  We discuss effective field theory treatments of the problem of three
particles interacting via short-range forces (range R >> a_2, with a_2 the
two-body scattering length). We show that forming a once-subtracted scattering
equation yields a scattering amplitude whose low-momentum part is
renormalization-group invariant up to corrections of O(R^3/a_2^3). Since
corrections of O(R/a_2) and O(R^2/a_2^2) can be straightforwardly included in
the integral equation's kernel, a unique solution for 1+2 scattering phase
shifts and three-body bound-state energies can be obtained up to this accuracy.
We use our equation to calculate the correlation between the binding energies
of Helium-4 trimers and the atom-dimer scattering length. Our results are in
excellent agreement with the recent three-dimensional Faddeev calculations of
Roudnev and collaborators that used phenomenological inter-atomic potentials.
