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Extraction of the pion-nucleon coupling constant using the effective-range expansion with the left-hand cut

Bo-Yang Liu, Bing Wu, Ji-Wei Fu, Meng-Lin Du, Feng-Kun Guo, Ulf-G. Meißner

TL;DR

This work extends the effective-range expansion by including the left-hand cut from one-pion exchange, enabling a near-threshold description of $np$ scattering in the $^1S_0$ and $^3S_1$ channels. Using a dispersion-based $N/D$ construction, the authors fit phase-shift data with polynomial expansions and extract the pole positions, scattering lengths, effective ranges, and the pseudoscalar pion-nucleon coupling $g_{\pi N}^2/(4\pi)$, noting that the latter carries larger uncertainties. The results show stable low-energy parameters across expansion orders and reproduce the known amplitude zero near $k\simeq 0.35$ GeV, while the $g_{\pi N}^2/(4\pi)$ values are consistent with, but slightly below, some high-precision determinations, highlighting the role of residual beyond-OPE effects. Overall, the generalized ERE with the left-hand cut proves to be a viable framework for probing long-range singularities in near-threshold hadronic interactions and can be connected to chiral EFT in appropriate regions.

Abstract

We apply the generalized effective-range expansion of Phys. Rev. Lett. 135, 011903(2025), which incorporates the left-hand cut from one-pion exchange, to low-energy neutron-proton scattering in the $^1S_0$ and $^3S_1$ channels. The amplitude zero for the center-of-mass momentum near 0.35 GeV in the $^1S_0$ channel is naturally accommodated within this framework. We extract the pole position, scattering length, effective range, and the pseudoscalar pion-nucleon coupling constant $g_{πN}^2/(4π)$ at different expansion orders. The low-energy parameters are stable and consistent with established values, while $g_{πN}^2/(4π)$ exhibits larger uncertainties. The extraction of $g_{πN}^2/(4π)$ is data-driven, relying on the analytic constraints from the left-hand cut and phase-shift data within the one-pion-exchange approximation. Despite larger uncertainties compared to high-precision extractions, the consistency with established values demonstrates that this framework can probe the left-hand-cut singularity.

Extraction of the pion-nucleon coupling constant using the effective-range expansion with the left-hand cut

TL;DR

This work extends the effective-range expansion by including the left-hand cut from one-pion exchange, enabling a near-threshold description of scattering in the and channels. Using a dispersion-based construction, the authors fit phase-shift data with polynomial expansions and extract the pole positions, scattering lengths, effective ranges, and the pseudoscalar pion-nucleon coupling , noting that the latter carries larger uncertainties. The results show stable low-energy parameters across expansion orders and reproduce the known amplitude zero near GeV, while the values are consistent with, but slightly below, some high-precision determinations, highlighting the role of residual beyond-OPE effects. Overall, the generalized ERE with the left-hand cut proves to be a viable framework for probing long-range singularities in near-threshold hadronic interactions and can be connected to chiral EFT in appropriate regions.

Abstract

We apply the generalized effective-range expansion of Phys. Rev. Lett. 135, 011903(2025), which incorporates the left-hand cut from one-pion exchange, to low-energy neutron-proton scattering in the and channels. The amplitude zero for the center-of-mass momentum near 0.35 GeV in the channel is naturally accommodated within this framework. We extract the pole position, scattering length, effective range, and the pseudoscalar pion-nucleon coupling constant at different expansion orders. The low-energy parameters are stable and consistent with established values, while exhibits larger uncertainties. The extraction of is data-driven, relying on the analytic constraints from the left-hand cut and phase-shift data within the one-pion-exchange approximation. Despite larger uncertainties compared to high-precision extractions, the consistency with established values demonstrates that this framework can probe the left-hand-cut singularity.
Paper Structure (9 sections, 24 equations, 4 figures, 1 table)

This paper contains 9 sections, 24 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formalism
  3. ERE with the lhc
  4. The $S$-wave $NN$ OPE potential and unitarized amplitude
  5. Extraction of low-energy parameters
  6. Results and discussion
  7. Channel-by-channel Fit
  8. Joint analysis of the 1S0 and 3S1 channels
  9. Summary

Figures (4)

  • Figure 1: $^1S_0$$np$ scattering phase shift. Data points are from the following references: SP82 (Ref. Arndt:1982ep), Nijm I, Nijm II, Reid93, and Nijm93 (Ref. Stoks:1994wp), ESC96 (Ref. Rijken:1995pw), ChEFT (Ref. Reinert:2017usi), SP40 (Ref. Arndt:2000xc), and PWA93 (Ref. Stoks:1993tb).
  • Figure 2: Feynman diagram for the $NN\to NN$ process with OPE.
  • Figure 3: Comparison of $k \cot \delta$ between the best fit and data for the $np$$|^1S_0,I=1\rangle$ channel (fit performed with the generalized ERE \ref{['equ:kcotdeltawithlhc']} on scattering phase shift from Ref. Stoks:1993tb).
  • Figure 4: Joint fit to the data from Ref. Stoks:1993tb using the generalized ERE in Eq. \ref{['equ:kcotdeltawithlhc']}.