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

TL;DR

This paper calculates advanced nuclear matrix elements for neutrinoless double-beta decay in heavy nuclei using chiral effective field theory, revealing significant N$^2$LO contributions and uncertainties across different nuclear models.

Contribution

It provides the first comprehensive N$^2$LO calculations of NMEs for key isotopes using both pnQRPA and shell model, including loop diagrams often neglected.

Findings

N$^2$LO contributions are around -5% to +15%.

Sign discrepancy between models suggests different intermediate state behaviors.

Ultrasoft NME sizes are comparable to beyond closure approximation contributions.

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''.