Uncertainty quantification in the nuclear shell model

The uncertainty quantifications of theoretical results are of great importance in making meaningful comparisons of those results with experimental data and making predictions in experimentally unknown regions. By quantifying uncertainties, one can make more solid statements about, e.g., origins of discrepancy in some quantities between theory and experiment. We propose a novel method for uncertainty quantification for the effective interactions of nuclear shell-model calculations as an example. The effective interaction is specified by a set of parameters, and its probability distribution in the multidimensional parameter space is considered. This enables us to quantify the agreement with experimental data in a statistical manner and the resulting confidence intervals show unexpectedly large variations. Moreover, we point out that a large deviation of the confidence interval for the energy in shell-model calculations from the corresponding experimental data can be used as an indicator of some exotic property, e.g., $\alpha$ clustering. Other possible applications and impacts are also discussed.

Figure 1

Marginal distributions of 17 parameters for the $p$ shell, in MeV. Red curves and blue bars denote the $3\sigma$ and $1\sigma$ confidence intervals, respectively. Each plot is scaled to the same height for aesthetic purposes. White dots denote the medium of the distribution and, namely, the ${\mathbf{\theta }}_{\mathrm{MAP}}$. The TBMEs are abbreviated as $\mathrm{v}(abcd;JT)$, and the orbits are labeled by $1=p_{1/2}$ and $3=p_{3/2}$.

Figure 2

Excitation energies and energies of ${}^{12}\mathrm{C}$ and ${}^{14}\mathrm{C}$. Each state is classified by its $(J^{\pi }T,N_{JT})$, where $(J^{\pi }T,N_{JT})$ stands for the $N\mathrm{th}$ lowest states with the total angular momentum $J^{\pi }$ and total isospin $T$. Experimental data on ENSDF [33] are shown by horizontal dotted lines in red. Corresponding theoretical results with the LA samples are summarized in violin plots in green or pink. Here green and pink one corresponds to the states in fit and not in fit, respectively. In each violin plot, mean values and $1\overline{\sigma }$ errors are shown by white horizontal lines and white error bars, respectively, where $\overline{\sigma }$ means standard deviations of all results. Results by a CK interaction and the VS-IMSRG are also shown.