Quantifying truncation errors in effective field theory

TL;DR

This paper adapts Bayesian methods to quantify truncation errors in effective field theory, providing a statistical framework that incorporates prior beliefs about coefficient naturalness and validating it with examples including neutron-proton scattering.

Contribution

It introduces a Bayesian approach to estimate EFT truncation errors, linking naturalness priors to standard error estimation procedures and exploring their robustness.

Findings

Bayesian intervals align with traditional error estimates.

Naturalness priors produce consistent 68% DOB intervals.

Convergence properties can validate EFT expansion assumptions.

Abstract

Bayesian procedures designed to quantify truncation errors in perturbative calculations of quantum chromodynamics observables are adapted to expansions in effective field theory (EFT). In the Bayesian approach, such truncation errors are derived from degree-of-belief (DOB) intervals for EFT predictions. Computation of these intervals requires specification of prior probability distributions ("priors") for the expansion coefficients. By encoding expectations about the naturalness of these coefficients, this framework provides a statistical interpretation of the standard EFT procedure where truncation errors are estimated using the order-by-order convergence of the expansion. It also permits exploration of the ways in which such error bars are, and are not, sensitive to assumptions about EFT-coefficient naturalness. We first demonstrate the calculation of Bayesian probability distributions for the EFT truncation error in some representative examples, and then focus on the application of chiral EFT to neutron-proton scattering. Epelbaum, Krebs, and Mei{\ss}ner recently articulated explicit rules for estimating truncation errors in such EFT calculations of few-nucleon-system properties. We find that their basic procedure emerges generically from one class of naturalness priors considered, and that all such priors result in consistent quantitative predictions for 68% DOB intervals. We then explore several methods by which the convergence properties of the EFT for a set of observables may be used to check the statistical consistency of the EFT expansion parameter.