Energy spectrum of $\beta$ electrons in neutrinoless double-$\beta$ decay including the excitation of the electron shell of atoms

Double-$\beta$ decay is accompanied with a high probability by the excitation of the electron shell of the daughter atom; as a result, the energy carried away by $\beta$ electrons decreases. The mean value and standard deviation of the excitation energy of the electron shell of the daughter atom in the double-$\beta$ decay of germanium $_{32}^{76}\mathrm{Ge} \rightarrow \ _{34}^{76}\mathrm{Se}^* + 2\beta^-(+~2\bar{\nu_e})$ have been determined within the Thomas–Fermi and relativistic Dirac–Hartree–Fock methods. Using the estimates thus obtained, a two-parameter model of the energy spectrum of $\beta$ electrons in the neutrinoless mode has been developed including the redistribution of the reaction energy between the decay products. The shift of the total energy of $\beta$ electrons does not exceed $50$ eV with a probability of $90\%$. However, the mean excitation energy is $\sim400$ eV, i.e., an order of magnitude higher, whereas the standard deviation is $\sim2900$ eV, which is apparently due to a significant contribution from inner electron levels to the energy characteristics of the process. The distortion of the shape of the peak of the $0\nu2\beta$ decay should be taken into account when analyzing the data of detectors with a resolution of $\sim100$ eV or higher.