1. Introduction
Tiny but nonzero neutrino masses explaining neutrino oscillation experiments [1] are unambiguous evidence of new physics (NP) beyond the standard model (SM). It is because in SM there are only left-handed neutrinos so therefore the neutrinos can acquire neither Dirac masses nor Majorana masses. Hence to explore any mechanism inducing the tiny neutrino masses, as well as relevant phenomenology, is an important direction to search for NP. The simple extension of SM is to introduce three right-handed neutrinos in a singlet of the gauge group SU(2) additionally, where the neutrinos acquire Dirac masses, and to fit the neutrino oscillation and nuclear decay experiments as well as astronomy observations, the corresponding Yukawa couplings of Higgs to the neutrinos are requested so tiny as ≲ 10−12, that is quite unnatural.
However, neutrino(s) may acquire masses naturally by introducing Majorana mass terms in extended SMs. Once a Majorana mass term is introduced, certain interesting physics arise. One of the consequences is that the lepton-number violation (LNV) processes, e.g. the nuclear neutrinoless double beta decays (0ν2β), etc may occur. Of them, the 0ν2β decays are especially interesting, because they may tell us sensitively about the nature of the neutrinos, whether Dirac [2] or Majorana [3]. When the decays 0ν2β are observed in experiments, most likely, the neutrinos contain Majorana components. Thus studying 0ν2β decays is attracting special attention.
Nowadays there are several experiments running to observe the 0ν2β decays, and the most stringent experimental bounds on the processes are obtained by GERDA [4, 5] and KamLAND-Zen [6, 7]. They adopt suitable approaches and nuclei such as 76Ge and 136Xe respectively. Now the latest experimental lower bound on the decay half-life given by GERDA experiments is ${T}_{1/2}^{0\nu }\gt 1.8\times {10}^{26}$ years (90% C.L.) for nucleus 76Ge [8], and in the near future, the sensitivity can reach up to 1028 years [9]. For the nucleus 136Xe, the most stringent lower bound on the decay half-life is ${T}_{1/2}^{0\nu }\,\gt 1.07\times {10}^{26}$ years (90% C.L.) given by KamLAND-Zen [6], and the corresponding future sensitivity can reach up to 2.4 × 1027 years [10]. Moreover, underground experiments PANDAX, CDEX etc, which are originally designed for searching for WIMP dark matter, are also planning to seek the 0ν2β decays i.e. they may observe the 0ν2β decays with the sensitivity which at least will set a fresh lower bound.
In literature, there are a lot of theoretical analyses on the 0ν2β decays. The analyses are carried out generally by dividing the estimation of the 0ν2β decays into three ‘factors’: one is at quark level to evaluate the amplitude for the ‘core’ process d + d → u + u + e + e of the decays; the second one is, from quark level to nucleon level, to involve the quark process into the relevant nucleon one i.e. the ‘initial’ quarks d and d involve into the two neutrons in the initial nucleus and the ‘final’ quarks u and u involve into the two protons in the final nucleus; the third one is, from nucleon level to nucleus level, the relevant nucleons involve in the initial nucleus and the final nucleus properly. For the ‘core’ process, in [11] a general Lorentz-invariant effective Lagrangian is constructed by dimension-9 operators, and in [12, 13] the QCD corrections to all of these dimension-9 operators are calculated. In [14] the short-range effects at quark level are considered, the analyses of the decay rates in the SM effective field theory are presented in [15–17], the 0ν2β decay rates are derived in [18], the corresponding nuclear matrix elements (NME) and phase-space factors (PSF) for the second and the third factors of the decays i.e. from quark level to nucleon and nucleus levels, are considered in [19–29], some theoretical predictions on the 0ν2β for certain models are presented in [30–34], and the theoretical analyses on the decays are reviewed in [35–37].
In this work, we are investigating the constrains from the 0ν2β decays for the B-L supersymmetric model (B-LSSM) and for the TeV scale left–right symmetric model (LRSM) comparatively. It is because the two models are typical: both have an LNV source but the mechanisms which give rise to the Majorana mass terms are different [38–51]. In the B-LSSM, the tiny neutrino masses are acquired naturally through the so-called type-I seesaw mechanism which is proposed firstly by Weinberg [52]. In the LRSM [53, 54], the tiny neutrino masses are acquired by both type-I and type-II seesaw mechanisms, in addition, the new right-handed gauge bosons ${W}_{R}^{\pm }$ is introduced in this model, then both left-handed and right-handed currents cause the 0ν2β decays [55–72]. As a result, the computations of the decays are much more complicated in the LRSM than those in the B-LSSM. Hence these two models, being representatives of NP models, are typical for the 0ν2β decays, and one may learn the mechanisms in the models well via analyzing the 0ν2β decays comparatively.
In the study here, we will mainly focus on the first ‘factor’ about the quark level i.e. the core process, which relates to the applied specific model closely. We will evaluate the Wilson coefficients of the operators relevant to the core process d + d → u + u + e + e etc on the models, whereas the estimation of the other two ‘factors’, i.e. to evaluate ‘NME’ and ‘PSF’ etc, we will follow the literature [29–33]. With respect to the ‘core’ process d + d → u + u + e + e, all of the contributions in the B-LSSM can be deduced directly quite well, while the contributions in LRSM cannot be so. As shown in [73], the calculations in the LRSM are much more complicated and the interference effects are quite hard to be considered well. In this work, a new approximation, i.e. the momenta of the two involved quarks inside the initial or final nuclei is tried to be set equal, is made so as may reduce all contributions in the LRSM quite similar to the case of B-LSSM. Then the calculations in LRSM are simplified quite a lot and under the approximation, the interference effects can be treated comparatively well. Finally, for comparison, we also present the results obtained by the traditional method [65].
The paper is organized as follows: In section 2 , for B-LSSM, the seesaw mechanisms which give rise to the tiny neutrino masses, the heavy neutral leptons as well, the relevant interactions etc, the calculations of the 0ν2β decay half-lives of the nuclei are given. Similarly, in section 3 , for LRSM, the seesaw mechanisms which give rise to the tiny neutrino masses, as well the heavy neutral leptons, the relevant interactions, and the calculations of the 0ν2β decay half-lives of the nuclei are given. In sections 4.1 and 4.2 the numerical results for B-LSSM and LRSM are presented respectively. Finally, in section 5 brief discussions and conclusions are given. In the appendix , the needed QCD corrections to the effective Lagrangian which contains the dimension-9 operators in the region from the energy scale μ ≃ MW to the energy scale μ ≃ 1.0 GeV are collected.
