Correlating the 0νββ-Decay Amplitudes of ¹³⁶Xe with the Ordinary Muon Capture (OMC) Rates of ¹³⁶Ba

1. Introduction

The theoretical study of the hypothesized rare neutrinoless double beta ($0\nu \beta \beta$) decay is a challenging, yet among the most promising avenues of searches of physics beyond the standard model [1,2,3,4,5]. The complexity in the study of $0\nu \beta \beta$ decay stems from the involvement of nuclear-structure effects/correlations from low to high momentum-exchange scales ($q\le 100-200$ MeV) and nuclear states of high energy and/or multipolarity ($J^{\pi }$). Experimental nuclear-structure data at medium and high momentum scales is seldom available and is almost entirely uncharted territory, making it difficult for nuclear models and hence the computed $0\nu \beta \beta$ nuclear matrix elements (NMEs) to be improved upon being tuned to such data. Given that $0\nu \beta \beta$ decay has not been measured, accurate nuclear modeling of this process for various $0\nu \beta \beta$-decay candidates is essential for determining the sensitivity of experiments designed to detect this rare decay [6]. There are significant discrepancies in the $0\nu \beta \beta$-decay NMEs computed in various nuclear-model frameworks [1], and imperfect nuclear-structure calculations demand the use of an effective value of the axial vector coupling ($g_A^{eff}$) [5,6,7,8]. Discrepancies in $0\nu \beta \beta$-decay NMEs across nuclear models and uncertainty in the value of $g_A^{eff}$ propagate in the 2^nd and 4^th power, respectively, to the computed/predicted half-lives.

Ordinary muon capture (OMC) is a seemingly miraculous process in that it is the only known practical way to systematically investigate the nuclear structure experimentally at momentum scales relevant to the physics of $0\nu \beta \beta$ decay [9,10,11,12]. OMC can also populate all the nuclear states that are intermediate states of the odd-odd nucleus, via which the $0\nu \beta \beta$ decay proceeds [9,10,11,12]. This means that OMC can and is used to access decay amplitudes of one leg, involving either the daughter or the mother nucleus of the "two-step" rare transition, depending on whether $0\nu \beta \beta$ is of ${\beta }^{-}$ or ${\beta }^{+}$/EC type, respectively [9,10,11,12,13]. Involvement of common decay amplitudes in this way for the computed OMC NMEs/rates and $0\nu \beta \beta$-decay NMEs leads us to look for connections between the two. Towards both ends of computing more accurate $0\nu \beta \beta$-decay NMEs and determination of $g_A^{eff}$, the OMC process is a gift from nature as it can help address both of these goals. The study of OMC is the best-known way to test the fitness of nuclear models and improve their accuracy, by closely tuning them to experimental data for computing physically relevant OMC NMEs. Tuning the nuclear models this way makes them optimized to compute $0\nu \beta \beta$-decay amplitudes and ultimately NMEs due to similar nuclear and weak-interaction contributions involved in the two processes [10,11,12,14]. As an example, OMC can give us access to the value of the particle-particle interaction parameter ($g_{pp}$) in the pnQRPA (proton-neutron quasiparticle random-phase approximation) framework [10,11,12], and $g_A^{eff}$ and/or the effective value of the pseudoscalar coupling ($g_P$) [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] at momentum scales relevant for $0\nu \beta \beta$ decay.

Only in the recent past, major experimental efforts have been made to leverage OMC to illuminate further the mystery of $0\nu \beta \beta$ decay, by measuring (partial) OMC rates as being done in present state-of-the-art experiments such as the MONUMENT experiment [30]. Such experiments will make available invaluable experimental constraints for grounding the theoretical modeling of OMC processes, and offering a tangible map for the improvement of nuclear models, leading the way to more accurate computed OMC/$0\nu \beta \beta$-decay NMEs.

