  We present a new rearrangement of short-range interactions in the $^1S_0$
nucleon-nucleon channel within Chiral Effective Field Theory. This is intended
to reproduce the amplitude zero (scattering momentum $\simeq$ 340 MeV) at
leading order, and it includes subleading corrections perturbatively in a way
that is consistent with renormalization-group invariance. Systematic
improvement is shown at next-to-leading order, and we obtain results that fit
empirical phase shifts remarkably well all the way up to the pion-production
threshold. An approach in which pions have been integrated out is included,
which allows us to derive analytic results that also fit phenomenology
surprisingly well.
