The renormalization of the shell-model neutrinoless double-beta decay operator starting from effective field theory at next-to-next-to-leading order
This paper is the continuation of a previous study, where shell-model calculations of the matrix elements for the neutrinoless double-beta decay were performed within a fully-consistent framework, starting from expressions of the nuclear Hamiltonian and decay operators which have been derived through chiral perturbation theory. The decays under our investigation are those involving 48Ca, 76Ge, and 82Se, as in our earlier analyses of electroweak nuclear decays with Hamiltonians and operators derived within effective field theory. In this work we focus on perturbative properties of the chiral expansion of the decay operator, namely in assessing the weight of terms beyond the leading order on the nuclear matrix elements of medium-mass candidates. To this end, we have constructed effective shell-model decay operators starting from two-body currents up to next-to-next-to-leading order and, for the first time, we introduce for nuclei of experimental interest contributions of the three-body current operators, which emerge at next-to-next-to-next-to-leading order of the chiral perturbation theory. Our investigation is aimed to evaluate the theoretical uncertainties affecting the calculated nuclear matrix elements of the neutrinoless double-beta decay, by conjugating the estimate of the inaccuracy related to the finiteness of the degrees of freedom of the nuclear model and the truncation of the chiral expansion of the decay operator.