Detecting prolonged activity minima in binary stars. The case of $\zeta^{2}$ Ret

Context. It is well known that from 1645 to 1715 solar activity was notably low and the number of sunspots was extremely reduced. This epoch is known as the Maunder Minimum (MM). The study of stars in prolonged activity minima like the MM could help to shed light on this enigmatic epoch. Aims. To explore if the star $\zeta^{2}$ Ret, which belongs to a binary system, is in a state similar to the MM. We have collected more than 430 spectra acquired between 2000 and 2019 with the HARPS, REOSC, UVES, and FEROS spectrographs. Methods. We performed a detailed long-term activity study of both components using the Mount Wilson index, which is obtained from the Ca II H\&K lines. To search for signs of an activity cycle, we analyzed the resulting time-series with the Generalized Lomb-Scargle and CLEAN periodograms. Results. Our spectroscopic analysis shows a high activity level for zet01 Ret and a significant decrease in the magnetic activity cycle amplitude of $\zeta^{2}$ Ret. Thus, the activity difference between both components has been slightly increased ($\Delta$log (RHK) $\sim$0.24 dex), when compared to the previously reported value. The long series analyzed here allowed us to recalculate and constrain the period of $\zeta^{2}$ Ret to $\sim$7.9 yr. We also detected a long-term activity cycle of $\sim$4.2 yr in $\zeta^{1}$ Ret. Conclusions. By analogy with the scenario that proposes a weak solar cycle during the MM, we suggest that activity signatures showed by $\zeta^{2}$ Ret, i.e., a very low activity level when compared to its stellar companion, a notably decreasing amplitude ($\sim$47%), and a cyclic behaviour, are possible evidence that this star could be in a MM state. It is, to our knowledge, the first MM candidate star detected through a highly discrepant activity behaviour in a binary system.