Background: The quantum-number projected generator coordinate method (PGCM) has gained increasing attention, in part due to its combination with the in-medium similarity renormalization group (IMSRG) to describe collective excitations in medium-mass deformed nuclei, as well as nuclear matrix elements (NMEs) of neutrinoless double-$\beta$ ($0\nu \beta \beta$) decay.

Purpose: Extending the PGCM to Gamow-Teller (GT) transitions and two-neutrino double-$\beta$ ($2\nu \beta \beta$) decay is nontrivial, as it requires an accurate description of not only nuclear ground states but also a large number of excited states. In this work, we aim to achieve a minimal extension of the PGCM to describe GT transition strengths in even-even nuclei and to compute the NME of $2\nu \beta \beta$ decay.

Method: Within the PGCM framework, the wave functions of odd-odd nuclei are constructed as superpositions of neutron and proton quasiparticle configurations built on quasiparticle vacua constrained to have, on average, odd neutron and odd proton particle numbers. The angular momentum and particle numbers associated with the underlying mean-field states are restored through projection techniques. Using a shell-model Hamiltonian defined in the $fp$ shell, we assess the validity of this approach by benchmarking GT transitions in calcium and titanium isotopes, as well as the $2\nu \beta \beta$ decay of ${}^{48}\mathrm{Ca}$ to ${}^{48}\mathrm{Ti}$, against exact solutions. For comparison, we also confront our results with those obtained from configuration-interaction calculations employing different particle-hole truncation schemes, both with and without IMSRG evolution.

Results: The PGCM generally reproduces GT transitions to both low-lying and giant resonance states in ${}^{42–48}\mathrm{Ca}$ and ${}^{42–48}\mathrm{Ti}$. Although the agreement with exact results deteriorates as the number of valence nucleons increases, the overall description remains robust. The NME of the $2\nu \beta \beta$ decay from ${}^{48}\mathrm{Ca}$ to ${}^{48}\mathrm{Ti}$ is calculated without invoking the closure approximation. We find that the PGCM overestimates this matrix element by about 51%, primarily due to an overestimation of the GT transition strength from ${}^{48}\mathrm{Ti}$ to the first excited state of ${}^{48}\mathrm{Sc}$. Overall, the performance of the PGCM is comparable to, and in some cases exceeds, that of the CI calculation with the two-particle–two-hole truncation for the nuclei concerned.

Conclusions: The currently implemented PGCM framework provides a reliable description of GT transitions from even-even nuclei to low-lying states of odd-odd nuclei in regions not far from closed shells. As the number of valence nucleons increases, deviations from exact results become more noticeable, reflecting the growing importance of complex many-body correlations. These discrepancies are expected to be reduced by extending the set of generator coordinates and by incorporating the IMSRG evolution, which together offer a promising route toward further enhancing the predictive power of the PGCM framework.

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• Electroweak interactions in nuclear physics
• Nuclear structure & decays
• Many-body techniques