Neutrinoless ββ decay nuclear matrix elements complete up to N2LO in heavy nuclei

We evaluate all nuclear matrix elements (NMEs) up to next-to-next-to leading order (N 2 LO) in chiral effective field theory ( χ EFT) for the neutrinoless double-beta ( 0 ν β β ) decay of the nuclei most relevant for experiments, including 76 Ge, 100 Mo, and 136 Xe. We use the proton-neutron quasiparticle random-phase approximation (pnQRPA) and the nuclear shell model to calculate the N 2 LO NMEs from very low-momentum (ultrasoft) neutrinos and from loop diagrams usually neglected in 0 ν β β studies. Our results indicate that the overall N 2 LO contribution is centered around − ( 5 - 10 ) % for the shell model and + ( 10 - 15 ) % for the pnQRPA, with sizable uncertainties due to the scale dependence of the ultrasoft NMEs and the short-range nature of the loop NMEs. The sign discrepancy between many-body methods is common to all studied nuclei and points to the different behaviour of the intermediate states of the 0 ν β β decay. Within uncertainties, our results for the ultrasoft NME are of similar size as contributions usually referred to as “beyond the closure approximation”.