Towards complete leading-order predictions for neutrinoless double $\beta$ decay

Abstract

The amplitude for the neutrinoless double $\beta$ ($0\nu \beta \beta$) decay of the two-neutron system, $nn\to pp{e}^{-}{e}^{-}$, constitutes a key building block for nuclear-structure calculations of heavy nuclei employed in large-scale $0\nu \beta \beta$ searches. Assuming that the $0\nu \beta \beta$ process is mediated by a light-Majorana-neutrino exchange, a systematic analysis in chiral effective field theory shows that already at leading order a contact operator is required to ensure renormalizability. In this work, we develop a method to estimate the numerical value of its coefficient in analogy to the Cottingham formula and validate the result by reproducing the charge-independence-breaking contribution to the nucleon-nucleon scattering lengths. Our central result, while derived in the $\overline{\text{MS}}$ scheme, is given in terms of the renormalized amplitude ${\mathcal{A}}_{\nu }(|\mathbf{p}|,|{\mathbf{p}}^{\prime }|)$, matching to which will allow one to determine the contact-term contribution in regularization schemes employed in nuclear-structure calculations. Our results thus greatly reduce a crucial uncertainty in the interpretation of searches for $0\nu \beta \beta$ decay.

Comments: 7 pages, 3 figures; journal version. See arXiv:2102.03371 for an extended version