Neutrinoless $\beta \beta$ decay nuclear matrix elements complete up to N${}^2$LO in heavy nuclei

Abstract

We evaluate all nuclear matrix elements (NMEs) up to next-to-next-to leading order (N${}^2$LO) in chiral effective field theory ($\chi$EFT) for the neutrinoless double-beta ($0\nu \beta \beta$) 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 $0\nu \beta \beta$ 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 $0\nu \beta \beta$ decay. Within uncertainties, our results for the ultrasoft NME are of similar size as contributions usually referred to as ``beyond the closure approximation''.

10 pages, 4 figures, and 2 tables. Corrected a typo in Eq. (6)

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