  Background: Neutrinoless double beta decay, if observed, would reveal physics
beyond the Standard Model (SM) of particle physics, namely it would prove that
neutrinos are Majoran fermions and that the lepton number is not conserved.
Purpose: The analysis of the results of neutrinoless double beta decay
observations requires an accurate knowledge of several nuclear matrix elements
(NME) for different mechanism that may contribute to the decay. We provide a
complete 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 effective
two-body interactions that were fine-tuned to describe the low-energy
spectroscopy of $pf$-shell nuclei. We checked our model by calculating the
two-neutrino transition probability to the g.s. of $^{48}$Ti. We also make
predictions for the transition to the first excited $0^+_2$ state of $^{48}$Ti.
Results: We present results for all NME relevant for the neutrinoless
transitions to the $0^+_1$ and $0^+_2$ states, and using the lower experimental
limit for the g.s. to g.s. half-life we extract upper limits for the neutrino
physics parameters. Conclusions: We provide accurate NME for the two-neutrino
and neutrinoless double beta decay transitions in A=48 system, which can be
further used to analyze the experimental results of double beta decay
experiments when they become available.
