Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions

Chiral effective field theory ($\chi \mathrm{EFT}$) provides a systematic approach to describe low-energy nuclear forces. Moreover, $\chi \mathrm{EFT}$ is able to provide well-founded estimates of statistical and systematic uncertainties—although this unique advantage has not yet been fully exploited. We fill this gap by performing an optimization and statistical analysis of all the low-energy constants (LECs) up to next-to-next-to-leading order. Our optimization protocol corresponds to a simultaneous fit to scattering and bound-state observables in the pion-nucleon, nucleon-nucleon, and few-nucleon sectors, thereby utilizing the full model capabilities of $\chi \mathrm{EFT}$. Finally, we study the effect on other observables by demonstrating forward-error-propagation methods that can easily be adopted by future works. We employ mathematical optimization and implement automatic differentiation to attain efficient and machine-precise first- and second-order derivatives of the objective function with respect to the LECs. This is also vital for the regression analysis. We use power-counting arguments to estimate the systematic uncertainty that is inherent to $\chi \mathrm{EFT}$, and we construct chiral interactions at different orders with quantified uncertainties. Statistical error propagation is compared with Monte Carlo sampling, showing that statistical errors are, in general, small compared to systematic ones. In conclusion, we find that a simultaneous fit to different sets of data is critical to (i) identify the optimal set of LECs, (ii) capture all relevant correlations, (iii) reduce the statistical uncertainty, and (iv) attain order-by-order convergence in $\chi \mathrm{EFT}$. Furthermore, certain systematic uncertainties in the few-nucleon sector are shown to get substantially magnified in the many-body sector, in particular when varying the cutoff in the chiral potentials. The methodology and results presented in this paper open a new frontier for uncertainty quantification in ab initio nuclear theory.

Popular Summary

Quantifying the unknown is an outstanding question in all fields of science. Reliable theoretical errors make it possible to infer the significance of a disagreement between experiment and theory, which may hint at new physics. These errors are also useful for identifying relevant new experiments. Here, we develop a framework for uncertainty quantification in ab initio nuclear theory and propagate errors to several nuclear observables.

We provide, for the first time, a common statistical regression analysis of two key frameworks in theoretical nuclear physics: ab initio many-body methods and chiral effective field theory. Our approach is computationally feasible, which allows us to incorporate systematic uncertainties and to establish a credible program for assessing the total error budget. We provide a set of mathematically optimized nuclear interaction models with known statistical properties and demonstrate methods to propagate this information to various nuclear observables. We use brute-force Monte Carlo sampling with 100,000 sampling points to demonstrate the accuracy of our error-propagation methods. We show that the theoretical uncertainties in nuclear physics very often dominate over experimental uncertainties and that systematic errors are amplified in heavier nuclei such as ${}^{16}O$; while the systematic uncertainty is found to be 5% for the binding energy of ${}^4\mathrm{H}e$, it increases to 30% for ${}^{16}O$.

We expect that our results and methodology can be readily applied to explore uncertainties in the field of ab initio nuclear theory.

Figure 1

Schematic overview of the Feynman diagrams present at leading order (LO), next-to-leading order (NLO), and next-to-next-to-leading order (NNLO). Nucleons (pions) are represented by solid (dashed) lines. The $NNN$ interaction enters at NNLO. A circle, diamond, and square represent a vertex of order $\mathrm{\Delta }=0$, 1, and 2, respectively.

Figure 2

Residual distribution for the NNLOsim potential, with a sample mean and standard deviation of $-0.04(1)$ and 0.977(9), respectively. The deviations from normality, as discussed in the text, are mainly due to the model error of $\chi \mathrm{EFT}$.

Figure 3

Saclay amplitudes $a$ to $e$ at a ${\theta }_{\mathrm{cm}}=45°$ scattering angle for the potential NNLOsim. The model error bands were extracted according to the discussion in the text.

