Reexamining the light neutrino exchange mechanism of the $0\nu \beta \beta$ decay with left- and right-handed leptonic and hadronic currents

The extension of the Majorana neutrino mass mechanism of the neutrinoless double-beta decay $\left(0\nu \beta \beta \right)$ with the inclusion of right-handed leptonic and hadronic currents is revisited. While only the exchange of light neutrinos is assumed, the $s_{1/2}$ and $p_{1/2}$ states of emitted electrons as well as recoil corrections to the nucleon currents are taken into account. Within the standard approximations the decay rate is factorized into a sum of products of kinematical phase-space factors, nuclear matrix elements, and the fundamental parameters that characterize the lepton number violation. Unlike in the previous treatments, the induced pseudoscalar term of hadron current is included, resulting in additional nuclear matrix elements. An improved numerical computation of the phase-space factors is presented, based on the exact Dirac wave functions of the $s_{1/2}$ and $p_{1/2}$ electrons with finite nuclear size and electron screening taken into account. The dependence of values of these phase-space factors on the different approximation schemes used in evaluation of electron wave functions is discussed. The upper limits for effective neutrino mass and the parameters $\langle \lambda \rangle$ and $\langle \eta \rangle$ characterizing the right-handed current mechanism are deduced from data on the $0\nu \beta \beta$ decay of ${}^{76}\text{Ge}$ and ${}^{136}\text{Xe}$ using nuclear matrix elements calculated within the nuclear shell model and quasiparticle random phase approximation. The differential decay rates, i.e., the angular correlations and the single electron energy distributions for various combinations of the total lepton number violating parameters that can help to disentangle the possible mechanism, are described and discussed.

Figure 1

The radial wave functions of an electron in the $s_{1/2}$ wave state, $g_{-1}\left(\varepsilon \right)$ and $f_{+1}\left(\varepsilon \right)$ (upper panels), and in the $p_{1/2}$ wave state, $g_{+1}\left(\varepsilon \right)$ and $f_{-1}\left(\varepsilon \right)$ (lower panels), as function of the electron kinetic energy $\varepsilon -m_e$. Results are presented for an electron emitted in the double-$\beta$ decay of ${}^{150}\text{Nd}$. In an evaluation of radial wave functions (w.f.) four different approximation schemes are considered [see Sec. (2) for details]: (A) the standard approximation of Doi et al. [6]. (B) An analytical solution of Dirac equations assuming a pointlike nucleus. (C) An exact solution of Dirac equations for a uniform charge distribution in the nucleus, considered at the nuclear surface. (D) The same as the preceding case but the electron screening is taken into account [3].

Figure 2

The phase factors $G_{0k}\left(k=1,\cdots ,11\right)$ in units of $y{r}^{-1}$ for the $0\nu \beta \beta$ decay of ${}^{150}\text{Nd}$. Results are presented for approximate electron wave functions (type A [6]) and exact Dirac wave functions with finite nuclear size and electron screening (type D [3]). The exponents $p$ for $k=1,\cdots ,11$ are $14,14,15,15,13,12,10,11,10,15,15$, respectively.

Figure 3

The normalized $r_1\left(r_2\right)$ dependence of $M^{0\nu }$ for ${}^{76}\text{Ge}$ and ${}^{136}\text{Xe}$. $r_1$ and $r_2$ are absolute values of a position vector of the $\beta$-decaying nucleon in a nucleus.

Figure 4

The decomposition of coefficients $C_I[I=mm$, $m\lambda$, $m\eta$, $\lambda \lambda$, $\eta \eta$, and $\lambda \eta$; see Eqs. (33) and (37)] on partial contributions $C_I^{0k}$ associated with phase-space factors $G_{0k}\left(k=1,\cdots ,11\right)$. The symbols standing for index $I$ are shown on the $x$ axis. The partial contributions are identified by index $k$, whose value is shown by the corresponding bar. The contributions from largest to the third largest are displayed in red, blue, and orange colors, respectively. Ratios $C_I^{0k}/C_I$ calculated with the ISM and QRPA matrix elements are presented with left and right bars for each value of index $I$, respectively. Results for ${}^{76}\text{Ge}$ and ${}^{136}\text{Xe}$ are presented in the lower (b) and upper (a) panels, respectively.

