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

Abstract

The potential correlation between the ordinary muon capture (OMC) on ¹³⁶Ba and $0\nu \beta \beta$ decay of ¹³⁶Xe is explored. For this, we compute $0\nu \beta \beta$-decay amplitudes for intermediate states in ¹³⁶Cs below 1 MeV of excitation and for angular-momentum values $J\le 5$ by using the proton–neutron quasiparticle random-phase approximation (pnQRPA) and nuclear shell model (NSM). We compare these amplitudes with the corresponding OMC rates, computed in a previous Universe article (Universe 2023, 9, 270) for the same energy and angular-momentum ranges. The obtained results suggest that an extension of the present analysis to a wider energy and angular-momentum region could be highly beneficial for probing the $0\nu \beta \beta$-decay nuclear matrix elements using experimental data on OMC rates to intermediate states of $0\nu \beta \beta$ decays.

1. Introduction

The theoretical study of the hypothesized rare neutrinoless double beta ($0\nu \beta \beta$) decay is challenging, yet it is among the most promising avenues of physics research 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 are seldom available and are 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 2nd and 4th powers, 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. Recent calculations of OMC rates were performed in [30].

Only in the recent past have major experimental efforts been made to leverage OMC to illuminate further the mystery of $0\nu \beta \beta$ decay, by measuring (partial) OMC rates, as performed in present state-of-the-art experiments such as the MONUMENT experiment [31]. Such experiments will make available invaluable experimental constraints for grounding the theoretical modeling of OMC processes, offering a tangible map for the improvement of nuclear models, and 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 [10,11,12,32]. Such connections can be used to determine the accuracy of the computed $0\nu \beta \beta$-decay NMEs. In Reference [32], 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 [32], 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 [33], 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) [34] 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 [33]. 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 [33]. As in [33], the decay amplitudes are computed for states with excitation energy $\le 1$ MeV, an energy range relevant to present-day MONUMENT experiment [31].

In order to give the reader a brief glimpse of the parameters used in the calculations, we repeat here some of the information given in full length in Ref. [33]. For the NSM calculations, we chose the jj55pn model space with the $2s$-$1d$-$0g_{7/2}$-$0h_{11/2}$ set of single-particle orbitals for both protons and neutrons. We use the sn100pn interaction [35] and we use the quenching factor $q=0.74$ benchmarked by the works [36,37,38], leading to the effective axial coupling of $g_A^{eff}=0.93$. We determine the value of the pseudoscalar coupling $g_P$ by using the Godberger–Treiman partially conserved axial vector current (PCAC) hypothesis $g_P/g_A\approx 6.8$. For more details, see [33].

For the pnQRPA, we use the no-core valence space of Ref. [39] based on the Woods–Saxon parameters of Ref. [40] modified slightly at the proton and neutron Fermi surfaces in order to better reproduce the single-quasiparticle type of spectra of the neighboring odd-A nuclei. We use the Bonn-A G-matrix interaction [41] with the BCS (Bardeen–Cooper–Schriefer) mean field defined by the pairing-parameter values $g_p^{pair}=0.83$ for protons and $g_n^{pair}=0.87$ for neutrons, fitted to reproduce the available phenomenological proton and neutron separation energies in ¹³⁶Ba [34]. The beyond-BCS mean field effective residual interaction is defined using the particle-hole parameter $g_{ph}=1.18$, fitted to the experimental energy of the Gamow–Teller giant resonance. For the particle–particle channel, we use the renormalization scheme introduced in [42] by dividing the corresponding strength parameter to the isoscalar $g_{pp}^{T=0}=0.7$ and isovector $g_{pp}^{T=1}=0.7$ parts, the values taken from Ref. [39]. For the weak axial coupling, we use the value $g_A^{eff}=0.83$ taken from the systematics obtained in [43]. For the weak pseudoscalar coupling, we use the recipe adopted for the NSM above.

An interesting additional point could be raised here: Our adopted pnQRPA framework is based on a spherical single-particle mean field. This should be contrasted with the fact that the nucleus ¹³⁶Xe is spherical but the nucleus ¹³⁶Ba shows signs of deformation, as seen, e.g., in its excitation spectrum. This mismatch of shapes of the $0\nu \beta \beta$ mother and daughter nuclei is a well-known driver of the suppression of the magnitude $0\nu \beta \beta$ NME through the pnQRPA overlap factor present in Equation (8) below [44,45,46]. This feature naturally affects the accuracy of both the OMC rate calculations and the right-leg virtual amplitudes of the $0\nu \beta \beta$ decay. However, as both the OMC and $0\nu \beta \beta$ calculations are affected by the same inaccuracy, the correlations found between the two processes are most likely not altered much. A similar argument supports the view that also the correlations found in the NSM calculations are robust and are not affected by the difference in deformations of the two nuclei.

