Nuclear Transition Matrix Elements for Double-β Decay Within PHFB Model

Employing the projected-Hartree-Fock-Bogoliubov (PHFB) approach, nuclear transition matrix elements (NTMEs) have been calculated to study the three complementary modes of β⁻β⁻ decay, namely two neutrino β⁻β⁻ (2νβ⁻β⁻) decay, neutrinoless β⁻β⁻ (0νβ⁻β⁻) decay within mass mechanism and Majoron accompanied 0νβ⁻β⁻ (0νβ⁻β⁻χ) decay. Reliability of HFB wave functions generated with four different parametrizations of the pairing plus multipolar type of effective two-body interaction has been ascertained by comparing a number of nuclear observables with the available experimental data. Specifically, the calculated NTMEs M^(2ν) of 2νβ⁻β⁻ decay have been compared with the observed data. Effects due to different parametrizations of effective two-body interactions, form factors and short-range correlations have been studied. It has also been observed that deformation plays a crucial role in the nuclear structure aspects of 0νβ⁻β⁻ decay. Uncertainties in NTMEs calculated with wave functions generated with four different parametrizations of the pairing plus multipolar type of effective two-body interaction, dipole form factor and three different parametrizations of Jastrow type of short-range correlations within mechanisms involving light Majorana neutrinos, heavy Majorana neutrinos, sterile neutrinos and Majorons have been statistically estimated.

1. Introduction

The story of neutrinos as well as weak interaction is rather checkered and quite exciting due to their enigmatic nature. Presently, a number of projects, namely ⁴⁸Ca (CANDLES), ⁷⁶Ge (GERDA, MAJORANA, LEGEND), ⁸²Se (SuperNEMO, Lucifer), ¹⁰⁰Mo (MOON, AMoRE), ¹¹⁶Cd (COBRA), ¹³⁰Te (CUORE, CUPID, SNO+), ¹³⁶Xe (XMASS, EXO, KamLAND-Zen, NEXT), ¹⁵⁰Nd (SNO++, SuperNEMO, DCBA/MTD) are dedicated (already designed/planned) [1, 2] to observe the possible occurrence of neutrinoless double beta (0νββ) decay for establishing the Majorana nature of neutrinos. In addition, information on the lepton number violation, possible hierarchies in the neutrino mass spectrum, the origin of neutrino mass and CP violation in the leptonic sector can be inferred [3, 4]. Over the past years, the four experimentally distinguishable modes ββ decay, namely two neutrino double beta (2νββ) decay, neutrinoless double beta (0νββ) decay, single Majoron accompanied neutrinoless double beta (0νββϕ) and double Majoron accompanied neutrinoless double beta (0νββϕϕ) decay have been studied extensively both experimentally [1, 2] and theoretically [3, 4].

The possible occurrence of 0νββ decay has been investigated within a number of mechanisms, namely the exchange of light as well as heavy neutrinos, and the right handed currents in the left-right symmetric model (LRSM), the exchange of sleptons, neutralinos, squarks, and gluinos in the R_p-violating minimal super symmetric standard model, existence of heavy sterile neutrinos, Majoron models, compositeness, the exchange of leptoquarks, and extradimensional scenarios. Arguably, the observation of 0νββ decay due to any mechanism would imply the Majorana nature of neutrinos/sneutrinos [5, 6]. The main objective of the nuclear structure calculations is to provide reliable nuclear transition matrix elements (NTMEs) for extracting gauge theoretical parameters with minimum uncertainty.

In the conventional nuclear models, there are three basic ingredients, namely the model space, the single particle energies (SPEs) and the effective two body interaction. Usually, they are chosen on the basis of practical considerations albeit a consistent procedure is available for their choice. Due to the rare nature of nuclear ββ decay, strongly suppressed channels play a crucial role in the evolution of NTMEs and hence, NTMEs are quite sensitive to the details of wave functions of the parent, intermediate and daughter nuclei. Remarkably, the observed suppression of M^(2ν) was first explained in the quasiparticle random phase approximation (QRPA) approach [7, 8]. Over the years, the QRPA as well as its extensions [9–11], and QRPA with isospin restoration [12] have emerged as the most successful models. In spite of the impressive success of shell-model approach [13–26], presently available numerical facilities highly limit its application to the description of medium and heavy mass nuclei.

In deformed QRPA formalism, the effect of suppression of NTMEs for the 2νβ⁻β⁻ as well as 0νβ⁻β⁻ decay due to different deformations of parent and daughter nuclei has been reported in Rodríguez et al. [27], Simkovic et al. [28], Pacearescu et al. [29], Yousef et al. [30], and Fang et al. [31]. In the shell model [16–18], the effect of deformation on the NTMEs has been investigated and it has been shown that the NTMEs are the largest for equal deformation of parent and daughter nuclei. Proper consideration of nuclear deformation in the evaluation of NTMEs has resulted in the implementation of deformed QRPA [32–36], projected Hartree-Fock-Bogoliubov (PHFB) model [37–39], energy density functional (EDF) approach [40], covariant density functional theory (CDFT) [41, 42], and interacting boson model (IBM) [43–46] with isospin restoration [47]. By adjusting the free parameters of the models, the observed half-lives of 2νβ⁻β⁻ decay have been reproduced in all models. However, different predictions are obtained for other observables. Specifically, the calculated NTMEs M^(0ν) differ by a factor of 2–3.

In the theoretical study of $0\nu {\beta }^{-}{\beta }_{}^{-}$decay, three different approaches have been adopted sofar to estimate the uncertainties in NTMEs. The spread between all the calculated NTMEs for a particular ββ emitter [48] has been translated to theoretical uncertainty by estimating the average and standard deviation of NTMEs [49, 50]. As a model independent approach to check the accuracy of NTMEs, the ratios of calculated NTMEs-squared have been compared with the ratios of observed half-lives $T_{1/2}^{0\nu }$ [51]. With the consideration of two models, namely QRPA and RQRPA, three sets of basis states and three realistic two-body effective interactions based on the charge dependent Bonn, Argonne, and Nijmen potentials, theoretical uncertainties have been estimated by Rodin et al. [52]. Interestingly, the results of QRPA and RQRPA are not only close but the variances are also substantially smaller than the average values. Further, the approach of Rodin et al. [52] has been preferred in Hyvärinen et al. [53] following an extensive analysis by Suhonen [54] and Rodin et al. [55]. In addition, uncertainties in NTMEs due to short range correlations (SRC) have been estimated using the unitary correlation operator method (UCOM) [56, 57], self-consistent coupled cluster method (CCM) [58] and PHFB approach [38]. Recently, it has been suggested by Engel [59] that the systematic errors can be reduced by including all physics known to be important in the error analysis following Reference [60].

In addition to successfully reproducing experimental data on the yrast spectra, reduced B(E2 : 0⁺ → 2⁺) transition probabilities, quadrupole moments Q(2⁺), gyromagnetic factors g(2⁺), the PHFB approach was employed to study the effect of deformation on the 0⁺ → 0⁺ [61] and the 0⁺ → 2⁺ [62] transitions of 2νβ⁻β⁻ decay in ¹⁰⁰Mo. In conjunction with the summation method [63–65], the PHFB model has already been applied to study the 0⁺ → 0⁺ [66, 67] and 0⁺ → 2⁺ transition [68] of 2νβ⁻β⁻ decay of nuclei in the mass range 90 ≤ A ≤ 150 as well as 2νe⁺ββ decay modes of ⁹⁶Ru, ¹⁰²Pd, ¹⁰⁶Cd and ¹⁰⁸Cd isotopes [69, 70]. The effects of nuclear deformation inhibiting 2νββ decay [67, 71], 0νβ⁻β⁻ decay [72] and 0νe⁺ββ modes [73, 74] in the Majorana neutrino mass mechanism have been reported. The influence of hexadecapolar interaction on the ββ decay matrix elements has also been analyzed [75].

Uncertainties in NTMEs calculated with the PHFB approach using four different parametrizations of a nuclear Hamiltonian and three different short-range correlations, was first performed for the 0νβ⁻β⁻ decay in the light Majorana neutrino mass mechanism [37]. Including pseudoscalar and weak magnetism terms in the nucleonic current, uncertainties in the NTMEs of 0νβ⁻β⁻ decay have been estimated in the mechanisms involving light Majorana neutrinos, sterile neutrinos, classical Majorons [38], heavy Majorana neutrinos [39], and new Majorons [76]. In Rath et al. [38], no difference has been observed in the effects due to finite size of nucleons (FNS) either by employing dipole form factors or form factors taking the structure of nucleons into account. In addition, uncertainties in the NTMEs of 0νe⁺β⁺β⁺ modes [74] due to the exchange of light and heavy Majorana neutrinos have been reported.

