Right-handed weak currents in neutrinoless $\beta \beta $ decays and ton scale $\beta\beta$ detectors

Right handed weak-currents (RHCs) in the left-right (L-R) symmetric model for neutrinoless double beta decays (DBDs) of both the $0^+~\to~0^+$ and $0^+~\to~2^+$ transitions are discussed from both theoretical and experimental view points. $<\lambda>$ and $<\eta>$-terms are related by $<\lambda>/<\eta>\approx \tan\beta$, which is constrained in the regions of $1-60$ for SUSY grand unified theories (GUTs) and of $1-165$ for non-SUSY GUTs. The enhancement mechanisms of the $<\eta>$ term over the $<\lambda>$ term in the $0^+$ transition are shown, and the $\Delta$ isobar contribution to the NME for the transition to the 2$^+$ state is found to be of the order of $20\%$ of the NME with the quenched weak coupling. The new and interesting RHC regions of $<\lambda>\approx 5\times10^{-8}$ and $<\eta>\approx 1.5\times 10^{-10}$ are shown to be exclusively explored by measuring both the $\beta\beta $ and $\gamma$ rays associated with the ground and excited DBDs by means of the ton-scale DBD detectors for the IH (inverted hierarchy) $\nu$-masses. The actual RHCs to be studied depend on the RHC NMEs.