A Consistency Test of EFT Power Countings from Residual Cutoff Dependence

Abstract: A method to quantitatively assess the consistency of power-counting proposals in Effective Field Theories (EFT) which are non-perturbative at leading order is presented. The Renormalisation Group evolution of an observable predicts the functional form of its residual cutoff dependence on the breakdown scale of an EFT, on the low-momentum scales, and on the order of the calculation. Passing this test is a necessary but not sufficient consistency criterion for a suggested power counting whose exact nature is disputed. In Chiral Effective Field Theory (ChiEFT) with more than one nucleon, a lack of universally accepted analytic solutions obfuscates the convergence pattern in results. This led to proposals which predict different sets of Low Energy Coefficients (LECs) at the same chiral order, and at times even predict a different ordering long-range contributions. The method may independently check whether an observable is renormalised at a given order, and proves estimates of both the breakdown scale and the momentum-dependent order-by-order convergence pattern. Conversely, it helps identify those LECs (and long-range pieces) which ensure renormalised observables at a given order. I also discuss assumptions and the relation to Wilson's Renormalisation Group; useful observable and cutoff choices; the momentum window with likely best signals; its dependence on the values and forms of cutoffs as well as on the EFT parameters; the impact of fitting LECs to data; and caveats as well as limitations. Since the test is designed to minimise the use of data, it quantitatively falsifies if the EFT has been renormalised consistently. This complements other tests which quantify how an EFT compares to experiment. Its application in particular to the 3P0 and P2-3F2 partial waves of NN scattering in ChiEFT may elucidate persistent power-counting issues.