2. Reliability of HFB Wave Functions

With the assumption that the nucleus consists of non-relativistic point nucleons and neglecting many-nucleon forces, the conventional nuclear many-body Hamiltonian H in an appropriate model space M is given by

$\begin{array}{l}H={\displaystyle \sum _{\alpha \beta }}\langle \alpha |H_0|\beta \rangle a_{\alpha }^{†}{a}_{\beta }+\frac{1}{4}{\displaystyle \sum _{\alpha \beta \gamma \delta }}\langle \alpha \beta |V_{eff}|\gamma \delta \rangle a_{\alpha }^{†}{a}_{\beta }^{†}{a}_{\delta }{a}_{\gamma },\end{array}$

where the effective two-body interaction V_eff is usually derived from nucleon-nucleon force by the use of Brandow's linked cluster diagram expansion [77] and the energy dependence of the V_eff can be eliminated by Kuo's folded diagram expansion [78]. Usually, the “realistic interactions” obtained from the above theoretical procedure are not quite successful in reproducing spectroscopic properties of nuclei. Hence, “empirical effective interactions” and “schematic effective interactions” are used quite often in reproducing the observed spectroscopic data [79, 80]. An effective operator O_eff is also to be defined such that the same observable are obtained in a finite dimensional model space M as reproduced by a bare operator O acting in an infinite dimensional Hilbert space. The problems associated with the numerical implementation of such effective operators in perturbative approach have been discussed by Haxton and Lau [81].

In the HFB theory [82], the HF field and the pairing interaction are treated simultaneously and on equal footing. Further, the particle coordinates are transformed to quasiparticle coordinates through general Bogoliubov transformation

$\begin{array}{ll}{q}_{\alpha }^{†}={\displaystyle \sum _{\beta }}\left(u_{\alpha \beta }{a}_{\beta }^{†}+{\upsilon }_{\alpha \beta }{a}_{\beta }\right),& (1)\end{array}$

$\begin{array}{ll}{q}_{\alpha }={\displaystyle \sum _{\beta }}\left(u_{\alpha \beta }^{*}{a}_{\beta }+{\upsilon }_{\alpha \beta }^{*}{a}_{\beta }^{†}\right),& (2)\end{array}$

such that the interaction between quasiparticles is relatively weak. Essentially, the Hamiltonian H is expressed as

$\begin{array}{ll}H=H_0+H_{qp}+H_{qp-qp},& (3)\end{array}$

where H₀ is the energy of the quasiparticle vacuum, H_qp is the elementary quasiparticle excitations and H_qp−qp is a weak interaction between the quasiparticles. In the HFB theory, the interaction between the quasiparticles is usually neglected and the Hamiltonian H is approximated by independent quasiparticle Hamiltonian. In TDHFB and QRPA, the effects of quasiparticle interaction are included to certain extent.

In the PHFB model, a state with good angular momentum J is obtained from the HFB intrinsic state |Φ₀〉 using the standard projection technique [83].

$\begin{array}{ll}\begin{array}{lll}|{\Psi }_0^J\rangle \hfill & =\hfill & P_{00}^J|{\Phi }_0\rangle \hfill \\ \hfill & =\hfill & \left[\frac{\left(2J+1\right)}{8{\pi }^2}\right]{\displaystyle \int D_{00}^J}(\Omega )R(\Omega )|{\Phi }_0\rangle d\Omega .\hfill \end{array}& (4)\end{array}$

The axially symmetric HFB intrinsic state |Φ₀〉 with K = 0 can be formulated as

$\begin{array}{ll}|{\Phi }_0\rangle ={\displaystyle \prod _{im}\left(u_{im}+{\upsilon }_{im}{b}_{im}^{†}{b}_{i\overline{m}}^{†}\right)}|0\rangle & (5)\end{array}$

$\begin{array}{ll}=N\exp \left(\frac{1}{2}{\displaystyle \sum _{\alpha \beta }{f}_{\alpha \beta }}{a}_{\alpha }^{†}{a}_{\beta }^{†}|0\rangle \right),& (6)\end{array}$

where the creation operators $b_{im}^{†}$ and $b_{i\overline{m}}^{†}$ are given by

$\begin{array}{ll}{b}_{im}^{†}={\displaystyle \sum _{\alpha }}{C}_{i\alpha ,m}{a}_{\alpha m}^{†}\text{ and }{b}_{i\overline{m}}^{†}={\displaystyle \sum _{\alpha }}{(-1)}^{l+j-m}{C}_{i\alpha ,m}{a}_{\alpha ,-m}^{†},& (7)\end{array}$

with

$\begin{array}{ll}{f}_{\alpha \beta }={\displaystyle \sum _i}{C}_{i{m}_{\alpha }{j}_{\alpha }}{C}_{i{m}_{\beta }{j}_{\beta }}\left(\frac{{\upsilon }_{im\alpha }}{u_{im\alpha }}\right){\delta }_{m_{\alpha }-m_{\beta }},& (8)\end{array}$

and N is a normalization constant.

In Rath et al. [37], Chandra et al. [66], Singh et al. [67], and Chandra et al. [75], the model space, SPE's, parameters of pairing plus multipolar type of effective two-body interaction have already been discussed. Specifically, the wave functions are obtained by minimizing the expectation value of the effective Hamiltonian consisting of single particle Hamiltonian H_sp, and pairing V(P) plus multipolar (quadrupole-quadrupole V(QQ) and hexadecapole-hexadecapole V(HH) parts) type of effective two-body interaction [75]

$\begin{array}{ll}H=H_{sp}+V(P)+V(QQ)+V(HH).& (9)\end{array}$

In the V(QQ), the strengths of proton-proton, neutron-neutron and proton-neutron interactions are denoted by χ_2pp, χ_2nn and χ_2pn, respectively. By reproducing the experimental excitation energy $E_{2^{+}}$ either by taking χ_2pp = χ_2nn and varying χ_2pn or by considering χ_2pp = χ_2nn = χ_2pn/2 and adjusting the three parameters together, two different parameterizations of V(QQ), namely PQQ1 and PQQ2 [75] were obtained. Additional consideration of the hexadecapolar V(HH) part of the effective interaction provided two more parametrizations, namely PQQHH1 and PQQHH2 [37].

In Rath et al. [37] and Chandra et al. [75], the reliability of wave functions generated with four different parametrizations of the effective two-body interaction, namely PQQ1, PQQHH1, PQQ2, and PQQHH2 has been established by comparing the theoretically calculated results for a number of spectroscopic properties, namely the yrast spectra, reduced B(E2:0⁺ → 2⁺) transition probabilities, deformation parameters β₂, quadrupole moments Q(2⁺) and g-factors g(2⁺) of ^{94, 96}Zr, ^{94, 96, 100}Mo, ¹⁰⁰Ru, ¹¹⁰Pd, ¹¹⁰Cd, ^{128, 130}Te, ^{128, 130}Xe, ¹⁵⁰Nd, and ¹⁵⁰Sm isotopes with the available experimental data.

By adjusting the strength parameter χ_2pn or χ_2pp, the experimental excitation energies $E_{2^{+}}$ [84] have been reproduced to about 98% accuracy. With respect to PQQ1 interaction, the maximum change in excitation energies $E_{4^{+}}$ and $E_{6^{+}}$ is about 8 and 31%, respectively [66, 67]. The reduced B(E2:0⁺ → 2⁺) transition probabilities, deformation parameters β₂, static quadrupole moments Q(2⁺) and gyromagnetic factors g(2⁺) differ by about 20%, 10% (except for ⁹⁴Zr and PQQ2 interaction), 27 and 12% (except for ^{94, 96}Zr and PQQ2 interaction), respectively, and there is an overall agreement between the theoretically calculated and experimentally observed data [85, 86]. Employing wave functions generated with the above mentioned four parametrizations of effective two-body interaction, the calculated [75] and experimental deformation parameters β₂ [85] of parent and daughter nuclei involved in the ββ decay of ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes have been presented in Table 1. Interestingly, the experimentally observed deformation parameters β₂ have been well reproduced.

TABLE 1

Table 1. Theoretically calculated deformation parameters β₂ of nuclei participating in ββ decay with PQQ1, PQQ2, PQQHH1, and PQQHH2 parametrizations of effective two-body interaction along with experimental recommended values [85].

