Context. The high number of super-Earth and Earth-like planets in the habitable zone (HZ) detected around M-dwarf stars in the last years has revealed these stellar objects to be the key for planetary radial velocity (RV) searches. Aims. Using the HARPS-N spectrograph within The HArps-n red Dwarf Exoplanet Survey (HADES) we reach the precision needed to detect small planets with a few Earth masses using the RV technique. Methods. We obtained 138 HARPS-N RV measurements between 2013 May and 2020 September of GJ 720 A, classified as an M0.5V star located at a distance of 15.56 pc. To characterize the stellar variability and to discern the periodic variation due to the Keplerian signals from those related to stellar activity, the HARPS-N spectroscopic activity indicators and the simultaneous photometric observations were analyzed. The combined analysis of HARPS-N RVs and activity indicators let us to address the nature of the periodic signals. The final model and the orbital planetary parameters were obtained by fitting simultaneously the stellar variability and the Keplerian signal using a Gaussian process regression and following a Bayesian criterion. Results. The HARPS-N RV periodic signals around 40 d and 100 d have counterparts at the same frequencies in HARPS-N activity indicators and photometric light curves. Then we attribute these periodicities to stellar activity the former period being likely associated with the stellar rotation. GJ 720 A shows the most significant signal at 19.466$pm$0.005 d with no counterparts in any stellar activity indices. We hence ascribe this RV signal, having a semiamplitude of 4.72$pm$0.27 m/s , to the presence of a sub-Neptune mass planet. The planet GJ 720 Ab has a minimum mass of 13.64$pm$0.79 M$_{oplus}$, it is in circular orbit at 0.119$pm$0.002 AU from its parent star, and lies inside the inner boundary of the HZ around its parent star.
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