# Shell Model Studies of Competing Mechanisms to the Neutrinoless Double-Beta Decay in124Sn, 130Te, and 136Xe

Authors: Andrei Neacsu; Mihai Horoi
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

Neutrinoless double-beta decay is a predicted beyond Standard Model process that could clarify some of the not yet known neutrino properties, such as the mass scale, the mass hierarchy, and its nature as a Dirac or Majorana fermion. Should this transition be observed, there are still challenges in understanding the underlying contributing mechanisms. We perform a detailed shell model investigation of several beyond Standard Model mechanisms that consider the existence of right-handed currents. Our analysis presents different venues that can be used to identify the dominant mechanisms for nuclei of experimental interest in the massA~130 region (124Sn, 130Te, and 136Xe). It requires accurate knowledge of nine nuclear matrix elements that we calculate in addition to the associated energy-dependent phase space factors.


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## Body

## 1. Introduction

Should the neutrinoless double-beta decay (0νββ) be experimentally observed, the lepton number conservation is violated by two units and the back-box theorems [1–4] predict the neutrino to be a Majorana particle. In addition to the nature of the neutrino (whether a Dirac or a Majorana fermion), there are other unknown properties of the neutrino that could be investigated via 0νββ, such as the mass scale, the absolute mass, or the underlying neutrino mass mechanism. There are several beyond Standard Model mechanisms that could compete and contribute to this process [5, 6]. Reliable calculations of the nuclear matrix elements (NME) are necessary to perform an appropriate analysis that could help evaluate the contribution of each mechanism.The most commonly investigated neutrinoless0νββ mechanism is the so-called mass mechanism involving the exchange of light left-handed neutrinos, for which the NME were calculated using many nuclear structure methods. Calculations that consider the contributions of heavy, mostly sterile, right-handed neutrinos have become recently available, while left-handed heavy neutrinos have been shown to have a negligible effect [7, 8] and their contribution is generally dismissed. A comparison of the recent mass mechanism results obtained with the most common methods can be seen in Figure 6 of [9], where one can notice the differences that still exist among these nuclear structure methods. Figure 7 of [9] shows the heavy neutrino results for several nuclear structure methods, and the differences are even larger than those in the light neutrino case because of the uncertainties related to the short-range correlation (SRC) effects. There are efforts to reduce these uncertainties by the development of an effective transition operator that treats the SRC consistently [10].Because shell model calculations were successful in predicting two-neutrino double-beta decay half-lives [11] before experimental measurements and as shell model calculations of different groups largely agree with each other without the need to adjust model parameters, we calculate our nuclear matrix elements using shell model techniques and Hamiltonians that reasonably describe the experimental spectroscopic observables.Experiments such as SuperNEMO [12, 13] could track the outgoing electrons and help distinguish between the mass mechanism (ν) and so-called λ and η mechanisms [14, 15]. This would also provide complementary data at low energies for testing the existence of right-handed contributions predicted by left-right symmetric models [15–19], currently investigated at high energies in colliders and accelerators such as LHC [20]. To distinguish the possible contribution of the heavy right-handed neutrino using shell model nuclear matrix elements, measurements of lifetimes for at least two different isotopes are necessary, ideally that of an A~80 isotope and another lifetime of an A~130 isotope, as discussed in Section V of [21]. It is expected that if the neutrinoless double-beta decay is confirmed in any of the experiments, more resources and upgrades could be dedicated to boost the statistics and to reveal more information on the neutrino properties.Following our recent study for82Se in [21], which is the baseline isotope of SuperNEMO, we extend our analysis of λ and η mechanisms to other nuclei of immediate experimental interest: 124Sn, 130Te, and 136Xe. These isotopes are under investigation by the TIN.TIN [22] (124Sn), CUORE [23, 24], SNO+ [25] (130Te), NEXT [26], EXO [27], and KamLAND-Zen [28] (136Xe) experiments. For the mass region A~130, we perform calculations in jj55 model space consisting of 0g7/2,1d5/2,1d3/2,2s1/2 and 0h11/2 valence orbitals using the SVD shell model Hamiltonian [29] that was fine-tuned with experimental data from Sn isotopes. Our tests of this Hamiltonian include energy levels, B(E2)↑ transitions, occupation probabilities, Gamow-Teller strengths, and NME decomposition for configurations of protons/neutrons pairs coupled to some spin (I) and some parity (positive or negative), called I-pair decomposition. These tests and validations of the SVD Hamiltonian can be found in [9] for 124Sn and in [30] for 130Te and 136Xe. Calculations of NME in larger model spaces (e.g., jj77 model space that includes 0g9/2 and 0h9/2 orbitals missing in jj55 models space) were successfully performed for 136Xe [31], but for 124Sn and 130Te they are much more difficult and would require special truncations.In this work, assuming the detection of several tens of0νββ decay events, we present a possibility to identify right-handed contributions from λ and η mechanisms by analyzing the two-electron angular and energy distributions that could be measured.We organize this paper as follows: Section2 shows a brief description of the neutrinoless double-beta decay formalism considering a low-energy Hamiltonian that takes into account contributions from right-handed currents. Section 3 presents an analysis of the half-lives and of the two-electron angular and energy distributions results for 124Sn, 130Te, and 136Xe. Finally, we dedicate Section 4 to conclusions.