3. Two Neutrino Double Beta Decay

Incorporating the summation method [63–65], the PHFB model has already been applied to study the 0⁺ → 0⁺ [66, 67] and 0⁺ → 2⁺ [68] transitions of 2νβ⁻β⁻ decay. The inverse half-life for the 0${}^{+}\to J_F^{+}$ transition of the $2\nu {\beta }^{-}{\beta }_{}^{-}$ decay can be written as

$\begin{array}{ll}{\left[T_{1/2}^{(2\nu )}(0^{+}\to J_F^{+})\right]}^{-1}=G_{2\nu }(J_F^{+})|M^{(2\nu )}(J_F^{+}){|}^2,& (10)\end{array}$

and the integrated kinematical factor $G_{2\nu }(J_F^{+})$ has been calculated with good accuracy employing the exact Dirac wave functions in conjunction with finite nuclear size and screening effects [87–89]. Further, the model dependent NTME M^(2ν)(J⁺) is defined as

$\begin{array}{ll}{M}^{(2\nu )}(J_F^{+})=\sqrt{\frac{1}{J_F+1}}{\displaystyle \sum _N}\frac{\langle J_F^{+}\parallel \sigma {\tau }^{+}\parallel 1_N^{+}\rangle \langle 1_N^{+}\parallel \sigma {\tau }^{+}\parallel 0_I^{+}\rangle }{{\left[E_0+E_N-E_I\right]}^{J_F+1}},& (11)\end{array}$

where

$\begin{array}{ll}{E}_0=\frac{1}{2}(E_I-E_F)=\frac{1}{2}{Q}_{{\beta }^{-}{\beta }^{-}}+m_e.& (12)\end{array}$

Further, $|0_I^{+}\rangle ,|J_F^{+}\rangle$ and $|1_N^{+}\rangle$ are the initial, final and virtual intermediate states, respectively. By performing the summation over intermediate states using the summation method [63–65], the NTME $M^{(2\nu )}(J_F^{+})$ is written as

$\begin{array}{ll}{M}^{(2\nu )}(J_F^{+})=\sqrt{J_F+3}{\displaystyle {\displaystyle \sum }_{\pi ,\nu }}\frac{\langle J_F^{+}\Vert {[\sigma \otimes \sigma ]}^{(J)}{\tau }^{+}{\tau }^{+}\Vert 0_I^{+}\rangle }{{[E_0+{\Delta }_{\beta }(k)]}^{J_F+1}},& (13)\end{array}$

with Δ_β(k) = ε(n_π, l_π, j_π)−ε(n_ν, l_ν, j_ν). Notably, this expression is same as reported by Hirsch et al. [90].

In the energy denominator, the difference in single particle energies Δ_β(k) of protons in the intermediate nucleus and neutrons in the parent nucleus is mainly due to the difference in Coulomb energies. Hence

$\begin{array}{ll}{\Delta }_{\beta }(k)=\{\begin{array}{llll}{\Delta }_C\hfill & \hfill & for\hfill & n_{\nu }=n_{\pi },l_{\nu }=l_{\pi },j_{\nu }=j_{\pi }\hfill \\ {\Delta }_C+\Delta E_{s.o.splitting}\hfill & \hfill & for\hfill & n_{\nu }=n_{\pi },l_{\nu }=l_{\pi },j_{\nu }\ne j_{\pi }\hfill \end{array},\hfill & (14)\end{array}$

where the Coulomb energy difference Δ_C is given by Bohr and Mottelson [91].

$\begin{array}{ll}{\Delta }_C=\frac{0.70}{A^{1/3}}\left[(2Z+1)-0.76\left\{{(Z+1)}^{4/3}-Z^{4/3}\right\}\right]\text{ MeV.}& (15)\end{array}$

Presently, each proton-neutron excitation is treated according to its spin-flip or non–spin-flip nature and the spin-orbit splitting is explicitly included in the energy denominator. Hence, the use of summation method goes beyond the closure approximation and previously employed summation method in the pseudo SU(3) model [90, 92].

In the PHFB model, the NTMEs corresponding to a transition operator $O_{\alpha }^{(k)}(J)$ (k = 2ν or 0ν) for the 0${}^{+}\to J_F^{+}$ transition of β⁻β⁻ decay are obtained using

$\begin{array}{ll}\begin{array}{lll}{M}_{\alpha }^{(k)}(J_F^{+})\hfill & =\hfill & \langle {\Psi }_{00}^{J_f}\Vert O_{\alpha }^{(k)}(J){\tau }^{+}{\tau }^{+}\Vert {\Psi }_{00}^{J_i}\rangle \hfill \\ \hfill & =\hfill & {\left[n_{(Z,N)}^{Ji}{n}_{(Z+2,N-2)}^{J_F}\right]}^{-1/2}{\displaystyle \underset{0}{\overset{\pi }{\int }}{n}_{(Z,N),(Z+2,N-2)}(\theta )}\hfill \\ \hfill & \hfill & \times {\displaystyle \sum _{\mu }\left[\begin{array}{lll}{J}_i\hfill & J\hfill & J_f\hfill \\ -\mu \hfill & \mu \hfill & 0\hfill \end{array}\right]}{d}_{\mu 0}^{J_i}\left(\theta \right){\displaystyle \sum _{\alpha \beta \gamma \delta }\langle \alpha \beta \left|O_{\alpha }^{(k)}(J){\tau }^{+}{\tau }^{+}\right|\gamma \delta \rangle }\hfill \\ \hfill & \hfill & \times {\displaystyle \sum _{\epsilon \eta }{\left[\left(1+F_{Z,N}^{(\pi )}(\theta )f_{Z+2,N-2}^{(\pi )\ast }\right)\right]}_{\epsilon \alpha }^{-1}}{\left(f_{Z+2,N-2}^{(\pi )\ast }\right)}_{\epsilon \beta }\hfill \\ \hfill & \hfill & \times {\left[\left(1+F_{Z,N}^{(\nu )}(\theta )f_{Z+2,N-2}^{(\nu )\ast }\right)\right]}_{\gamma \eta }^{-1}{\left(F_{Z,N}^{(\nu )\ast }\right)}_{\eta \delta }sin\theta d\theta .\hfill \end{array}& (16)\end{array}$

In Chandra et al. [66] and Singh et al. [67], the required expressions to evaluate amplitudes (u_im, v_im) and expansion coefficients C_{ij, m} of axially symmetric HFB intrinsic state |Φ₀〉 with K = 0, n^J, n_{(Z, N), (Z+2, N−2)}(θ), f_{Z, N} and F_{Z, N}(θ) have been presented.

In Table 2, the theoretically calculated NTMEs M^(2ν)(0⁺) using wave functions generated with PQQ1, PQQ2, PQQHH1 and PQQHH2 parameterizations of effective two-body interaction, along with their averages ${\overline{M}}_{eff}^{(2\nu )}(0^{+})=g_A^2{\overline{M}}^{(2\nu )}(0^{+})$ (g_A = 1.2701) and recommended values of Barabash [93] have been displayed. With different parametrizations, the theoretically calculated NTMEs M^(2ν) for the 0⁺ → 0⁺ transition change up to approximately 11% except ⁹⁴Zr, ^{128, 130}Te, and ¹⁵⁰Nd isotopes, for which the changes are approximately 42, 21, 26, and 18%, respectively.

TABLE 2

Table 2. Theoretically calculated NTMEs M^(2ν) with four different parametrizations of the effective two-body interaction along with experimental recommended values [93].

With the observation that the inclusion of deformation in the mean field can suppress the NTMEs M^(2ν)(2⁺) calculated employing pnQRPA by about a factor of 341 [94], the 0⁺ → 2⁺ transition of 2νβ⁻β⁻ decay has been studied in the PHFB approach [68]. With respect to NTMEs M^(2ν) (2⁺) calculated using pnQRPA, the average NTMEs ${\overline{M}}^{(2\nu )}$ (2⁺) estimated employing the PHFB model are further reduced by a factor between 1 and 150 corresponding to ⁹⁶Zr and ¹²⁸Te isotopes, respectively. Further, the compiled theoretical and experimental results [68] suggest that the observation of the 0⁺ → 2⁺ transition of 2νβ⁻β⁻ decay may be possible in ⁹⁶Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ¹³⁰Te and ¹⁵⁰Nd isotopes.

