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

The subprocess $nn\to pp{e}^{-}{e}^{-}$ is a key ingredient in the interpretation of nuclear neutrinoless double-beta decay. Intermediate $\mathrm{\Delta }$ resonances may provide additional enhancements to this transition. We take a first step toward a $\mathrm{\Delta }$-full description of $nn\to pp{e}^{-}{e}^{-}$ by investigating the neutrinoless double-beta decay ${\mathrm{\Delta }}^{-}\to p{e}^{-}{e}^{-}$ 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 $\mathrm{\Delta }$-nucleon mass limit and provide the corresponding long-range prediction.