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 (N2LO) in chiral effective field theory (χEFT) for the neutrinoless double-beta (0νββ) decay of the nuclei most relevant for experiments, including 76Ge, 100Mo, and 136Xe. We use the proton-neutron quasiparticle random-phase approximation (pnQRPA) and the nuclear shell model to calculate the N2LO NMEs from very low-momentum (ultrasoft) neutrinos and from loop diagrams usually neglected in 0νββ studies. Our results indicate that the overall N2LO 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 behavior of the intermediate states of the 0νββ decay. Within uncertainties, our results for the ultrasoft NME are of similar size as contributions usually referred to as “beyond the closure approximation”.