Renormalized approach to neutrinoless double-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> decay

The process at the heart of neutrinoless double-$\ensuremath{\beta}$ decay, $nn\ensuremath{\rightarrow}pp\phantom{\rule{0.16em}{0ex}}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}}$ induced by a light Majorana neutrino, is investigated in pionless and chiral effective field theory. We show in various regularization schemes the need to introduce a short-range lepton-number-violating operator at leading order, confirming earlier findings. We demonstrate that such a short-range operator is only needed in spin-singlet $S$-wave transitions, while leading-order transitions involving higher partial waves depend solely on long-range currents. Calculations are extended to include next-to-leading-order corrections in perturbation theory, where to this order no additional undetermined parameters appear. We establish a connection based on chiral symmetry between neutrinoless double-$\ensuremath{\beta}$ decay and nuclear charge-independence breaking induced by electromagnetism. Data on the latter confirm the need for a leading-order short-range operator but do not allow for a full determination of the corresponding lepton-number-violating coupling. Using a crude estimate of this coupling, we perform ab initio calculations of the matrix elements for neutrinoless double-$\ensuremath{\beta}$ decay for $^{6}\mathrm{He}$ and $^{12}\mathrm{Be}$. We speculate on the phenomenological impact of the leading short-range operator on the basis of these results.