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Two-neutrino double-beta decay of certain nuclear isotopes is one of therarest Standard Model processes observed in nature. Its neutrinolesscounterpart is an exotic lepton-number nonconserving process that is widelysearched for to determine if the neutrinos are Majorana fermions. In order toconnect the rate of these processes to the Standard Model and beyond theStandard Model interactions, it is essential that the corresponding nuclearmatrix elements are constrained reliably from theory. Lattice quantumchromodynamics (LQCD) and low-energy effective field theories (EFTs) areexpected to play an essential role in constraining the matrix element of thetwo-nucleon subprocess, which could in turn provide the input into ab initionuclear-structure calculations in larger isotopes. Focusing on the two-neutrinoprocess $nn \to pp \, (ee \bar{\nu}_e\bar{\nu}_e)$, the amplitude isconstructed in this work in pionless EFT at next-to-leading order,demonstrating the emergence of a renormalization-scale independent amplitudeand the absence of any new low-energy constant at this order beyond thosepresent in the single-weak process. Most importantly, it is shown how a LQCDfour-point correlation function in Euclidean and finite-volume spacetime can beused to constrain the Minkowski infinite-volume amplitude in the EFT. The sameformalism is provided for the related single-weak process, which is an input tothe double-$\beta$ decay formalism. The LQCD-EFT matching procedure outlinedfor the double-weak amplitude paves the road toward constraining thetwo-nucleon matrix element entering the neutrinoless double-beta decayamplitude with a light Majorana neutrino.

Two-neutrino double-beta decay in pionless effective field theory from a Euclidean finite-volume correlation function