  We present the first chiral effective theory derivation of the neutrinoless
double beta-decay $nn\rightarrow pp$ potential induced by light Majorana
neutrino exchange. The effective-field-theory framework has allowed us to
identify and parameterize short- and long-range contributions previously missed
in the literature. These contributions can not be absorbed into
parameterizations of the single nucleon form factors. Starting from the quark
and gluon level, we perform the matching onto chiral effective field theory and
subsequently onto the nuclear potential. To derive the nuclear potential
mediating neutrinoless double beta-decay, the hard, soft and potential neutrino
modes must be integrated out. This is performed through next-to-next-to-leading
order in the chiral power counting, in both the Weinberg and pionless schemes.
At next-to-next-to-leading order, the amplitude receives additional
contributions from the exchange of ultrasoft neutrinos, which can be expressed
in terms of nuclear matrix elements of the weak current and excitation energies
of the intermediate nucleus. These quantities also control the two-neutrino
double beta-decay amplitude. Finally, we outline strategies to determine the
low-energy constants that appear in the potentials, by relating them to
electromagnetic couplings and/or by matching to lattice QCD calculations.
