Properties of Active-Neutrino Oscillations and Double-Beta Decay in the Presence of Sterile-Neutrino Contributions

Abstract

The contributions of light sterile neutrinos to the properties of active-neutrino oscillations and the properties of double-beta decay are estimated under the assumption of the Majorana nature of neutrinos on the basis of a phenomenological model involving three active and three sterile neutrinos. The appearance and survival probabilities for active neutrinos are determined upon taking into account sterile-neutrino contributions in order to explain all known anomalies in neutrino data at short distances by employing the same test values of the model parameters. Modified graphical dependences of the survival and appearance probabilities for electron neutrinos/antineutrinos in beams of muon neutrinos/antineutrinos on the distance and other model parameters at various neutrino energies, as well as on the ratio of the distance to the neutrino energy, are presented. A significant distinction between the probability curves within the neutrino model being considered and the simple sinusoidal curves based on the neutrino model involving only one sterile neutrino is found. The effective electron-neutrino masses for beta decay and for neutrinoless double-beta decay are estimated in the presence of the sterile-neutrino contributions. In addition, the properties of selenium-82 two-neutrino double-beta decay are calculated. These results can be used in interpreting and predicting the results of ground-based experiments aimed at searches for sterile neutrinos and neutrinoless double-beta decay.

Appendices

Appendix A

AMPLITUDE FOR TWO-NEUTRINO DOUBLE-BETA DECAY OF SELENIUM-82

We will now calculate theoretically the total and differential intensities of ${}^{82}$Se two-neutrino double-beta decay. In order to calculate the intensity of two-neutrino transitions, it is necessary to perform summation over all possible $1^{+}$ states of the intermediate nucleus [57, 58]. For this, we need the absolute values and phases of the matrix elements $\langle 0_{f}^{+}||\hat{\beta}^{-}||1_{N}^{+}\rangle$ and $\langle 1_{N}^{+}||\hat{\beta}^{-}||0_{i}^{+}\rangle$ (where $\hat{\beta}^{-}=\sigma\tau^{-}$), which appear in the expression for $T_{1/2}^{2\nu 2\beta}$:

$$\left[T_{1/2}^{2\nu 2\beta}\left(0_{i}^{+}\to 0_{f}^{+}\right)\right]^{-1}=\frac{G_{\beta}^{4}g_{A}^{4}}{32\pi^{7}\ln 2}$$

(A.1)

$${}\times\int\limits_{m_{e}}^{T+m_{e}}{d\varepsilon_{1}}\int\limits_{m_{e}}^{T+2m_{e}-\varepsilon_{1}}d\varepsilon_{2}\int\limits_{0}^{T+2m_{e}-\varepsilon_{1}-\varepsilon_{2}}d\omega_{1}$$

$${}\times F(Z_{f},\varepsilon_{1})F(Z_{f},\varepsilon_{2})p_{1}\varepsilon_{1}p_{2}\varepsilon_{2}\omega_{1}^{2}\omega_{2}^{2}A_{0_{f}^{+}}.$$

The expression for $A_{0_{f}^{+}}$ has the form

$$4A_{0_{f}^{+}}=\bigg{|}\sum\limits_{N}\left\langle 0_{f}^{+}\left|\left|\hat{\beta}^{-}\right|\right|1_{N}^{+}\right\rangle$$

(A.2)

$${}\times\left\langle 1_{N}^{+}\left|\left|\hat{\beta}^{-}\right|\right|0_{i}^{+}\right\rangle(K_{N}+L_{N})\bigg{|}^{2}$$

$${}+\frac{1}{3}\bigg{|}\sum\limits_{N}\left\langle 0_{f}^{+}\left|\left|\hat{\beta}^{-}\right|\right|1_{N}^{+}\right\rangle$$

$${}\times\left\langle 1_{N}^{+}\left|\left|\hat{\beta}^{-}\right|\right|0_{i}^{+}\right\rangle(K_{N}-L_{N})\bigg{|}^{2}.$$

Here, $p_{1}$ and $p_{2}$ are the momenta of the electrons involved, while $\varepsilon_{1}$ and $\varepsilon_{2}$ are their energies; $\omega_{1}$ and $\omega_{2}=T+2m_{e}-\varepsilon_{1}-\varepsilon_{2}-\omega_{1}$ are the antineutrino energies; $T=E_{i}-E_{f}-2m_{e}=Q_{\beta\beta}$ is the total kinetic energy of final-state leptons; and $E_{i}$ ($E_{f}$) is the mass of the parent (daughter) nucleus. Further, $F(Z_{f},\varepsilon)$ is a Coulomb factor that takes into account the effect of the electrostatic field of the nucleus on the emitted electrons, while the quantities $K_{N}$ and $L_{N}$ contain the energy denominators characteristic of second-order perturbation theory; that is,

$$K_{N}=\frac{1}{\mu_{N}+\left(\varepsilon_{1}+\omega_{1}-\varepsilon_{2}-\omega_{2}\right)/2}$$

(A.3a)

$${}+\frac{1}{\mu_{N}-\left(\varepsilon_{1}+\omega_{1}-\varepsilon_{2}-\omega_{2}\right)/2},$$

$$L_{N}=\frac{1}{\mu_{N}+\left(\varepsilon_{1}+\omega_{2}-\varepsilon_{2}-\omega_{1}\right)/2}$$

(A.3b)

$${}+\frac{1}{\mu_{N}-\left(\varepsilon_{1}+\omega_{2}-\varepsilon_{2}-\omega_{1}\right)/2},$$

where $\mu_{N}=E_{N}-(E_{i}+E_{f})/2$, $E_{N}$ being the energy of the $N$th $1^{+}$ state of the intermediate nucleus.