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 [33]. The OMC formalism of Reference [33] is an extended Morita–Fujii formalism described in detail in [47,48]. Lately, the use of realistic muon wave functions has been implemented [49], and up-to-date computations for OMC rates of ¹³⁶Ba are presented in Reference [33]. 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 follows:

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

(1)

where a negative muon (${\mu }^{-}$) is captured by the atomic 1s ground state of ¹³⁶Ba, leading to final spin-parity 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 follows:

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

(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 [50]. We present here key relations. Assuming light Majorana neutrino exchange [4,50], 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 follows:

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

(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 follows:

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

(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 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 as their energy is ≤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. Phenomenological pnQRPA-computed $0\nu \beta \beta$-decay amplitudes of the NMEs of Equation (6). The amplitudes are given in units of ${10}^{-3}$.

${\mathit{J}}^{\mathit{\pi }}$	${\mathit{E}}_{\mathbf{exc}}$ [keV]	${\mathit{M}}_{\mathbf{F}}$	${\mathit{M}}_{\mathbf{GT}}$	${\mathit{M}}_{\mathbf{T}}$	${\mathit{M}}^{\left(0\mathit{\nu }\right)}$
$5_1^{+}$	0	0	−0.18	0.040	−0.14
$3_1^{+}$	102	0	−0.28	0.037	−0.24
$2_1^{+}$	120	0.20	−0.13	−0.034	−0.46
$4_1^{+}$	154	0.07	−0.02	−0.006	−0.13
$1_1^{+}$	193	0	−1.61	0.021	−1.59
$4_2^{+}$	203	4.96	−0.32	−0.13	−7.65
$3_2^{+}$	264	0	−51.6	16.3	−35.2
$3_3^{+}$	281	0	−10.1	0.88	−9.20
$3_4^{+}$	338	0	−9.46	−3.27	−12.7
$2_2^{+}$	367	0.01	−0.16	−0.0041	−0.22
$3_1^{-}$	458	0.02	−0.16	−0.0060	−0.25
$4_3^{+}$	494	−0.007	−0.24	−0.089	−0.32
$5_1^{-}$	515	0.03	−0.11	−0.050	−0.21
$4_1^{-}$	558	0	−0.42	0.068	−0.35
$2_3^{+}$	561	5.29	−24.5	−8.22	−40.4
$5_2^{-}$	637	0.07	−0.01	−0.004	−0.12
$4_2^{-}$	695	0	−0.24	0.084	−0.16
$2_1^{-}$	704	0	−0.49	0.097	−0.40
$3_2^{-}$	926	0.23	−0.02	−0.008	−0.36

and

Table 2. Phenomenological NSM-computed $0\nu \beta \beta$-decay amplitudes of the NMEs of Equation (6). The amplitudes are given in units of ${10}^{-3}$.

${\mathit{J}}^{\mathit{\pi }}$	${\mathit{E}}_{\mathbf{exc}}$ [keV]	${\mathit{M}}_{\mathbf{F}}$	${\mathit{M}}_{\mathbf{GT}}$	${\mathit{M}}_{\mathbf{T}}$	${\mathit{M}}^{\left(0\mathit{\nu }\right)}$
$5_1^{+}$	0	0	−4.355	1.411	−2.944
$3_1^{+}$	23	0	−5.695	1.445	−4.250
$4_1^{+}$	39	1.230	−0.153	−0.058	−1.633
$2_1^{+}$	83	6.993	−5.160	−1.433	−14.677
$3_2^{+}$	181	0	−2.269	−0.555	−2.824
$2_2^{+}$	224	0.894	−4.481	−1.748	−7.262
$3_3^{+}$	244	0	−1.647	−0.059	−1.706
$4_2^{+}$	323	1.989	−2.575	−1.019	−5.895
$4_3^{+}$	498	2.582	−3.525	−1.578	−8.088
$3_4^{+}$	517	0	6.540	−0.289	6.251
$5_1^{-}$	522	0.723	−2.483	−1.048	−4.366
$3_1^{-}$	545	0.379	−4.893	−1.701	−7.033
$1_1^{+}$	545	0	33.820	−1.602	32.218
$4_1^{-}$	547	0	5.565	0.589	6.154
$2_3^{+}$	615	−1.426	2.804	0.780	5.232
$5_2^{-}$	670	1.313	−0.511	−0.210	−2.239
$1_2^{+}$	752	0	6.066	−0.064	6.003
$4_2^{-}$	760	0	−1.281	0.309	−0.972
$2_4^{+}$	803	0.290	−0.889	−0.251	−1.475
$4_4^{+}$	885	0.082	−0.108	−0.044	−0.248
$2_1^{-}$	1016	0	−44.144	0.371	−43.772