2. The B-LSSM for 0ν2β decays
In the B-LSSM, the local gauge group is SU(3)C ⨂ SU(2)L ⨂ U(1)Y ⨂ U(1)B−L, where B, L denote the baryon number and lepton number respectively, and the details about the gauge fields, their breaking, extra lepton and Higgs fields etc can be found in [42–51]. In this model, tiny neutrino masses are acquired by the so-called type-I seesaw mechanism, when the U(1)B−L symmetry is broken spontaneously by the two U(1)B−L singlet scalars (Higgs). The mass matrix for neutrinos and neutral heavy leptons in the model can be expressed as
$\begin{eqnarray}\left(\begin{array}{cc}0 & {M}_{D}^{{\rm{T}}}\\ {M}_{D} & {M}_{R}\end{array}\right),\end{eqnarray}$
and the mass matrix can be diagonalized in terms of a unitary matrix Uν as follows: $\begin{eqnarray}{U}_{\nu }^{{\rm{T}}}\left(\begin{array}{cc}0 & {M}_{D}^{{\rm{T}}}\\ {M}_{D} & {M}_{R}\end{array}\right){U}_{\nu }=\left(\begin{array}{cc}{\hat{m}}_{\nu } & 0\\ 0 & {\hat{M}}_{N}\end{array}\right),\end{eqnarray}$
where the neutrino masses ${\hat{m}}_{\nu }=\mathrm{diag}({m}_{{\nu }_{1}},{m}_{{\nu }_{2}},{m}_{{\nu }_{3}})$, the masses of the heavy neutral leptons ${\hat{M}}_{N}=\mathrm{diag}({M}_{{N}_{1}},{M}_{{N}_{2}},{M}_{{N}_{3}})$ and Uν is a matrix of 6 × 6 which can be rewritten as $\begin{eqnarray}{U}_{\nu }=\left(\begin{array}{c}U\quad S\\ T\quad V\end{array}\right),\end{eqnarray}$
where U, S, T, V are matrices of 3 × 3.The interactions, being applied later on, in the model are
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{I} & = & \displaystyle \frac{{\rm{i}}{g}_{2}}{\sqrt{2}}\sum _{j=1}^{3}\left[{U}_{{ij}}{\bar{e}}_{i}{\gamma }^{\mu }{P}_{L}{\nu }_{j}{W}_{L,\mu }^{-}\right.\\ & & \left.+{S}_{{ij}}{\bar{e}}_{i}{\gamma }^{\mu }{P}_{L}{N}_{j}{W}_{L,\mu }^{-}+{\rm{h}}.{\rm{c}}\right],\end{array}\end{eqnarray}$
where ν, N are the four-component fermion fields of the light and heavy neutral leptons respectively.The Feynman diagrams which are responsible for the dominated contributions to the 0ν2β decays in the B-LSSM are plotted in figure 1. Note here that the contributions from charged Higgs exchange(s) are ignored safely as they are highly suppressed by the charged Higgs masses and Yukawa couplings, thus in figure 1 the charged Higgs exchange does not appear at all. To evaluate the contributions corresponding to the Feynman diagrams for the 0ν2β decays, and to consider the roles of the neutral leptons (light neutrinos and heavy neutral leptons) in the Feynman diagrams, the useful formulae are collected below so as to deal with the neutrino propagator sandwiched by various chiral project operators ${P}_{L,R}=\tfrac{1}{2}(1\mp {\gamma }_{5})$:
$\begin{eqnarray}\begin{array}{rcl}{P}_{L}\displaystyle \frac{k/+m}{{k}^{2}-{m}^{2}}{P}_{L} & = & \displaystyle \frac{m}{{k}^{2}-{m}^{2}}{P}_{L},\\ {P}_{R}\displaystyle \frac{k/+m}{{k}^{2}-{m}^{2}}{P}_{R} & = & \displaystyle \frac{m}{{k}^{2}-{m}^{2}}{P}_{R},\\ {P}_{L}\displaystyle \frac{k/+m}{{k}^{2}-{m}^{2}}{P}_{R} & = & \displaystyle \frac{k/}{{k}^{2}-{m}^{2}}{P}_{R},\\ {P}_{R}\displaystyle \frac{k/+m}{{k}^{2}-{m}^{2}}{P}_{L} & = & \displaystyle \frac{k/}{{k}^{2}-{m}^{2}}{P}_{L},\end{array}\end{eqnarray}$
and the propagator $\begin{eqnarray}\displaystyle \frac{k/+m}{{k}^{2}-{m}^{2}}\simeq \left\{\begin{array}{l}-\displaystyle \frac{1}{m},\,\,\,{m}^{2}\gg {k}^{2}\\ \displaystyle \frac{k/+m}{{k}^{2}},\,\,\,{m}^{2}\ll {k}^{2}\end{array}\right..\end{eqnarray}$
Figure 1. The dominant Feynman diagrams for the 0ν2β decays in the B-LSSM Model. (a) The contributions from the heavy neutral lepton exchanges, (b) The contributions from the light neutrino exchange. |
Relating to the exchanges of the heavy neutral leptons (the virtual neutral lepton momentum k has $| k| \,\simeq 0.10\,\mathrm{GeV}\ll {M}_{{N}_{i}}$) for the decays, from figure 1(a) the effective Lagrangian at the energy scale $\mu \simeq {M}_{{W}_{L}}$ can be read out as3 ).
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{2{m}_{p}}{{G}_{F}^{2}\cos {\theta }_{C}^{2}}{{ \mathcal L }}_{\mathrm{eff}}^{\mathrm{BL}}(N)=\sum _{i}\displaystyle \frac{2{m}_{p}}{{M}_{{N}_{i}}}{\left({S}_{1i}\right)}^{2}\\ \quad \times [4(\bar{u}{\gamma }_{\mu }{P}_{L}d)(\bar{u}{\gamma }^{\mu }{P}_{L}d)\bar{e}{P}_{R}{e}^{c}]\\ \quad \equiv {C}_{3R}^{{LL}}(N){{ \mathcal O }}_{3R}^{{LL}},\\ \quad {C}_{3R}^{{LL}}(N)=\sum _{i}\displaystyle \frac{2{m}_{p}}{{M}_{{N}_{i}}}{\left({S}_{1i}\right)}^{2},\\ \quad {{ \mathcal O }}_{3Z}^{{XY}}=8(\bar{u}{\gamma }_{\mu }{P}_{X}d)(\bar{u}{\gamma }^{\mu }{P}_{Y}d)\bar{e}{P}_{Z}{e}^{c},\end{array}\end{eqnarray}$
where X, Y, Z = L, R, θC is the Cabibbo angle, mp is proton mass introduced for normalization of the effective Lagrangian, and S1i is the matrix elements in equation (Since the nuclear 0ν2β decays take place at the energy scale of about μ ≈ 0.10 GeV, obviously we need to consider the QCD corrections for the effective Lagrangian obtained at the energy scale $\mu \simeq {M}_{{W}_{L}}$ in equation (7 ) i.e. to evolute the effective Lagrangian in terms of renormalization group equation (RGE) method from the energy scale $\mu \simeq {M}_{{W}_{L}}$ to that μ ≃ 1.0 GeV first, where the corrections are in perturbative QCD (pQCD) region. Thus, for completeness, the QCD corrections to all of the possible dimension-9 operators which may contribute to the nuclear 0ν2β decays, are calculated by the RGE method, and the details of computations are collected in the appendix . Whereas the QCD corrections in the energy scale region μ ≃ 1.0 GeV ∼ μ ≃ 0.10 GeV, being in the non-perturbative QCD region, we take them into account by inputting the experimental measurements for the relevant current matrix elements of nucleons, which emerge when calculating the amplitude based on the effective Lagrangian at μ ≃ 0.10 GeV.
For the decays when considering contributions from the neutrino (${m}_{{\nu }_{i}}\ll | k| $) exchanges as described by the Feynman diagram figure 1(b), and the interferences between the light neutrinos’ and heavy neutral leptons’ contributions, to derive the effective Lagrangian for the neutrino contributions is better at the energy scale μ ≃ 1.0 GeV as heavy neutral leptons too. At this energy scale the effective Lagrangian can be written down according to the Feynman diagram figure 1(b) as below:8 ) may be involved like in the above case for the heavy neutral lepton, via inputting the experimental measurements for the relevant current matrix elements of nucleons, which emerge when calculating the matrix elements in the amplitudes at μ ≈ 0.10 GeV.
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{2{m}_{p}}{{G}_{F}^{2}\cos {\theta }_{C}^{2}}{{ \mathcal L }}_{\mathrm{eff}}^{\mathrm{BL}}(\nu )=\displaystyle \frac{{m}_{{\nu }_{i}}}{{m}_{e}}{\left({U}_{1i}\right)}^{2}\\ \quad \times \displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}{{ \mathcal O }}_{3R}^{{LL}}\equiv {C}_{3R}^{{LL}}(\nu )\displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}{{ \mathcal O }}_{3R}^{{LL}},\\ \quad {C}_{3R}^{{LL}}(\nu )=\displaystyle \frac{{m}_{{\nu }_{i}}}{{m}_{e}}{\left({U}_{1i}\right)}^{2}.\end{array}\end{eqnarray}$
Since the light neutrino exchange is of long range, the QCD corrections in the region μ ≃ 1.0 GeV ∼ μ ≃ 0.10 GeV to the coefficients in equation (To evaluate the half-life ${T}_{1/2}^{0\nu }\equiv \mathrm{ln}2/{\rm{\Gamma }}$ of the 0ν2β decays, the contributions from the heavy neutral leptons (figure 1(a)) and those from the neutrinos (figure 1(b)) should be summed up for the amplitudes. The half-life ${T}_{1/2}^{0\nu }\equiv \mathrm{ln}2/{\rm{\Gamma }}$ of the 0ν2β decays can be written as [28]10 ) that the factor $\tfrac{2{m}_{p}{m}_{e}}{-{k}^{2}}$ in equation (8 ) is absorbed into the so-called ‘neutrino potential’ which is used to compute the long range NME. AndA60 )), ${M}_{3}^{{XX}}(N)=-200\pm 56(-111\pm 31.08)$ [19, 28] for 76Ge(136Xe) is the NME corresponding to short range contributions which is defined as
$\begin{eqnarray}\displaystyle \frac{1}{{T}_{1/2}^{0\nu }}={G}^{0\nu }| {M}^{0\nu }{| }^{2}{\left|\displaystyle \frac{{m}_{{ee}}^{\mathrm{BL}}}{{m}_{e}}\right|}^{2},\end{eqnarray}$
where G0ν = 2.36 × 10−15 (14.56 × 10−15) yr−1 [28] for 76Ge(136Xe) is the PSF, M0ν = −6.64 ± 1.06 (−3.60 ± 0.58) [19, 28] for 76Ge(136Xe)4(4The NMEs adopted here are obtained within the framework of the microscopic interacting meson model for nuclei [28], and the uncertainties for NMEs calculations from the various nuclear structure models are quite wild, here we remind only that the NMEs obtained by various approaches are varying by a factor of (2–3) roughly [74].) is the NME corresponding to the long range contributions which is defined as $\begin{eqnarray}\begin{array}{l}{M}^{0\nu }\equiv \langle {{ \mathcal O }}_{F}^{+}| \displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}[4(\bar{u}{\gamma }_{\mu }{P}_{L}d)\\ \quad \times (\bar{u}{\gamma }^{\mu }{P}_{L}d)]| {{ \mathcal O }}_{I}^{+}\rangle \end{array}\end{eqnarray}$
with $| {{ \mathcal O }}_{I}^{+}\rangle $, $\langle {{ \mathcal O }}_{F}^{+}| $ denoting the initial and final nuclear states respectively. Note that in equation ( $\begin{eqnarray}{m}_{{ee}}^{\mathrm{BL}}\equiv {U}_{3}^{{XX}}{C}_{3R}^{{LL}}(N)\displaystyle \frac{{M}_{3}^{{XX}}(N)}{{M}^{0\nu }}+{C}_{3R}^{{LL}}(\nu ),\end{eqnarray}$
where U3XX is the QCD running factor from $\mu \simeq {M}_{{W}_{L}}$ to μ ≃ 1.0 GeV (the numerical result of U3XX can be found in equation ( $\begin{eqnarray}{M}_{3}^{{XX}}(N)\equiv \langle {{ \mathcal O }}_{F}^{+}| [4(\bar{u}{\gamma }_{\mu }{P}_{X}d)(\bar{u}{\gamma }^{\mu }{P}_{X}d)]| {{ \mathcal O }}_{I}^{+}\rangle .\end{eqnarray}$
3. The LRSM
For the model LRSM, the gauge fields are SU(3)C ⨂SU(2)L ⨂ SU(2)R ⨂ U(1)B−L, and the details about the gauge fields and their breaking can be found in [38–41]. In this model, the tiny neutrino masses are obtained by both of type-I and type-II seesaw mechanisms due to introducing the right-handed neutral leptons and two triplet Higgs (scalars) accordingly. In the model, the mass matrix for the neutral leptons generally is written as13 )5(5The matrix equation (13 ) with ML = 0 indicates the masses are acquired by type-I seesaw mechanism; it with MD = 0 indicates the masses are acquired by type-II seesaw mechanism; it in general feature indicates the masses are acquired by type-I+II seesaw mechanism.) can be diagonalized in terms of a unitary matrix Uν, whereas the matrix Uν can be expressed similarly as that in the case of the B-LSSM equation (3 ).
