Ab initio uncertainty quantification of neutrinoless double-beta decay in $^{76}$Ge

The observation of neutrinoless double-beta ($0\nu\beta\beta$) decay would offer proof of lepton number violation, demonstrating that neutrinos are Majorana particles, while also helping us understand why there is more matter than antimatter in the Universe. If the decay is driven by the exchange of the three known light neutrinos, a discovery would, in addition, link the observed decay rate to the neutrino mass scale through a theoretical quantity known as the nuclear matrix element (NME). Accurate values of the NMEs for all nuclei considered for use in $0\nu\beta\beta$ experiments are therefore crucial for designing and interpreting those experiments. Here, we report the first comprehensive ab initio uncertainty quantification of the $0\nu\beta\beta$-decay NME, in the key nucleus $^{76}$Ge. Our method employs nuclear strong and weak interactions derived within chiral effective field theory and recently developed many-body emulators. Our result, with a conservative treatment of uncertainty, is an NME of $2.60^{+1.28}_{-1.36}$, which, together with the best-existing half-life sensitivity and phase-space factor, sets an upper limit for effective neutrino mass of $187^{+205}_{-62}$ meV. The result is important for designing next-generation germanium detectors aiming to cover the entire inverted hierarchy region of neutrino masses.