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We use an effective field theory for short-range forces (SREFT) to analyzesystems of three identical bosons interacting via a two-body potential thatgenerates a scattering length, $a$, which is large compared to the range of theinteraction, $\ell$. The amplitude for the scattering of one boson off a boundstate of the other two is computed to next-to-next-to-leading order (N$^2$LO)in the $\ell/a$ expansion. At this order, two pieces of three-body data arerequired as input in order to renormalize the amplitude (for fixed $a$). Weapply our results to a model system of three Helium-4 atoms, which are assumedto interact via the TTY potential. We generate N$^2$LO predictions foratom-dimer scattering below the dimer breakup threshold using the bound-stateenergy of the shallow Helium-4 trimer and the atom-dimer scattering length asour two pieces of three-body input. Based on the convergence pattern of theSREFT expansion, as well as differences in the predictions of tworenormalization schemes, we conclude that our N$^2$LO phase- shift predictionswill receive higher-order corrections of $< 0.2$%. In contrast, the predictionof SREFT for the binding energy of the "deep" trimer of Helium-4 atoms displayspoor convergence.

Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order