$\begin{eqnarray}\left(\begin{array}{c}{M}_{L}\,\,\,\,{M}_{D}^{{\rm{T}}}\\ {M}_{D}\,\,\,\,{M}_{R}\end{array}\right)\end{eqnarray}$
and the mass matrix equation (For the model LRSM, if the left–right symmetry is not broken manifestly but spontaneously, i.e. gL = gR ≡ g2 and as one consequence, the mass terms of W bosons can be written as
$\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{{M}_{W}}=\displaystyle \frac{{g}_{2}^{2}}{4}\left(\begin{array}{c}{W}_{L}^{+},\,\,{W}_{R}^{+}\end{array}\right)\\ \,\times \left(\begin{array}{c}{v}_{1}^{2}+{v}_{2}^{2}+2{v}_{L}^{2}\quad 2{v}_{1}{v}_{2}\\ 2{v}_{1}{v}_{2}\quad {v}_{1}^{2}+{v}_{2}^{2}+2{v}_{R}^{2}\end{array}\right)\left(\begin{array}{c}{W}_{L}^{-}\\ {W}_{R}^{-}\end{array}\right),\end{array}\end{eqnarray}$
where v1, v2, vL, vR (vL ≪ vR) are the VEVs of new scalars (Higgs) in the LRSM. Then the physical masses of the W bosons can be obtained [72] $\begin{eqnarray}\begin{array}{rcl}{M}_{{W}_{1}} & \simeq & \displaystyle \frac{{g}_{2}}{2}{\left({v}_{1}^{2}+{v}_{2}^{2}\right)}^{1/2},\\ {M}_{{W}_{2}} & \simeq & \displaystyle \frac{{g}_{2}}{\sqrt{2}}{v}_{R}.\end{array}\end{eqnarray}$
The mass eigenstates ${W}_{1,2}^{\pm }$ are related to the interaction eigenstates ${W}_{L,R}^{\pm }$ by ζ $\begin{eqnarray}\left(\begin{array}{c}{W}_{1}^{\pm }\\ {W}_{2}^{\pm }\end{array}\right)=\left(\begin{array}{c}\cos \zeta ,\,\,\sin \zeta \\ -\sin \zeta ,\,\,\cos \zeta \end{array}\right)\left(\begin{array}{c}{W}_{L}^{\pm }\\ {W}_{R}^{\pm }\end{array}\right),\end{eqnarray}$
where $\tan 2\zeta =\tfrac{2{v}_{1}{v}_{2}}{{v}_{R}^{2}-{v}_{L}^{2}}$.The interactions, being applied later on, in the model are
$\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{I}^{{LRSM}}=\displaystyle \frac{{\rm{i}}{g}_{2}}{\sqrt{2}}\sum _{j=1}^{3}\left[{\bar{e}}_{i}(\cos \zeta {U}_{{ij}}{\gamma }^{\mu }{P}_{L}\right.\\ \quad +\sin \zeta {T}_{{ij}}^{* }{\gamma }^{\mu }{P}_{R}){\nu }_{j}{W}_{1,\mu }^{-}\\ \quad +{\bar{e}}_{i}(\cos \zeta {T}_{{ij}}^{* }{\gamma }^{\mu }{P}_{R}-\sin \zeta {U}_{{ij}}{\gamma }^{\mu }{P}_{L}){\nu }_{j}{W}_{2,\mu }^{-}\\ \quad +{\bar{e}}_{i}(\cos \zeta {S}_{{ij}}{\gamma }^{\mu }{P}_{L}+\sin \zeta {V}_{{ij}}^{* }{\gamma }^{\mu }{P}_{R}){N}_{j}{W}_{1,\mu }^{-}\\ \quad +{\bar{e}}_{i}(\cos \zeta {V}_{{ij}}^{* }{\gamma }^{\mu }{P}_{R}-\sin \zeta {S}_{{ij}}{\gamma }^{\mu }{P}_{L}){N}_{j}{W}_{2,\mu }^{-}\\ \quad +\bar{u}(\cos \zeta {\gamma }^{\mu }{P}_{L}+\sin \zeta {\gamma }^{\mu }{P}_{R}){{dW}}_{1,\mu }^{-}\\ \quad \left.+\bar{u}(\cos \zeta {\gamma }^{\mu }{P}_{R}-\sin \zeta {\gamma }^{\mu }{P}_{L}){{dW}}_{2,\mu }^{-}+{\rm{h}}.{\rm{c}}\right],\end{array}\end{eqnarray}$
where the definitions for U, S, T, V, ν, N are the same as the ones in the B-LSSM.In the LRSM, the dominant contributions to the 0ν2β decays are represented by Feynman diagrams figure 2.
Figure 2. The dominant Feynman diagrams for the 0ν2β decays in the LRSM. (a) The contributions from the heavy neutral lepton exchanges, (b) the contributions from the light neutrino exchanges. |
In comparison with those of the B-LSSM, the contributions from the Higgs exchanges can be ignored so in figure 2 there is no Higgs exchange at all, but besides the left-handed gauge boson components ${W}_{L}^{\pm }$, there are right-handed components ${W}_{R}^{\pm }$ in ${W}_{1}^{\pm }$ and ${W}_{2}^{\pm }$ gauge bosons. Therefore the situation in determining the effective Lagrangian for the 0ν2β decays is different and comparatively complicated than that for the B-LSSM. In the case of the contributions from the heavy neutral lepton exchanges shown in figure 2 (a), considering the fact that the heavy neutral leptons propagator $\tfrac{{M}_{{N}_{i}}+k/}{{k}^{2}-{M}_{{N}_{i}}^{2}}\approx \tfrac{-1}{{M}_{{N}_{i}}}$ (${M}_{{N}_{i}}\geqslant {M}_{{W}_{1}}$) and $\sin \zeta ,{S}_{1i},{T}_{1i}\ll 1.0$ in the decays, by using equations (5 ), (6 ) the effective Lagrangian at the energy scale $\mu \simeq {M}_{{W}_{1}}$ (W1 is the lighter one boson between W1,2) can be written down as follows18 ) are neglected. However, since $\tan 2\zeta =\tfrac{2{v}_{1}{v}_{2}}{{v}_{R}^{2}-{v}_{L}^{2}}$, i.e. $\zeta \approx {{xM}}_{{W}_{1}}^{2}/{M}_{{W}_{2}}^{2}$ and when x ≡ v2/v1 > 0.02 [75], the terms with ${C}_{3L}^{{RL}}(N)$ and ${C}_{3L}^{{LL}}(N)$ can also make essential contributions compared with the terms with ${C}_{3L}^{{RR}}(N)$, ${C}_{3R}^{{LL}}(N)$. Thus in this work, when evaluating the 0ν2β decays we would like to keep the contributions from the terms of ${C}_{3L}^{{RL}}(N)$, ${C}_{3L}^{{LL}}(N)$, and consider the QCD corrections to the effective Lagrangian equation (18 ) in a similar way as that in the B-LSSM.