The connection of having common decay amplitudes in the computed NMEs of the two processes prompted the search for the potential correlations and trends between $0\nu \beta \beta$-decay NMEs and OMC rates/NMEs as presented in References [10,11,12,31]. Such connections can be used to determine the accuracy of the computed $0\nu \beta \beta$-decay NMEs. In Reference [31], average OMC NMEs and $0\nu \beta \beta$-decay NMEs for key $0\nu \beta \beta$-decay candidates including ¹³⁶Xe (the focus of this work) were compared in the framework of pnQRPA, and systematic correspondences were observed between the two. The NMEs were compared in order to minimize the kinematic and phase-space effects in the anticipated correlations. Given the trends observed in Reference [31], we anticipate seeing this correspondence map to correspondences between OMC rates and $0\nu \beta \beta$-decay NMEs. Evidence for such a connection is foreshadowed from trends between OMC rates and $2\nu \beta \beta$-decay NMEs as presented in References [10,11,12]. Correlations between OMC rates and $0\nu \beta \beta$-decay NMEs are of high interest as this can be a direct bridge between experimental OMC rates and theoretical $0\nu \beta \beta$-decay NMEs.

The focus of this work is to further elucidate this bridge in the context of $0\nu \beta \beta$ decaying ¹³⁶Xe, using for the first time the pnQRPA and nuclear shell model (NSM) together in OMC and non-closure $0\nu \beta \beta$ formalisms. We compute the $0\nu \beta \beta$-decay amplitudes in the above frameworks and compare them with results obtained for the OMC rates of ¹³⁶Ba, as presented in Reference [32], using the same nuclear models. In using different nuclear models, one can see if the potential correlations are model-independent and follow the similarities and differences in the trends that emerge.

2. Theory

2.1. Nuclear-Model Calculations

In the present work we adopt the nuclear shell model (NSM) and the proton-neutron quasiparticle random-phase approximation (pnQRPA) [33] as the basic nuclear-model frameworks. We compute the wave functions of the states of the odd-odd nucleus ¹³⁶Cs by using these models in order to access the $0\nu \beta \beta$-decay amplitudes in a non-closure approach and compare them with the corresponding OMC rates computed in Reference [32]. In both the OMC and $0\nu \beta \beta$ calculations we use the phenomenological NSM (sm-phen) and pnQRPA (qrpa-phen) approach, with the relevant parameters defined in Table 1 of Reference [32]. As in [32], the decay amplitudes are computed for states with excitation energy $\le 1$ MeV, an energy range relevant to present-day MONUMENT experiment [30].

2.2. Ordinary Muon Capture (OMC)

The OMC is a well-studied nuclear process, both experimentally and theoretically [9]. In this work we compare our calculated $0\nu \beta \beta$-decay amplitudes with the OMC rates of Reference [32]. The OMC formalism of Reference [32] is an extended Morita-Fujii formalism described in detail in [34,35]. Lately, the use of realistic muon wave functions has been implemented [36], and up-to-date computations for OMC rates of ¹³⁶Ba have been presented in Reference [32]. We refer readers to these results as we use them for the purposes of this work. For completeness, we present here some key relations for computing the OMC rates, where OMC of ¹³⁶Ba proceeds as:

$${\mu }^{-}{+}^{136}\mathrm{Ba}\left(0_{g.s.}^{+}\right)\to {\nu }_{\mu }{+}^{136}\mathrm{Cs}\left(J_f^{\pi }\right),$$

(1)

where a negative muon (${\mu }^{-}$) is captured by the ground state of ¹³⁶Ba, leading to final states $J_f^{\pi }$ in ¹³⁶Cs. At the same time, a muon neutrino (${\nu }_{\mu }$) is emitted. The general expression of the OMC rate is given as:

$$W=2P(2J_f+1)\left(1-{\displaystyle \frac{q}{m_{\mu }+AM}}\right)q^2,$$

(2)

where the momentum exchange q is expressed as

$$q=(m_{\mu }-W_0)\left(1-{\displaystyle \frac{m_{\mu }}{2(m_{\mu }+AM)}}\right).$$

(3)

Here, $J_f$ is the final-state spin-parity, M is the average nucleon mass, A is the nuclear mass number, and $m_{\mu }$ ($m_e$) is the rest mass of the muon (electron). The threshold energy is given by

$$W_0=M_f-M_i+m_e+E_X,$$

(4)

where $M_i$ and $M_f$ are the masses of the initial and final nuclei, and $E_X$ is the excitation energy of the final nuclear state, in our case of ¹³⁶Cs. The rate function P contains the NMEs, phase-space factors, and combinations of weak couplings $g_A$ (axial-vector), $g_P$ (induced pseudoscalar), and $g_M=1+{\mu }_p-{\mu }_n$ (induced weak-magnetism), with ${\mu }_p$ and ${\mu }_n$ being the anomalous magnetic moments of the proton and neutron, respectively.

