  The process at the heart of neutrinoless double-beta decay, $nn \rightarrow p
p\, e^- e^-$ 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
corrections in perturbation theory, where to this order no additional
undetermined parameters appear. We establish a connection based on chiral
symmetry between neutrinoless double-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-beta decay for $^6$He and $^{12}$Be. We speculate on the
phenomenological impact of the leading short-range operator on the basis of
these results.
