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

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 (LEFT) 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 LEFT 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 1S0 neutron-neutron interaction channel. While doing so we account for the LEFT 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 1S0 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 1S0 channel are consistent with the PDFs obtained in our Bayesian analysis and that, when accounting for LEFT truncation errors, one must go to next-to-next-to-leading order to confidently detect isospin breaking effects.