<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Δ</mml:mi></mml:math>-isobar resonance effects studied by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>τ</mml:mi><mml:mi>σ</mml:mi></m

Nuclear matrix elements (NMEs) for neutrinoless $\ensuremath{\beta}\ensuremath{\beta}$ decays and inverse-$\ensuremath{\beta}$ decays are crucial for studying $\ensuremath{\nu}$ properties beyond the standard model and astro-$\ensuremath{\nu}$ nuclear interactions. The NMEs consist mainly of the isospin ($\ensuremath{\tau}$) spin ($\ensuremath{\sigma}$) component, and the $\ensuremath{\tau}\ensuremath{\sigma}$ strength (square of the $\ensuremath{\tau}\ensuremath{\sigma}$ NME) is studied experimentally by charge exchange reactions (CERs). The summed Gammow-Teller ($\ensuremath{\tau}\ensuremath{\sigma}$) and spin dipole ($\ensuremath{\tau}\ensuremath{\sigma}{Y}_{1}$) strengths measured by CERs are shown to be reduced half with respect to the nucleon-based sum-rule limits. The $\ensuremath{\tau}\ensuremath{\sigma}$ NMEs are shown to be quenched due to the non-nucleonic $\mathrm{\ensuremath{\Delta}}$-isobar resonance effect on the basis of the measured strengths and the quasiparticle random-phase approximation analysis with effective nucleon-nucleon and $N\mathrm{\ensuremath{\Delta}}$ $\ensuremath{\tau}\ensuremath{\sigma}$ interactions. The quenching effect is incorporated by using an effective $\ensuremath{\tau}\ensuremath{\sigma}$ coupling of ${g}_{\ensuremath{\tau}\ensuremath{\sigma}}^{\mathrm{\ensuremath{\Delta}}}/{g}_{\ensuremath{\tau}\ensuremath{\sigma}}$ $\ensuremath{\approx}0.7$ with ${g}_{\ensuremath{\tau}\ensuremath{\sigma}}$ being the coupling for a free nucleon. The quenching effect is applied to the $\ensuremath{\tau}\ensuremath{\sigma}$ components of the weak, electromagnetic and nuclear interaction NMEs. Impact of the $\mathrm{\ensuremath{\Delta}}$-resonance effect on $\ensuremath{\nu}$ studies in nuclei is discussed.