for the aforementioned computational scheme in Section 2.1.

Computed OMC rates for individual $J_k^{\pi }$ states can be found in Reference [33]. 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 [32]. Another important consideration is that, given that the nuclear-structure calculations are not perfect, we only consider the trends within the same nuclear model. Therefore, for the purposes of this paper, we look at trends in pnQRPA (qrpa-phen) and NSM (sm-phen) results independently. In

Table 3. Cumulative $0\nu \beta \beta$-decay amplitudes $M^{\left(0\nu \right)}$($J^{\pi }$) and OMC rates OMC($J^{\pi }$) for multipoles $J^{\pi }\le 5$. OMC rates and the decay amplitudes are given in units of ${10}^3$ 1/s and ${10}^{-3}$, respectively.

	qrpa-phen		sm-phen
$J^{\pi }$	$M^{\left(0\nu \right)}$($J^{\pi }$)	OMC($J^{\pi }$)	$M^{\left(0\nu \right)}$($J^{\pi }$)	OMC($J^{\pi }$)
$5^{+}$	−0.14	0.50	−2.944	0.0
$4^{+}$	−8.10	20.4	−15.864	9.8
$3^{+}$	−57.4	87.8	−2.529	29.9
$2^{+}$	−41.0	201.1	−18.182	34.9
$1^{+}$	−1.59	206.9	38.221	5.2
$5^{-}$	−0.33	0.80	−6.605	0.6
$4^{-}$	−0.52	21.2	5.182	15.0
$3^{-}$	−0.61	38.7	−7.033	9.7
$2^{-}$	−0.40	14.2	−43.772	44.4

, 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 are 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 $M^{\left(0\nu \right)}$($J^{\pi }$) appear to be roughly “mirror reflections” 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 for the Gamow–Teller giant resonance region in the case of the pnQRPA.

In order to shed further light on the comparison of the OMC and $0\nu \beta \beta$ results, one can plot the absolute values of $0\nu \beta \beta$ amplitudes $|M^{\left(0\nu \right)}|$ ($0\nu \beta \beta$ NME for short) against the absolute values of the average OMC nuclear matrix elements $|M^{\mu }{|}_{\mathrm{ave}}$ (OMC NME for short), defined in [32] as

$$W=2\pi |M^{\mu }{|}_{\mathrm{ave}}^2{q}^2{\displaystyle \frac{dq}{d{E}_f}},$$

(9)

with the detailed expression of $|M^{\mu }{|}_{\mathrm{ave}}$ given in [32]. As performed in [32], we can plot the cumulative percentages of $|M^{\left(0\nu \right)}|$ and $|M^{\mu }{|}_{\mathrm{ave}}$ as functions of the excitation energy of the individual $J^{\pi }$ states below 1 MeV of excitation in ¹³⁶Cs. This is shown in Figure 3.

As can be seen in Figure 3, both for the NSM and the pnQRPA, the two absolute NMEs follow each other’s trends rather closely. This is particularly true for the NSM, where the big jump in the cumulative absolute values happens at the same energy for both models. For the pnQRPA, two almost vertical jumps are visible for the $0\nu \beta \beta$ NME, whereas for the OMC NME, the corresponding jumps are softer but are still located in the same energy regions as those of the $0\nu \beta \beta$ NME. This comparison could be extended to higher intermediate energies once data on OMC rates become available. Through the potential data, one can access the OMC NME $|M^{\mu }{|}_{\mathrm{ave}}$ as a function of the excitation energy and, in principle, correct $|M^{\left(0\nu \right)}|$ accordingly at large jumps of the OMC NME.

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, in addition to rather strong correlations in the cumulative values of the OMC and $0\nu \beta \beta$ “nuclear matrix elements”. 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.