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We present a method to determine the leading-order (LO) contact term contributing to the $nn to pp e^-e^-$ amplitude through the exchange of light Majorana neutrinos. Our approach is based on the representation of the amplitude as the momentum integral of a known kernel (proportional to the neutrino propagator) times the generalized forward Compton scattering amplitude $n(p_1) n(p_2) W^+ (k) to p(p_1^prime) p(p_2^prime) W^- (k)$, in analogy to the Cottingham formula for the electromagnetic contribution to hadron masses. We construct model-independent representations of the integrand in the low- and high-momentum regions, through chiral EFT and the operator product expansion, respectively. We then construct a model for the full amplitude by interpolating between these two regions, using appropriate nucleon factors for the weak currents and information on nucleon-nucleon ($N! N$) scattering in the $^1S_0$ channel away from threshold. By matching the amplitude obtained in this way to the LO chiral EFT amplitude we obtain the relevant LO contact term and discuss various sources of uncertainty. We validate the approach by computing the analog $I = 2$ $N! N$ contact term and by reproducing, within uncertainties, the charge-independence-breaking contribution to the $^1S_0$ $N! N$ scattering lengths. While our analysis is performed in the $overline{rm MS}$ scheme, we express our final result in terms of the scheme-independent renormalized amplitude ${cal A}_ u(|{bf p}|,|{bf p}^prime|)$ at a set of kinematic points near threshold. We illustrate for two cutoff schemes how, using our synthetic data for ${cal A}_ u$, one can determine the contact-term contribution in any regularization scheme, in particular the ones employed in nuclear-structure calculations for isotopes of experimental interest.
The amplitude for the neutrinoless double $beta$ ($0 ubetabeta$) decay of the two-neutron system, $nnto ppe^-e^-$, constitutes a key building block for nuclear-structure calculations of heavy nuclei employed in large-scale $0 ubetabeta$ searches. Ass
We present the first ab initio calculations of neutrinoless double beta decay matrix elements in $A=6$-$12$ nuclei using Variational Monte Carlo wave functions obtained from the Argonne $v_{18}$ two-nucleon potential and Illinois-7 three-nucleon inte
Observation of neutrinoless double beta decay, a lepton number violating process that has been proposed to clarify the nature of neutrino masses, has spawned an enormous world-wide experimental effort. Relating nuclear decay rates to high-energy, bey
Within the framework of chiral effective field theory we discuss the leading contributions to the neutrinoless double-beta decay transition operator induced by light Majorana neutrinos. Based on renormalization arguments in both dimensional regulariz
Accurate nuclear matrix elements (NMEs) for neutrinoless double beta decays of candidate nuclei are important for the design and interpretation of future experiments. Significant progress has been made in the modeling of these NMEs from first princip