  Two-neutrino double-beta decay of certain nuclear isotopes is one of the
rarest Standard Model processes observed in nature. Its neutrinoless
counterpart is an exotic lepton-number nonconserving process that is widely
searched for to determine if the neutrinos are Majorana fermions. In order to
connect the rate of these processes to the Standard Model and beyond the
Standard Model interactions, it is essential that the corresponding nuclear
matrix elements are constrained reliably from theory. Lattice quantum
chromodynamics (LQCD) and low-energy effective field theories (EFTs) are
expected to play an essential role in constraining the matrix element of the
two-nucleon subprocess, which could in turn provide the input into ab initio
nuclear-structure calculations in larger isotopes. Focusing on the two-neutrino
process $nn \to pp \, (ee \bar{\nu}_e\bar{\nu}_e)$, the amplitude is
constructed in this work in pionless EFT at next-to-leading order,
demonstrating the emergence of a renormalization-scale independent amplitude
and the absence of any new low-energy constant at this order beyond those
present in the single-weak process. Most importantly, it is shown how a LQCD
four-point correlation function in Euclidean and finite-volume spacetime can be
used to constrain the Minkowski infinite-volume amplitude in the EFT. The same
formalism is provided for the related single-weak process, which is an input to
the double-$\beta$ decay formalism. The LQCD-EFT matching procedure outlined
for the double-weak amplitude paves the road toward constraining the
two-nucleon matrix element entering the neutrinoless double-beta decay
amplitude with a light Majorana neutrino.