## 2. Brief Formalism of0νββ

The existence of right-handed currents and their contributions to the neutrinoless double-beta decay rate has been considered for a long time [14, 32], but most frequent calculations treat only the light left-handed neutrino-exchange mechanism (commonly referred to as “the mass mechanism”). One model that considers the right-handed currents contributions and includes heavy particles that are not part of the Standard Model is the left-right symmetric model [17, 18]. Within the framework of the left-right symmetric model, the neutrinoless double-beta decay half-life expression is(1)T1/20ν-1=G010νgA4M0νην+M0NηNRL+ηNRR+ηλXλ+ηηXη+⋯2,where ην, ηNRL, ηNRR, ηλ, and ηη are neutrino physics parameters defined in [15] (see also Appendix A of [21]), M0ν and M0N are the light and heavy neutrino-exchange nuclear matrix elements [5, 6], and Xλ and Xη are combinations of NME and phase space factors, which are calculated in this paper. G010ν is a phase space factor [33] that one can calculate [34] with good precision for most cases [35–37]. The “⋯” sign represents other possible contributions, such as those of R-parity violating SUSY particle exchange [5, 6], Kaluza-Klein modes [6, 38, 39], violation of Lorentz invariance, and equivalence principle [40–42], which we neglected here. ηNRL term also exists in the seesaw type I mechanisms but its contribution is negligible if the heavy mass eigenstates are larger than 1 GeV [8]. We consider a seesaw type I dominance [43] and we will neglect this contribution here.For an easier read, we perform the following change of notation:ν=ην, λ=ηλ, and η=ηη.In this paper, we provide an analysis of the two-electron relative energy and angular distributions for124Sn, 130Te, and 136Xe using shell model NME that we calculate. The purpose of this analysis is to identify the relative contributions of ηλ and ηη terms in (1). A similar analysis for 82Se was done using QRPA NME in [12] and with shell model NME in [21]. As in [21], we start from the classic paper of Doi et al. [14], describing the neutrinoless double-beta decay process using a low-energy effective Hamiltonian that includes the effects of the right-handed currents. By simplifying some notations and ignoring the contribution from ηNRR term, which has the same energy and angular distribution as ην term, the half-life expression [14] is written as(2)T1/20ν-1=MGT0ν2Cν2+Cνλcos⁡ϕ1+Cνηcos⁡ϕ2+Cλ2+Cη2+Cληcos⁡ϕ1-ϕ2,where ϕ1 and ϕ2 are the relative CP-violating phases (Eq. A7 of [21]) and MGT0ν is the Gamow-Teller contribution of the light neutrino-exchange NME. Cα are contributions from different mechanisms: Cν2 are from the left-handed leptonic currents, Cλ2 are from the right-handed leptonic and right-handed hadronic currents, and Cη2 are from the right-handed leptonic and left-handed hadronic currents. Cνλ, Cνη, and Cλη contain the interference between these terms. These are defined as(3)Cν2=C1ν2,Cνλ=C2νλ,Cνη=C3ην,Cλ2=C4λ2,Cη2=C5η2,Cλη=C6ηλ,where C1–6 are combinations of nuclear matrix elements and phase space factors (PSF). Their expressions can be found in Appendix B, Eqs. (B1) of [21]. MGT0ν and the other nuclear matrix elements that appear in the expressions of the Ci factors are presented in Eq. (B4) of [21].We write the differential decay rate for0+→0+0νββ transition as(4)d2W0+→0+0νdϵ1dcos⁡θ12=a0νω0νϵ12meR2Aϵ1+Bϵ1cos⁡θ12.Here, ϵ1 is the energy of one electron in units of mec2, R is the nuclear radius (R=r0A1/3, with r0 = 1.2 fm), θ12 is the angle between the outgoing electrons, and the expressions for the constant a0ν and the function ω0ν are given in Appendix C, Eqs. (C2) and (C3) of [34], respectively. The functions A(ϵ) and B(ϵ) are defined as combinations of factors that include PSF and NME: (5a)Aϵ1=N1ϵ12+N2ϵ12+N3ϵ12+N4ϵ12,(5b)Bϵ1=-2Re⁡N1⋆ϵ1N2ϵ1+N3⋆ϵ1N4ϵ1.The detailed expressions of N1–4(ϵ1) components are presented in Eqs. (B7) of [21].We can express the half-life as follows:(6)T1/20ν-1=1ln⁡2∫dW0+→0+0ν=a0νln⁡2meR2·∫1T+1Aϵ1ω0νϵ1dϵ1,with the normalized kinetic energy T defined as(7)T=Qββmec2,where Qββ is the Q-value of the decay.The integration of (4) over ϵ1 provides the angular distribution of the electrons that we write as(8)dW0+→0+0νdΩ=a0ν4πmeR2∫1T+1Aϵ1ω0νϵ1dϵ1+dΩ2π∫1T+1Bϵ1ω0νϵ1dϵ1,where dΩ=2πdcos⁡θ12.Integrating (4) over cos⁡θ12 provides the single-electron spectrum. As in [21], we express the decay rate as a function of the difference in the energy of the two outgoing electrons: Δt=(ϵ1-ϵ2)mec2, where ϵ2=T+2-ϵ1 is the kinetic energy of the second electron. We can write the energy of one electron as(9)ϵ1=T+2+Δt/mec22. Changing the variable, the energy distribution as a function of Δt is(10)2dW0+→0+0νdΔt=2a0νmeR2ω0νΔtmec2AΔt.


