Deciphering the evolution of the Milky Way discs: Gaia APOGEE Kepler giant stars and the Besan\c{c}on Galaxy Model

We investigate the properties of the double sequences of the Milky Way discs visible in the [$\alpha$/Fe] vs [Fe/H] diagram. In the framework of Galactic formation and evolution, we discuss the complex relationships between age, metallicity, [$\alpha$/Fe], and the velocity components. We study stars with measured chemical, seismic and astrometric properties from the APOGEE survey, the Kepler and Gaia satellites, respectively. We separate the [$\alpha$/Fe]-[Fe/H] diagram into 3 stellar populations: the thin disc, the high-$\alpha$ metal-poor thick disc and the high-$\alpha$ metal-rich thick disc and characterise each of these in the age-chemo-kinematics parameter space. We compare results obtained from different APOGEE data releases and using two recent age determinations. We use the Besancon Galaxy model (BGM) to highlight selection biases and mechanisms not included in the model. The thin disc exhibits a flat age-metallicity relation while [$\alpha$/Fe] increases with stellar age. We confirm no correlation between radial and vertical velocities with [Fe/H], [$\alpha$/Fe] and age for each stellar population. Considering both samples, V$_\varphi$ decreases with age for the thin disc, while it increases with age for the h$\alpha$mp thick disc. Although the age distribution of the h$\alpha$mr thick disc is very close to that of the h$\alpha$mp thick disc between 7 and 14 Gyr, its kinematics seems to follow that of the thin disc. This feature, not predicted by the hypotheses included in the BGM, suggests a different origin and history for this population. Finally, we show that there is a maximum dispersion of the vertical velocity, $\sigma_Z$, with age for the h$\alpha$mp thick disc around 8 Gyr. The comparisons with the BGM simulations suggest a more complex chemo-dynamical scheme to explain this feature, most likely including mergers and radial migration effects