Metallicity of Active Galactic Nuclei from ultraviolet and optical emission lines-II. Revisiting the $C43$ metallicity calibration and its implications

In this study, a new semi-empirical calibration is proposed between ultraviolet emission lines (\ion{C}{iii}]$\lambda1909$, \ion{C}{iv}$\lambda1549$, \ion{He}{ii}]$\lambda1640$) of type~2 AGNs and their metallicity ($Z$). This calibration is derived by comparing a large sample of 106 objects (data taken from the literature) located over a wide range of redshifts ($0 \: \lesssim \: z \: \lesssim \: 4.0$) with predictions from photoionization models that adopt a recent C/O-O/H relation derived via estimates using the $T_{\rm e}$ method, which is considered the most reliable method. We found that the new calibration produces $Z$ values in agreement (within an uncertainty of $\pm 0.1$ dex) with those from other calibrations and from estimates via the $T_{\rm e}$-method. We find also that AGN metallicities are already high at early epochs, with no evidence for monotonic evolution across the redshift range $0 \: \lesssim \: z \: \lesssim \: 12$. Notably, the highest metallicities in our sample, reaching up to $\rm 4\: Z_{\odot}$, are found in objects at $2 \lesssim z \lesssim 3$. This redshift range coincides with the peak of the cosmic star formation rate history, suggesting a strong connection between the major epoch of star formation, black hole growth, and rapid metal enrichment in the host galaxies of AGNs. Furthermore, our analysis reveals no significant correlation between AGN metallicity and radio properties (radio spectral index or radio luminosity) or host galaxy stellar mass. The lack of a clear mass-metallicity relation, consistent with findings for local AGNs, suggests that the chemical evolution of the nuclear gas is decoupled from the global properties of the host galaxy.