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-byorder convergence pattern of the EFT. A first application was made to chiral EFT for neutronproton scattering using the semi-local 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.

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Effective field theories (EFTs) organize the description of complex systems into an infinite sequence of decreasing importance. Predictions are made with a finite number of terms, which induces a truncation error that is often left unquantified. We formalize the notion of EFT convergence and propose a Bayesian truncation error model for predictions that are correlated across the independent variables, e.g., energy or scattering angle. Central to our approach are Gaussian processes that encode both the naturalness and correlation structure of EFT coefficients. Our use of Gaussian processes permits efficient and accurate assessment of credible intervals, allows EFT fits to easily include correlated theory errors, and provides analytic posteriors for physical EFT-related quantities such as the expansion parameter. We demonstrate that model-checking diagnostics-applied to the case of multiple curves-are powerful tools for EFT validation. As an example, we assess a set of nucleon-nucleon scattering observables in chiral EFT. In an effort to be self contained, appendices include thorough derivations of our statistical results. Our methods are packaged in Python code, called gsum [1], that is available for download on GitHub. * melendez.27@osu.edu † furnstahl.1@osu.edu ‡ phillid1@ohio.edu § mpratola@stat.osu.edu ¶ scwesolowski@salisbury.eduOur assumption is that the divergence occurs sufficiently beyond the kth order that the truncation model error estimate up to order ∞ is a good approximation.

We recently developed a Bayesian framework for parameter estimation in general effective field theories. Here we present selected results from using that framework to estimate parameters with a nucleon-nucleon (N N ) potential derived using chiral effective field theory (χEFT): the semi-local N N potential of Epelbaum, Krebs, and Meißner (EKM). There are many N N scattering data, up to high energies, and with rather small errors, so imposing a penalty for unnatural low-energy constants (LECs) usually has a small effect on the fits. In contrast, we have found that including an estimate of higher orders in χEFT plays an important role in robust parameter estimation. We present two case studies where our Bayesian machinery illuminates physics issues. The first involves the EKM potential at fourth order in the χEFT expansion: the two-dimensional posterior probability density function (pdf) for the fourth-order s-wave LECs obtained from the Nijmegen PWA93 phase shifts indicates these parameters in the N N potential are degenerate. We trace this feature of the pdf to the presence of an operator in the fourth-order N N potential that vanishes on-shell. The second case study examines the stability of LEC extractions as more data at higher energies are included in the fit. We show that as long as χEFT truncation errors are properly accounted for in the parameter estimation, the LEC values extracted using our Bayesian approach are not sensitive to the maximum energy chosen for the fit. Uncorrelated and fully correlated models for the truncation errors are compared, pointing the way to the use of Gaussian processes to more generally model the correlation structure. arXiv:1808.08211v1 [nucl-th] 24 Aug 2018 ‡ This phenomenon was also studied in the recent chiral interaction of Reinert, Krebs, and Epelbaum [18].

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