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

TL;DR

This paper applies a Bayesian framework to estimate parameters in chiral effective field theory for nucleon-nucleon interactions, revealing degeneracies and stability issues in low-energy constants with respect to data and higher-order corrections.

Contribution

It introduces a Bayesian approach to parameter estimation in chiral EFT, highlighting the importance of including higher-order uncertainties and analyzing parameter degeneracies.

Findings

Degeneracy in fourth-order LECs due to on-shell vanishing operators

Parameter estimates are stable when truncation errors are properly modeled

Comparison of uncorrelated and correlated error models suggests Gaussian processes for future work

Abstract

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.