The calculation of the nuclear matrix elements $\langle 0_{f}^{+}||\hat{\beta}^{-}||1_{N}^{+}\rangle$ and $\langle 1_{N}^{+}||\hat{\beta}^{-}||0_{i}^{+}\rangle$ is a very difficult theoretical challenge [59]. At the same time, it can be assumed that, for some isotopes, a dominant contribution to the sums over $N$ in expression (A.2) comes from the ground state of the intermediate nucleus, provided that the spin–parity of this state is $J^{\pi}=1^{+}$. This mechanism of two-neutrino double-beta decay corresponds to the hypothesis of dominance of the ground state of the intermediate nucleus (SSD mechanism—dominance of a single state [60, 61]). The situation in question prevails for ${}^{100}$Mo, in which case one can treat, with a high precision, the two-neutrino double-beta transition as a two-step process that connects the initial (${}^{100}$Mo) and final (${}^{100}$Ru) states of the process through the $1^{+}$ ground state of the ${}^{100}$Tc intermediate nucleus. The nuclear matrix elements $M_{1}^{I}=\langle 1_{\textrm{g.s.}}^{+}||\hat{\beta}^{-}||0_{i}^{+}\rangle$ and $M_{1}^{F}=\langle 0_{f}^{+}||\hat{\beta}^{-}||1_{\textrm{g.s.}}^{+}\rangle$ can be found from the values of the transition strength $ft$ for electron capture or single beta-decay process. Here, $ft$ is the product of the phase-space factor and the half-life with respect to the corresponding single beta process. For example, we have

$$M_{1}^{I}=\frac{1}{g_{A}}\sqrt{\frac{3D}{ft_{\textrm{EC}}}},\quad M_{1}^{F}=\frac{1}{g_{A}}\sqrt{\frac{3D}{ft_{\beta^{-}}}},$$

(A.4)

where $g_{A}=1.27561$, $D=\frac{2\pi^{3}\ln 2}{G_{\beta}^{2}m_{e}^{5}}=6288.6$ s, $G_{\beta}=G_{F}\cos\theta_{C}$, $G_{F}=1.166378\times 10^{-5}$ GeV${}^{-2}$, and $\cos\theta_{C}=0.97425$.

If the SSD hypothesis is valid under the condition that the ground state of the intermediate nucleus has the spin–parity of $J^{\pi}=1^{+}$, the intensity of the two-neutrino transition is determined exclusively by the intensities of individual beta processes, which are characterized by the factors $ft_{\beta^{-}}$ and $ft_{\textrm{EC}}$ [62]. As a result, it is independent of $G_{\beta}$ and $g_{A}$ and has the form

$$T_{1/2}^{(2\nu)}(0^{+}\to 0^{+})$$

(A.5)

$${}=\frac{16\pi^{2}ft_{\textrm{EC}}ft_{\beta^{-}}}{3\ln 2(\lambda_{C}/c)H(T,0_{f}^{+})}$$

$${}=2.997\times 10^{14}{\text{yr}}\times\frac{10^{\log ft_{\textrm{EC}}+\log ft_{\beta^{-}}}}{H(T,0_{f}^{+})},$$

where

$$H(T,0_{f}^{+})=\int\limits_{1}^{T+1}d\varepsilon_{1}$$

(A.6)

$${}\times\int\limits_{1}^{T+2-\varepsilon_{1}}d\varepsilon_{2}\int\limits_{0}^{T+2-\varepsilon_{1}-\varepsilon_{2}}d\omega_{1}F(Z_{f},\varepsilon_{1})$$

$${}\times F(Z_{f},\varepsilon_{2})p_{1}\varepsilon_{1}p_{2}\varepsilon_{2}\omega_{1}^{2}\omega_{2}^{2}(K^{2}+KL+L^{2}).$$

The value of $\log ft_{\beta^{-}}$ is well known from ${}^{100}$Tc beta decay. It is $4.59$, which corresponds to $M_{1}^{F}=0.546$. A determination of $\log ft_{\textrm{EC}}$ on the basis of experiments aimed at studying electron capture in ${}^{100}$Tc is a difficult experimental challenge. The currently most precise value of $\log ft_{\textrm{EC}}$ for electron capture in ${}^{100}$Tc was obtained in [63]. It is $\log ft_{\textrm{EC}}=4.29_{-0.07}^{+0.08}$.

