Relation between the $0\nu \beta \beta$ and $2\nu \beta \beta$ nuclear matrix elements revisited

Abstract

We show that the dominant Gamow-Teller part, $M_{GT}^{0\nu }$, of the nuclear matrix element governing the neutrinoless $\beta \beta$ decay is related to the matrix element $M_{cl}^{2\nu }$ governing the allowed two-neutrino $\beta \beta$ 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 $\beta \beta$ decay. Analyzing this relation allows us to understand the contrasting behavior of these matrix elements when $A$ and $Z$ is changed; while $M_{GT}^{0\nu }$ changes slowly and smoothly, $M^{2\nu }$ has pronounced shell effects. We also discuss the possibility of phenomenological determination of the $M_{cl}^{2\nu }$ and from them of the $M_{GT}^{0\nu }$ values from the experimental study of the ${\beta }^{\pm }$ strength functions.

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