Toward Complete Leading-Order Predictions for Neutrinoless Double <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi></mml:math> Decay
Vincenzo Cirigliano, Wouter Dekens, Jordy de Vries, Martin Hoferichter, Emanuele Mereghetti
Abstract
The amplitude for the neutrinoless double $\ensuremath{\beta}$ ($0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$) decay of the two-neutron system $nn\ensuremath{\rightarrow}pp{e}^{\ensuremath{-}}{e}^{\ensuremath{-}}$ constitutes a key building block for nuclear-structure calculations of heavy nuclei employed in large-scale $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ searches. Assuming that the $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ 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 Letter, we develop a method to estimate the numerical value of its coefficient (in analogy to the Cottingham formula for electromagnetic contributions to hadron masses) and validate the result by reproducing the charge-independence-breaking contribution to the nucleon-nucleon scattering lengths. Our central result, while derived in dimensional regularization, is given in terms of the renormalized amplitude ${\mathcal{A}}_{\ensuremath{\nu}}(|\mathbf{p}|,|{\mathbf{p}}^{\ensuremath{'}}|)$, 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 $0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta}$ decay.