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{2{m}_{p}}{{G}_{F}^{2}\cos {\theta }_{C}^{2}}{{ \mathcal L }}_{\mathrm{eff}}^{\mathrm{LR}}(N)={C}_{3R}^{{LL}}(N){{ \mathcal O }}_{3R}^{{LL}}\\ \quad +{C}_{3L}^{{LL}}(N){{ \mathcal O }}_{3L}^{{LL}}+{C}_{3L}^{{RL}}(N){{ \mathcal O }}_{3L}^{{RL}}\\ \quad +{C}_{3L}^{{RR}}(N){{ \mathcal O }}_{3L}^{{RR}};\\ \quad {C}_{3R}^{{LL}}(N)=\displaystyle \frac{2{m}_{p}}{{M}_{{N}_{i}}}{\cos }^{4}\zeta {S}_{1i}^{2},\\ \quad {C}_{3L}^{{LL}}(N)=\displaystyle \frac{2{m}_{p}}{{M}_{{N}_{i}}}{\cos }^{2}\zeta {\sin }^{2}\zeta {V}_{1i}^{* 2}.\\ \quad {C}_{3L}^{{RL}}(N)=\displaystyle \frac{2{m}_{p}}{{M}_{{N}_{i}}}{\cos }^{3}\zeta \sin \zeta {V}_{1i}^{* 2}{\left(\displaystyle \frac{{M}_{{W}_{L}}}{{M}_{{W}_{R}}}\right)}^{2},\\ \quad {C}_{3L}^{{RR}}(N)=\displaystyle \frac{2{m}_{p}}{{M}_{{N}_{i}}}{\cos }^{4}\zeta {V}_{1i}^{* 2}{\left(\displaystyle \frac{{M}_{{W}_{L}}}{{M}_{{W}_{R}}}\right)}^{4}.\end{array}\end{eqnarray}$
In the literature, due to small ζ, the contributions corresponding to ${C}_{3L}^{{RL}}(N)$, ${C}_{3L}^{{LL}}(N)$ in equation (As the next step, when considering the contributions from the light neutrino exchanges as figure 2(b), owing to the fact that ${W}_{1,2}^{\pm }$ contain both components ${W}_{L,R}^{\pm }$ in LRSM, according to equations (5 ), (6 ), the light neutrino propagators with chiral project operators ${P}_{L,R}\tfrac{{m}_{{\nu }_{i}}+k/}{{k}^{2}-{m}_{{\nu }_{i}}^{2}}{P}_{L,R}$ or ${P}_{L,R}\tfrac{{m}_{{\nu }_{i}}+k/}{{k}^{2}-{m}_{{\nu }_{i}}^{2}}{P}_{R,L}$, as central factors, finally contribute the factors as $\tfrac{{m}_{{\nu }_{i}}}{{k}^{2}}{P}_{L,R}$ or $\tfrac{k/}{{k}^{2}}{P}_{L,R}$ respectively to the results. Namely, the factors $\tfrac{k/}{{k}^{2}}{P}_{L,R}$ are new and substantial, and in B-LSSM they do not appear at all. Moreover when the contributions from the ‘higher order’ terms for $\sin \zeta ,\,{S}_{1i},\,{T}_{1i}$, such as those small terms proportional to ${\sin }^{2}\zeta ,\,{S}_{1i}^{2},\,{T}_{1i}^{2}$ etc, are ignored, then the effective Lagrangian at the energy scale μ ≃ 1.0 GeV may be read out from figure 2(b) as
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{2{m}_{p}}{{G}_{F}^{2}\cos {\theta }_{C}^{2}}{{ \mathcal L }}_{\mathrm{eff}}^{\mathrm{LR}}(\nu )\\ \quad ={\cos }^{4}\zeta \displaystyle \frac{{U}_{1i}^{2}{m}_{\nu i}}{{m}_{e}}\displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}{{ \mathcal O }}_{3R}^{{LL}}\\ \quad +\,{\cos }^{3}\zeta \sin \zeta {U}_{1i}{T}_{1i}^{* }[4(\bar{u}{\gamma }_{\mu }{P}_{L}d)(\bar{u}{\gamma }_{\nu }{P}_{L}d)\bar{e}{\gamma }^{\mu }\\ \quad \times \displaystyle \frac{2{m}_{p}k/}{{k}^{2}}{\gamma }^{\nu }{e}^{c}]+\\ \quad {\cos }^{4}\zeta {U}_{1i}{T}_{1i}^{* }\displaystyle \frac{{M}_{{W}_{L}}^{2}}{{M}_{{W}_{R}}^{2}}[4(\bar{u}{\gamma }_{\mu }{P}_{R}d)\\ \quad \times (\bar{u}{\gamma }_{\nu }{P}_{L}d)\bar{e}{\gamma }^{\mu }\displaystyle \frac{2{m}_{p}k/}{{k}^{2}}{\gamma }^{\nu }{P}_{R}{e}^{c}].\end{array}\end{eqnarray}$
The QCD corrections in the energy scale region μ ≃ 1.0 GeV to μ ≃ 0.10 GeV, being of non-perturbative QCD, are taken into account by inputting in the experimental measurements for the relevant current matrix elements of nucleons, which emerge at the effective Lagrangian at μ ≃ 0.10 GeV.In [55–66], the second term and the third term of equation (19 ) are defined as η, λ respectively. Extracting the factors
$\begin{eqnarray}\begin{array}{rcl}{C}_{\eta } & = & {\cos }^{3}\zeta \sin \zeta {U}_{1i}{T}_{1i}^{* },\\ {C}_{\lambda } & = & {\cos }^{4}\zeta {U}_{1i}{T}_{1i}^{* }\displaystyle \frac{{M}_{{W}_{1}}^{2}}{{M}_{{W}_{2}}^{2}},\end{array}\end{eqnarray}$
then the operators $[4(\bar{u}{\gamma }_{\mu }{P}_{L}d)(\bar{u}{\gamma }_{\nu }{P}_{L}d)\bar{e}{\gamma }^{\mu }\tfrac{2{m}_{p}k/}{{k}^{2}}{\gamma }^{\nu }{e}^{c}]$, $[4(\bar{u}{\gamma }_{\mu }{P}_{R}d)(\bar{u}{\gamma }_{\nu }{P}_{L}d)\bar{e}{\gamma }^{\mu }\tfrac{2{m}_{p}k/}{{k}^{2}}{\gamma }^{\nu }{P}_{R}{e}^{c}]$ are attributed to the calculations of NME and PSF.Whereas when calculating the NMEs and PSF, the interference effects among the contributions, especially to consider the contributions from the factors $\tfrac{k/}{{k}^{2}}{P}_{L,R}$ for the light neutrino exchanges, are complicated and hard (in the literature, to treat them even the Lorentz covariance is lost [76]). In this work to calculate NMEs and PSF, we try to make an additional approximation on the contributions relevant to the factors $\tfrac{k/}{{k}^{2}}{P}_{L,R}$ for the light neutrino exchanges, which we call a ‘frozen approximation’. Under the approximation, the momenta of the two involved quarks inside the initial nucleus and two involved quarks inside the final nucleus (figure 2) are assumed to be equal approximately:A60 )), ${C}_{3Z}^{{XY}}(N,\nu )$ is the coefficient defined in equation (23 ), ${G}_{11-}^{(0)}=-0.28\times {10}^{-15}(-1.197\,\times {10}^{-15}){\mathrm{years}}^{-1}$ [28] for 76Ge(136Xe) is the PSF, and ${M}_{3}^{{XY}}(\nu )=4.24\pm 0.68(2.17\pm 0.35)$, ${M}_{3}^{{XY}}(N)=99.8\,\pm 27.94(51.2\pm 14.34)$ [19, 28] for 76Ge(136Xe) are the NMEs corresponding to the exchange of light neutrinos, heavy neutral leptons respectively which are defined as22 ) is absorbed into the ‘neutrino potential’ which is used to compute the long range NME. In addition, we should note that in equation (25 ) the terms of ${C}_{5}^{{RR}}(\nu ),{C}_{5}^{{LL}}(\nu ),{C}_{5}^{{RL}}(\nu ),{C}_{5}^{{LR}}(\nu )$ in equation (23 ) do not make contributions to the 0ν2β decays for the reason below. They make contributions to 0ν2β decays in the form23 ), then equation (27 ) shows the contributions to the decays from the operators corresponding to the coefficients ${C}_{5}^{{RR}}(\nu )$, ${C}_{5}^{{LL}}(\nu )$, ${C}_{5}^{{RL}}(\nu )$, ${C}_{5}^{{LR}}(\nu )$ are cancelled completely.
