  A lattice quantum chromodynamics (LQCD) calculation of the nuclear matrix
element relevant to the $nn\to ppee\overline{\nu}_e\overline{\nu}_e$ transition
is described in detail, expanding on the results presented in Ref. [1]. This
matrix element, which involves two insertions of the weak axial current, is an
important input for phenomenological determinations of double-$\beta$ decay
rates of nuclei. From this exploratory study, performed using unphysical values
of the quark masses, the long-distance deuteron-pole contribution to the matrix
element is separated from shorter-distance hadronic contributions. This
polarizability, which is only accessible in double-weak processes, cannot be
constrained from single-$\beta$ decay of nuclei, and is found to be smaller
than the long-distance contributions in this calculation, but non-negligible.
In this work, technical aspects of the LQCD calculations, and of the relevant
formalism in the pionless effective field theory, are described. Further
calculations of the isotensor axial polarizability, in particular near and at
the physical values of the light-quark masses, are required for precise
determinations of both two-neutrino and neutrinoless double-$\beta$ decay rates
in heavy nuclei.
