HADES RV Programme with HARPS-N at TNG. XVI. A super-Earth in the habitable zone of the GJ 3998 multi-planet system

The low masses of M dwarfs create attractive opportunities for exoplanet radial-velocity (RV) detections. These stars, however, exhibit strong stellar activity that may attenuate or mimic planetary signals. We present a velocimetric analysis on one such M dwarf, GJ 3998 ($d=18.2\,\text{pc}$), with two published short-period super-Earths: GJ 3998 b and GJ 3998 c. We use additional data from the HARPS-N spectrograph to confirm these two planets and to look for more. We carry out joint modelling of: (i) RV planetary signals, (ii) stellar rotation in RV and activity indicators through Gaussian processes, (iii) long-term trends in RV and activity indicators. We constrain the rotational period of GJ 3998 to $P_\text{rot}=30.2\pm 0.3\,\text{d}$ and discover long-term sinusoidal imprints in RV and FWHM of period $P_\text{cyc}=316^{+14}_{-8}\,\text{d}$. We confirm GJ 3998 b and GJ 3998 c, and detect a third planet: GJ 3998 d, whose signal had been previously attributed to stellar activity. GJ 3998 d has an orbital period of $41.78\pm 0.05\,\text{d}$, a minimum mass of $6.07^{+1.00}_{-0.96}\,\text{M}_\oplus$ and a mean insolation flux of $1.2^{+0.3}_{-0.2}\,\Phi_\oplus$. This makes it one of the few known planets receiving Earth-like insolation flux.