HADES RV programme with HARPS-N at TNG: XII. The abundance signature of M dwarf stars with planets

Context. Most of our current knowledge on planet formation is still based on the analysis of main sequence, solar-type stars. Conversely, detailed chemical studies of large samples of M dwarfs hosting planets are still missing. Aims. Correlations exist between the presence of different types of planets around FGK stars and metallicity, individual chemical abundance, and stellar mass. We aim to test whether or not these correlations still hold for the less-massive M dwarf stars. Methods to determine stellar abundances of M dwarfs from high-resolution optical spectra in a consistent way are still missing. The present work is a first attempt to fill this gap. Methods. We analyse a large sample of M dwarfs with and without known planetary companions in a coherent and homogeneous way. We develop for the first time a methodology to determine stellar abundances of elements other than iron for M dwarf stars from high-resolution optical spectra. Our methodology is based on the use of a principal component analysis and sparse Bayesian methods. We made use of a set of M dwarfs orbiting around an FGK primary with known abundances to train our methods. We applied our methods to derive stellar metalliticies and abundances of a large sample of M dwarfs observed within the framework of current radial-velocity surveys. We then used a sample of nearby FGK stars to cross-validate our technique by comparing the derived abundance trends in the M dwarf sample with those found on the FGK stars. Results. The metallicity distribution of the different subsamples reveals a correlation between the metallicities of M dwarfs and their probability of hosting giant planets. We also find a correlation between this latter probability and stellar mass. M dwarfs hosting low-mass planets do not seem to follow the so-called planet-metallicity correlation. We also find that the frequency of low-mass planets does not depend on the mass of the stellar host. These results appear to be in agreement with those of previous works. However, we note that for giant-planet hosts our metallicities predict a weaker planet-host metallicity correlation but a stronger mass-dependency than corresponding values derived from photometric results. We show for the first time that there seems to be no differences between M dwarfs with and without known planets in terms of their abundance distributions of elements different from iron. Conclusions. Our data show that low-mass stars with planets follow the same metallicity, mass, and abundance trends as their FGK counterparts, which are usually explained within the framework of core-accretion models