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

We apply the generalized effective-range expansion of Du [], which incorporates the left-hand cut from one-pion exchange to low-energy neutron–proton scattering in the S 0 1 and S 1 3 channels. The amplitude zero for the center-of-mass momentum near 0.35 GeV in the S 0 1 channel is naturally accommodated within this framework. We extract the pole position, scattering length, effective range, shape parameter, and the pseudoscalar pion–nucleon coupling constant g π N 2 / ( 4 π ) at different expansion orders. The low-energy parameters are stable and consistent with established values, while g π N 2 / ( 4 π ) exhibits larger uncertainties. The extraction of g π N 2 / ( 4 π ) 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.