Challenging the AGN scenario for JWST/NIRSpec broad H$\alpha$ emitters/Little Red Dots in light of non-detection of NIRCam photometric variability and X-ray

JWST has uncovered a substantial population of high-$z$ ($z \gtrsim 4$) galaxies exhibiting broad H$\alpha$ emission line with a Full Width at Half Maximum exceeding $1,000$~km~s$^{-1}$. This population includes a subset known as 'Little Red Dots', characterized by their compact morphology and extremely red rest-frame optical colors. If all of these broad H$\alpha$ emitters were attributed to type $1-1.9$ Active Galactic Nuclei (AGNs), it would imply a significantly higher number density of low-luminosity AGNs than extrapolated from that of more luminous AGNs. Here, we have examined the rest-frame ultraviolet (UV)-optical flux variability of five JWST broad H$\alpha$ emitters using multi-epoch, multi-band JWST/NIRCam imaging data. The rest-frame temporal sampling interval of the NIRCam data ($\sim 400-500$~days$/(1+z)$) is comparable to typical variability timescales of AGNs with black hole (BH) masses of $M_{\text{BH}} \sim 10^{7}~M_{\odot}$; thus, the flux variations should be detectable if AGNs were present. However, no measurable flux variation over the rest-frame wavelength range of $\lambda_{\text{rest}} \sim 1,500-9,000$Å has been detected, placing stringent upper limits on the variability amplitudes. This result, combined with the X-ray faintness confirmed by the ultra-deep {\it Chandra} data, indicates that, under the AGN scenario, we need to postulate peculiar Compton-thick broad-line AGNs with either ($a$) an intrinsically non-variable AGN disk continuum, ($b$) a host galaxy-dominated continuum, or ($c$) scattering-dominated AGN emission. Alternatively, ($d$) they could be non-AGNs where the broad-line emission originates from unusually fast and dense/low-metallicity star-formation-driven outflows or inelastic Raman scattering of stellar UV continua by neutral hydrogen atoms.