The incidence of X-ray AGN and non-AGN galaxies in the far-infrared: insights into host galaxy properties and AGN obscuration

We investigate the far-infrared (far-IR) incidence of X-ray-selected active galactic nuclei (AGN) and non-AGN galaxies as a function of stellar mass (M$_*$), star formation rate (SFR), and specific black hole accretion rate ($\lambda_{\text{sBHAR}}$), using data from five extragalactic fields (COSMOS, XMM-LSS, Stripe82, ELAIS-S1, and CDFS-SWIRE). We construct spectral energy distributions (SEDs) from optical-to-far-IR photometry to derive host galaxy properties and assess AGN obscuration. X-ray absorption is quantified using the 4XMM-DR11s catalog. Our final sample includes 172,697 non-AGN galaxies (53% Herschel-detected) and 2,417 X-ray AGN (73% Herschel-detected), with $10 < log\,[M_*/M_\sun] < 12$ and $0 < z < 2$. X-ray AGN exhibit a relatively flat far-IR detection rate across stellar mass and specific SFR ($sSFR = SFR / M_*$), unlike non-AGN galaxies, where detection correlates strongly with SFR. Among AGN, far-IR detection declines with increasing $\lambda_{\text{sBHAR}}$, despite rising SFR. Our results suggest X-ray AGN are preferentially found in gas-rich environments, where star formation and black hole accretion coexist. Far-IR incidence remains high across all sSFR bins, supporting a scenario in which AGN feedback regulates, rather than abruptly quenches, star formation. Comparing AGN and non-AGN SFRs without separating Herschel-detected from non-detected sources introduces biases. Obscured AGN show ~10% higher far-IR detection rates than unobscured ones, yet at similar $\lambda_{\text{sBHAR}}$, unobscured AGN tend to have higher SFR. This may suggest obscured AGN inhabit dustier systems with moderate star formation contributing to the far-IR. Our findings support a regulatory AGN feedback mode operating over extended timescales in gas-rich galaxies.