4. Neutrinoless Double Beta Decay

In the left-right symmetric model [95–97], the possible mechanisms of $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay, namely the exchange of left and right handed Majorana neutrinos, involve strangeness conserving left handed and right handed charged currents. The detailed theoretical formalism required to study the $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay was developed in Doi and Kotani [95, 96], Haxton and Stephenson [98], and Tomoda [99] with the assumption that in the charged current weak processes, the vector and axial vector parts of the current-current interaction give dominant contribution and under the assumption of zero mass neutrinos, the other terms being proportional to the lepton mass squared are negligible. With the inclusion of pseudoscalar and weak magnetism terms in the nucleonic current, it has been shown by Šimkovic et al. [100] that the contribution of the pseudoscalar term is equivalent to a modification of the axial vector current due to Goldberger-Treiman PCAC relation and greater than the vector current. In the standard mass mechanism, the contributions of pseudoscalar and weak magnetism terms of the recoil current can change the NTMEs M^(0ν) up to about 30% in the QRPA [100, 101], 20% in the ISM [16, 17] and 15% in the IBM [43]. The change in M^(0N) is quite substantial. Presently, the calculation of NTMEs of $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay within mechanisms involving light Majorana neutrinos, heavy Majorana neutrinos, sterile neutrinos and Majorons has been discussed.

4.1. Light Majorana Neutrino Mass Mechanism

In the mechanism involving light Majorana neutrino mass, the half-life $T_{1/2}^{(0\nu )}$ for the 0⁺ → 0⁺ transition of $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay is given by [100, 101]

$\begin{array}{ll}{\left[T_{1/2}^{(0\nu )}(0^{+}\to 0^{+})\right]}^{-1}=G_{01}|\frac{\langle m_{\nu }\rangle }{m_e}{M}^{(0\nu )}{|}^2,& (17)\end{array}$

where

$\begin{array}{ll}\langle m_{\nu }\rangle ={{\displaystyle \sum }}_i^{\prime }{U}_{ei}^2{m}_i,m_i<10eV.& (18)\end{array}$

Incorporating the exact Dirac wave functions, finite nuclear size and screening effects, the phase space factors have been calculated recently to good accuracy [87, 88, 102] and in the closure approximation, the NTME M^(0ν) is defined as

$\begin{array}{ll}{M}^{(0\nu )}={\displaystyle \sum _{n,m}}\langle 0_F^{+}\parallel \left[-\frac{H_F(r_{nm})}{g_A^2}+{\sigma }_n·{\sigma }_m{H}_{GT}(r_{nm})+S_{nm}{H}_T(r_{nm})\right]{\tau }_n^{+}{\tau }_m^{+}\parallel 0_I^{+}\rangle ,& (19)\end{array}$

with

$\begin{array}{ll}{S}_{nm}=3\left({\sigma }_n·{\widehat{r}}_{nm}\right)\left({\sigma }_m·{\widehat{r}}_{nm}\right)-{\sigma }_n·{\sigma }_m.& (20)\end{array}$

The neutrino potentials associated with Fermi, Gamow-Teller (GT) and tensor operators are given by

$\begin{array}{ll}{H}_{\alpha }(r_{nm})=\frac{2R}{\pi }{\displaystyle \int \frac{f_{\alpha }\left(q{r}_{nm}\right)}{\left(q+\overline{A}\right)}}{h}_{\alpha }(q)qdq,& (21)\end{array}$

where f_α(qr_nm) = j₀(qr_nm) and f_α(qr_nm) = j₂(qr_nm) for α = Fermi/GT and tensor potentials, respectively. The effects due to FNS have been incorporated through two different parametrizations of the form factors, namely dipole form factor and an alternative parametrization of $g_V(q^2)$ and $g_M(q^2)$ with the consideration of internal structure of protons and neutrons [103]. The details about these form factors and form factor related functions h_F(q), h_GT(q) and h_T(q) have been given in Rath et al. [38].

4.2. Heavy Majorana Neutrino Mass Mechanism

In order to ascertain the dominant mechanism contributing to 0νβ⁻β⁻ decay [104–106], there has been an increased interest recently to calculate reliable NTMEs for 0νβ⁻β⁻ decay due to the exchange of heavy Majorana neutrinos. Due to the exchange of heavy Majorana neutrinos between nucleons having finite size, the half-life $T_{1/2}^{0\nu }$ for the 0⁺ → 0⁺ transition of 0νβ⁻β⁻ decay is given by

$\begin{array}{ll}{\left[T_{1/2}^{(0\nu )}\left(0^{+}\to 0^{+}\right)\right]}^{-1}=G_{01}|\left(\frac{m_p}{\langle M_N\rangle }\right)M^{(0N)}{|}^2,& (22)\end{array}$

where m_p is the proton mass and

$\begin{array}{ll}{\langle M_N\rangle }^{-1}={{\displaystyle \sum }}_i{U}_{ei}^2{m}_i^{-1},m_i>1\text{GeV}.& (23)\end{array}$

In the closure approximation, the NTMEs M^(0N) are of the form [32, 58, 107]

$\begin{array}{l}{M}^{(0N)}={\displaystyle \sum _{n,m}}\langle 0_F^{+}\parallel \left[-\frac{H_{Fh}(r_{nm})}{g_A^2}+{\sigma }_n·{\sigma }_m{H}_{GTh}(r_{nm})+S_{nm}{H}_{Th}(r_{nm})\right]{\tau }_n^{+}{\tau }_m^{+}\parallel 0_I^{+}\rangle .\end{array}$

The short ranged neutrino potentials due to the exchange of heavy Majorana neutrinos are given by

$\begin{array}{l}{H}_{\alpha h}(r_{nm})=\frac{2R}{(m_p{m}_e)\pi }{\displaystyle \int }{f}_{\alpha h}(q{r}_{nm})h_{\alpha }(q)q^{2}dq,\end{array}$

with f_Fh(qr_nm) = f_GTh(qr_nm) = j₀(qr_nm) and f_Th(qr_nm) = j₂(qr_nm). The details regarding the h_F(q), h_GT(q) and h_T(q) are given in Rath et al. [39].

4.3. Mechanism Involving Sterile Neutrinos

The indication of ${\overline{\nu }}_{\mu }\to {\overline{\nu }}_e$ conversion in the short base line experiments [108, 109] has been explained with 0.2 eV < Δm² < 2 eV and 10⁻³ < sin²2θ < 4.10⁻². The short base line oscillation [110–112] is also supported by recent results of the reactor fluxes. All these observations, if confirmed, would imply the existence of more than three massive neutrinos [113]. Further, it has already been shown that the mixing of a light sterile neutrino (mass ≪ 1 eV) with a much heavier sterile neutrino (mass ≫ 1 GeV) would result in observable signals in current ββ decay experiments [114]. In addition, other interesting alternative scenarios are also possible [115, 116].

With the consideration of the exchange of a Majorana neutrino between two nucleons, the contribution of the sterile ν_h neutrino to the half-life $T_{1/2}^{(0\nu )}$ for the 0⁺ → 0⁺ transition of $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay [116] is given by

$\begin{array}{ll}{\left[T_{1/2}^{(0\nu )}(0^{+}\to 0^{+})\right]}^{-1}=G_{01}|U_{eh}^2\frac{m_h}{m_e}{M}^{(0\nu )}(m_h){|}^2,& (24)\end{array}$

where U_eh is the ν_h − ν_e mixing matrix element and the NTME $M^{(0\nu )}(m_h)$ is written as

$\begin{array}{ll}{M}^{(0\nu )}(m_h)=\langle 0_F^{+}\Vert \left[-\frac{H_F\left(m_{h,}r\right)}{g_A^2}+{\sigma }_n·{\sigma }_m{H}_{GT}\left(m_{h,}r\right)+S_{nm}{H}_T\left(m_{h,}r\right)\right]{\tau }_n^{+}{\tau }_m^{+}\Vert 0_I^{+}\rangle .& (25)\end{array}$

In Equation (25), the neutrino potentials are of the form

$\begin{array}{ll}{H}_{\alpha }\left(m_{h,}r\right)=\frac{2R}{\pi }{\displaystyle {\int }_0^{\infty }}\frac{f_{\alpha }\left(qr\right)h_{\alpha }\left(q^2\right)q^{2}dq}{\sqrt{q^2+m_h^2}\left(\sqrt{q^2+m_h^2}+\overline{A}\right)},& (26)\end{array}$

and the expressions for $h_{\alpha }(q^2)$ are given in Rath et al. [38]. In the limits m_h → 0, and m_h → large, NTMEs $M^{(0\nu )}(m_h)\to M^{(0\nu )}$ and $M^{(0\nu )}(m_h)\to (m_p{m}_e/m_h^2)M^{(0N)}$, respectively.

