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

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 $|\langle m_{\nu }\rangle |<0.06\mathrm{eV}$ for light neutrinos and $|\langle m_{{\nu }_h}^{-1}\rangle {|}^{-1}>3.065\times {10}^8\mathrm{GeV}$ for heavy neutrinos.

Figure 1

NME $M^{0\nu }$ for the $0\nu \beta \beta$ decay of ${}^{150}\mathrm{Nd}\to {}^{150}\mathrm{Sm}$ mediated by (a) light- and (b) heavy-neutrino exchange, with the total and the VV, AA, AP, PP, and MM components separately. Results are calculated within the $\mathrm{GCM}+\mathrm{PNAMP}$ scheme based on the CDFT using both the full relativistic (Rel.) and the nonrelativistic-reduced (NR) decay operators with (SRC) and without (bare) the Argonne-parametrized SRCs.

Figure 2

The function $H_i\left(q\right)$ with (SRC) and without (bare) the Argonne-SRC modification for the VV, AA, AP, PP, and MM channels in the $0\nu \beta \beta$ NME of light- and heavy-neutrino exchange, respectively.

Figure 3

The $q$-space distribution of the $0\nu \beta \beta$ NME with heavy-neutrino exchange. Comparisons are made between the calculations using both the full relativistic (Rel.) and the nonrelativistic-reduced (NR) decay operators with (SRC) and without (bare) the Argonne-parametrized SRCs. Particle-number projected spherical wave functions are used in this calculation for the initial nucleus ${}^{150}\mathrm{Nd}$ and the final nucleus ${}^{150}\mathrm{Sm}$.

Figure 4

Comparison of the NMEs $M^{0\nu }$ of the $0\nu \beta \beta$ decay from different model calculations, which include the EDF [58], IBM [55], PHFB [50, 51], QRPA-NC [42], QRPA-Tü [45], and CSM [15] calculations, as well as the CDFT calculation in this paper with the $\mathrm{GCM}+\mathrm{PNAMP}$ wave functions and the Argonne-parametrized SRCs. The CDFT results without considering the SRC effect [6] is also shown for the light-neutrino exchange mode by the dashed line in panel (a).