We describe here microscopic calculations performed on the dominant forbidden transitions in reactor antineutrino spectra above 4 MeV using the nuclear shell model. By taking into account Coulomb corrections in the most complete way, we calculate the shape factor with the highest fidelity and show strong deviations from allowed approximations and previously published results. Despite small differences in the ab initio electron cumulative spectra, large differences on the order of several percent are found in the antineutrino spectra. Based on the behavior of the numerically calculated shape factors we propose a parametrization of forbidden spectra. Using Monte Carlo techniques we derive an estimated spectral correction and uncertainty due to forbidden transitions. We establish the dominance and importance of forbidden transitions in both the reactor anomaly and spectral shoulder analysis with their respective uncertainties. Based on these results, we conclude that a correct treatment of forbidden transitions is indispensable in both the normalization anomaly and spectral shoulder.

Evolution of the electron spectra with the effective value of the weak axial-vector coupling constant $g_{\mathrm{A}}$ was followed for 26 first-, second-, third-, fourth- and fifth-forbidden ${\beta }^{-}$ decays of odd-$A$ nuclei by calculating the involved nuclear matrix elements (NMEs) in the framework of the microscopic quasiparticle-phonon model (MQPM). The next-to-leading-order terms were included in the $\beta$-decay shape factor of the electron spectra. The spectrum shapes of third- and fourth-forbidden nonunique decays were found to depend strongly on the value of $g_{\mathrm{A}}$, while first- and second-forbidden decays were mostly unaffected by the tuning of $g_{\mathrm{A}}$. The $g_{\mathrm{A}}$-driven evolution of the normalized $\beta$ spectra was found to be quite universal, largely insensitive to the small changes in the nuclear mean field and the adopted residual many-body Hamiltonian producing the excitation spectra of the MQPM. This makes the comparison of experimental and theoretical electron spectra, coined “the spectrum-shape method” (SSM), a robust tool for extracting information on the effective values of the weak coupling constants. In this exploratory work two new experimentally interesting decays for the SSM treatment were discovered: the ground-state-to-ground-state decays of ${}^{99}\mathrm{Tc}$ and ${}^{87}\mathrm{Rb}$. Comparing the experimental and theoretical spectra of these decays could shed light on the effective values of $g_{\mathrm{A}}$ and $g_{\mathrm{V}}$ for second- and third-forbidden nonunique decays. The measurable decay transitions of ${}^{135}\mathrm{Cs}$ and ${}^{137}\mathrm{Cs}$, in turn, can be used to test the SSM in different many-body formalisms. The present work can also be considered as a (modest) step towards solving the $g_{\mathrm{A}}$ problem of the neutrinoless double beta decay.

The evolution of the shape of the electron spectra of 16 forbidden ${\beta }^{-}$ decays as a function of $g_{\mathrm{A}}$ was studied using the nuclear shell model in appropriate single-particle model spaces with established, well-tested nuclear Hamiltonians. The $\beta$ spectra of ${}^{94}\mathrm{Nb}\left(6^{+}\right)\to {}^{94}\mathrm{Mo}\left(4^{+}\right)$ and ${}^{98}\mathrm{Tc}\left(6^{+}\right)\to {}^{98}\mathrm{Ru}\left(4^{+}\right)$ were found to depend strongly on $g_{\mathrm{A}}$, which makes them excellent candidates for the determination of the effective value of $g_{\mathrm{A}}$ with the spectrum-shape method (SSM). A strong $g_{\mathrm{A}}$ dependence is also seen in the spectrum of ${}^{96}\mathrm{Zr}\left(0^{+}\right)\to {}^{96}\mathrm{Nb}\left(6^{+}\right)$. This decay could be used for determining the quenching of $g_{\mathrm{A}}$ in sixth-forbidden decays in the future, when the measurement of the spectrum becomes experimentally feasible. The calculated shell-model electron spectra of the ground-state-to-ground-state decays of ${}^{87}\mathrm{Rb}$, ${}^{99}\mathrm{Tc}$, and ${}^{137}\mathrm{Cs}$ and the decay of ${}^{137}\mathrm{Cs}$ to the isomeric $11/2^{-}$ state in ${}^{137}\mathrm{Ba}$ were found to be in excellent agreement with the spectra previously calculated using the microscopic quasiparticle-phonon model. This is further evidence of the robust nature of the SSM observed in the previous studies.

We have performed a systematic study of the $logft$ values, shape factors, and electron spectra for the second-forbidden nonunique ${\beta }^{-}$ decays of ${}^{24}\mathrm{Na}\left(4^{+}\right)\to {}^{24}\mathrm{Mg}\left(2^{+}\right)$ and ${}^{36}\mathrm{Cl}\left(2^{+}\right){\to }^{36}\mathrm{Ar}\left(0^{+}\right)$ transitions under the framework of the nuclear shell model. We have performed the shell model calculations in the $sd$ model space, using more recent microscopic effective interactions such as Daejeon16, chiral N3LO, and JISP16. These interactions are derived from the no-core shell model wave functions using Okubo-Lee-Suzuki transformation. For comparison, we have also shown the results obtain from the phenomenological USDB interaction. To test the predictive power of these interactions first we have computed low-lying energy spectra of parent and daughter nuclei involved in these transitions. The computed results for energy spectra, nuclear matrix elements, $logft$ values, shape factors, electron spectra, and decomposition of the integrated shape factor are reported and compare with the available experimental data.