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Neutrinoless double-$\beta$ ($0\nu\beta\beta$) decay of certain atomicisotopes, if observed, will have significant implications for physics ofneutrinos and models of physics beyond the Standard Model. In the simplestscenario, if the mass of the light neutrino of the Standard Model has aMajorana component, it can mediate the decay. Systematic theoretical studies ofthe decay rate in this scenario, through effective field theories matched to\emph{ab initio} nuclear many-body calculations, are needed to draw conclusionsabout the hierarchy of neutrino masses, and to plan the design of futureexperiments. However, a recently identified short-distance contribution atleading order in the effective field theory amplitude of the subprocess $nn \topp\,(ee)$ remains unknown, and only lattice quantum chromodynamics (QCD) candirectly and reliably determine the associated low-energy constant. While thenumerical computations of the correlation function for this process areunderway with lattice QCD, the connection to the physical amplitude, and hencethis short-distance contribution, is missing. A complete framework that enablesthis complex matching is developed in this paper. The complications arisingfrom Euclidean and finite-volume nature of the corresponding correlationfunction are fully resolved, and the value of the formalism is demonstratedthrough a simple example. The result of this work, therefore, fills the gapbetween first-principle studies of the $nn \to pp\,(ee)$ amplitude from latticeQCD and those from effective field theory, and can be readily employed in theongoing lattice-QCD studies of this process.

The path from lattice QCD to the short-distance contribution to $0\nu\beta\beta$ decay with a light Majorana neutrino