$\begin{eqnarray}{p}_{1}\simeq {p}_{2}\equiv \bar{p},\,\,{k}_{1}\simeq {k}_{2}\equiv \overline{k}.\end{eqnarray}$
With the ‘frozen approximation’ and ‘the on-shell approximation’ onto the momenta for the out legs as well, all contributions corresponding to figure 2(b) can be well-deduced and the final results can be collected as $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{2{m}_{p}}{{G}_{F}^{2}\cos {\theta }_{C}^{2}}{{ \mathcal L }}_{\mathrm{eff}}^{\mathrm{LR}}(\nu )=\displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}[{C}_{3R}^{{LL}}(\nu ){{ \mathcal O }}_{3R}^{{LL}}\\ \quad +{C}_{3L}^{{LL}}(\nu ){{ \mathcal O }}_{3L}^{{LL}}+{C}_{3L}^{{RL}}(\nu ){{ \mathcal O }}_{3L}^{{RL}}+{C}_{3R}^{{RL}}(\nu ){{ \mathcal O }}_{3R}^{{RL}}\\ \quad +{C}_{5}^{{LL}}(\nu ){{ \mathcal O }}_{5}^{{LL}}+{C}_{5}^{{RR}}(\nu ){{ \mathcal O }}_{5}^{{RR}}\\ \quad +{C}_{5}^{{LR}}(\nu ){{ \mathcal O }}_{5}^{{LR}}+{C}_{5}^{{RL}}(\nu ){{ \mathcal O }}_{5}^{{RL}}],\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal O }}_{5}^{{XY}} & = & 4(\bar{u}{\gamma }_{\mu }{P}_{X}d)(\bar{u}{P}_{Y}d)\bar{e}{\gamma }_{\mu }{\gamma }^{5}{e}^{c},\\ {C}_{3R}^{{LL}}(\nu ) & = & \displaystyle \frac{1}{{m}_{e}}{\cos }^{3}\zeta {U}_{1i}({m}_{{\nu }_{i}}\cos \zeta {U}_{1i}\\ & & -{m}_{e}\sin \zeta {T}_{1i}^{* }),\\ {C}_{3L}^{{LL}}(\nu ) & = & -{\cos }^{3}\zeta \sin \zeta {U}_{1i}{T}_{1i}^{* },\\ {C}_{3L}^{{RL}}(\nu ) & = & {C}_{3R}^{{RL}}(\nu )=-\displaystyle \frac{1}{2}{\cos }^{4}\zeta {U}_{1i}{T}_{1i}^{* }{\left(\displaystyle \frac{{M}_{{W}_{1}}}{{M}_{{W}_{2}}}\right)}^{2},\\ {C}_{5}^{{RR}}(\nu ) & = & -{C}_{5}^{{LL}}(\nu )\\ & = & \displaystyle \frac{{m}_{u}-{m}_{d}}{{m}_{e}}{\cos }^{3}\zeta \sin \zeta {U}_{1i}{T}_{1i}^{* }\\ & & -\displaystyle \frac{{m}_{d}}{{m}_{e}}{\cos }^{4}\zeta {U}_{1i}{T}_{1i}^{* }{\left(\displaystyle \frac{{M}_{{W}_{1}}}{{M}_{{W}_{2}}}\right)}^{2},\\ {C}_{5}^{{LR}}(\nu ) & = & -{C}_{5}^{{RL}}(\nu )\\ & = & \displaystyle \frac{{m}_{d}-{m}_{u}}{{m}_{e}}{\cos }^{3}\zeta \sin \zeta {U}_{1i}{T}_{1i}^{* }\\ & & -\displaystyle \frac{{m}_{u}}{{m}_{e}}{\cos }^{4}\zeta {U}_{1i}{T}_{1i}^{* }{\left(\displaystyle \frac{{M}_{{W}_{1}}}{{M}_{{W}_{2}}}\right)}^{2}.\end{array}\end{eqnarray}$
Then in LRSM, the half-life of 0ν2β decays can be written as [28] $\begin{eqnarray}\displaystyle \frac{1}{{T}_{1/2}^{0\nu }}={G}^{0\nu }| {M}^{0\nu }{| }^{2}{\left|\displaystyle \frac{{m}_{{ee}}^{\mathrm{LR}}}{{m}_{e}}\right|}^{2},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}{m}_{{ee}}^{\mathrm{LR}}={m}_{e}\left\{\left|[{C}_{3L}^{{RR}}(N)+{C}_{3L}^{{LL}}(N)]{U}_{3}^{{XX}}\displaystyle \frac{{M}_{3}^{{XX}}(N)}{{M}^{0\nu }}\right.\right.\\ \quad +{C}_{3L}^{{RL}}(N){U}_{(31)11}^{{XY}}\displaystyle \frac{{M}_{3}^{{XY}}(N)}{{M}^{0\nu }}\\ \quad {\left.+{C}_{3L}^{{LL}}(\nu )+{C}_{3L}^{{RL}}(\nu )\displaystyle \frac{{M}_{3}^{{XY}}(\nu )}{{M}^{0\nu }}\right|}^{2}\\ \quad +{\left|{C}_{3R}^{{LL}}(N){U}_{3}^{{XX}}\displaystyle \frac{{M}_{3}^{{XX}}(N)}{{M}^{0\nu }}+{C}_{3R}^{{LL}}(\nu )\right|}^{2}\\ \quad +2\displaystyle \frac{{G}_{11-}^{(0)}}{{G}^{0\nu }}\left[{C}_{3L}^{{RR}}(N)+{C}_{3L}^{{LL}}(N)]{U}_{3}^{{XX}}\displaystyle \frac{{M}_{3}^{{XX}}(N)}{{M}^{0\nu }}\right.\\ \quad +{C}_{3L}^{{RL}}(N){U}_{(31)11}^{{XY}}\displaystyle \frac{{M}_{3}^{{XY}}(N)}{{M}^{0\nu }}\\ \quad \left.+{C}_{3L}^{{LL}}(\nu )+{C}_{3L}^{{RL}}(\nu )\displaystyle \frac{{M}_{3}^{{XY}}(\nu )}{{M}^{0\nu }}\right]\\ \quad {\left.\times \left[{C}_{3R}^{{LL}}(N){U}_{3}^{{XX}}\displaystyle \frac{{M}_{3}^{{XX}}(N)}{{M}^{0\nu }}+{C}_{3R}^{{LL}}(\nu )\right]\right\}}^{1/2},\end{array}\end{eqnarray}$
where ${U}_{(31)}^{{XY}}$ is the 2 × 2 RGE evolution matrix from $\mu \simeq {M}_{{W}_{1}}$ to μ ≃ 1.0 GeV (the numerical result of ${U}_{(31)}^{{XY}}$ can be found in equation ( $\begin{eqnarray}\begin{array}{rcl}{M}_{3}^{{XY}}(\nu ) & \equiv & \langle {{ \mathcal O }}_{F}^{+}| \displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}[4(\bar{u}{\gamma }_{\mu }{P}_{X}d)(\bar{u}{\gamma }^{\mu }{P}_{Y}d)]| {{ \mathcal O }}_{I}^{+}\rangle ,\\ {M}_{3}^{{XY}}(N) & \equiv & \langle {{ \mathcal O }}_{F}^{+}| [4(\bar{u}{\gamma }_{\mu }{P}_{X}d)(\bar{u}{\gamma }^{\mu }{P}_{Y}d)]| {{ \mathcal O }}_{I}^{+}\rangle .\end{array}\end{eqnarray}$
Similar to the case of B-LSSM, the factor $\tfrac{2{m}_{p}{m}_{e}}{-{k}^{2}}$ in equation ( $\begin{eqnarray}\begin{array}{l}{M}_{5}^{{XX}}(\nu )[{C}_{5}^{{RR}}(\nu )+{C}_{5}^{{LL}}(\nu )]\\ \quad +{M}_{5}^{{XY}}(\nu )[{C}_{5}^{{RL}}(\nu )+{C}_{5}^{{LR}}(\nu )],\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{M}_{5}^{{XX}}(\nu ) & \equiv & \langle {{ \mathcal O }}_{F}^{+}| \displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}[4(\bar{u}{\gamma }_{\mu }{P}_{X}d)(\bar{u}{P}_{X}d)]| {{ \mathcal O }}_{I}^{+}\rangle ,\\ {M}_{5}^{{XY}}(\nu ) & \equiv & \langle {{ \mathcal O }}_{F}^{+}| \displaystyle \frac{2{m}_{p}{m}_{e}}{-{k}^{2}}[4(\bar{u}{\gamma }_{\mu }{P}_{X}d)(\bar{u}{P}_{Y}d)]| {{ \mathcal O }}_{I}^{+}\rangle .\end{array}\end{eqnarray}$
Since ${C}_{5}^{{RR}}(\nu )=-{C}_{5}^{{LL}}(\nu )$, ${C}_{5}^{{RL}}(\nu )=-{C}_{5}^{{LR}}(\nu )$ as shown in equation (4. Numerical results
With the formulas obtained above, now in this section, we do the numerical calculations and present the results on the 0ν2β decays for the nuclei 76Ge and 136Xe accordingly. In our numerical calculations, the parameters are taken as the weak boson mass: ${M}_{{W}_{L}}=80.385\,\mathrm{GeV}$ for B-LSSM and ${M}_{{W}_{1}}\,=80.385\,\mathrm{GeV}$ for LRSM, mb = 4.65 GeV for b-quark mass, mc = 1.275 GeV for c-quark mass, αem(mZ) = 1/128.9 for the coupling of the electromagnetic interaction, αs(mZ) = 0.118 for the coupling of the strong interaction. The constraints from available experimental data, such as the most stringent upper limit on the sum of neutrino masses by PLANK [77] ${\sum }_{i}{m}_{{\nu }_{i}}\lt 0.12\,\mathrm{eV};$ the neutrino mass-squared differences obtained via analyzing the solar and atmospheric neutrino oscillation data at 3σdeviations [77]3 ), being the Pontecorvo–Maki–Nakagawa–Sakata mixing matrix [1] to describe the mixing of the light neutrinos. Then we have
