[Paper Review] Assessing Theory Uncertainties in EFT Power Countings from Residual Cutoff Dependence

Harald W. Grießhammer|arXiv (Cornell University)|Nov 2, 2015

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper proposes a method to quantitatively assess the internal consistency of power-counting schemes in non-perturbative Effective Field Theories (EFTs), particularly in Chiral EFT with multiple nucleons. By analyzing the residual cutoff dependence of observables, it tests whether the observed functional form matches the predictions of the Renormalization Group, providing a data-minimal criterion to verify if an EFT is consistently renormalized at a given order, with applications to NN scattering partial waves like $^3P_0$ and $^3P_2$-$^3F_2$. The method helps identify properly renormalized LECs and estimates the breakdown scale and convergence pattern without relying on experimental data.

ABSTRACT

I summarise a method to quantitatively assess the consistency of power-counting proposals in Effective Field Theories which are non-perturbative at leading order. It uses the fact that the Renormalisation Group evolution of an observable predicts the functional form of its residual cutoff dependence on the EFT breakdown scale, low-momentum scales, and 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. For example, in ChiEFT with more than one nucleon, a lack of universally accepted analytic solutions obfuscates the relation between convergence pattern and numerical results, and led to proposals which predict different numbers of Low Energy Coefficients at the same chiral order. The method may provide an independent check whether an observable is renormalised at a given order, and of both the breakdown scale and the momentum-dependent order-by-order convergence pattern of an EFT. Conversely, it may help identify LECs which produce renormalised observables at a given order. I also discuss its underlying assumptions and relation to the RG Equation; useful choices for observables and cutoffs; the momentum window in which the test provides best signals; its dependence on the values and forms of cutoffs as well as on the EFT parameters; the impact of fitting Low Energy Coefficients to data in different or the same channel; caveats and limitations. Since the test is designed to minimise the use of data, it allows one to quantitatively falsify if the EFT has been renormalised consistently, rather than quantifying how an EFT compares to experiment. Its application in particular to the 3P0 and 3P2-3F2 partial waves of NN scattering in ChiEFT may elucidate persistent power-counting issues. Details and a better bibliography can be found in an upcoming publication.

Motivation & Objective

• To develop a quantitative, data-independent criterion for assessing the internal consistency of power-counting proposals in non-perturbative EFTs.
• To address the lack of universally accepted analytic solutions in multi-nucleon systems, where different power-counting schemes predict varying numbers of Low Energy Coefficients (LECs) at the same chiral order.
• To provide a falsifiable test for whether an EFT is consistently renormalized at a given order, independent of experimental comparison.
• To estimate the EFT breakdown scale and momentum-dependent convergence pattern from the functional form of residual cutoff dependence.
• To identify which LECs are necessary to absorb residual cutoff dependence and thus ensure consistent renormalization.

Proposed method

• The method tests whether the residual cutoff dependence of an observable matches the functional form predicted by the Renormalization Group (RG) evolution of the EFT.
• It uses the fact that the RG predicts a specific power-law dependence of observables on the cutoff, the low-momentum scales, and the order of the calculation.
• The test involves fitting the observed cutoff dependence to a power law and checking if the slope matches the expected $n+1$ scaling for order $Q^n$.
• It applies to observables in the momentum window where the EFT is expected to be valid, minimizing reliance on experimental data.
• The method is applied to partial waves in NN scattering, such as $^3P_0$ and $^3P_2$-$^3F_2$, to probe persistent power-counting ambiguities.
• It uses various cutoff forms and fit windows to test robustness and reduce false positives from fine-tuning or anomalous coefficients.

Experimental results

Research questions

• RQ1 Does the residual cutoff dependence of an observable in a non-perturbative EFT follow the functional form predicted by the Renormalization Group at a given order?
• RQ2 Can this cutoff dependence test falsify inconsistent power-counting schemes without relying on experimental data?
• RQ3 To what extent can the method estimate the EFT breakdown scale and momentum-dependent convergence pattern?
• RQ4 How do different cutoff forms and regulator choices affect the reliability of the consistency test?
• RQ5 Which Low Energy Coefficients are essential for renormalizing the theory at a given order, as revealed by their role in canceling cutoff dependence?

Key findings

• The residual cutoff dependence of an observable must scale as $Q^{n+1}$ at order $Q^n$ if the power counting is consistent with RG predictions.
• A failure to observe this scaling indicates that the EFT is not consistently renormalized at that order, providing a falsifiable criterion.
• The method can estimate the EFT breakdown scale $\overline{\Lambda}_{\text{EFT}}$ and the momentum-dependent convergence pattern from numerical results alone.
• The test is robust when the $n+1$ scaling is consistently observed across multiple observables, cutoff forms, and fit windows.
• The method identifies which LECs are necessary to absorb residual cutoff dependence, thereby indicating their role in consistent renormalization.
• Application to $^3P_0$ and $^3P_2$-$^3F_2$ partial waves may resolve long-standing ambiguities in Chiral EFT power counting.