4.4. Majoron Accompanied Neutrinoless Double Decay

The proposed nine classical [117–121] and new Majoron models [114, 122–124] may be distinguished as single Majoron emitting 0νββϕ decay and double Majoron emitting 0νββϕϕ decay modes. Over the past decades, experimental studies devoted to 0νββχ decay have provided stringent limits on the Majoron coupling constant < g > ~10⁻⁵ and information on the sensitivities of the ongoing experiments to different new Majoron models [125, 126]. In the QRPA [127] and PHFB [76] approach, the observability of nine Majoron models has been investigated theoretically.

The half-life $T_{1/2}^{(0\nu \chi )}$ for the 0⁺ → 0⁺ transition of $0\nu {\beta }^{-}{\beta }^{-}{\chi }_{}$ decay is written by

$\begin{array}{ll}{\left[T_{1/2}^{(0\nu \chi )}(0^{+}\to 0^{+})\right]}^{-1}=\left|\langle g_{\alpha }\rangle {|}^m{G}_{\alpha }^{(\chi )}\right|M_{\alpha }^{(\chi )}{|}^2.& (27)\end{array}$

The symbol χ denotes modes involving single Majoron ϕ or two Majorons ϕϕ and the index m = 2 and 4 for the $0\nu {\beta }^{-}{\beta }^{-}{\varphi }_{}$ and 0νβ⁻β⁻ϕϕ decay modes, respectively. Further, the index α indicates different Majoron models as given in Table 3. The phase space factors have been calculated to good accuracy by Kotila et al. [128]. In the same Table 3, NTMEs $M_{\alpha }^{(\chi )}$ corresponding to different Majoron models are also presented. In the closure approximation, the NTMEs $M_{\alpha }^{(\chi )}$ are defined as

$\begin{array}{ll}{M}_{m_{\nu }}^{\left(\chi \right)}=M^{\left(0\nu \right)},& (28)\end{array}$

$\begin{array}{ll}{M}_{CR}^{\left(\chi \right)}=\left(\frac{g_V}{g_A}\right)\left(\frac{f_W}{3}\right){\displaystyle \sum _{n,m}\langle 0_F^{+}\Vert {\sigma }_n.{\sigma }_m{H}_R(r,\overline{A}){\tau }_n^{+}{\tau }_m^{+}\Vert 0_I^{+}\rangle },& (29)\end{array}$

$\begin{array}{ll}{M}_{{\omega }^2}^{\left(\chi \right)}={\displaystyle \sum _{n,m}\langle 0_F^{+}\Vert \left[{\left(\frac{g_V}{g_A}\right)}^2-{\sigma }_n.{\sigma }_m\right]H_{{\omega }^2}(r,\overline{A}){\tau }_n^{+}{\tau }_m^{+}\Vert 0_I^{+}\rangle },& (30)\end{array}$

where NTMEs $M_{m_{\nu }}^{(\chi )}$ of the classical Majoron models and NTMEs M^(0ν) of the light Majorana neutrino mass mechanism are identical. Retaining only the central part of the recoil term in $H_R(r,\overline{A})$ following Hirsch et al. [127], the neutrino potentials $H_R(r,\overline{A})$ and $H_{{\omega }^2}(r,\overline{A})$ required for the calculation of the other two matrix elements $M_{CR}^{(\chi )}$ and $M_{{\omega }^2}^{(\chi )}$, respectively, are defined as

$\begin{array}{llll}{H}_R(r,\overline{A})& =& \frac{1}{4{\pi }^{2}M}{\displaystyle \int e^{iq·r}}\left[\frac{\overline{A}+2q}{q{(q+\overline{A})}^2}\right]{\left(\frac{{\Lambda }^2}{q^2+{\Lambda }^2}\right)}^4{d}^{3}q,& (31)\end{array}$

$\begin{array}{llll}{H}_{{\omega }^2}(r,\overline{A})& =& \frac{m_e^{2}R}{16{\pi }^2}{\displaystyle \int e^{iq·r}}\left[\frac{3{\overline{A}}^2+9\overline{A}q+8q^2}{q^3{(q+\overline{A})}^3}\right]{\left(\frac{{\Lambda }^2}{q^2+{\Lambda }^2}\right)}^4{d}^{3}q,& (32)\end{array}$

and all the details about the required parameters have been given in Rath et al. [76]. The qualitative dependence of different neutrino potentials $H_R(r,\overline{A})$ and $H_{{\omega }^2}(r,\overline{A})$ on momentum transfer q is of different nature. In contrast to the neutrino potential $H_R(r,\overline{A})$, $H_{{\omega }^2}(r,\overline{A})$ is quite singular in nature. In the evolution of $M_{{\omega }^2}^{(\chi )}$, the contributions from low momentum q are crucial and the accuracy of the calculated NTMEs is quite uncertain. In accordance with Hirsch et al. [127], the magnitudes of $M_{{\omega }^2}^{(\chi )}$ are uncertain by about one order magnitude.

TABLE 3

Table 3. Different Majoron models according to Bamert et al. [114].

5. Nuclear Structure Aspects of Transition Matrix Elements

In addition to the spin-isospin dependence as in case of $M_{}^{(2\nu )}$, the NTMEs for 0νββ decay are momentum dependent. In the evaluation of reliable NTMEs, the FNS, SRC and the renormalized value of axial vector coupling constant g_A [47, 129–132] play a decisive role. Employing two different parametrizations of the form factors, namely dipole form factor and an alternative parametrization with the consideration of internal structure of protons and neutrons [38], it has been shown that the difference in NTMEs due to these different form factors is almost negligible. Presently, we consider the calculation of NTMEs with dipole form factor only.

Due to the exchange of ρ and ω mesons, the repulsive nucleon-nucleon potentials result in SRC, which has been incorporated through phenomenological Jastrow type of correlations with Miller-Spencer parametrization [133]. In addition, the use of effective transition operator [134], the exchange of ω-meson [135], UCOM [32, 56, 57] and the self-consistent CCM [58] have also been considered. In the self-consistent CCM [58], effects of Argonne and CD-Bonn two nucleon potentials have been parametrized by Jastrow type of correlations within a few percent accuracy. Explicitly,

$\begin{array}{ll}f(r)=1-c{e}^{-a{r}^2}(1-b{r}^2),& (33)\end{array}$

where a = 1.1, 1.59 and 1.52 fm⁻², b = 0.68, 1.45 and 1.88 fm⁻² and c = 1.0, 0.92 and 0.46 for Miller-Spencer parametrization, CD-Bonn and Argonne V18 NN potentials, denoted as SRC1, SRC2 and SRC3, respectively.

In order to estimate theoretical uncertainties in NTMEs statistically, sets of twelve NTMEs M^(0ν), M^(0N), $M^{(0\nu )}(m_h)$, $M_{m_{\nu }}^{(\chi )}$, $M_{CR}^{(\chi )}$, and $M_{{\omega }^2}^{(\chi )}$ for the $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay of ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes have been calculated with the consideration of four different parametrizations of effective two-body interaction, dipole form factor (FNS) and three different parametrizations of the Jastrow type of SRC (FNS+SRC) using Equation (16).

5.1. Effects Due to FNS and SRC

In Table 4, the relative changes (in %) in NTMEs M^(0ν), M^(0N), $M_{CR}^{(\chi )}$, and $M_{{\omega }^2}^{(\chi )}$ calculated with four different parametrizations of the effective two-body interaction due to the inclusion of FNS, FNS+SRC (FNS+SRC1, FNS+SRC2, and FNS + SRC3) have been presented. Considering $M_{VV}^{(0\nu )}+M_{AA}^{(0\nu )}$ as point nucleon case, the NTMEs M^(0ν) are reduced by about 18–23% in the case of FNS. With the addition of SRC1, SRC2, and SRC3, the NTMEs M^(0ν) are further reduced by 12–17%, 1.0–2.0%, and 2.4–3.0% approximately relative to the FNS case. Due to FNS, the maximum change in the values of $M_{CR}^{(\chi )}$ are in between 13 and 16%. With the inclusion of SRC, the NTMEs $M_{CR}^{(\chi )}$ change by about 12–15%, less than 1% and 3–4% due to SRC1, SRC2, and SRC3, respectively. It is noteworthy that the NTMEs $M_{{\omega }^2}^{(\chi )}$ change negligibly due to FNS and SRC.

