Deviations in the shape of the \(2\nu \beta \beta \) energy spectra can provide hints of new physics. Below we report on results of searches for physics beyond the Standard Model that can modify the two-electron energy sum distribution of the \(^{100}\)Mo \(2\nu \beta \beta \) decay due to emission of Majoron bosons, the existence of a bosonic component in the neutrino states and possible Lorentz invariance violation.
The shape of the two-electron energy sum distribution in various types of decays is characterized by the spectral index n [39], being determined by the phase space \(G \sim (Q_{\beta \beta }-T)^n\), where \(Q_{\beta \beta }\) is the full energy released in the decay minus two electron masses and T is the sum of kinetic energies of two emitted electrons. The ordinary \(2\nu \beta \beta \) decay has a spectral index of \(n=5\). Any modification from this functional form can be an indication of new physics.
A number of grand unification theories predict the existence of a massless or light boson which couples to the neutrino. Neutrinoless \(\beta \beta \) decay can proceed with the emission of one or two Majoron bosons resulting in a continuous energy sum spectrum with spectral index \(n \ne 5\). The decay accompanied by a single Majoron emission has \(n=1,2\) and 3, while models with two Majoron emissions predict \(n=3\) and 7 (see [40] and references therein). The results for the neutrinoless \(\beta \beta \) decay with the emission of a Majoron corresponding to the spectral index \(n=1\) have already been published in [18, 19]. The Majoron-accompanied \(0\nu \beta \beta \) decay modes with spectral indices \(n=2,3\) and 7 are considered here.
It was noted in [41] that violation of the Pauli exclusion principle resulting in a bosonic component in the neutrino states can be tested by looking at the shape of the energy and angular distributions of the electrons emitted in \(\beta \beta \) decay. For the two-electron energy sum distribution the corresponding index would be \(n=6\).
Lorentz invariance is a fundamental symmetry. However, new physics at very high energies close to the Planck scale can manifest itself in small effects at low energies, including Lorentz invariance violation. Consequently, searches for non-Lorentz invariant effects have attracted active theoretical and experimental effort [42,43,44,45]. The possibility to test Lorentz invariance with \(\beta \beta \) decay was discussed in [46, 47]. In case of \(2\nu \beta \beta \) decay the Lorentz invariance violation may be manifested as a modification of the conventional electron sum spectrum due to an additional contribution of the Lorentz-violating perturbation with a spectral shape of \(n=4\).
Table 3 Summary of systematic uncertainties on the measured \(2\nu \beta \beta \) \(^{100}\)Mo half-life
The theoretical distributions of the two-electron energy sum for different modes of \(^{100}\)Mo \(\beta \beta \) decay discussed above are shown in Fig. 8. The difference in the shape of the distributions due to different spectral indices n is used to evaluate possible contributions from physics beyond the Standard Model. No significant deviations from the expected \(^{100}\)Mo \(2\nu \beta \beta \) spectral shape (\(n=5\)) have been observed and therefore limits on new physics parameters have been set using the full energy sum spectrum of the full \(^{100}\)Mo data set. The contributions of the \(\beta \beta \) decay modes with spectral indices \(n=2,3,6,7\) are constrained with a modified frequentist \(CL_s\) method [48, 49] using a profile likelihood fitting technique (COLLIE software package [50]). A profile likelihood scan is used for the distribution with the spectral index \(n=4\) in order to explore possibility of negative as well as positive Lorentz-violating perturbation.
The systematic uncertainties on background contributions discussed in Sect. 4.2, the 5% uncertainty on the detector acceptance and selection efficiency for signal, a possible distortion in the shape of the two-electron energy sum spectrum due to the energy calibration accuracy, as well as a 5% error on the modelling of the energy loss of electrons are taken into account in limit setting without imposing a constraint on the normalization of standard \(2\nu \beta \beta \) contribution.
The limits on the half-lives for different \(0\nu \beta \beta \) modes with Majoron(s) emission, and for the bosonic neutrino admixture obtained with the \(CL_s\) method are given in Table 4.
Table 4 Lower bounds on half-lives (\(\times 10^{21}\) year) at \(90\%\) C.L. from \(0\nu \beta \beta \) searches with Majoron emission (spectral indices \(n=2,3,7\)), and searches for the bosonic neutrino admixture. The ranges in the expected half-life limits are from the \(\pm 1\sigma \) range of the systematic uncertainties on the background model, signal efficiency and distortions in the shape of the energy spectrum
The half-life limits on the Majoron \(0\nu \beta \beta \) modes are translated into the upper limits on the lepton number violating parameter \(\langle g_{ee}\rangle \), which is proportional to the coupling between the neutrino and the Majoron boson, using the relation,
$$\begin{aligned} 1/T_{1/2} = |\langle g_{ee}\rangle |^m G |M|^2, \end{aligned}$$
(4)
where G is the phase space (which includes the axial-vector coupling constant \(g_A\)), M is the nuclear matrix element, and \(m=2(4)\) is the mode with the emission of one (two) Majoron particle(s). The M and G values are taken from [51]. For the single Majoron emission and \(n=3\), M and G are taken from [52]. There are no NME and phase space calculations available for \(n=2\).
