nearly unity covering factor without requiring a gas density increase may result from a significant increase in the BLR filling factor or size, corresponding to a ``stuffed BLR" or ``giant BLR," respectively.