TABLE 4

Table 4. Relative changes (in %) in NTMEs M^(0ν), M^(0N), $M_{CR}^{(\chi )}$, and $M_{{\omega }^2}^{(\chi )}$ calculated with four different parametrizations of the effective two-body interaction due to the inclusion of FNS, FNS+SRC (FNS+SRC1, FNS+SRC2, and FNS + SRC3), and average energy denominator $\overline{A}/2$.

It is observed [39] that the Fermi matrix element M_Fh = M_Fh−VV contributes about 20% to the total NTME. Due to the inclusion of the pseudoscalar and weak magnetism terms in the hadronic currents, the Gamow-Teller matrix element is noticeably modified. The absolute value of M_GT−AA is increased by M_GT−PP and the contribution of M_GT−AP is significant with an opposite sign. The contribution of M_GT−MM is quite small and with the inclusion of SRC, a change of sign is noticed. The contribution of tensor matrix elements is less that 2%. With the consideration of SRC, the Fermi and GT matrix elements change significantly and the tensor matrix element is the least affected. With respect to the point nucleon case, the NTMEs $M_N^{(0\nu )}$ are reduced by about 40% due to FNS. Due to SRC1, the NTMEs are reduced to about one third of its original value and the other two parameterizations of the SRC, namely SRC2 and SRC3, have a sizable effect, albeit much smaller than SRC1.

5.2. Validity of Closure Approximation

In order to test the validity of closure approximation, the NTMEs $M_{\alpha }^{(k)}$ are calculated for $\overline{A}/2$ in the energy denominator and the changes in NTMEs due to four different parametrizations of the effective two-body interaction and three different parameterizations of SRC are given in Table 4. In the case of light Majorana neutrino exchange, the relative change in NTMEs M^(0ν), by changing the energy denominator to $\overline{A}/2$ instead of $\overline{A}$ is in between 8.7–12.6%. In the Majoron accompanied 0νβ⁻β⁻χ decay, the relative changes in NTMEs $M_{CR}^{(\chi )}$ due to the use of $\overline{A}/2$ instead of $\overline{A}$ in the energy denominator is at the most about 16.0%. As the $\overline{A}$ is reduced by a factor of 2, the NTMEs $M_{{\omega }^2}^{(\chi )}$, however, change appreciably. It may be mentioned that the structure of neutrino potentials associated with different matrix elements is reflected in the observed sensitivities of different NTMEs to the magnitudes of $\overline{A}$.

5.3. Effects Due to PQQ1, PQQHH1, PQQ2, and PQQHH2 Parametrizations

The evaluated NTMEs M^(0ν) of considered nuclei but for ¹²⁸Te with PQQ1 and PQQ2 parameterizations are quite close and with the inclusion of the hexadecapolar term, they are reduced in magnitude, depending specifically on the structure of nuclei. With respect to PQQ1, the maximum variation in M^(0ν) due to the PQQHH1, PQQ2, and PQQHH2 parameterizations lies between 20–25% [38]. Due to PQQHH1, PQQ2, and PQQHH2 parametrizations, the maximum variations in M^(0N) with reference to PQQ1 interaction, are about 24, 18, and 26%, respectively [39]. With respect to PQQ1 parametrization, NTMEs $M_{CR}^{(\chi )}$ and $M_{{\omega }^2}^{(\chi )}$ due to other three parametrizations change up to 25 and 34%.

5.4. Radial Evolutions of NTMEs

The best possible way of studying the role of the FNS and SRC is to display the radial evolution of NTMEs $M_{\alpha }^{(k)}$ defined by [32]

$\begin{array}{ll}{M}_{\alpha }^{(k)}={\displaystyle \int }{C}_{\alpha }^{(k)}(r)dr.& (34)\end{array}$

By studying the radial evolution of NTMEs M^(0ν) due to the exchange of light Majorana neutrino, it has been shown in the QRPA [32] and ISM [18] that the magnitude of M^(0ν) for all nuclei undergoing $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay exhibits a maximum at about the internucleon distance r ≈ 1 fm, and the contributions of decaying pairs coupled to J = 0 and J > 0 almost cancel out beyond r ≈ 3 fm. Within the PHFB approach, similar observations on the radial evolution of NTMEs M^(0ν) due to the exchange of light [38] Majorana neutrinos have also been reported.

In Figure 1, the radial distributions of C^(0ν), C^(0N), $C_{CR}^{(\chi )}$, and $C_{{\omega }^2}^{(\chi )}$ of ¹³⁰Te nuclei with the PQQ1 parameterization of the effective two body interaction in four cases, namely FNS, FNS+SRC1, FNS+SRC2 and FNS+SRC3 have been plotted. As noticed, the distribution of C^(0ν) are peaked at r = 1.0 fm for FNS and the peak is shifted to 1.25 fm with the addition of SRC1 and SRC2. With the inclusion of SRC3, the position of the peak remains, however, unchanged at r = 1.0 fm. The distribution of $C_{CR}^{(\chi )}$ in the case of FNS is peaked at r = 1.0 fm and with the addition of SRC1, the peak shifts to 1.4 fm. The position of the peak with the inclusion of SRC2 and SRC3, is however, changed to r = 1.2 fm. Although, the radial distributions of C^(0ν) and $C_{CR}^{(\chi )}$ extend up to about 10 fm, the radial evolutions of M^(0ν) and $M_{CR}^{(\chi )}$ result from distributions of C^(0ν) and $C_{CR}^{(\chi )}$ up to 3 fm. The distributions $C_{{\omega }^2}^{(\chi )}$ are oscillating in nature with decreasing amplitudes, albeit, the first peak is similar to the distributions of NTMEs M^(0ν) and $M_{CR}^{(\chi )}$. The total distribution extending up to 15 fm contributes to the evolution of matrix element $M_{{\omega }^2}^{(\chi )}$.

FIGURE 1

Figure 1. Radial dependence of (A) C^(0ν)(r), (B) $C_N^{(0\nu )}(r)$, (C) $C_{CR}^{(\chi )}(r)$, and (D) $C_{{\omega }^2}^{(\chi )}(r)$ for the 0νβ⁻β⁻ and 0νβ⁻β⁻χ decay modes of ¹³⁰Te isotope.

With the exchange of heavy Majorana neutrino, the distribution $C_N^{(0\nu )}$ in the case of FNS is peaked at r≈0.5 fm and the peak shifts to about 0.8 fm with the addition of SRC1 and SRC2. With SRC3, the position of peak is shifted to 0.7 fm. In the evolution of matrix element M^(0N), the total distribution extending up to 2 fm contributes. Remarkably, the above observations regarding the radial distributions of C^(0ν), C^(0N), $C_{CR}^{(\chi )}$, and $C_{{\omega }^2}^{(\chi )}$ remain valid in the cases of PQQ2, PQQHH1, and PQQHH2 parametrizations for all considered seven nuclei, namely ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes.

5.5. Deformation Effects

In Auerbach et al. [136, 137] and Troltenier et al. [138], an inverse correlation between the GT strength and quadrupole moment has been reported. Using a quasiparticle Tamm-Dancoff approximation (TDA) based on deformed Hartree-Fock (DHF) calculations with Skyrme interactions [139], a deformed self consistent HF+RPA method with Skyrme type interactions [140–142], the effect of deformation on the distribution of the GT and β decay properties have been studied. The comparison between the experimental GT strength distribution B(GT) and the results of QRPA calculations was implemented as a novel method of deducing the deformation of N = Z nucleus ⁷⁶Sr [143].

In the PHFB model [66, 67, 69, 70, 72], the effect of deformation on the NTMEs of ββ decay has been studied and the role of quadrupolar correlations has also been investigated. The quenching of NTMEs seems to be closely related with the explicit inclusion of deformation effects, which are absent in the other models. Out of several possibilities, the quadrupole moment of the intrinsic state $\langle Q_2^0\rangle$ and quadrupole deformation parameter β₂ have been taken as a quantitative measure of the deformation. The variation of the $\langle Q_2^0\rangle$, β₂, and M^(K) with respect to the change in strength of QQ interaction ζ_qq has been investigated in order to understand the role of deformation on the NTMEs M^(K) [66, 67, 72].

