Benchmarking the projected generator coordinate method for nuclear Gamow-Teller transitions

In this work, we aim to achieve a minimal extension of the quantum-number projected generator coordinate method (PGCM) to describe Gamow-Teller (GT) transition strengths in even-even nuclei and to compute the NME of $2νββ$ decay. 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 $2νββ$ decay of $^{48}$Ca to $^{48}$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 in-medium similarity renormalization group (IMSRG) evolution.