The upper limits on the Majoron-neutrino coupling constant \(\langle g_{ee}\rangle \) are shown in Table 5. One can see that the NEMO-3 results presented here are the current best limits for \(n=3\) and the single Majoron emission mode and are comparable with the world’s best results from the EXO-200 [53] and GERDA [54] experiments for the other two modes.
Table 5 Upper limits on the Majoron-neutrino coupling constant \(\langle g_{ee}\rangle \) from NEMO-3 (\(^{100}\)Mo, this work) and EXO-200 (\(^{136}\)Xe) [53] and GERDA (\(^{76}\)Ge) [54] experiments. All limits are at 90% C.L. The ranges are due to uncertainties in NME calculations
The contribution of bosonic neutrinos to the \(2\nu \beta \beta \)-decay rate can be parametrised as [41]:
$$\begin{aligned} W_{tot} = \cos ^4 \chi W_f + \sin ^4 \chi W_b, \end{aligned}$$
(5)
where \(W_f\) and \(W_b\) are the weights in the neutrino wave-function expression corresponding to the two fermionic and two bosonic antineutrino emission respectively. The purely fermionic, \(T_{1/2}^{f}\), and purely bosonic, \(T_{1/2}^{b}\), half-lives are calculated under the SSD model to be [41] :
$$\begin{aligned} T_{1/2}^f(0^+g.s. )= & {} 6.8\times 10^{18}~\text{ year },\nonumber \\ T_{1/2}^b(0^+g.s.)= & {} 8.9\times 10^{19}~\text{ year }. \end{aligned}$$
(6)
Using the NEMO-3 half-life limit of \(T_{1/2}^b(0^+g.s.) > 1.2\times 10^{21}\) year (Table 4) an upper limit on the bosonic neutrino contribution to the \(^{100}\)Mo \(2\nu \beta \beta \) decay to the ground state can be evaluated as:
$$\begin{aligned} \sin ^2 \chi < 0.27 \,(90\%\,\text{ C.L. }). \end{aligned}$$
(7)
Although this limit is stronger than the bound obtained earlier in [41], the \(2\nu \beta \beta \) transition of \(^{100}\)Mo to the ground state is not very sensitive to bosonic neutrino searches due to a small value of the expected bosonic-to-fermionic decay branching ratio \(r_0 (0^+g.s. ) = 0.076\). The \(^{100}\)Mo \(2\nu \beta \beta \) decay to the first excited \(2^+_1\) state has a branching ratio of \(r_0 (2^+_1 ) = 7.1\) [41] and is therefore potentially more promising despite a lower overall decay rate. The current best experimental limit for this process is \(T_{1/2}(2^+_1) > 2.5\times 10^{21}\) year [55]. This bound is still an order of magnitude lower than the theoretically expected half-life value of \(T_{1/2}^b(2^+_1) = 2.4\times 10^{22}\) year for purely bosonic neutrino, and two orders of magnitude lower than the corresponding expected value for purely fermionic neutrino, \(T_{1/2}^f(2^+_1) = 1.7\times 10^{23}\) year [41].
The Standard Model Extension (SME) provides a general framework for Lorentz invariance violation (LIV) [42]. In this model, the size of the Lorentz symmetry breakdown is controlled by SME coefficients that describe the coupling between standard model particles and background fields. Experimental limits have been set on hundreds of these SME coefficients from constraints in the matter, photon, neutrino and gravity sectors [42]. The first search for LIV in \(2\nu \beta \beta \) decay was carried out in [56]. The two-electron energy sum spectrum of \(^{136}\)Xe was used to set a limit on the parameter \(\mathring{a}^{(3)}_{of}\), which is related to a time-like component of this LIV operator. The value of this parameter was constrained to be \(-2.65\times 10^{-5}\) GeV \(< \mathring{a}^{(3)}_{of} < 7.6\times 10^{-6}\) GeV by looking at deviations from the predicted energy spectrum of \(^{136}\)Xe \(2\nu \beta \beta \) decay [56].
In this work we adopt the same method, using the phase space calculations from [57], and perform a profile likelihood scan over positive and negative contributions of LIV to two-electron events by altering the \(^{100}\)Mo \(2\nu \beta \beta \) energy sum spectrum with positive and negative values of \(\mathring{a}^{(3)}_{of}\). The result of this scan is shown in Fig. 9.
The minimum of the profile log-likelihood function corresponds to \(-135\) counts and is not statistically significant even at 1\(\sigma \) level. The 90% CL exclusion limit is shown in Fig. 9 with the dashed line and gives \(-1798\) and 1527 events for negative and positive contributions to the deviation from the \(^{100}\)Mo \(2\nu \beta \beta \) energy sum spectrum respectively. The corresponding constraint on \(\mathring{a}^{(3)}_{of}\) is calculated using equations (2)–(6) in [56]. The result for \(^{100}\)Mo obtained with a full set of NEMO-3 data is
$$\begin{aligned} -4.2\times 10^{-7}~\text{ GeV }< \mathring{a}^{(3)}_{of} < 3.5\times 10^{-7}~\text{ GeV }\,(90\%\,\text{ C.L. }).\nonumber \\ \end{aligned}$$
(8)
A summary of the best available constraints on LIV and CPT violation parameters can be found in compilation [42].