It is noticed that there is an anticorrelation between the deformation parameter and the NTMEs M^(K) in general. The effect of deformation on M^(K) is quantified by the quantity D^(K) defined as the ratio of M^(K) at zero deformation (ζ_qq = 0) and full deformation (ζ_qq = 1) and is given by [72, 73]

$\begin{array}{ll}{D}^{(K)}=\frac{M^{(K)}({\zeta }_{qq}=0)}{M^{(K)}({\zeta }_{qq}=1)}.& (35)\end{array}$

In Table 5, the values of deformation ratios D^(2ν), D^(0ν), D^(0N), $D_{CR}^{(\chi )}$ and $D_{{\omega }^2}^{(\chi )}$ are presented for ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd nuclei. Owing to deformation effects, the NTMEs M^(2ν), M^(0ν), M^(0N), $M_{CR}^{(\chi )}$, and $M_{{\omega }^2}^{(\chi )}$ are suppressed by factor of about 2–6 in the mass range 90 ≤ A ≤ 150, and the deformation ratios are independent of underlying mechanisms. Although the results are presented for PQQ1 parametrization with SRC1, the deformation ratios are independent of used SRC.

TABLE 5

Table 5. Deformation ratios D^(2ν), D^(0ν), D^(0N), $D_{CR}^{(\chi )}$, and $D_{{\omega }^2}^{(\chi )}$ for PQQ1 parametrization with SRC1.

5.6. Uncertainties in NTMEs

Employing sets of twelve NTMEs M^(K) calculated for ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes, averages ${\overline{M}}^{(K)}$ and variances $\Delta {\overline{M}}^{(K)}$have been statistically estimated using

$\begin{array}{ll}{\overline{M}}^{(K)}=\frac{{\displaystyle {\sum }_{i=1}^N{M}_i^{(K)}}}{N},& (36)\end{array}$

and

$\begin{array}{ll}\Delta {\overline{M}}^{(K)}=\frac{1}{\sqrt{N-1}}{\left[{\displaystyle \sum _{i=1}^N}{\left({\overline{M}}^{(K)}-M_i^{(K)}\right)}^2\right]}^{1/2}.& (37)\end{array}$

By evaluating the same mean ${\overline{M}}^{(K)}$ and standard deviations $\Delta {\overline{M}}^{(K)}$ for eight NTMEs calculated using SRC2 and SRC3 parameterizations, the role of Miller-Spencer parameterization of Jastrow type of SRC has been ascertained. In Table 6, the estimated averages ${\overline{M}}^{(0\nu )}$, ${\overline{M}}^{(0N)}$, ${\overline{M}}_{CR}^{(\chi )}$, and ${\overline{M}}_{{\omega }^2}^{(\chi )}$ along with their respective variances $\Delta {\overline{M}}^{(0\nu )}$, $\Delta {\overline{M}}^{(0N)}$, $\Delta {\overline{M}}_{CR}^{(\chi )}$, and $\Delta {\overline{M}}_{{\omega }^2}^{(\chi )}$ for ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes have been presented. The calculated NTMEs due to exchange of light Majorana neutrino ${\overline{M}}^{(0\nu )}$ as well as heavy Majorana neutrino ${\overline{M}}^{(0N)}$ within PHFB model have been compared with the results of other models in Table 7.

TABLE 6

Table 6. Average values of NTMEs ${\overline{M}}^{(0\nu )}$, ${\overline{M}}^{(0N)}$, ${\overline{M}}_{CR}^{(\chi )}$, and ${\overline{M}}_{{\omega }^2}^{(\chi )}$ for ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes (a) with and (b) without SRC1 [39, 76].

TABLE 7

Table 7. Comparison of NTMEs due to light Majorana neutrino exchange M^(0ν) and heavy Majorana neutrino exchange M^(0N) calculated within PHFB model (case (a)) for ⁹⁶Zr, ¹⁰⁰Mo, ^{128, 130}Te, and ¹⁵⁰Nd isotopes with those of other models.

In general, the uncertainties $\Delta {\overline{M}}^{(0\nu )}$ displayed in Table 7, are of the order of 10%. In the case of ¹³⁰Te and ¹⁵⁰Nd isotopes, the uncertainties $\Delta {\overline{M}}^{(0\nu )}$ are about 12 and 15%, respectively. Exclusion of NTMEs due to SRC1 in the statistical analysis reduce the uncertainties by 1.5–5%. Without (with) SRC1, estimated uncertainties $\Delta {\overline{M}}_{CR}^{(\chi )}$ are about 4–14% (9–15%). Due to the identical form of corresponding operators, the uncertainties in both the NTMEs ${\overline{M}}^{(0\nu )}$ and $M_{CR}^{(\chi )}$ are of the same order. Uncertainties $\Delta {\overline{M}}_{{\omega }^2}^{(\chi )}$ in NTMEs ${\overline{M}}_{{\omega }^2}^{(\chi )}$ exhibit a negligible dependence on SRC and are about 7.0–21% (7.5–21%) without (with) SRC1. Estimated uncertainties $\Delta {\overline{M}}^{(0N)}$ in average NTMEs ${\overline{M}}^{(0N)}$ for the $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay of ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te and ¹⁵⁰Nd isotopes are about 35%. With the exclusion of NTMEs due to SRC1, uncertainties in NTMEs are reduced to about 16–20%. Without (with) SRC1, uncertainties $\Delta {\overline{M}}^{(0\nu )}(m_h)$ in ${\overline{M}}^{(0\nu )}(m_h)$ are about 4% (9%)–20% (36%) depending on the considered mass of the sterile neutrinos [38].

5.7. Nuclear Sensitivities

The nuclear sensitivity ξ^(K) is related to the sensitivity of effective parameter of underlying mechanism and has been defined by [100]

$\begin{array}{ll}{\xi }^{(K)}=10^8\sqrt{G_{01}}\left|M^{(K)}\right|,& (38)\end{array}$

with an arbitrary normalization factor 10⁸ so that the nuclear sensitivities turn out to be order of unity. In Table 8, the nuclear sensitivities ξ^(K), with K = 0ν, 0N, ξ^(χ)(ββϕ), ξ^(χ)(ββϕϕ) for the $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay of ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes are presented. It is observed that the nuclear sensitivities for 0νβ⁻β⁻ decay of ¹⁰⁰Mo, ¹⁵⁰Nd, ¹³⁰Te, ¹¹⁰Pd, ⁹⁶Zr, ⁹⁴Zr, and ¹²⁸Te isotopes are in the decreasing order of their magnitudes. However, the nuclear sensitivities ξ^(χ) for the $0\nu {\beta }^{-}{\beta }^{-}{\chi }_{}$decay of ¹⁵⁰Nd, ¹⁰⁰Mo, ⁹⁶Zr, ¹³⁰Te, ¹¹⁰Pd, ⁹⁴Zr, and ¹²⁸Te isotopes are in the decreasing order of their magnitudes. In comparison with new Majoron models, the nuclear sensitivities of the promising nuclei for the classical Majoron models are larger by about a factor of 10³⁻⁴. Remarkably, the nuclear sensitivities ξ^(K) of different nuclei are mode dependent.

TABLE 8

Table 8. Nuclear sensitivities ξ^(0ν), ξ^(0N), ${\xi }_{\alpha }^{(\chi )}(\beta \beta \varphi )$ and ${\xi }_{\alpha }^{(\chi )}(\beta \beta \varphi \varphi )$.

5.8. Extraction of Gauge Theoretical Parameters

Limits on the effective mass of light Majorana neutrino 〈m_ν〉, effective mass of heavy Majorana neutrino 〈M_N〉, the ν_h−ν_e mixing matrix element U_eh, and effective Majoron-neutrino coupling constants 〈g_α〉 have been extracted from the largest observed limits on half-lives $T_{1/2}^{(0\nu )}$ of $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay of ^{94, 96}Zr, ^{98, 100}Mo, ^{128, 130}Te, and ¹⁵⁰Nd isotopes and given in Table 9. In Table 10, the effective parameters 〈m_ν〉, 〈M_N〉, and 〈g_α〉 in classical and new Majoron models have been presented. In the case of ¹³⁰Te nuclei, the most stringent limits on 〈m_ν〉 and 〈M_N〉 are < 0.17 eV and >1.12 × 10⁸ GeV, respectively. Within classical Majoron model, the most stringent extracted limit on $\langle g_{\alpha }\rangle <1.89\times 10^{-5}$ is obtained for ¹⁰⁰Mo isotope. As the NTMEs and phase space factors of the new Majoron models are smaller than those of classical Majoron models, the extracted limits on 〈g_α〉 of new Majoron models are large by a factor of 10⁴⁻⁵ than those of classical Majoron models. In Figure 2, the extracted limits on the ν_h−ν_e mixing matrix element U_eh from the largest observed limits on the half-lives $T_{1/2}^{0\nu }$ of 0νβ⁻β⁻ decay are displayed. In comparison to laboratory experiments, astrophysical and cosmological observations [156, 157], the extracted limits on the ν_h−ν_e mixing matrix element U_eh span a wider region of ν_h mass m_h and are comparable to the limits obtained in Blennow et al. [158].