In calculating the half-life with respect to the two-neutrino double-beta transition, it is frequently assumed that the kinetic energies of emitted leptons are approximately equal [57, 58, 64]. We then have $K\approx L\approx 2/\mu$. This is the situation where the expression for $T_{1/2}^{2\nu 2\beta}$ is dominated by the contributions associated with intermediate states of the nucleus at high excitation energies (HSD mechanism—dominance of highly lying excited states). As was shown in [65], the theoretical value of $T_{1/2}^{2\nu 2\beta}$ is overestimated within this approach, where the lepton-energy dependence of $K$ and $L$ is disregarded. In the case of the $0^{+}\to 0_{\textrm{g.s.}}^{+}$ transition in ${}^{100}$Mo, the effect is about 25$\%$. The inclusion of the lepton-energy dependence of $K$ and $L$ on the basis of the SSD mechanism makes it possible to obtain the differential intensity with respect to the energy of a single electron, $P(\varepsilon)=d\ln I/d\varepsilon$, for the two-neutrino double-beta decay of the isotope ${}^{100}$Mo [66, 67]. This corresponds to the data of the NEMO-3 experiment [68].

In the case of ${}^{82}$Se double-beta decay, the ground state of the intermediate nucleus, ${}^{82}$Br${}_{\textrm{g.s.}}(5^{-})$, has a mass that is less by $423$ keV than the mass of the ${}^{82}$Se parent nucleus. A virtual Gamow–Teller transition is possible through the first excited $1^{+}$ state of the ${}^{82}$Br nucleus at $E_{x}=75$ keV. Accordingly, $M(^{82}\textrm{Se})-M(^{82}\textrm{Br}^{\ast},1_{1}^{+})=348$ keV, and there are grounds to believe that the SSD mechanism will be operative in ${}^{82}$Se two-neutrino double-beta decay.

The excited state of bromine-82 (${}^{82}$Br${}^{\ast}$, $1_{1}^{+}$) at $E_{x}=75$ keV was found in the experiment devoted to studying the charge-exchange reaction ${}^{82}$Se (${}^{3}$He, t)${}^{82}$Br and reported in [69]. This state is characterized by a large Gamow–Teller strength of $B({\textrm{GT}})=0.338$. It is worth noting that higher lying excited $1_{1}^{+}$ states of bromine-82 at energies $E_{x}$ below $2$ MeV correspond to order-of-magnitude smaller transition strengths. From the SSD hypothesis, it follows that one should only take into account the contribution of the $1_{1}^{+}$ state of the ${}^{82}$Br intermediate nucleus to the sum over $N$ in the expression in (A.2). If, alternatively, the transition proceeds through many higher lying intermediate excited states, then the HSD mechanism is responsible for the two-neutrino double-beta transition in ${}^{82}$Se. The choice of model is an important physics issue since it has an effect on the differential intensities in the two-neutrino channel and, hence, on estimations of the background. The results of the measurements performed with the NEMO-3 detector [50] for the distribution of the intensity with respect to the energy of a single electron favor the SSD approach. Also, investigations performed in the CUPID-0 experiment show that the SSD approach provides a better description of the distribution with respect to the total energy of the electrons than the HSD approach does [53]. The NEMO-3 detector and the future SuperNEMO detectors involving a tracker and a calorimeter would permit fully reconstructing the topology of $\beta\beta$ events. Thus, we have seen that the distribution of the single-electron energy is sensitive to the nuclear mechanism, so that an accurate statistical investigation of this distribution could lead to discriminating between the two theoretical approaches in question [50].

The nuclear matrix element $M_{1}^{I}=\langle 1_{1}^{+}||\hat{\beta}^{-}||0_{i}^{+}\rangle$ is determined from the Gamow–Teller strength of $B({\textrm{GT}})=0.338$ [69] according to the relation $\left|M({\textrm{GT}})\right|^{2}=B({\textrm{GT}})$, which is valid for the $0^{+}\to 0^{+}$ two-neutrino double-beta transition. As a result, we obtain $M_{1}^{I}=0.581$. In the future, it will be possible to determine the matrix element $M_{1}^{F}=\langle 0_{f}^{+}||\hat{\beta}^{-}||1_{1}^{+}\rangle$ from an investigation that is devoted to the charge-exchange reaction ${}^{82}$Kr(d, ${}^{2}$He)${}^{82}$Br, but which has not yet been completed. Nevertheless, one can find the matrix element $M_{1}^{F}$ on the basis of Eqs. (A.1) and (A.2) by employing the $T_{1/2}^{2\nu 2\beta}$ value obtained in the NEMO-3 experiment. In the case where the SSD mechanism governs ${}^{82}$Se two-neutrino double-beta decay, we arrive at the half-life value of $T_{1/2}^{2\nu 2\beta}=9.39\times 10^{19}$ yr [50]. The corresponding value of $M_{1}^{F}$ is $0.23$ at $B({\textrm{GT}}^{+})=0.0529$.

Figure 4 gives the distributions with respect to the energy of a single electron that correspond to the HSD and SSD mechanisms. Of course, it would be interesting to compare the theoretical distributions that we obtained with the results of future measurements in the SuperNEMO experiment.