## 3. Results

The formalism used in this paper is taken from [21], where it was used to analyze the two-electron angular and energy distributions for 82Se, the baseline isotope of the SuperNEMO experiment [12, 13]. It was adapted from [14, 33] with some changes for simplicity and consistency and updated with modern notations. Here we use it to analyze in detail 0νββ decay two-electron angular and energy distributions for 124Sn, 130Te, and 136Xe. The nine NME required are calculated in this paper using the SVD shell model Hamiltonian [29] in jj55 model space which was thoroughly tested and validated for 124Sn in [9] and for 130Te and 136Xe in [30]. For an easier comparison to other results, we use gA value of 1.254; we include short-range correlations with CD-Bonn parametrization, finite nucleon size effects, and higher-order corrections of the nucleon current [44]. Should one change to the newer recommended gA value of 1.27 [45], the NME results would change by only 0.5% [46] and the effective PSF (multiplied by gA4) would change by 5%. This is negligible when compared to the uncertainties in the NME. gA quenching is not considered here and an extended justification for this decision is given in [21].In Table1, we present the nine dimensionless NME for 124Sn, 130Te, and 136Xe calculated in this work using an optimal closure energy E = 3.5 MeV which was obtained using a recently proposed method [47]. By using an optimal closure energy obtained for this Hamiltonian, we get 0νββ NME results in agreement with beyond closure approaches [48]. The definition of these NME and the details regarding their calculations are given in Appendix B of [21].Table 1
The nine NME of124Sn, 130Te, and 136Xe.
M G T M F M G T ω M F ω ⁢MGTq M F q M T M R M P 124Sn 1.85 - 0.47 2.05 - 0.46 1.79 - 0.27 0.0 1 2.66 - 2 . 06 130Te 1.66 - 0.44 1.86 - 0.43 1.59 - 0.25 - 0.0 1 2.56 - 1 . 73 136Xe 1.50 - 0.40 1.68 - 0.39 1.44 - 0.23 - 0.0 1 2.34 - 1 . 62The integrated PSF of the outgoing electrons, denoted byG1–G9, which enter Cα terms of (2), depend on the Q-value of the transition, the mass, and the charge of the final nucleus. We calculate these integrated PSF using a new effective method [34] which was in agreement with the latest results and was tested for 11 nuclei. Their complete expressions can also be found in Appendix C of [21]. The largest difference between our PSF and those of [35], among our three isotopes of interest, is of about 16% for G8 of 136Xe. Should one use the older formalism of [14], differences of about 88% are expected in the case of G8 for 136Xe. One should keep in mind that the expressions for the two-electron angular and energy distributions (A and B terms in (5a) and (5b)) contain energy-dependent (unintegrated) PSF and not the integrated PSF that are found in tables throughout the literature. Eqs. (B7) of [21] provide the details of their expressions. The values for the nine integrated PSF are presented in Table 2. The results shown include gA4=1.254 constant, such that G1=G010νgA4 in (1) and G1,9=G01,09gA4 of [35].Table 2
The nine PSF expressed inyr-1.
