Low-energy theorems for neutron-proton scattering in $\chi$EFT using a perturbative power counting

Low-energy theorems (LETs) for effective-range parameters in nucleon-nucleon scattering encode properties of the long-range part of the nuclear force. We compute LETs for S-wave neutron-proton scattering using chiral effective field theory with a modified version of Weinberg power counting. Corrections to the leading order amplitude are included in distorted-wave perturbation theory and we incorporate contributions up to the third order in the power counting. We find that LETs in the $^1S_0$ and $^3S_1$ partial waves agree well with empirical effective-range parameters. At the same time, phase shifts up to laboratory scattering energies of about 100 MeV can be reproduced. We show that it is important to consider the pion mass splitting in the one-pion exchange potential in the $^1S_0$ partial wave while the effect is negligible in the $^3S_1$ partial wave. We conclude that pion exchanges, as treated in this power counting, accurately describe the long-range part of the $S$-wave nuclear interaction.