[Paper Review] Comment on "Determination of the chiral coupling constants c_3 and c_4 in new pp and np partial-wave analyses"

D. R. Entem, R. Machleidt|ArXiv.org|Mar 8, 2003

Physics of Superconductivity and Magnetism3 citations

TL;DR

This paper critiques the Nijmegen group's method for determining chiral low-energy constants $c_3$ and $c_4$ in nucleon-nucleon interactions, arguing that their use of a model-dependent, cutoff-dependent $r$-space potential at next-to-next-to-leading order (NNLO) of chiral perturbation theory introduces a systematic error of up to 90% in the two-pion-exchange amplitude. The authors assert that the correct, model-independent NNLO amplitudes—uniquely determined by chiral symmetry and known constants—yield significantly different predictions, rendering the Nijmegen extraction of $c_3$ and $c_4$ unreliable.

ABSTRACT

In a recent study [M.C.M. Rentmeester et al., Phys. Rev. C 67, 044001 (2003)], the Nijmegen group reports on the determination of the chiral low-energy constants (LEC), c_3 and c_4, involved in the 2-pi-exchange part of the NN amplitude at next-to-next-to-leading order (NNLO) of chiral perturbation theory. This analysis does not apply the uniquely-determined and model-independent NN amplitudes at NNLO and uses, instead, amplitudes that are up 90% smaller. We point out that this flaw produces a large systematic error, rendering the Nijmegen method unsuitable for a reliable determination of the LEC.

Motivation & Objective

• To challenge the reliability of the Nijmegen group's determination of chiral low-energy constants $c_3$ and $c_4$ in nucleon-nucleon scattering.
• To identify the core flaw in their method: the use of a model-dependent, cutoff-dependent $r$-space potential instead of the unique, model-independent amplitudes predicted by chiral perturbation theory at NNLO.
• To emphasize that chiral perturbation theory at NNLO provides exact, unique predictions for $D$-wave and higher partial waves, which should be used as a benchmark for LEC extraction.
• To caution against extrapolating NNLO chiral perturbation theory beyond its valid energy regime, especially in higher partial waves.
• To advocate for model-independent, theory-based extraction of low-energy constants to ensure accuracy and consistency.

Proposed method

• The authors compare the Nijmegen group's $r$-space potential with the theoretically unique, model-independent two-pion-exchange amplitude derived from chiral perturbation theory at NNLO.
• They use known values of $g_A = 1.29$ and $f_\pi = 92.4$ MeV, and the educated estimate $c_1 = -0.76$ GeV$^{-1}$, to compute the correct NN amplitudes for $D$- and $F$-wave partial waves.
• The comparison is visualized in Fig. 1, where the solid lines represent the unique NNLO predictions and the dashed lines show the Nijmegen amplitudes, revealing a discrepancy of up to 90% in magnitude.
• The analysis highlights that contact terms at NNLO do not contribute to $L \geq 2$ partial waves, so the $D$- and $F$-wave amplitudes are fully determined by one-pion and two-pion exchange at ${\cal O}(Q^3)$, with no ambiguity.
• They argue that the Nijmegen method introduces artificial model dependence by truncating the potential at $r = 1.6$ fm, which is inconsistent with the momentum-space expansion of chiral perturbation theory.
• The authors stress that chiral perturbation theory is a fundamental quantum field theory with exact predictions at each order, and thus should not be replaced by phenomenological model-building.

Experimental results

Research questions

• RQ1 Why does the Nijmegen group's determination of the chiral low-energy constants $c_3$ and $c_4$ produce a large systematic error in the two-pion-exchange amplitude?
• RQ2 How does the use of a cutoff-dependent $r$-space potential in the Nijmegen analysis violate the principles of chiral perturbation theory at NNLO?
• RQ3 What is the correct, model-independent prediction for the $NN$ amplitude in $D$- and $F$-wave partial waves at NNLO, and how does it differ from the Nijmegen result?
• RQ4 To what extent is chiral perturbation theory at NNLO valid for $D$- and $F$-wave scattering, and why is extrapolation to 500 MeV problematic?
• RQ5 Can the constants $c_3$ and $c_4$ be reliably extracted from $NN$ phase shifts if the underlying amplitude is not uniquely determined by theory?

Key findings

• The Nijmegen group's two-pion-exchange amplitude at NNLO differs from the uniquely determined theoretical prediction by up to 90%, indicating a large systematic error in their method.
• The correct NNLO amplitudes for $D$- and $F$-wave partial waves are uniquely determined by one-pion exchange and two-pion exchange at ${\cal O}(Q^3)$, with no ambiguity, provided $g_A$, $f_\pi$, and $c_1$ are known.
• The use of a cutoff at $r = 1.6$ fm in the $r$-space potential introduces model dependence, which is inconsistent with the fundamental principles of chiral perturbation theory.
• The Nijmegen method is unsuitable for reliable determination of $c_3$ and $c_4$ because it does not use the unique, model-independent amplitudes predicted by chiral perturbation theory at NNLO.
• Chiral perturbation theory at NNLO is only valid up to about 50 MeV in $D$ waves and 150 MeV in $F$ waves, making extrapolation to 500 MeV inadvisable.
• The authors conclude that the Nijmegen group's extraction of $c_3$ and $c_4$ with 2–5% accuracy is likely compromised by this systematic error, rendering the results unreliable.