Tensor force role in β decays analyzed within the Gogny-interaction shell model

Background: The half-life of the famous $^{14}\mathrm{C}\ensuremath{\beta}$ decay is anomalously long, with different mechanisms: the tensor force, cross-shell mixing, and three-body forces, proposed to explain the cancellations that lead to a small transition matrix element.Purpose: We revisit and analyze the role of the tensor force for the $\ensuremath{\beta}$ decay of $^{14}\mathrm{C}$ as well as of neighboring isotopes.Methods: We add a tensor force to the Gogny interaction, and derive an effective Hamiltonian for shell-model calculations. The calculations were carried out in a $p\text{\ensuremath{-}}sd$ model space to investigate cross-shell effects. Furthermore, we decompose the wave functions according to the total orbital angular momentum $L$ in order to analyze the effects of the tensor force and cross-shell mixing.Results: The inclusion of the tensor force significantly improves the shell-model calculations of the $\ensuremath{\beta}$-decay properties of carbon isotopes. In particular, the anomalously slow $\ensuremath{\beta}$ decay of $^{14}\mathrm{C}$ can be explained by the isospin $T=0$ part of the tensor force, which changes the components of $^{14}\mathrm{N}$ with the orbital angular momentum $L=0,1$, and results in a dramatic suppression of the Gamow-Teller transition strength. At the same time, the description of other nearby $\ensuremath{\beta}$ decays are improved.Conclusions: Decomposition of wave function into $L$ components illuminates how the tensor force modifies nuclear wave functions, in particular suppression of $\ensuremath{\beta}$-decay matrix elements. Cross-shell mixing also has a visible impact on the $\ensuremath{\beta}$-decay strength. Inclusion of the tensor force does not seem to significantly change, however, binding energies of the nuclei within the phenomenological interaction.