$\begin{eqnarray}\begin{array}{l}{\rm{\Delta }}{m}_{12}^{2}\equiv {m}_{{\nu }_{2}}^{2}-{m}_{{\nu }_{1}}^{2}=(7.4\pm 0.61)\times {10}^{-5}\,{\mathrm{eV}}^{2},\\ \quad \left\{\begin{array}{l}{\rm{\Delta }}{m}_{13}^{2}\equiv {m}_{{\nu }_{3}}^{2}-{m}_{{\nu }_{1}}^{2}\approx (2.526\pm 0.1)\times {10}^{-3}\,{\mathrm{eV}}^{2}\qquad \qquad (\mathrm{NH})\\ {\rm{\Delta }}{m}_{32}^{2}\equiv {m}_{{\nu }_{2}}^{2}-{m}_{{\nu }_{3}}^{2}\approx (2.508\pm 0.1)\times {10}^{-3}\,{\mathrm{eV}}^{2}\qquad \qquad (\mathrm{IH}),\end{array}\right.\end{array}\end{eqnarray}$
etc, are well-considered. Owing to the fact that the hierarchy of neutrino masses has not been fixed yet, we take the two possibilities below to carry out the analyses i.e. the normal hierarchy (NH) ${m}_{{\nu }_{1}}\lt {m}_{{\nu }_{2}}\lt {m}_{{\nu }_{3}}$ and the inverse hierarchy (IH) ${m}_{{\nu }_{3}}\lt {m}_{{\nu }_{1}}\lt {m}_{{\nu }_{2}}$. Moreover, we use the matrix U, the upper-left sub-matrix of the whole matrix Uν in equation ( $\begin{eqnarray}\begin{array}{l}{\left({U}_{1i}\right)}^{2}{m}_{\nu i}={c}_{12}^{2}{c}_{13}^{2}{{\rm{e}}}^{i\alpha }{m}_{{\nu }_{1}}\\ \quad +{c}_{13}^{2}{s}_{12}^{2}{{\rm{e}}}^{i\beta }{m}_{{\nu }_{2}}+{s}_{13}^{2}{m}_{{\nu }_{3}},\end{array}\end{eqnarray}$
where α, β are Majorana CP violating phases, ${c}_{\mathrm{12,13}}\,\equiv \cos {\theta }_{\mathrm{12,13}},{s}_{\mathrm{12,13}}\equiv \sin {\theta }_{\mathrm{12,13}}$ are the neutrino oscillation parameters and at 3σ error level they can be written as $\begin{eqnarray}\begin{array}{rcl}{s}_{12}^{2} & = & 0.3125\pm 0.0375,\\ {s}_{13}^{2} & = & 0.022405\pm 0.001965.\end{array}\end{eqnarray}$
The direct searching for the right-handed gauge boson sets the lower bound for the mass of W2 boson as ${M}_{{W}_{2}}\gtrsim 4.8\,\mathrm{TeV}$ [78–81], and the WR, WL mixing angle as ζ ≲ 7.7 × 10−4 [65]. Finally, we have x ≡ v2/v1 > 0.02 [75] required by the CP-violation data for the K and B mesons, where v1, v2 are the VEVs of the two Higgs particles in the LRSM. Equations (11 ) and (25 ) show meeLR depends the NME and phase space factor of the chosen nuclei. However these factors appear in the expression of meeLR in the form of some ratios, and these ratios for 76Ge are equal to those for 136Xe roughly, hence we can use the same expression of meeLR for 76Ge, 136Xe in the analyses later on.
4.1. The numerical results for the B-LSSM
According to the above analysis, in B-LSSM the contributions from the heavy neutral leptons shown in equation (7 ) are highly suppressed (S1i ≈ 10−7) for the TeV-scale heavy neutral leptons, hence the dominant contributions to the 0ν2β decays come from the light neutrinos. Then with the neutrino mixing parameters29 ) at 3σ error level, ${m}_{{ee}}^{\mathrm{BL}}$ versus mν-lightest (the lightest neutrino mass) for the B-LSSM is plotted in figure 3, where the green (red) points denote the NH (IH) results, the blue (purple) solid line denotes the constraints from the lower 0ν2β decay half-life bound of 76Ge (136Xe), the blue (purple) dashed line denotes the experimental ability of 76Ge (136Xe) for the next generation of experiments, the orange solid (dashed) line denotes the constraints from PLANK 2018 for NH (IH) neutrino masses (the meaning for the red points, green points, blue lines, purple lines, orange lines is also adopted accordingly in the figures later on). In figure 3, ${m}_{{ee}}^{\mathrm{BL}}$ is well below the experimental upper bounds in the cases of NH and IH, and there is not tighter restriction on the mν-lightest than that offered by PLANK. Additionally, the blue dashed line shows that there is certainly opportunity to observe the 0ν2β decays in the next generation of experiments, whereas if any of the decays is not observed in the next generation of experiments, then the IH neutrino masses will be excluded completely by the blue dashed line in figure 3.
$\begin{eqnarray}\begin{array}{rcl}{s}_{12}^{2} & = & (0.275\sim 0.35),\,\,{s}_{13}^{2}=(0.02044\sim 0.024\,37),\\ \alpha & = & (0\sim 2\pi ),\,\,\beta =(0\sim 2\pi )\end{array}\end{eqnarray}$
and the neutrino mass-squared differences as those in equation (
Figure 3. ${m}_{{ee}}^{\mathrm{BL}}$ versus mν-lightest under scanning the neutrino mass-squared differences equation ( |
4.2. The numerical results for the LRSM
In the LRSM owing to the bosons WL, WR and their mixing, the situation is much more complicated than that in the B-LSSM, and both the light neutrinos and heavy neutral leptons make substantial contributions to the 0ν2β decays. For TeV-scale heavy neutral leptons in the LRSM, the consequences of the type II seesaw dominance are similar to those of the type I seesaw dominance, while the consequences are very different from the ones of type I+II seesaw dominance, because the light-heavy neutral lepton mixing is not much suppressed in this case. Hence we do the numerical computations for the type I and type I+II seesaw dominance cases in our analyses. For simplicity and not losing general features, we assume that there is only one heavy neutral lepton making substantial contributions to the model. It indicates that there are no off-diagonal elements in the matrix MR in equation (13 ), i.e. we have ${M}_{R}\simeq {\hat{M}}_{N}\,=\mathrm{diag}({M}_{{N}_{1}},{M}_{{N}_{2}},{M}_{{N}_{3}})$. Then under the dominance of either the type I seesaw or the type I+II seesaw mass mechanisms, we compute ${m}_{{ee}}^{\mathrm{LR}}$ numerically in turn.
4.2.1. The results under type I seesaw dominance
Under the type I seesaw dominance and due to the tiny neutrino masses, the sub-matrix T in equation (3 ) has T1j ≪ 1 (j = 1, 2, 3). It indicates that the contributions from the terms proportional to ${T}_{1i}^{* }$ as shown in equation (23 ) are highly suppressed. Then scanning the parameter space in equations (29 ), (32 ) and in the parameter space
$\begin{eqnarray}\begin{array}{rcl}x & = & (0.02\sim 0.5),\,\,{M}_{{W}_{2}}=(4.8\sim 10.0)\mathrm{TeV},\\ {M}_{{N}_{1}} & = & (0.10\sim 3.0)\mathrm{TeV},\end{array}\end{eqnarray}$
we plot ${m}_{{ee}}^{\mathrm{LR}}$ versus mν-lightest in figure 4.