Figure 5

Limits on the effective neutrino mass $m_{\beta \beta }$ and right-handed parameters $\eta$ (left panels, $\langle \lambda \rangle =0$) and $\lambda$ (right panels, $\langle \eta \rangle =0$) implied by the constraints on the $0\nu \beta \beta$ decay of ${}^{76}\text{Ge}$ (lower panels, $T_{1/2}^{0\nu }\ge 3.{010}^{25}$ yr [25]) and ${}^{136}\text{Xe}$ (upper panels, $T_{1/2}^{0\nu }\ge 3.{410}^{25}$ yr [26]). To derive the bounds, the values of nuclear matrix elements calculated within the ISM [23] and the QRPA [22] are used. Results are presented for approximate electron wave functions (type A) and exact Dirac wave functions with finite nuclear size and electron screening (type D). Ellipses show the boundaries of the allowed domains.

Figure 6

The single-electron differential decay rate normalized to the total decay rate vs the electron energy $\tilde{\varepsilon }[\tilde{\varepsilon }=(\varepsilon -m_e)/Q_{\beta \beta }]$ for $0\nu \beta \beta$ decay of ${}^{76}\text{Ge}$ [left panels (a), (c), and (e)] and ${}^{136}\text{Xe}$ [right panels (b), (d), and (f)]. Results are presented for mechanism determined by factors $|m_{\beta \beta }{|}^2\left(\langle \lambda \rangle =\langle \eta \rangle =0\right)$ [panels (e) and (f)], ${\left|\langle \lambda \rangle \right|}^2(m_{\beta \beta }=\langle \eta \rangle =0$) [panels (c) and (d)], and ${\left|\langle \eta \rangle \right|}^2(m_{\beta \beta }=\langle \lambda \rangle =0$) [panels (e) and (f)]. Approximate electron wave functions (w.f. A) and exact Dirac wave functions with finite nuclear size and electron screening (w.f. D) are considered. Nuclear matrix elements calculated within the ISM [23] and the QRPA [22] are used in calculations.

Figure 7

The angular correlation factor [see Eq. (55)] vs the electron energy $\tilde{\varepsilon }[\tilde{\varepsilon }=(\varepsilon -m_e)/Q_{\beta \beta }]$ for $0\nu \beta \beta$ decay of ${}^{76}\text{Ge}$ [left panels (a), (c), and (e)] and ${}^{136}\text{Xe}$ [right panels (b), (d), and (f)]. Results are presented for mechanism determined by factors $|m_{\beta \beta }{|}^2\left(\langle \lambda \rangle =\langle \eta \rangle =0\right)$ [panels (e) and (f)], ${\left|\langle \lambda \rangle \right|}^2\left(m_{\beta \beta }=\langle \eta \rangle =0\right)$ [panels (c) and (d)], and ${\left|\langle \eta \rangle \right|}^2\left(m_{\beta \beta }=\langle \lambda \rangle =0\right)$ [panels (e) and (f)]. Approximate electron wave functions (w.f. A) and exact Dirac wave functions with finite nuclear size and electron screening (w.f. D) are considered. Nuclear matrix elements calculated within the ISM [23] are used in calculations.

Figure 8

The angular correlation factor [see Eq. (55)] vs the electron energy $\tilde{\varepsilon }[\tilde{\varepsilon }=(\varepsilon -m_e)/Q_{\beta \beta }]$ for $0\nu \beta \beta$ decay of ${}^{136}\text{Xe}$. Results are presented for the ${\left|\langle \lambda \rangle \right|}^2$ term ($m_{\beta \beta }=\langle \eta \rangle =0$). Approximate electron wave functions (w.f. A) and exact Dirac wave functions with finite nuclear size and electron screening (w.f. D) are considered. Nuclear matrix elements calculated within the ISM [23] and the QRPA [22] are used in calculations.