Quantifying statistical uncertainties in ab initio nuclear physics using Lagrange multipliers

Theoretical predictions need quantified uncertainties for a meaningful comparison to experimental results. This is an idea which presently permeates the field of theoretical nuclear physics. In light of the recent progress in estimating theoretical uncertainties in ab initio nuclear physics, I here present and compare methods for evaluating the statistical part of the uncertainties. A special focus is put on the (for the field) novel method of Lagrange multipliers (LM). Uncertainties from the fit of the nuclear interaction to experimental data are propagated to a few observables in light-mass nuclei to highlight any differences between the presented methods. The main conclusion is that the LM method is more robust, while covariance-based methods are less demanding in their evaluation.

Figure 1

Illustration of the LM method. A two-parameter chi-squared surface is shown as a filled surface, with the edge corresponding to $\mathrm{\Delta }{\chi }^2=L$. The solid lines are contour levels of an observable $O$ with central value 0. The statistical uncertainty of the LM method is given by all contour lines that crosses the filled surface. In this case this results in an uncertainty of $\pm 2$. The dashed line represents the parameter values obtained for various fixed values of the observable $O$. The dotted line is a contour line for the chi-squared surface. As expected, the dashed line crosses the contour lines of $O$ at the point where the chi-squared value is lowest.

Figure 2

(Upper) Propagated statistical uncertainties for the helium-4 binding energy, in keV. The methods, from left to right, are (i) Monte Carlo sampling using the covariance matrix, (ii) the same as (i), using a quadratic approximation of the LEC dependence of the observable, (iii) the same as (ii), using instead a linear approximation, (iv) the same as (iii), except using an approximate covariance matrix, (v) the LM method, and (vi) a quadratic approximation to (v). See text for details. (Lower) The variation in ${\chi }^2$ as a function of the observable value, as obtained using the LM method. The solid line shows the quadratic approximation of the LM calculations.

Figure 3

The same as in Fig. 2, for the helium-4 point-proton radius, in fm.

Figure 4

The same as in Fig. 2, for the deuteron binding energy, in keV.

Figure 5

The same as in Fig. 2, for the neutron-proton analyzing power at laboratory scattering energy $175.26\mathrm{MeV}$ and center-of-mass scattering angle 97.63 degrees.

Figure 6

The same as in Fig. 2, for the deuteron point-proton radius in fm, when using artificially enlarged uncertainties in the observables, $\gamma {\sigma }_n^{\left(\mathrm{tot}\right)}$ with $\gamma =10$. This results in larger statistical uncertainties and a nonquadratic chi-squared surface around the minimum. The error bar with a square is a modified quadratic propagation to account for higher-order terms in the chi-squared surface; see the text for details.