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 (AGNs) and galaxies that do not host an AGN (non-AGNs) as a function of the stellar mass (M_*), star formation rate (SFR), and specific black hole accretion rate (λ_ sBHAR), using data from five well-characterized extragalactic fields (COSMOS, XMM-LSS, Stripe82, ELAIS-S1, and CDFS-SWIRE). We constructed spectral energy distributions (SEDs) using optical-to-far-IR photometry to derive host galaxy properties and assess AGN obscuration, while X-ray absorption was quantified using the 4XMM-DR11s catalogue. Our final sample comprises 172, 697 non-AGN galaxies (53% Herschel-detected) and 2, 417 X-ray AGNs (73% Herschel-detected), with ̊m 10 < log M_*/M_ sun < 12 and ̊m 0 < z < 2. We find that X-ray AGNs exhibit a relatively flat far-IR detection rate across stellar mass and specific SFR (̊m sSFR = SFR / M_*), unlike non-AGN galaxies, where detection correlates strongly with star formation. Far-IR detection among AGNs decreases with increasing λ_ sBHAR even as their SFR rises. Our results suggest that X-ray AGNs are preferentially found in gas-rich environments, where both star formation and black hole accretion are fuelled by the presence of cold gas. The far-IR incidence of X-ray AGNs remains high across all sSFR bins, indicating that these AGNs can coexist with ongoing star formation for extended periods, in line with a scenario in which AGNs feedback regulates rather than abruptly quenches star formation. We also find that comparing AGNs and non-AGN SFRs without separating Herschel-detected from non-detected sources introduces biases. Obscured AGNs show 10% higher far-IR detection rates than unobscured ones, yet at similar λ_ sBHAR unobscured AGNs tend to have higher SFR. This may indicate that obscured AGNs reside in dustier environments where moderate star formation still contributes to far-IR emission. Our results support a scenario in which AGNs and star formation coexist in gas-rich galaxies, with AGNs feedback acting as a regulatory process over extended timescales and not necessarily quenching.