G 1 ⋅ 1 0 14 G 2 ⋅ 1 0 14 G 3 ⋅ 1 0 14 G 4 ⋅ 1 0 15 G 5 ⋅ 1 0 13 G 6 ⋅ 1 0 12 G 7 ⋅ 1 0 10 G 8 ⋅ 1 0 11 G 9 ⋅ 1 0 11 124Sn 1.977 4.184 1.248 3.909 6.685 3.648 2.749 2.585 0.731 130Te 3.122 8.026 2.092 6.267 10.18 5.335 4.340 4.261 1.106 136Xe 3.188 7.798 2.114 6.372 10.79 5.464 4.458 4.522 1.099C i factors (i=1,…,6) of (3), representing combinations of NME and PSF, are presented in Table 3. As one can clearly see, C5 term that appears in η mechanism is the largest. This is because of the large magnitude of G7, G8, and G9 PSF displayed in Table 2.Table 3
C i factors (i=1,…,6) corresponding to (3) expressed in yr-1.
C 1 ⋅ 1 0 14 C 2 ⋅ 1 0 14 C 3 ⋅ 1 0 1 1 C 4 ⋅ 1 0 13 C 5 ⋅ 1 0 9 C 6 ⋅ 1 0 13 124Sn 2.67 - 1 . 43 0 . 69 0.54 1.34 - 0.71 130Te 4.25 - 2 . 41 1 . 08 1.64 2.26 - 2.34 136Xe 4.36 - 2 . 44 1 . 30 1.04 2.24 - 1.34To test the possibility of disentangling the right-handed contributions in the framework of the left-right symmetric model, we consider three theoretical cases: the case of the mass mechanism denoted withν and presented with the black color in the figures, the case of λ mechanism dominance in competition with ν denoted with λ and displayed with the blue color, and the case of η mechanism dominance in competition with ν denoted with η and displayed with the red color. This color choice is consistent throughout all the figures.Considering the latest experimental limits [15, 35] from 76Ge 0νββ half-life, we select a value for the mass mechanism parameter ν which corresponds to a light neutrino mass of about 1 meV, while the values for λ and η effective parameters are chosen to barely dominate over the mass mechanism. Should their values be reduced four times, their contributions would not be distinguishable from the mass mechanism. Table 4 shows the values of these parameters used in the analysis.Table 4
The neutrino parameter values chosen for0νββ mechanisms described in the text.
ν λ η Mass mechanism (ν) 2 ⋅ 1 0 - 7 0 0 Lambda mechanism (λ) 2 ⋅ 1 0 - 7 2 ⋅ 1 0 - 7 0 Eta mechanism (η) 2 ⋅ 1 0 - 7 0 2 ⋅ 1 0 - 9We consider four combinations for the CP phasesϕ1 and ϕ2 (each one being 0 or π) which can influence the half-lives and the two-electron distributions. The maximum difference arising from the interference of these phases produces the uncertainties that are displayed as bands in the figures, changing the amplitudes and the shapes. The color convention for Figures 2–9 assigns red bands with a wavy pattern (lighter grey in black and white print) to η mechanism and blue bands without a pattern (darker grey in black and white print) to λ mechanism. As the mass mechanism does not depend on ϕ1 and ϕ2, there is no interference, and it is represented by a single thick black line. Because the mass mechanism is the most studied case in the literature, one may consider it as the reference case.The calculated half-lives of124Sn, 130Te and 136Xe are presented in Table 5. Their values can be obtained either from (2) or from (6). The maximum differences from the interference phases produce the intervals. For an easier comparison of the half-lives and the uncertainties, we also plot them in Figure 1. One can notice that the inclusion of λ or η contributions reduces the half-lives.Table 5
Calculated half-lives(T1/2) intervals for each mechanism expressed in years. The range of the interval corresponds to the uncertainty in the CP phases ϕ1 and ϕ2 in (2).
