Finite-volume formalism in the $2\underset{}{\overset{H_I+H_I}{\to }}2$ transition: an application to the lattice QCD calculation of double beta decays

We present the formalism for connecting a second-order electroweak $2\underset{}{\overset{H_I+H_I}{\to }}2$ transition amplitudes in the finite volume (with two hadrons in the initial and final states) to the physical amplitudes in the infinite volume. Our study mainly focus on the case where the low-lying intermediate state consists of two scattering hadrons. As a side product we also reproduce the finite-volume formula for $2\underset{}{\overset{H_I}{\to }}2$ transition, originally obtained by Brice~no and Hansen. With the available finite-volume formalism, we further discuss how to treat with the finite-volume problem in the double beta decays $nn\to ppee\bar{\nu }\bar{\nu }$ and $nn\to pp ee$.