2.3. $0\nu \beta \beta$ Decay

The computational scheme used here is presented in detail in Reference [37]. We present here key relations. Assuming light Majorana neutrino exchange [4,37], the inverse half-life can be written as

$${\left[t_{1/2}^{\left(0\nu \right)}(0_i^{+}\to 0_f^{+})\right]}^{-1}=g_A^4{G}_{0\nu }{\left|M^{\left(0\nu \right)}\right|}^2{\left|\langle m_{\nu }\rangle \right|}^2,$$

(5)

where $G_{0\nu }$ is the phase-space factor for the final-state leptons, $g_A$ is the axial vector coupling constant, $\langle m_{\nu }\rangle$ is the effective neutrino mass, and $M^{\left(0\nu \right)}$ is the nuclear matrix element (NME). The $M^{\left(0\nu \right)}$ NME can be decomposed as:

$$M^{\left(0\nu \right)}=M_{\mathrm{GT}}^{\left(0\nu \right)}-{\left({\displaystyle \frac{g_V}{g_A}}\right)}^2{M}_{\mathrm{F}}^{\left(0\nu \right)}+M_{\mathrm{T}}^{\left(0\nu \right)},$$

(6)

where $M_{\mathrm{GT}}^{\left(0\nu \right)}$, $M_{\mathrm{F}}^{\left(0\nu \right)}$, and $M_{\mathrm{T}}^{\left(0\nu \right)}$ are the Gamow-Teller, Fermi, and Tensor components of the NME, respectively, and $g_V$ is the vector coupling constant. Contribution from various multipoles constituting all the intermediate transitions is given as :

$$M_K^{\left(0\nu \right)}=\sum _{J^{\pi }}{M}_K^{\left(0\nu \right)}\left(J^{\pi }\right),$$

(7)

where $K=\mathrm{GT},\mathrm{F},\mathrm{T}$, and $M_K^{\left(0\nu \right)}\left(J^{\pi }\right)$ are the contributions from all the states i of an the intermediate multipole $J^{\pi }$. Each multipole contribution is, in turn, decomposed in terms of the two-particle transition matrix elements and one-body transition densities. In the pnQRPA calculations the two-particle transition matrix element reads

$$\begin{array}{c}\hfill M_K^{\left(0\nu \right)}\left(J^{\pi }\right)=\sum _{k_1,k_2,J^{\prime }}\sum _{p{p}^{\prime },n{n}^{\prime }}{(-1)}^{j_n+j_{p^{\prime }}+J+J^{\prime }}\sqrt{2J^{\prime }+1}\left\{\begin{array}{ccc}{j}_p& j_n& J\\ j_{n^{\prime }}& j_{p^{\prime }}& J^{\prime }\end{array}\right\}\times \\ \hfill \left(p{p}^{\prime }:J^{\prime }\left|O_K\right|n{n}^{\prime }:J^{\prime }\right)\left(0_f^{+}\left|\right|{\left[c_{p^{\prime }}^{†}{\tilde{c}}_n\right]}_J\left|\right|J_{k_1}^{\pi }\right)\langle J_{k_1}^{\pi }|J_{k_2}^{\pi }\rangle \left(J_{k_2}^{\pi }\left|\right|{\left[c_p^{†}{\tilde{c}}_n\right]}_J\left|\right|0_i^{+}\right),\end{array}$$

(8)

where $k_1,k_2$ label the pnQRPA solutions for a given multipole $J^{\pi }$, starting from the final ($k_1$) and initial ($k_2$) nuclei, and $p,p^{\prime },n,n^{\prime }$ denote the proton and neutron single-particle quantum numbers. The operator $O_K$ contains the neutrino potentials, the characteristic two-particle operators for the different K components, and short-range correlation effects. The quantities $\left(0_f^{+}\left|\right|{\left[c_{p^{\prime }}^{†}{\tilde{c}}_n\right]}_J\left|\right|J_{k_1}^{\pi }\right)$ and $\left(J_{k_2}^{\pi }\left|\right|{\left[c_p^{†}{\tilde{c}}_n\right]}_J\left|\right|0_i^{+}\right)$ are the corresponding decay amplitudes and $\langle J_{k_1}^{\pi }|J_{k_2}^{\pi }\rangle$ is an overlap factor connecting the two branches of pnQRPA solutions for the ¹³⁶Cs wave functions.

