Relation between the 0νββ and 2νββ nuclear matrix elements reexamined

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^± strength functions