Assessing Theory Errors from Residual Cutoff Dependence

A recent editorial [Phys.~Rev.~A 83, 040001] emphasised the need to quantify theoretical uncertainties. Ideally, ``double-blind'' calculations would assess theory-errors based on input and method, and not by comparison to data. This is particularly important if data is absent or its consistency is checked. Effective Field Theories (EFTs) promise a well-defined scheme to provide such reproducible, objective, quantitative error estimates. But how can one validate the expansion? This is a particularly nagging question in Nuclear Physics, where a fully consistent chiral EFT is still under development since the $NN$ interaction is non-perturbative. One can indeed quantify the consistency of an EFT from the dependence of observables ${\cal O}(k;\lambda)$ at low momentum $k$ on the cutoff $\lambda$ employed in numerical calculations. The power-counting in the small, dimension-less quantity $Q\propto k$ of an EFT quanitiatively predicts $1-{\cal O}(k;\lambda_1)/{\cal O}(k;\lambda_2)\propto k^{n+1}$ for a calculation at order $Q^n$. The slope of a double-logarithmic plot of this quantity against $k$ reveals thus the order of accuracy $n$. In contradistinction to a method proposed by Lepage, this approach does not compare to data to assess uncertainties. Examples are given.