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Background: Neutrinoless double beta decay, if observed, would reveal physicsbeyond the Standard Model (SM) of particle physics, namely it would prove thatneutrinos are Majoran fermions and that the lepton number is not conserved.Purpose: The analysis of the results of neutrinoless double beta decayobservations requires an accurate knowledge of several nuclear matrix elements(NME) for different mechanism that may contribute to the decay. We provide acomplete analysis of these NME for the decay of the ground state (g.s.) of$^{48}$Ca to the g.s. $0^+_1$ and first excited $0^+_2$ state of $^{48}$Ti.Method: For the analysis we used the nuclear shell model with effectivetwo-body interactions that were fine-tuned to describe the low-energyspectroscopy of $pf$-shell nuclei. We checked our model by calculating thetwo-neutrino transition probability to the g.s. of $^{48}$Ti. We also makepredictions for the transition to the first excited $0^+_2$ state of $^{48}$Ti.Results: We present results for all NME relevant for the neutrinolesstransitions to the $0^+_1$ and $0^+_2$ states, and using the lower experimentallimit for the g.s. to g.s. half-life we extract upper limits for the neutrinophysics parameters. Conclusions: We provide accurate NME for the two-neutrinoand neutrinoless double beta decay transitions in A=48 system, which can befurther used to analyze the experimental results of double beta decayexperiments when they become available.

Shell model analysis of competing mechanism to the double-beta decay of $^{48}$Ca