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A consistency test of EFT power countings from residual cutoff dependence

  • Regular Article –Theoretical Physics
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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 breakdown scale of an Effective Field Theory (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. For example, in Chiral Effective Field Theory (\(\chi \hbox {EFT}\)) 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 (LECs) at the same chiral order, and at times even predicts a different ordering long-range contributions. The method may provide an independent check whether an observable is properly renormalised at a given order, and allows one to estimate both the breakdown scale and the momentum-dependent order-by-order convergence pattern of an EFT. Conversely, it may help identify those LECs (and long-range pieces) which produce renormalised observables at a given order. I also discuss its underlying assumptions and relation to the Wilsonian Renormalisation Group Equation; useful choices for observables and cutoffs; the momentum window in which the test likely provides 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 in different or the same channel; and caveats as well as 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. This complements other tests which quantify how an EFT compares to experiment. Its application in particular to the \({}^{3}\mathrm {P}_{0}\) and \({}^{3}\mathrm {P}_{2}\)–\({}^{3}\mathrm {F}_{2}\) partial waves of \({\mathrm {NN}}\) scattering in \(\chi \hbox {EFT}\) may elucidate persistent power-counting issues.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The preciously few data underlying this work are available in full upon request from the author.]

Notes

  1. This corrects an error in Ref. [14] and leads to a more nuanced presentation from here on.

  2. Some claim that renormalisability requires that \({\mathcal {O}}\) has a unique limit as \(\varLambda \rightarrow \infty \).

  3. One could adhere to the philosophy that cutoffs and breakdown scales should be similar.

  4. Aside from the comments in the preceding two footnotes, I am a follower of the “democratic principle” that any cutoff is equally legitimate and valid, as long as .

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Acknowledgements

I cordially thank the organisers of the workshop The Tower of Effective (Field) Theories and the Emergence of Nuclear Phenomena (EFT and Philosophy of Science) at CEA/SPhN Saclay in 2017 for making a teenager’s dream come true to discuss with Philosophers in France, and all participants for the profound insight they shared, as well as for enlightening and entertaining discussions. These notes grew out of the inspirational and intense discourses at the workshops Nuclear Forces from Effective Field Theory at CEA/SPhN Saclay in 2013, Bound States and Resonances in Effective Field Theories and Lattice QCD Calculations in Benasque (Spain) in 2014, Chiral Dynamics 2015 in Pisa (Italy), EMMI Rapid Reaction Task Force ER15-02: Systematic Treatment of the Coulomb Interaction in Few-Body Systems at Darmstadt (Germany) in 2016, and New Ideas in Constraining Nuclear Forces at the ECT* in Trento (Italy) in 2018. I am most grateful to all their organisers and participants. Since 2013, exchanges with M. C. Birse, B. Demissie, A. Ekström, E. Epelbaum, C. Forssen, R. J. Furnstahl, J. Holt, B. Long, M. Pavon Valderrama, D. R. Phillips, M. J. Savage, I. Tews, R. G. E. Timmermans, U. van Kolck and Ch.-J. Yang allowed me to develop these ideas into a sharper analysis tool. M. J. Birse, B. Demissie, E. Epelbaum, D. R. Phillips and Ch.-J. Yang suggested important improvements to this script. I am especially indebted to ceaseless insistence on clarity by many emerging researchers, and by both referees. Finally, my colleagues may forgive mistakes and omissions in referencing work and historical precedents, and graciously continue to point out necessary corrections. This work was supported in part by the US Department of Energy under contract DE-SC0015393, and by The George Washington University: by the Dean’s Research Chair programme and an Enhanced Faculty Travel Award of the Columbian College of Arts and Sciences; and by the Office of the Vice President for Research and the Dean of the Columbian College of Arts and Sciences; and was conducted in part in GW’s Campus in the Closet.

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  1. Department of Physics, Institute for Nuclear Studies, George Washington University, Washington, DC, 20052, USA

    Harald W. Grießhammer

  2. Department of Physics, Duke University, Box 90305, Durham, NC, 27708, USA

    Harald W. Grießhammer

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Grießhammer, H.W. A consistency test of EFT power countings from residual cutoff dependence. Eur. Phys. J. A 56, 118 (2020). https://doi.org/10.1140/epja/s10050-020-00129-5

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