Nuclear matrix elements (NMEs) for neutrinoless $\beta \beta$ decays and inverse-$\beta$ decays are crucial for studying $\nu$ properties beyond the standard model and astro-$\nu$ nuclear interactions. The NMEs consist mainly of the isospin ($\tau$) spin ($\sigma$) component, and the $\tau \sigma$ strength (square of the $\tau \sigma$ NME) is studied experimentally by charge exchange reactions (CERs). The summed Gammow-Teller ($\tau \sigma$) and spin dipole ($\tau \sigma Y_1$) strengths measured by CERs are shown to be reduced half with respect to the nucleon-based sum-rule limits. The $\tau \sigma$ NMEs are shown to be quenched due to the non-nucleonic $\mathrm{\Delta }$-isobar resonance effect on the basis of the measured strengths and the quasiparticle random-phase approximation analysis with effective nucleon-nucleon and $N\mathrm{\Delta }$ $\tau \sigma$ interactions. The quenching effect is incorporated by using an effective $\tau \sigma$ coupling of $g_{\tau \sigma }^{\mathrm{\Delta }}/g_{\tau \sigma }$ $\approx 0.7$ with $g_{\tau \sigma }$ being the coupling for a free nucleon. The quenching effect is applied to the $\tau \sigma$ components of the weak, electromagnetic and nuclear interaction NMEs. Impact of the $\mathrm{\Delta }$-resonance effect on $\nu$ studies in nuclei is discussed.

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• Beta decay
• Collective models
• Giant resonances
• Nuclear astrophysics
• Nuclear decay
• Nuclear structure & decays
• Nuclear transition rates
• Nucleus-neutrino interactions
• Sum rules