Relation between the $0νββ$ and $2νββ$ nuclear matrix elements revisited

We show that the dominant Gamow-Teller part, $M^{0ν}_{GT}$, of the nuclear matrix element governing the neutrinoless $ββ$ decay is related to the matrix element $M^{2ν}_{cl}$ governing the allowed two-neutrino $ββ$ decay. That relation is revealed when these matrix elements are expressed as functions of the relative distance $r$ between the pair of neutrons that are transformed into a pair of protons in the $ββ$ decay. Analyzing this relation allows us to understand the contrasting behavior of these matrix elements when $A$ and $Z$ is changed; while $M^{0ν}_{GT}$ changes slowly and smoothly, $M^{2ν}$ has pronounced shell effects. We also discuss the possibility of phenomenological determination of the $M^{2ν}_{cl}$ and from them of the $M^{0ν}_{GT}$ values from the experimental study of the $β^{\pm}$ strength functions.

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