ν λ η 124Sn 2.73 ⋅ 1 0 26 [ 7 . 41 , 1 0 . 44 ] ⋅ 1 0 2 5 [ 2 . 47 , 4 . 69 ] ⋅ 1 0 25 130Te 2.12 ⋅ 1 0 26 [ 5 . 07 , 6 . 94 ] ⋅ 1 0 25 [ 1 . 85 , 3 . 34 ] ⋅ 1 0 25 136Xe 2.53 ⋅ 1 0 26 [ 6 . 23 , 8 . 6 0 ] ⋅ 1 0 25 [ 2 . 20 , 4 . 00 ] ⋅ 1 0 25Figure 1
The calculated lifetimes and their uncertainties (the hatched bars) form the interference of the unknown CP phases. From left to right, the three vertical bars for each nucleus correspond to the mass mechanism,η mechanism, and λ mechanism, respectively.Figure 2
Electrons angular distribution for124Sn. The red band corresponds to η mechanism and the blue band corresponds to λ mechanism. The width of the bands represents the uncertainties arising from the unknown CP phases ϕ1 and ϕ2.Figure 3
Same as Figure2 for 130Te.Figure 4
Same as Figure2 for 136Xe.Figure 5
Electrons energy distribution for124Sn. The red band corresponds to η mechanism and the blue band corresponds to λ mechanism. The width of the bands represents the uncertainties arising from the unknown CP phases ϕ1 and ϕ2.Figure 6
Same as Figure5 for 130Te.Figure 7
Same as Figure5 for 136Xe.Figure 8
The angular correlation coefficient for124Sn. The red band corresponds to η mechanism and the blue band corresponds to λ mechanism. The width of the bands represents the uncertainties arising from the unknown CP phases ϕ1 and ϕ2.Figure 9
Same as Figure8 for 130Te.The shapes of the two-electron angular distributions of (8) could be used to distinguish between the mass mechanism and λ or η mechanisms. However, many recorded events (tens or more) are needed for a reliable evaluation, and even then one can face difficulties due to the unknown CP phases. 124Sn angular distribution is presented in Figure 2. One can see that λ (blue bands) and η (red bands) exhibit similar shapes, differing in amplitude and opposite to that of the mass mechanism (black line). In the case of 130Te, the same is to be expected, but λ and η bands overlap due to the unknown phases, as seen in Figure 3. 136Xe angular distribution is very similar to that of 124Sn and is presented in Figure 4.In principle,λ and η contributions could be identified in the shapes of the two-electron energy distributions. While the tails of the distributions (when the difference between the energy of one electron and that of the other is maximal) overlap, the starting points (when both electrons have almost equal energies) are very different for λ mechanism from η mechanism. Figure 5 shows the energy distribution for 124Sn. 130Te energy distribution is presented in Figure 6. For 136Xe (Figure 7), we find an energy distribution very similar to that of 124Sn, like in the case of the angular distributions.To further aid with the disentanglement ofλ and η mechanisms, we provide plots of the angular correlation coefficient: α=Bϵ/A(ϵ) in our (4). This may help reduce the uncertainties induced by the unknown CP phases (see, e.g., Figures 6.5–6.9 of [14] and Figure 7 of [35]). From α(Δt), one may also obtain clearer separation from the mass mechanism over a wide range of energies. The angular correlation coefficient for 124Sn is presented in Figure 8. The same behavior can be identified in Figure 9 for 130Te and in Figure 10 for 136Xe.Figure 10
Same as Figure8 for 136Xe.


