Title: Characterisation of the stellar activity of M dwarfs. II. Relationship between Ca, H$α$, and Na chromospheric emissions

Abstract: The chromospheric emission estimated in the core of different lines, such as Ca II H & K, Na D1 and D2, and Halpha, is not always correlated between lines, as the Ca II H & K and Halpha emission time series are anti-correlated for a few percent of the stars, contrary to what is observed on the Sun. This puzzling result has been observed for both solar-type and M stars. Our objective is to characterise these relationships in more detail using complementary criteria, based on a set of spectra obtained with HARPS for a large sample of M dwarfs, to evaluate whether or not additional processes are required to explain the observations. We analysed the time average and variability of the Ca, Na, and Halpha emissions for 177 M0-M8 stars. We computed synthetic Halpha time series based on different assumptions of plage properties. The statistical properties of our sample, in terms of correlations and slopes between indices at different timescales, differ from what we previously obtained for FGK stars, with fewer stars with a null correlation, and a weaker dependence of the correlations on timescale. However, there can be a large dispersion from one season to another for stars. We find a clear indication of a change in the slope sign from the relation between averaged Ca (or Na) and Halpha for the most massive M dwarfs, as well as a change in slope in the Na-Ca relation. At this stage, we are not able to find simple plage properties that, alone, are sufficient to reproduce the observations. Simulations shows that it is not straightforward to compare the temporal variability correlation and the integrated indices and demonstrate the need for complex activity patterns to explain some of the observations. We conclude that the relation between these indices exhibits a large diversity in behaviour. More detailed simulations with complex activity patterns are necessary to understand these observations.