Dark Matter Constraints on Neutrinoless Double Beta Decay in Left–Right Symmetric Models

We analyze neutrinoless double beta decay in the minimal left--right symmetric model using aunified LRSM$\to$SMEFT$\to$LEFT pipeline and evaluate decay rates with \texttt{\(\nu\)DoBe}(IBM-2 nuclear matrix elements, scheme~A).Right-handed neutrino dark matter is implemented \emph{operationally} by explicitly decouplingthe lightest state $N_1$: for each parameter point we rescale the Dirac column associated with $N_1$,$(m_D)_{\alpha 1}\to\kappa (m_D)_{\alpha 1}$ with $\kappa\ll1$, suppressing heavy--light mixing.We then compute a lifetime proxy $\tau_{N_1}$ from the mixing measure $U^2$ extracted from thedecoupled point, and a relic-density proxy $\Omega_{N_1}h^2$ from a $W_R$-mediated freeze-outcross-section estimate used solely for scan-level selection.For the benchmark overlay in Fig.~\ref{fig:fig4_bench_vs_dm} we fix$m_{W_R}=6.2~\mathrm{TeV}$, $g_R=0.65$, $m_{N_1}=10~\mathrm{TeV}$ and $\kappa=5.7\times10^{-25}$,and label points ``DM-allowed'' if they satisfy$\left|\Omega_{N_1}h^2-\Omega_{\rm obs}h^2\right|\le\Delta\Omega$ and $\tau_{N_1}\ge\tau_{\rm min}$.For the resulting DM-selected sample, the full prediction remains light-neutrino dominated in bothmass orderings: $|m_{\beta\beta}^{\rm eff}|$ tracks $m_{\beta\beta}$ and the deviation satisfies$\left|m_{\beta\beta}^{\rm eff}/m_{\beta\beta}\right|-1\sim 10^{-5}\text{--}10^{-4}$ for thebenchmark choices shown, while auxiliary dim-7-only and dim-9-only switches are used solely asdiagnostics.