## 4. Conclusions

In this paper, we report shell model calculations necessary to disentangle the mixed right-handed/left-handed currents contributions (commonly referred to asη and λ mechanisms) from the mass mechanism in the left-right symmetric model. We perform an analysis of these contributions by considering three theoretical scenarios, one for the mass mechanism, one for λ dominance in competition with the mass mechanism, and one where η mechanism dominates in competition with the mass mechanism.The figures presented support the conclusions [14, 21] that one can distinguish λ or η dominance over the mass mechanism from the shape of the two-electron angular distribution, while one can discriminate λ from η mechanism using the shape of the energy distribution and that of the angular coefficient. The tables and the figures presented also show the uncertainties related to the effects of interference from the unknown CP-violating phases.We show our results for phase space factors, nuclear matrix elements, and lifetimes for0νββ transitions of 124Sn, 130Te, and 136Xe to ground states. In the case of the mass mechanism nuclear matrix elements, we obtain results which are consistent with previous calculations [21, 30], where the same SVD Hamiltonian was used. Similar to the case of 82Se [21], the inclusion of η and λ mechanisms contributions tends to decrease the half-lives.The phase space factors included in the analysis of lifetimes and two-electron distributions are calculated using a recently proposed accurate effective method [9] that provides results very close to those of [35]. Reference [35] takes into account consistently the effects of the realistic finite size proton distribution in the daughter nucleus, but it does not provide all the energy-dependent phase space contributions necessary for our analysis.Consistent with the calculations and the conclusions we obtained for82Se [21], if η mechanism exists, it may be favored to compete with the mass mechanisms because of the larger contribution from the phase space factors.Finally, we conclude that, in experiments where outgoing electrons can be tracked, such as SuperNEMO [12, 13] and NEXT [49], this analysis is possible if enough data is collected, generally of the order of a few tens of events. This may be beyond the capabilities of some of the current experiments, but should a positive neutrinoless double-beta decay measurement be achieved, it is expected that more resources could be allocated to improve the design, the statistics, and the variety of the investigated isotopes of the experiments which have realistic tracking capabilities.



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