Correlations and neutrinoless $\beta\beta$ decay nuclear matrix elements of $pf$-shell nuclei

We calculate the nuclear matrix elements (NMEs) for neutrinoless double-beta decays ($0\nu\beta\beta$) of $pf$-shell nuclei using the shell model (SM) and energy density functional (EDF) methods. The systematic study of non-physical decays (except for $^{48}$Ca) of Ca$\rightarrow$Ti, Ti$\rightarrow$Cr and Cr$\rightarrow$Fe allows for a detailed comparison between the two nuclear structure approaches. We observe that while the dominant Gamow-Teller part of the NME differs roughly by a factor of two between SM and EDF, when we restrict the calculations to spherical EDF states and seniority-zero SM configurations, the NMEs obtained by both methods are strikingly similar. This points out to the important role of nuclear structure correlations for $0\nu\beta\beta$ decay NMEs. We identify correlations associated with high-seniority components in the initial and final states of the decay as one of the reasons for the discrepancies between SM and EDF results. We also explore exact projection to good isospin, and conclude that it has only a moderate effect in the Gamow-Teller part of the NMEs but strongly affects the Fermi contribution. This work opens up the door for NME benchmarks between different theoretical approaches, and constitutes a step forward towards more reliable estimations of the NMEs.