## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(J^{\pi}\) & \(\mathbf{E_{\text{res}}}\) [keV] & \(\mathbf{M_{\text{F}}}\) & \(\mathbf{M_{\text{GT}}}\) & \(\mathbf{M_{\text{T}}}\) & \(\mathbf{M^{(br)}}\) \\ \hline \(5^{+}_{1}\) & 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 & \(-5.160\) & \(-1.433\) & \(-14.677\) \\ \(3^{+}_{3}\) & 181 & 0 & \(-2.269\) & \(-0.555\) & \(-2.824\) \\ \(2^{+}_{2}\) & 224 & 0.894 & \(-4.481\) & \(-1.748\) & \(-7.262\) \\ \(3^{+}_{3}\) & 244 & 0 & \(-1.647\) & \(-0.059\) & \(-1.706\) \\ \(4^{+}_{4}\) & 323 & 1.989 & \(-2.575\) & \(-1.019\) & \(-5.895\) \\ \(4^{+}_{4}\) & 498 & 2.582 & \(-3.525\) & \(-1.578\) & \(-0.088\) \\ \(3^{+}_{4}\) & 517 & 0 & 6.540 & \(-0.289\) & \(6.251\) \\ \(5^{-}_{5}\) & 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 & \(5.232\) \\ \(5^{-}_{5}\) & 670 & 1.313 & \(-0.511\) & \(-0.210\) & \(-2.239\) \\ \(1^{+}_{1}\) & 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 & \(-43.772\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Phenomenological NSM-computed \(0\
u\beta\beta\)-decay amplitudes of the NMEs of Equation (6). The amplitudes are given in units of \(10^{-3}\). In order to shed further light on the comparison of the OMC and \(0\
u\beta\beta\) results, one can plot the absolute values of \(0\
u\beta\beta\) amplitudes \(|M^{(0\
u)}|\) (\(0\
u\beta\beta\) NME for short) against the absolute values of the average OMC nuclear matrix elements \(|M^{\mu}|_{\rm{ave}}\) (OMC NME for short), defined in [32] as  
\[W=2\pi|M^{\mu}|_{\rm{ave}}^{2}q^{2}\frac{dq}{dE_{f}}\,, \tag{9}\]  
with the detailed expression of \(|M^{\mu}|_{\rm{ave}}\) given in [32]. As performed in [32], we can plot the cumulative percentages of \(|M^{(0\
u)}|\) and \(|M^{\mu}|_{\rm{ave}}\) as functions of the excitation energy of the individual \(J^{\pi}\) states below 1 MeV of excitation in \({}^{136}\)Cs. 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 \(0\
u\beta\beta\) NME, whereas for the OMC NME, the corresponding jumps are softer but are still located in the same energy regions as those of the \(0\
u\beta\beta\) 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 \(|M^{\mu}|_{\rm{ave}}\) as a function of the excitation energy and, in principle, correct \(|M^{(0\
u)}|\) accordingly at large jumps of the OMC NME. Figure 1: Cumulative \(0\
u\beta\beta\)-decay amplitudes and OMC rates for \(J^{\pi}\) states computed using pnQRPA (**qrpa-phen**). The decay amplitudes and OMC rates are scaled appropriately for optimal comparison. (**a**) \(M^{(0\
u)}(J^{+})\) (in units of \(10^{-3}\)) and OMC(\(J^{+}\)) (in units of \(10^{3}\) 1/s) vs. \(J^{+}\); (**b**) \(M^{(0\
u)}(J^{-})\) (in units of \(10^{-3}\)) and OMC(\(J^{-}\)) (in units of \(10^{5}\) 1/s) vs. \(J^{-}\). Figure 2: Cumulative \(0\
u\beta\beta\)-decay amplitudes and OMC rates for \(J^{\pi}\) states computed using NSM (**smphen**). The decay amplitudes and OMC rates have been scaled appropriately for optimal comparison.