Search for the in-situ production of $^{77}$Ge in the GERDA neutrinoless double-beta decay experiment

The beta decay of $^{77}$Ge and $^{77m}$Ge, both produced by neutron capture on $^{76}$Ge, is a potential background for Germanium based neutrinoless double-beta decay search experiments such as GERDA or the LEGEND experiment. In this work we present a search for $^{77}$Ge decays in the full GERDA Phase II data set. A delayed coincidence method was employed to identify the decay of $^{77}$Ge via the isomeric state of $^{77}$As (9/2$^{+}$, 475 keV, T$_{1/2}$ = 114 μs, $^{77m}$As). New digital signal processing methods were employed to select and analyze pile-up signals. No signal was observed, and an upper limit on the production rate of $^{77}$Ge was set at < 0.216 nuc/(kg $\cdot$ yr) (90% CL). This corresponds to a total production rate of $^{77}$Ge and $^{77m}$Ge of < 0.38 nuc/(kg $\cdot$ yr) (90% CL), assuming equal production rates. A previous Monte Carlo study predicted a value for in-situ $^{77}$Ge and $^{77m}$Ge production of (0.21 ± 0.07) nuc/(kg $\cdot$ yr), a prediction that is now further corroborated by our experimental limit. Moreover, tagging the isomeric state of $^{77m}$As can be utilised to further suppress the $^{77}$Ge background. Considering the similar experimental configurations of LEGEND-1000 and GERDA, the cosmogenic background in LEGEND-1000 at LNGS is estimated to remain at a sub-dominant level.