Figure 4. ${m}_{{ee}}^{\mathrm{LR}}$ versus mν-lightest with scanning the parameter space in equations ( |
Comparing figure 4 with figure 3, the red points show that the range of ${m}_{{ee}}^{\mathrm{LR}}$ is similar to the range of ${m}_{{ee}}^{\mathrm{BL}}$ in the case of IH neutrino masses, but in the case of NH neutrino masses, there are points with ${m}_{{ee}}^{\mathrm{LR}}\gt {m}_{{ee}}^{\mathrm{BL}}$. For IH neutrino masses, the contributions to ${m}_{{ee}}^{\mathrm{LR}}$ are dominated by the terms proportional to light neutrino masses, hence ${m}_{{ee}}^{\mathrm{LR}}$ depends on $x,{M}_{{W}_{2}},{M}_{{N}_{1}}$ negligibly, which leads to the range of reds points being similar to the results of ${m}_{{ee}}^{\mathrm{BL}}$. Additionally, ${m}_{{ee}}^{\mathrm{LR}}\gt {m}_{{ee}}^{\mathrm{BL}}$ shown as green points indicate the contributions to ${m}_{{ee}}^{\mathrm{LR}}$ can be dominated by heavy neutral leptons for appropriate values of $x,{M}_{{W}_{2}},{M}_{{N}_{1}}$.
In order to see the effects of $x,{M}_{{W}_{2}},{M}_{{N}_{1}}$ clearly, we take ${m}_{{\nu }_{1}}=0.001\,\mathrm{eV}$ for the NH neutrino masses, ${s}_{12},{s}_{13},{\rm{\Delta }}{m}_{12}^{2},{\rm{\Delta }}{m}_{13}^{2}$ at the corresponding center values and the CP violation phases α = β = 0. Then taking ${M}_{{W}_{2}}=5.0\,\mathrm{TeV}$, we plot ${m}_{{ee}}^{\mathrm{LR}}$ versus ${M}_{{N}_{1}}$ in figure 5(a), where the solid, dashed, and dotted lines denote the results for x = 0.10, 0.25, 0.40 respectively. Similarly, ${m}_{{ee}}^{\mathrm{LR}}$ versus ${M}_{{W}_{2}}$ for ${M}_{{N}_{1}}=0.20\,\mathrm{TeV}$ is plotted in figure 5(b).
Figure 5. With ${m}_{{\nu }_{1}}=0.001\,\mathrm{eV}$ for NH neutrino masses, (a): ${m}_{{ee}}^{\mathrm{LR}}$ versus ${M}_{{N}_{1}}$ for ${M}_{{W}_{2}}=5.0\,\mathrm{TeV}$, (b): ${m}_{{ee}}^{\mathrm{LR}}$ versus ${M}_{{W}_{2}}$ for ${M}_{{N}_{1}}=0.2\,\mathrm{TeV}$. The solid, dashed, dotted lines denote the obtained ${m}_{{ee}}^{\mathrm{LR}}$ for x = 0.10, 0.25, 0.40 respectively. |
From figure 5, one may see the fact that the obtained ${m}_{{ee}}^{\mathrm{LR}}$ decreases with the increasing of ${M}_{{N}_{1}}$, ${M}_{{W}_{2}}$, then ${m}_{{ee}}^{\mathrm{LR}}$ approaches to a constant when ${M}_{{N}_{1}}$ or ${M}_{{W}_{2}}$ becomes large. According to the definition of WL–WR mixing parameter ζ, the coefficient ${C}_{3R}^{{RL}}(N)$ in equation (18 ) increases with increasing $x\equiv \tfrac{{v}_{2}}{{v}_{1}}$, which leads to that ${m}_{{ee}}^{\mathrm{LR}}$ increases with x increasing as shown in the figures. Additionally one may see that ${m}_{{ee}}^{\mathrm{LR}}$ depends on the values of $x,\,{M}_{{W}_{2}},\,{M}_{{N}_{1}}$ mildly when ${M}_{{N}_{1}}$ or ${M}_{{W}_{2}}$ is large. It is because the contributions from heavy neutral lepton(s) are highly suppressed when its mass or right-handed boson mass becomes large, thus the contributions from the light neutrinos become dominant and proportional merely to the light neutrino masses in this case.
4.2.2. The results under the type I+II seesaw dominance
As pointed out above, the mixing parameters of the light neutrinos and the heavy neutral leptons are not tiny under the type I+II seesaw dominance, hence the Dirac mass matrix MD can also affect the numerical results via the mixing of the light neutrinos and the heavy neutral leptons. For simplicity and not losing general features, we assume that the mass matrix MD in equation (13 ) is diagonal as MD = diag (MD11, MD22, MD33). Taking the assumption that only one generation of heavy neutral leptons makes substantial contributions, i.e. MD22 and MD33 do not affect the results. Scanning the ranges determined by equations (29 ), (32 ) and the parameter space
$\begin{eqnarray}\begin{array}{rcl}x & = & (0.02\sim 0.5),{M}_{{W}_{2}}=(4.8\sim 10)\mathrm{TeV},\\ {M}_{{N}_{1}} & = & (0.10\sim 3.0)\mathrm{TeV},\\ {M}_{D11} & = & (0.10\sim 100)\mathrm{MeV},\end{array}\end{eqnarray}$
${m}_{{ee}}^{\mathrm{LR}}$ versus mν-lightest as the results is plotted in figure 6.
Figure 6. ${m}_{{ee}}^{\mathrm{LR}}$ versus mν-lightest with scanning the parameter space in equations ( |
Comparing figure 6 with figure 4 (also figure 3), a larger ${m}_{{ee}}^{\mathrm{LR}}$ can be reached in the case of type I+II seesaw dominance. Because the contributions are dominated by the terms proportional to S1i or ${T}_{1i}^{* }$ when MD11 is large (shown in equations (18 ), (23 )), and larger ${m}_{{ee}}^{\mathrm{LR}}$ is reached in this case. As indicated in figure 6 there are some points that are smaller than the points in figure 4 both for NH and IH neutrino masses, that is due to the existence of the cancellation effect, i.e. the contributions from the terms proportional to MD11 can be cancelled some amount of the contributions from the other terms The parameter MD11 represents the strength of light-heavy neutral lepton mixing, that we can introduce a familiar parameter ${S}_{e}^{2}\equiv {\sum }_{i=1}^{3}| {S}_{1i}{| }^{2}$ for the following analysis to describe the light-heavy neutral lepton mixing.
In order to show the effects of light-heavy mixing parameter Se2 and the cancellation effect well, we take ${M}_{{N}_{1}}=0.2\,\mathrm{TeV}$, ${m}_{{\nu }_{1}}=0.001\,\mathrm{eV}$ for the NH neutrino masses, ${s}_{12}^{2},{s}_{13}^{2},{\rm{\Delta }}{m}_{12}^{2},{\rm{\Delta }}{m}_{13}^{2}$ with the center values and the CP violation phases α = β = 0 accordingly. The effects with the parameters being fixed above are presented a similar behavior as the case of type I seesaw dominance, hence we do not repeat the study of the effects on these parameters. With ${M}_{{W}_{2}}=5.0\,\mathrm{TeV}$, in figure 7 (a) we plot ${m}_{{ee}}^{\mathrm{LR}}$ versus Se2, and the solid, dashed, dotted lines denote the results for x = 0.02, 0.06, 0.1 respectively. Similarly, with x = 0.02, ${m}_{{ee}}^{\mathrm{LR}}$ versus Se2 is plotted in figure 7(b), where the solid, dashed, and dotted lines denote the results for ${M}_{{W}_{2}}=5.0,6.0,7.0\,\mathrm{TeV}$ respectively.
