Nuclear matrix element of neutrinoless double-$\beta$ decay: Relativity and short-range correlations

Abstract

Background: The discovery of neutrinoless double-beta ($0\nu \beta \beta$) decay would demonstrate the nature of neutrinos, have profound implications for our understanding of matter-antimatter mystery, and solve the mass hierarchy problem of neutrinos. The calculations for the nuclear matrix elements $M^{0\nu }$ of $0\nu \beta \beta$ decay are crucial for the interpretation of this process. Purpose: We study the effects of relativity and nucleon-nucleon short-range correlations on the nuclear matrix elements $M^{0\nu }$ by assuming the mechanism of exchanging light or heavy neutrinos for the $0\nu \beta \beta$ decay. Methods: The nuclear matrix elements $M^{0\nu }$ are calculated within the framework of covariant density functional theory, where the beyond-mean-field correlations are included in the nuclear wave functions by configuration mixing of both angular-momentum and particle-number projected quadrupole deformed mean-field states. Results: The nuclear matrix elements $M^{0\nu }$ are obtained for ten $0\nu \beta \beta$-decay candidate nuclei. The impact of relativity is illustrated by adopting relativistic or nonrelativistic decay operators. The effects of short-range correlations are evaluated. Conclusions: The effects of relativity and short-range correlations play an important role in the mechanism of exchanging heavy neutrinos though the influences are marginal for light neutrinos. Combining the nuclear matrix elements $M^{0\nu }$ with the observed lower limits on the $0\nu \beta \beta$-decay half-lives, the predicted strongest limits on the effective masses are $|⟨m_{\nu }⟩|<0.06\mathrm{eV}$ for light neutrinos and $|⟨m_{{\nu }_h}^{-1}⟩{|}^{-1}>3.065\times 10^8\mathrm{GeV}$ for heavy neutrinos.

24 pages, 4 figures

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