Exploring Bayesian parameter estimation for chiral effective field theory using nucleon–nucleon phase shifts

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 (NN) potential derived using chiral effective field theory ($\chi$EFT): the semi-local NN potential of Epelbaum, Krebs, and Mei{\ss}ner (EKM). There are many NN 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 $\chi$EFT 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 $\chi$EFT 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 NN potential are degenerate. We trace this feature of the pdf to the presence of an operator in the fourth-order NN 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 $\chi$EFT 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.