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

Understand

We present the formalism for connecting a second-order electroweak $2\xrightarrow[]{H_I+H_I}2$ 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 $2\xrightarrow[]{H_I}2$ 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 $nn\to pp ee\bar{\nu}\bar{\nu}$ and $nn\to pp ee$.