In the NSM calculations we use one unique set of states in ¹³⁶Cs so that the sum over $k_1,k_2$ in Equation (8) is replaced by a sum over a single state number k and the overlap factor is not needed.

3. Results and Discussion

In order to compare the OMC rates and $0\nu \beta \beta$-decay amplitudes for $J_k^{\pi }$ states in a meaningful way, we consider the following physical assumptions: both quantities depend on the energy ($E_k\left(J_k^{\pi }\right)$), multipolarity ($J^{\pi }$), and nuclear-structure content of the $J_k^{\pi }$ (virtual) states being populated in the process. In the case of OMC rates, phase-space factors contribute directly to OMC rates. For $0\nu \beta \beta$-decay NMEs, the dependence of neutrino potentials on energy, affects the concerned decay amplitudes. For our analysis, we can assume the energy dependence to be a constant for all the states since their energy is $\le 1$ MeV. Therefore, our analysis simplifies, and we attribute the observed trends in OMC rates and $0\nu \beta \beta$-decay amplitudes to multipolarity ($J^{\pi }$) and nuclear-structure content of $J_k^{\pi }$ states. We see effects of such dependencies in the computed $0\nu \beta \beta$-decay amplitudes presented in Table 1 and Table 2 for the aforementioned computational scheme in Section 2.1.

Computed OMC rates for individual $J_k^{\pi }$ states can be found in Reference [32]. In order to smooth out the variations of the OMC rates and $0\nu \beta \beta$-decay amplitudes from one individual $J^{\pi }$ state to the other, we study the combined contribution to a given multipole $J^{\pi }$, an effective strategy already implemented in Reference [31]. Another important consideration is that given that the nuclear-structure calculations are not perfect, we only consider the trends within the same nuclear model. So for the purposes of this paper, we look at trends in pnQRPA (qrpa-phen) and NSM (sm-phen) results independently. In Table 3, the cumulative OMC rates OMC($J^{\pi }$) and amplitude contributions to $M^{\left(0\nu \right)}$($J^{\pi }$) for multipole $J^{\pi }$ are given, and also plotted in Figure 1 and Figure 2.

In the figures positive and negative multipoles are plotted separately for clarity, and the quantities are scaled appropriately for optimal comparison. As seen from the plots in Figure 1 and Figure 2, regular variations between the quantities are observed. The variation of OMC rates ($J^{\pi }$) and $M^{\left(0\nu \right)}$($J^{\pi }$) look roughly "mirror reflection" of each other, both in the context of pnQRPA, and NSM. Further conclusions cannot be made given the limited number of states and multipolarity, but the results look promising, giving impetus to a larger-scale study involving a larger set of states for each multipolarity, and covering a larger range of multipolarities, possibly to the Gamow-Teller giant resonance region in the case of the pnQRPA.

4. Conclusions

Correlations between the ordinary muon capture (OMC) on ¹³⁶Ba and $0\nu \beta \beta$ decay of ¹³⁶Xe were searched for using $0\nu \beta \beta$-decay intermediate states in ¹³⁶Cs below 1 MeV of excitation and for angular-momentum values $J\le 5$. We computed $0\nu \beta \beta$-decay amplitudes through these intermediate states by using the proton-neutron quasiparticle random-phase approximation (pnQRPA) and nuclear shell model (NSM). Comparison with a corresponding earlier OMC calculation suggests that there are "mirror type of" correlations between the $0\nu \beta \beta$-decay amplitudes and the OMC rates. These correlations suggest that an extension of the present analysis to a wider energy and angular-momentum region could lead to a practical way to probe the $0\nu \beta \beta$-decay nuclear matrix elements using experimental data on OMC rates to intermediate states of $0\nu \beta \beta$ decays.