Nuclear matrix elements for $\lambda$ mechanism of $0\nu\beta\beta$ of $^{48}$Ca in nuclear shell-model: Closure versus nonclosure approach

The $\lambda$ and $m_{\beta\beta}$ mechanisms of neutrinoless double beta decay ($0\nu\beta\beta$) occur with light neutrino exchange via $W_L-W_R$, and $W_L-W_L$ mediation, respectively. In the present study, we calculate the nuclear matrix elements (NMEs) for the $m_{\beta\beta}$ and $\lambda$ mechanisms of $0\nu\beta\beta$, which has origin in the left-right symmetric model with right-handed gauge boson at TeV scale. The NMEs are calculated for one of the $0\nu\beta\beta$ decaying isotope $^{48}$Ca in the interacting nuclear shell-model using the GXPF1A effective interaction of $pf$-shell. The NMEs are calculated in both closure and nonclosure approaches using four different methods: closure, running closure, running nonclosure, and mixed methods. All the NMEs are calculated incorporating the effects of the finite size of nucleons and the revisited higher order terms such as isoscalar and weak magnetism terms of the nucleon currents. Inclusion of the short-range nature of nucleon-nucleon interaction in Miller-Spencer, CD-Bonn, and AV18 parametrizations is also taken care of. The comparative dependence of the running closure and running nonclosure NMEs with the spin-parity of the allowed states of intermediate nucleus $^{48}$Sc, the coupled spin-parity of the two initial decaying neutrons and the final two protons, the cutoff excitation energy of $^{48}$Sc, the cutoff number of states of $^{48}$Sc are also examined. Results show that there are about 2-20\% enhancements in different types of total NMEs, calculated in the nonclosure approach as compared to the closure approach. The significant enhancements are found in the $M_{qGT}$ and $M_{qT}$ type NMEs for the inclusion of the higher-order terms of the nucleon currents.