The Three-Boson System at Next-To-Next-To-Leading Order

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.