Bayesian estimation of the low-energy constants up to fourth order in the nucleon-nucleon sector of chiral effective field theory

TL;DR

This paper employs Bayesian methods and Hamiltonian Monte Carlo to infer low-energy constants in chiral effective field theory for nucleon-nucleon interactions, accounting for uncertainties and charge-dependent effects.

Contribution

It introduces a Bayesian inference framework with HMC for high-dimensional LECs in chiral EFT, including error modeling and empirical data integration.

Findings

Posterior PDFs for LECs are obtained with HMC sampling.

Charge-dependent LECs are inferred using importance sampling and empirical data.

Results suggest next-to-next-to-leading order is needed to detect isospin breaking.

Abstract

We use Bayesian methods and Hamiltonian Monte Carlo (HMC) sampling to infer the posterior probability density function (PDF) for the low-energy constants (LECs) up to next-to-next-to-next- to-leading order (N3LO) in a chiral effective field theory ($\chi$EFT) description of the nucleon-nucleon interaction. In a first step, we condition the inference on neutron-proton and proton-proton scattering data and account for uncorrelated $\chi$EFT truncation errors. We demonstrate how to successfully sample the 31-dimensional space of LECs at N3LO using a revised HMC inference protocol. In a second step we extend the analysis by means of importance sampling and an empirical determination of the neutron-neutron scattering length to infer the posterior PDF for the leading charge-dependent contact LEC in the $^{1}S_0$ neutron-neutron interaction channel. While doing so we account for the $\chi$EFT truncation error via a conjugate prior. We use the resulting posterior PDF to sample the posterior predictive distributions for the effective range parameters in the $^{1}S_0$ wave as well as the strengths of charge-symmetry breaking and charge-independence breaking. We conclude that empirical point-estimate results of isospin breaking in the $^{1}S_0$ channel are consistent with the PDFs obtained in our Bayesian analysis and that, when accounting for $\chi$EFT truncation errors, one must go to next-to-next-to-leading order to confidently detect isospin breaking effects.