Figure 4

Comparison between calculated first and second derivatives of an objective function using finite differences (third order) with different step sizes (filled lines) and automatic differentiation (dashed lines). The calculation is done at a minimum where the first derivatives should be approximately zero. Because of cancellation effects, the finite-difference method cannot correctly reproduce the low values of the derivatives for any step size.

Figure 5

(a) Cumulative ${\chi }^2/N_{\mathrm{dof}}$ for $NN$ scattering data including the model error (see Sec. 2e). Note that the amplitude of the model error is chosen so that ${\chi }^2/N_{\mathrm{dof}}=1$ when all data up to $T_{\mathrm{lab}}=290\mathrm{MeV}$ are included. (b,c) Cumulative ${\chi }^2/N_{\mathrm{dof}}$ without the model error for $NN$ and $\pi N$ scattering data, respectively.

Figure 6

Graphical representation of the linear correlation matrix for NNLOsep (left panel) and NNLOsim (right panel) including selected LECs. The separately optimized NNLOsep potential does not probe the statistical correlation between LECs entering different optimization stages. It is striking that there are almost no correlations for the NNLOsep potential, while for the NNLOsim potential the situation is quite the opposite.

Figure 7

Histograms (filled green area) for the sampled probability distribution of the $nn$ (b,d) and $pp$ (a,c) scattering lengths (including Coulomb) using the NNLO potentials: NNLOsep (a,b) and NNLOsim (c,d). The dashed (solid) lines show error estimates from the sample assuming that the scattering length depends linearly (quadratically) on the fitting parameters. The final theory result (red square) from Eq. (34) agrees well with the sampled distribution.

Figure 8

Joint statistical probability distribution for $E({}^4\mathrm{He})$ and $r_{\mathrm{pt}\text{−}\mathrm{p}}({}^2\mathrm{H})$ for (a) NNLOsim and (b) NNLOsep obtained in a Monte Carlo sampling ($N_{\text{sample}}={10}^5$) as described in the text. Contour lines for this distribution are shown as black solid lines, while blue dotted (red dashed) contours are obtained assuming a linear (quadratic) dependence on the LECs for the observables.

Figure 9

Comparison between selected $NN$ and $\pi N$ experimental data sets and theoretical calculations for chiral interactions at LO, NLO, and NNLO. The bands indicate the total errors (statistical plus model errors). (a) $np$ total cross section for the sequentially optimized interactions with no clear signature of convergence with increasing chiral order. All other results are for the simultaneously optimized interactions: LOsim, NLOsim, and NNLOsim. (b) $np$ total cross section; (c) $np$ differential cross section; (d) $\pi N$ charge-exchange, differential cross section; (e) $\pi N$ elastic, differential cross section.

Figure 10

Predictions for the different reoptimizations of NNLOsim. On the $x$ axis is the maximum $T_{\mathrm{lab}}$ for the $NN$ scattering data used in the optimization. (a) Model error amplitude (20) reoptimized so that ${\chi }^2/N_{\mathrm{dof}}=1$ for the respective data subset. (b) Model prediction for the $np$ total cross section at $T_{\mathrm{lab}}=300\mathrm{MeV}$ with error bars representing statistical and model errors for the different reoptimizations.

Figure 11

Binding-energy predictions for (a) ${}^4\mathrm{He}$ and (b) ${}^{16}\mathrm{O}$ with the different reoptimizations of NNLOsim. On the $x$ axis is the employed cutoff $\mathrm{\Lambda }$. Vertically aligned red markers correspond to different $T_{\text{Lab}}^{\max }$ for the $NN$ scattering data used in the optimization. The experimental binding energies are $E({}^4\mathrm{He})\approx -28.30\mathrm{MeV}$, represented by a gray band in panel (a), and $E({}^{16}\mathrm{O})\approx -127.6\mathrm{MeV}$ [98]. Statistical error bars on the theoretical results are smaller than the marker size on this energy scale.