Figure 7. ${m}_{{ee}}^{\mathrm{LR}}$ versus Se2 with ${M}_{{N}_{1}}=0.2\,\mathrm{TeV}$, ${m}_{{\nu }_{1}}=0.001\,\mathrm{eV}$ for the NH neutrino masses, s12, s13, ${\rm{\Delta }}{m}_{12}^{2}$, ${\rm{\Delta }}{m}_{13}^{2}$ at the corresponding center values and the CP violation phases α = β = 0. (a): the results for ${M}_{{W}_{2}}=5.0\,\mathrm{TeV}$, and the solid, dashed, dotted lines denote the results for x = 0.02, 0.06, 0.10 respectively. (b): the results for x = 0.02, and the solid, dashed, dotted lines denote the results for ${M}_{{W}_{2}}=5.0,6.0,7.0\,\mathrm{TeV}$ respectively. |
From figures 7(a), (b) one may see the fact that with the mixing parameter Se2 increasing, ${m}_{{ee}}^{\mathrm{LR}}$ decreases to a minimum value and then increases. In the case of the type I+II seesaw dominance, i.e. S1i and T1i are not too small, the contributions from the light neutrinos are dominated over those terms which do not depend on the neutrino mass ${m}_{{\nu }_{i}}$ in equation (23 ). When Se2 is small, ${C}_{3L}^{{RR}}(N)$ in equation (18 ) plays a dominant role, and as Se2 increasing, the contributions from ${C}_{3L/R}^{{LL}}(\nu )$ in equation (23 ) become larger. The minimum values for ${m}_{{ee}}^{\mathrm{LR}}$ as shown in figure 7 is due to the opposite signs for ${M}_{3}^{{XX}}(N)$ and ${M}_{3}^{{XY}}(\nu )$, i.e. it is caused by cancellation of the contributions from ${C}_{3L}^{{RR}}(N)$ and ${C}_{3L/R}^{{RL}}(\nu )$. From the figures, the fact can be seen clearly that when Se2 is increasing, the contributions from ${C}_{3L/R}^{{RL}}(\nu )$ become dominant which leads to the increasing of ${m}_{{ee}}^{\mathrm{LR}}$ as long as Se2 becomes large enough. The results indeed show that with Se2 varying, the cancellation takes place in a proper manner. When the cancellation takes place, the value of Se2 depends on ${M}_{{W}_{2}}$ and x explicitly.
To compare the results obtained under the fresh approximation in this work with the ones obtained in the traditional way, we take the results from [65] and present ${m}_{{ee}}^{\mathrm{LR}}$ versus ${M}_{{W}_{2}}$ in figures 8(a), (b) for ${s}_{e}^{2}=0.25\times {10}^{-8},\,{10}^{-8}$ respectively, where the solid, dashed lines denote the results obtained under the fresh approximation in this work and those by the traditions way from in [65] respectively.
Figure 8. (a): ${m}_{{ee}}^{\mathrm{LR}}$ versus ${M}_{{W}_{2}}$ for ${S}_{e}^{2}=0.25\times {10}^{-8}$. (b): ${m}_{{ee}}^{\mathrm{LR}}$ versus ${M}_{{W}_{2}}$ for ${S}_{e}^{2}={10}^{-8}$. Both are with ${M}_{{N}_{1}}=0.2\,\mathrm{TeV}$, ${m}_{{\nu }_{1}}=0.01\,\mathrm{eV}$ for the NH neutrino masses. The solid, dashed lines denote the results obtained under the new approximation in this work and the traditions method shown in [65] respectively. |
In figure 8, we take ${M}_{{N}_{1}}=0.2\,\mathrm{TeV}$, ${m}_{{\nu }_{1}}=0.010\,\mathrm{eV}$, x = 0.020, ${s}_{12},{s}_{13},{\rm{\Delta }}{m}_{12}^{2},{\rm{\Delta }}{m}_{13}^{2}$ at the corresponding center values and the CP violation phases α = β = 0. Note that in the figures the results obtained in this work coincide well with the traditional ones when Se2 is small or ${M}_{{W}_{2}}$ is large in the chosen parameter space. In addition, the results obtained in this work, being approximate ones but having the interference effects considered better, are smaller than the ones obtained in the traditional way when W2 is not heavy enough, and the reduction factor depends on the parameters in LRSM completely and also comes from the uncertainties of NMEs partly. Considering the difficulties and the uncertainties etc in computing NMEs and the interference effects among the various contributions by the way in the literature and the approximation approach in this work, from figure 8, it seems that the results with very heavy W2 (${M}_{{W}_{2}}\geqslant 12$TeV) or small Se2 ( ≲ 10−9) may be convinced more as the results approach to coinciding with each other.
5. Summary and discussions
In the paper, we take the B-L supersymmetric standard model (B-LSSM) and TeV scale left–right symmetric model (LRSM) as two representations of two kinds of new physics models to study the nuclear neutrinoless double beta decays (0ν2β). As stated in the Introduction, the calculations are those about the factor on the ‘core’ factor of the decays: evaluating the process d + d → u + u + e + e by considering the effective Lagrangian containing the operators with the Wilson coefficients or considering the relevant amplitude etc. Whereas here the estimations of the other necessary ‘factors’ for the decays, i.e. to evaluate relevant nuclear matrix element (NME) and phase space factor (PSF) etc, that do not relate to the specific models, are treated by the following literature.
In the B-LSSM, all of the calculations can be well deduced and the results are dominated by the neutrino mass terms. However, in the LRSM, owing to the existence of right-handed gauge boson WR, the calculations are complicated, and the interference effects are hard and easily estimated. In this work, a new approximation, i.e. the momenta of the two involved quarks inside the initial nucleus and inside the final nucleus, are assumed to be equal approximately, is made, then all contributions in the LRSM can be well reduced and summed up all the contributions. Then the calculations in LRSM are simplified and the interference effects can be calculated comparatively easily. To see the consequences of the approximation, we compare the results obtained in this work with the results obtained by the traditional method numerically, and the results coincide with each other well when light-heavy neutral lepton mixing parameter Se2 is small or ${M}_{{W}_{2}}$ is large. For the effective dimension-9 contact interactions in these two models, the contributions from heavy neutral lepton exchange, the QCD corrections from the energy scale $\mu \simeq {M}_{{W}_{L}}$ (or ${M}_{{W}_{1}}$) to the energy scale μ ≃ 1.0 GeV to all dimension-9 operators in the effective Lagrangian which is responsible for the 0ν2β decays are calculated by the RGE method, and all the QCD corrections including the contributions from light neutrnos in the energy scale region μ ≃ 1.0 GeV to μ ≃ 0.10 GeV, being of non-perturbative QCD, are taken into account alternatively by inputting in the experimental measurements for the relevant current matrix elements of nucleons, which emerge at the effective Lagrangian at μ ≃ 0.10 GeV.
With necessary input parameters allowed by experimental data, the theoretical predictions on 0ν2β decay half-life 76Ge and 136Xe are obtained in these two models. In the B-LSSM, the contributions from heavy neutral leptons are highly suppressed by the tiny light-heavy neutral lepton mixing parameters for TeV-scale heavy neutral leptons. Hence ${m}_{{ee}}^{\mathrm{BL}}$ depends on the light neutrino masses mainly, and the numerical results show that the 0ν2β decays may be observed with quite a great opportunity in the near future. Whereas if the decays are not observed in the next generation of 0ν2β experiments, the IH neutrino masses are excluded completely by the lower bound on ${T}_{1/2}^{0\nu }{(}^{76}\mathrm{Ge})$ (as shown in figure 3).
In the LRSM, the situation is much more complicated than that in the B-LSSM, so we have gained a lot of experience in the study of the 0ν2β decays for the model. As for the type I seesaw dominance, the contributions from the terms proportional to ${T}_{1i}^{* }$ (see equation (23 )) are highly suppressed by the tiny neutrino masses. The numerical results of the contributions from light neutrinos are similar to the ones for B-LSSM, but the heavy neutral leptons can make comparatively large contributions through the right-handed current when ${M}_{{N}_{i}}$ ( ≲ 0.5 TeV) and ${M}_{{W}_{2}}$ ( ≲ 7 TeV) both are not too heavy. For the type I+II seesaw dominance, the terms which do not depend on the neutrino mass ${m}_{{\nu }_{i}}$ in equation (23 ) play the dominant roles. In this case, the contributions from ${C}_{3L/R}^{{RL}}(L)$ can be canceled somewhat by the contributions from ${C}_{3L}^{{RR}}(H)$ when the light-heavy mixing parameter Se2 is appropriate because the signs of the corresponding NMEs are opposite. Moreover, the effects of the cancellation are affected by Se2, x and the right-handed W-boson mass ${M}_{{W}_{2}}$ etc in a complicated way. In addition, figure 6 shows that the points either in green (NH) or in red (IH) spread out a lot, and there are many ‘exotic points’, that cannot be reached for the cases of the B-LSSM and type I seesaw dominance LRSM. Thus the characteristic feature on the distribution of ${m}_{{ee}}^{\mathrm{LR}}$ versus mν-lightest may help to realize whether the decays are caused by the LRSM in the type I+II seesaw dominance or not, particularly when the 0ν2β decays are observed and the points for ${m}_{{ee}}^{\mathrm{LR}}$ versus mν-lightest just fall on the exotic points.
Finally, according to the numerical results of the present comparative studies on the 0ν2β decays for the two typical models LRSM and B-LSSM, it may be concluded that the room for LRSM type models for the foreseeable future decay experiments is greater than that of B-LSSM type models, and having the right-handed gauge bosons, the feature of the LRSM type models is more complicated than that of the B-LSSM type models.