  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.
