Bayesian truncation errors in chiral effective field theory: Nucleon-nucleon observables

Chiral effective field theory (EFT) predictions are necessarily truncated at some order in the EFT expansion, which induces an error that must be quantified for robust statistical comparisons to experiment. In previous work, a Bayesian model for truncation errors of perturbative expansions was adapted to EFTs. The model yields posterior probability distribution functions (pdfs) for these errors based on expectations of naturalness encoded in Bayesian priors and the observed order-by-order convergence pattern of the EFT. A first application was made to chiral EFT for neutron-proton scattering using the semilocal potentials of Epelbaum, Krebs, and Meißner (EKM). Here we extend this application to consider a larger set of regulator parameters, energies, and observables as a general example of a statistical approach to truncation errors. The Bayesian approach allows for statistical validations of the assumptions and enables the calculation of posterior pdfs for the EFT breakdown scale. The statistical model is validated for EKM potentials whose convergence behavior is not distorted by regulator artifacts. For these cases, the posterior for the breakdown scale is consistent with EKM assumptions.

Figure 1

A Bayesian network [25, 26] for the ${\mathrm{\Delta }}_k$ truncation error model outlined in [17].

Figure 2

(a) Dimensionless coefficients as in Eq. (1) at each EFT order for the $np$ total cross section as a function of laboratory energy for EKM potentials with $R=0.9\mathrm{fm}$, and ${\mathrm{\Lambda }}_b=600\mathrm{MeV}$. Plot (b) uses the $p/{\mathrm{\Lambda }}_b$ expansion only.

Figure 3

Dimensionless coefficients as in Eq. (1) at each EFT order for the $np$ total cross section as a function of laboratory energy for (a) $R=0.8\text{−}\mathrm{fm}$ and (b) $R=1.0\text{−}\mathrm{fm}$ EKM potentials, both with ${\mathrm{\Lambda }}_b=600\mathrm{MeV}$.

Figure 4

Same as Fig. 3 but for EKM potentials with (a) $R=1.1\mathrm{fm}$ and ${\mathrm{\Lambda }}_b=500\mathrm{MeV}$, and (b) $R=1.2\mathrm{fm}$ and ${\mathrm{\Lambda }}_b=400\mathrm{MeV}$.

Figure 5

DoB intervals for the $np$ total cross section for $R=0.9\mathrm{fm}$ at each of the orders, using prior set ${\mathrm{C}}_{\epsilon }$.

Figure 6

Residuals defined in Eq. (11) at each order for the $np$ total cross section for $R=0.9\mathrm{fm}$, with DoB intervals calculated using prior set ${\mathrm{C}}_{\epsilon }$.

Figure 7

Residuals defined in Eq. (11) at each order for the $np$ total cross section for $R=0.9\mathrm{fm}$, with DoB intervals calculated using prior set ${\mathrm{C}}_{0.25\text{−}10}$.

Figure 8

Residuals defined in Eq. (11) at each order for ${[d\sigma /d\mathrm{\Omega }]}_{\text{res}}$ vs c.m. angle $\theta$ with $R=0.9$ fm, $E_{\mathrm{lab}}=96$ MeV, and error bands generated using ${\mathrm{C}}_{0.25\text{−}10}$.

Figure 9

Dimensionless coefficients as in Eq. (1) extracted at each $\theta$ from the differential cross section at $E_{\mathrm{lab}}=96$ MeV. $X_{\text{ref}}$ is chosen to be $X_0$, the leading order result, which is consistent with natural coefficients.

Figure 10

(a) Dimensionless coefficients as in Eqs. (1) and (13), and (b) residuals defined in Eq. (11) as a function of $\theta$ at $E_{\mathrm{lab}}=250$ MeV. (c) Dimensionless coefficients as a function of $E_{\mathrm{lab}}$ at $\theta ={120}^{\circ }$. All use the $R=0.9\mathrm{fm}$ EKM potentials. $X_{\text{ref}}=X_0$ for the differential cross section and $X_{\text{ref}}=1$ for the spin observables.

Figure 11

Consistency plot for the total cross section using the $R=0.9\mathrm{fm}$ EKM potential evaluated at $E=20,40,...,340$ MeV. Results were obtained using prior set ${\mathrm{C}}_{0.25\text{−}10}^{\left(1\right)}$ and are averaged over $\mathrm{NLO},\mathrm{N}{}^2\mathrm{LO}$, and $\mathrm{N}{}^3\mathrm{LO}$. The shaded bands represent 68% and 95% confidence intervals for the success rates (see text).

Figure 12

Consistency plot for the total cross section using the $R=0.9$ fm EKM potential at the recommended ${\mathrm{\Lambda }}_b=600$ MeV $\left(\lambda =1\right)$ and separated order by order. The DoBs were generated using ${\mathrm{C}}_{0.25\text{−}10}^{\left(1\right)}$ applied at energies $E_{\mathrm{lab}}=20,40,...,340$ MeV.

