Neutrinoless Double Beta Decay from Lattice QCD: The Long-Distance $π^{-} \rightarrow π^{+} e^{-} e^{-}$ Amplitude

Neutrinoless double beta decay ($0 νββ$) is a hypothetical nuclear decay mode with important implications. In particular, observation of this decay would demonstrate that the neutrino is a Majorana particle and that lepton number conservation is violated in nature. Relating experimental constraints on $0 νββ$ decay rates to the neutrino masses requires theoretical input in the form of non-perturbative nuclear matrix elements which remain difficult to calculate reliably. This work marks a first step toward providing a general lattice QCD framework for computing long-distance $0 νββ$ matrix elements in the case where the decay is mediated by a light Majorana neutrino. The relevant formalism is developed and then tested by computing the simplest such matrix element describing an unphysical $π^{-} \rightarrow π^{+} e^{-} e^{-}$ transition on a series of domain wall fermion ensembles. The resulting lattice data is then fit to next-to-leading-order chiral perturbation theory, allowing a fully-controlled extraction of the low energy constant governing the transition rate, $g_ν^{ππ}(μ= 770 \,\, \mathrm{MeV}) = -10.78(12)_{\rm stat}(51)_{\rm sys}$. Finally, future prospects for calculations of more complicated processes, such as the phenomenologically important $n^{0} n^{0} \rightarrow p^{+} p^{+} e^{-} e^{-}$ decay, are discussed.

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