Improving reddening estimates for RR Lyrae stars in the Gaia bands: a machine learning approach to the PAC(Z) relation

RR Lyrae stars are essential tracers of old stellar populations and distance indicators across the Milky Way and nearby galaxies. However, their use as standard candles is limited by uncertainties in extinction, especially in the Gaia bands. In Gaia DR3, absorption values (AG) for fundamental-mode RR Lyrae (RRab) were based on an empirical period-amplitude-color (PAC) relation calibrated on a small sample and on passband transformations. We aim to recalibrate extinction relations for RRab stars and establish, for the first time, analogous relations for first-overtone RR Lyrae (RRc) stars in the Gaia passbands. We used Gaia DR3 photometry and pulsation properties of an all-sky reference sample. Intrinsic colors (G - GRP)0 and (GBP - GRP)0 were derived through sequential feature selection combined with linear regression and bootstrap resampling, ensuring robust estimates of color excess and AG. We recalibrated PACZ relations for RRab stars and derived, for the first time, PAC relations for RRc stars in the Gaia bands. The new relations reproduce intrinsic colors with residual scatters of 0.03-0.04 mag and provide AG estimates across both pulsation types. Tests on RR Lyrae stars in Galactic globular clusters and dwarf galaxies show consistent AG values, with the PAC(Z) relations based on (GBP - GRP) being more stable than those using (G - GRP). The new relations provide reliable extinction estimates for both RRab and RRc stars, improving on the DR3 implementation and offering a valuable tool for Galactic studies in view of Gaia DR4.