The $^{150}$Nd($^3$He,$t$) reaction at 140 MeV/u and $^{150}$Sm($t$,$^3$He) reaction at 115 MeV/u were measured, populating excited states in $^{150}$Pm. The transitions studied populate intermediate states of importance for the (neutrinoless) $betabeta$ decay of $^{150}$Nd to $^{150}$Sm. Monopole and dipole contributions to the measured excitation-energy spectra were extracted by using multipole decomposition analyses. The experimental results were compared with theoretical calculations obtained within the framework of Quasiparticle Random-Phase Approximation (QRPA), which is one of the main methods employed for estimating the half-life of the neutrinoless $betabeta$ decay ($0 ubetabeta$) of $^{150}$Nd. The present results thus provide useful information on the neutrino responses for evaluating the $0 ubetabeta$ and $2 ubetabeta$ matrix elements. The $2 ubetabeta$ matrix element calculated from the Gamow-Teller transitions through the lowest $1^{+}$ state in the intermediate nucleus is maximally about half of that deduced from the half-life measured in $2 ubetabeta$ direct counting experiments and at least several transitions through $1^{+}$ intermediate states in $^{150}$Pm are required to explain the $2 ubetabeta$ half-life. Because Gamow-Teller transitions in the $^{150}$Sm($t$,$^3$He) experiment are strongly Pauli-blocked, the extraction of Gamow-Teller strengths was complicated by the excitation of the $2hbaromega$, $Delta L=0$, $Delta S=1$ isovector spin-flip giant monopole resonance (IVSGMR). However, the near absence of Gamow-Teller transition strength made it possible to cleanly identify this resonance, and the strength observed is consistent with the full exhaustion of the non-energy-weighted sum rule for the IVSGMR.
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