Neutrinoless double-beta decay of the $\Delta^-$ resonance

The subprocess $nn\to ppe^-e^-$ is a key ingredient in the interpretation of nuclear neutrinoless double-beta decay. Intermediate $\Delta$ resonances may provide additional enhancements to this transition. We take a first step toward a $\Delta$-full description of $nn\to ppe^-e^-$ by investigating the neutrinoless double-beta decay $\Delta^- \to p e^-e^-$ in the framework of chiral effective field theory. We systematically derive the long-range contribution from light-Majorana-neutrino exchange through loop diagrams and incorporate the short-range part through counterterms required by renormalization. We predict the pion-mass dependence of the decay amplitude in the kinematic configuration with collinear electrons. Furthermore, to facilitate lattice-QCD matching, we calculate the decay amplitude in the degenerate $\Delta$-nucleon mass limit and provide the corresponding long-range prediction.