# Bayesian Analysis of $\chi$EFT at Leading Order in a Modified Weinberg Power Counting Approach

## TL;DR

This paper applies Bayesian methods to analyze nucleon-nucleon interactions in chiral effective field theory at leading order, revealing insights into low-energy constants, cutoff dependence, and phase shifts, with implications for nuclear predictions.

## Contribution

It introduces a Bayesian framework for analyzing $ ext{chi}$EFT interactions at leading order, addressing cutoff dependence, limit-cycle behavior, and spurious bound states.

## Key findings

- No significant cutoff dependence in predictive distributions.
- Multimodal posterior with an insignificant bound state mode.
- Less constrained, more repulsive phase shifts compared to phase shift optimization.

## Abstract

We present a Bayesian analysis of renormalization-group invariant nucleon-nucleon interactions at leading order in chiral effective field theory ($\chi$EFT) with momentum cutoffs in the range 400--4000 MeV. We use history matching to identify relevant regions in the parameter space of low-energy constants (LECs) and subsequently infer the posterior probability density of their values using Markov chain Monte Carlo. All posteriors are conditioned on experimental data for neutron-proton scattering observables and we estimate the $\chi$EFT truncation error in an uncorrelated limit. We do not detect any significant cutoff dependence in the posterior predictive distributions for two-nucleon observables. For all cutoff values we find a multimodal LEC posterior with an insignificant mode harboring a bound $^1{S}_0$ state. The $^3P_0$ and $^3P_2$ phase shifts emerging from the Bayesian analysis are less constrained and typically more repulsive compared to the results of a phase shift optimization. We expect that our inference will impact predictions for nuclei. This work demonstrates how to perform inference in the presence of limit-cycle-like behavior and spurious bound states, and lays the foundation for a Bayesian analysis of renormalization-group invariant $\chi$EFT interactions beyond leading order.