Figure 13

Consistency plot as in Fig. 12 but generated using ${\mathrm{A}}_{\epsilon }^{\left(1\right)}$.

Figure 14

Consistency plot as in Fig. 12 but using $d\sigma /d\mathrm{\Omega }$. The observable is evaluated at $E_{\mathrm{lab}}=20,40,...,340$ MeV and $\theta ={40}^{\circ },{60}^{\circ },...,{140}^{\circ }$.

Figure 15

Consistency plots using ${\mathrm{C}}_{0.25\text{−}10}^{\left(1\right)}$ for the individual spin observables (a) $A_y$, (b) $A_{xx}$, and (c) $A_{yy}$ with $R=0.9$ fm and separated order by order. The observables are evaluated at $E_{\mathrm{lab}}=20,40,...,340$ MeV and $\theta ={40}^{\circ },{60}^{\circ },...,{140}^{\circ }$.

Figure 16

Consistency plots using ${\mathrm{C}}_{0.25\text{−}10}^{\left(1\right)}$ averaged over $\mathrm{NLO}–\mathrm{N}{}^3\mathrm{LO}$ results for (a) $\sigma$, (b) $d\sigma /d\mathrm{\Omega }$, and (c) the five selected spin observables with $R=0.9$ fm. The observables are evaluated at $E_{\mathrm{lab}}=96,143,200,300$ MeV and $\theta ={60}^{\circ },{120}^{\circ }$ if applicable.

Figure 17

Consistency plots as in Fig. 16, but with $R=0.8$ fm.

Figure 18

Consistency plots as in Fig. 16, but with $R=1.0$ fm.

Figure 19

Consistency plots with ${\mathrm{C}}_{0.25\text{−}10}^{\left(1\right)}$ and the $R=1.1$ fm EKM potential showing order-by-order results for (a) $\sigma$, (b) $d\sigma /d\mathrm{\Omega }$, and (c) the five considered spin observables evaluated at $E_{\mathrm{lab}}=20,40,...,340$ MeV and $\theta ={60}^{\circ },{120}^{\circ }$ if applicable.

Figure 20

Consistency plots as in Fig. 19, but with $R=1.2$ fm.

Figure 21

A Bayesian network that outlines the causal relationships between random variables when determining ${\mathrm{\Lambda }}_b$. For simplicity, only nodes for one ${\mathbf{\alpha }}_i$ are shown.

Figure 22

Posterior pdfs $pr\left({\mathrm{\Lambda }}_b\right|{\mathbf{b}}_k)$ for NN observables using the $R=0.9$ fm potential, at the kinematic points $E_{\mathrm{lab}}=96,143,200,300$ MeV and $\theta ={60}^{\circ },{120}^{\circ }$. We use prior set ${\mathrm{C}}_{\epsilon }$, and assume that ${\mathrm{\Lambda }}_{<}=300$ MeV and ${\mathrm{\Lambda }}_{>}=1500$ MeV. Thick and thin horizontal lines represent $68\%$ and $95\%$ DoBs, respectively, while the white dot signifies the median. $X_{pqik}$ stands for the combination of the five considered spin observables $A_y,A,D,A_{xx}$, and $A_{yy}$ treated as independent from one another. For $\sigma$ and $d\sigma /d\mathrm{\Omega },X_{\text{ref}}=X_0$, while $X_{\text{ref}}=1$ otherwise. For aesthetic purposes, each plot is scaled to the same height.

Figure 23

Posterior pdfs for ${\mathrm{\Lambda }}_b$ as in Fig. 22, except the observables $A_y,A_{xx}$, and $A_{yy}$ are considered individually at $E_{\mathrm{lab}}=20,40,...,340$ MeV and $\theta ={40}^{\circ },{60}^{\circ },...,{140}^{\circ }$.

Figure 24

Posterior pdfs for ${\mathrm{\Lambda }}_b$ as in Fig. 22, except with $E_{\mathrm{lab}}=50,96,143,200$ MeV.

Figure 25

Posterior pdfs for ${\mathrm{\Lambda }}_b$ as in Fig. 22, except with $E_{\mathrm{lab}}=20,40,...,340$ MeV and $\theta ={40}^{\circ },{60}^{\circ },...,{140}^{\circ }$.

Figure 26

Posterior pdfs for ${\mathrm{\Lambda }}_b$ as in Fig. 22, except for $R=1.2$ fm using ${\mathrm{\Lambda }}_{<}=100$ MeV and ${\mathrm{\Lambda }}_{>}=900$ MeV.

Figure 27

Consistency plots comparing the error band success rate when compared to NPWA data for $R=0.9$ fm and prior set ${\mathrm{C}}_{0.25\text{−}10}$. (a) The total cross section is evaluated using $E_{\mathrm{lab}}=20,40,...,340$ MeV, (b) the differential cross section, and (c) set of selected spin observables use $E_{\mathrm{lab}}=96,143,200,300$ MeV and $\theta ={40}^{\circ },{60}^{\circ },...,{140}^{\circ }$.

Figure 28

The kinematics for nucleon-nucleon scattering [43].