TABLE 9

Table 9. Experimentally observed half-lives $T_{1/2}^{(0\nu )}(yr)$, $T_{1/2}^{(0\nu \varphi )}$ (yr) and $T_{1/2}^{(0\nu \varphi \varphi )}$ (yr) of ⁹⁴Zr, ⁹⁶Zr, ¹⁰⁰Mo, ¹¹⁰Pd ^{128, 130}Te, and ¹⁵⁰Nd isotopes.

TABLE 10

Table 10. Extracted effective light Majorana mass < m_ν>, effective heavy Majorana mass < M_N>, effective Majoron-neutrino couplings 〈g_α〉.

FIGURE 2

Figure 2. Variation in extracted limits on the ν_h−ν_e mixing matrix element $|U_{eh}{|}^2$ with the mass m_h.

As the sensitivities of the ongoing double-β decay experiments juxtaposed with the new Majoron models are quite weak, a comparison between the expected half lives $T_{1/2}^{(0\nu \chi )}$ of ^{94, 96}Zr, ¹⁰⁰Mo, ^{128, 130}Te, and ¹⁵⁰Nd isotopes within classical and new Majoron models is of experimental interest. In Table 11, the predicted half-lives $T_{1/2}^{(0\nu \chi )}$ for spectral indices n = 1, 3 of $0\nu {\beta }^{-}{\beta }^{-}{\varphi }_{}$ and n = 3, 7 of 0νβ⁻β⁻ϕϕ decay modes with $\langle g_{\alpha }\rangle =10^{-6}$ and g_A = 1.254 are presented. In addition, the predicted half-lives $T_{1/2}^{(0\nu )}$ of $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay of ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes are also given for 〈m_ν〉 = 50 meV. The smallest and largest predicted half-lives $T_{1/2}^{(0\nu \chi )}$ correspond to ¹⁵⁰Nd and ¹²⁸Te isotopes, respectively. It is noticed that the predicted half-lives $T_{1/2}^{(0\nu \chi )}$ of classical Majoron models for the potential nuclei are within the reach of planned experiments and in the near future, no observable decay signal of new Majoron models can be detected.

TABLE 11

Table 11. Predicted half-lives $T_{1/2}^{(0\nu )}(yr)$, $T_{1/2}^{(0\nu \varphi )}$ (yr) and $T_{1/2}^{(0\nu \varphi \varphi )}$ (yr) of ⁹⁴Zr, ⁹⁶Zr, ¹⁰⁰Mo, ¹¹⁰Pd ^{128, 130}Te, and ¹⁵⁰Nd isotopes with m_ν = 0.05 eV and 〈g〉 = 10⁻⁶.

6. Conclusions

In the PHFB model, sets of four HFB intrinsic wave functions have been generated with four different parameterizations of pairing plus multipole effective two body interaction. These sets of four HFB intrinsic wave functions reasonably reproduced the observed spectroscopic properties, namely the yrast spectra, deformation parameters β₂ (the reduced B(E2:0⁺ → 2⁺) transition probabilities), static quadrupole moments Q(2⁺), g-factors g(2⁺) of participating nuclei in $2\nu {\beta }^{-}{\beta }_{}^{-}$ decay, and the NTMEs M_2ν [66, 67]. With the consideration of three different parameterizations of Jastrow type of SRC, sets of twelve NTMEs have been calculated using dipole form factor to study the 0νβ⁻β⁻ decay of ^{94, 96}Zr, ¹⁰⁰Mo, ¹¹⁰Pd, ^{128, 130}Te, and ¹⁵⁰Nd isotopes within mechanisms involving light as well as heavy Majorana neutrino, sterile neutrinos, and Majorons.

By using $\overline{A}/2$ in place of $\overline{A}$, relative changes in NTMEs M^(0ν) and $M_{CR}^{(\chi )}$ are less than 16%, and the NTMEs $M_{{\omega }^2}^{(\chi )}$ vary by a factor of 2. Due to FNS and SRC, changes in NTMEs M^(0ν) and $M_{CR}^{(\chi )}$ are below 23 and 17%, respectively, and NTMEs $M_{{\omega }^2}^{(\chi )}$ remain almost unchanged. In the case of heavy Majorana neutrino exchange, the NTMEs M^(0N) are reduced by about 38–42% due to FNS and with the addition of SRC1, SRC2 and SRC3, NTMEs are further reduced by 65–68%, 40–42%, and 18–20%, respectively. Due to deformation, the NTMEs M^(0ν), M^(0N), $M_{CR}^{(\chi )}$, and $M_{{\omega }^2}^{(\chi )}$ change by a factor of 2–6.

With (without) SRC1, estimated uncertainties $\Delta {\overline{M}}^{(0\nu )}$ and $\Delta {\overline{M}}_{CR}^{(\chi )}$ are about 8–15% (4–13%) and 9–15% (4–14%), respectively. Uncertainties $\Delta {\overline{M}}_{{\omega }^2}^{(\chi )}$ in NTMEs ${\overline{M}}_{{\omega }^2}^{(\chi )}$ exhibit a negligible dependence on SRC and are about 7.0–21% (7.5–21%) without (with) SRC1. Further, uncertainties in NTMEs $M_N^{(0\nu )}$ are about 35% and by excluding NTMEs due to SRC1 corresponding to Miller-Spencer parametrization of Jastrow type of SRC, the maximum uncertainty is reduced to less than 20%. In the case of sterile neutrinos, the uncertainties in NTMEs without (with) SRC1 are in between 4–20% (8–36%) depending on the considered mass of the sterile neutrinos.

The nuclear sensitivities ξ^(χ) for new Majoron models are smaller by 3 to 4 order of magnitudes than those of classical Majoron models. The study of sensitivities of considered nuclei suggests that to extract the effective mass of light Majorana neutrino 〈m_ν〉, ¹⁰⁰Mo is the preferred isotope and ¹⁵⁰Nd is the favorable isotope to extract the 〈g_M〉. Thus, the sensitivities of different nuclei are mode dependent. The best limit on the effective neutrino mass 〈m_ν〉 < 0.17 eV is obtained from the observed limit on the half-live $T_{1/2}^{0\nu }$ > 1.5 × 10²⁵ yr of $0\nu {\beta }^{-}{\beta }_{}^{-}$ decay of ¹³⁰Te isotope [151]. Using average NTMEs ${\overline{M}}_N^{(0\nu )}$, the extracted stringent limit on the effective heavy Majorana neutrino mass 〈M_N〉 from available limits on experimental half-lives $T_{1/2}^{0\nu }$ is >1.12 × 10⁸ GeV for ¹³⁰Te isotope. It has been observed that in comparison to laboratory experiments, astrophysical and cosmological observations, the extracted limits on the sterile neutrino ν_h−ν_e mixing matrix element U_eh extend over a wider region of mass m_h. In the case of ¹⁰⁰Mo isotope, one obtains the best limit on the classical Majoron-neutrino coupling constant 〈g_M〉 < 1.89 × 10⁻⁵. Extracted effective Majoron-neutrino coupling constants 〈g_α〉 for new Majoron models are 3–4 order of magnitude larger than those of classical Majoron models. The calculated half lives $T_{1/2}^{(0\nu \chi )}$ for < g> = 10⁻⁶ suggest that although in the ongoing/planned ββ experiments, the classical Majoron accompanied 0νβ⁻β⁻ϕ decay might be observed, it would be very difficult to observe the Majoron accompanied 0νβ⁻β⁻χ decay within new Majoron models in the near future.

Author Contributions

PRat substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work. RC drafting the work and analysis of data for the work. KC revising it critically for important intellectual content. PRai revising it critically for important intellectual content.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We thank all our collaborators, especially Dr. J. G. Hirsch, for providing valuable suggestions through his active participation over the years. This work has been partially supported by DST-SERB, India (Grant No. SR/FTP/PS-085/2011), DST-SERB (Grant No. SB/S2/HEP-007/2013), the Council of Scientific and Industrial Research (CSIR), India (Sanction No. 03(1216)/12/EMR-II), and the Indo-Italian Collaboration DST-MAE project via Grant No. INT/Italy/P-7/2012 (ER).

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