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We treat low-energy $^3$He-$\alpha$ elastic scattering in an Effective FieldTheory (EFT) that exploits the separation of scales in this reaction. Wecompute the amplitude up to Next-to-Next-to-Leading Order (NNLO), developing ahierarchy of the effective-range parameters that contribute at various orders.We use the resulting formalism to analyze data for recent measurements atcenter-of-mass energies of 0.38-3.12 MeV using the SONIK gas target at TRIUMFas well as older data in this energy regime. We employ a likelihood functionthat incorporates the theoretical uncertainty due to truncation of the EFT anduse Markov Chain Monte Carlo sampling to obtain the resulting posteriorprobability distribution. We find that the inclusion of a small amount of dataon the analysing power $A_y$ is crucial to determine the sign of the p-wavesplitting in such an analysis. The combination of $A_y$ and SONIK dataconstrains all effective-range parameters up to $O(p^4)$ in both s- and p-wavesquite well. The ANCs and s-wave scattering length are consistent with a recentEFT analysis of the capture reaction ${}^3$He($\alpha$,$\gamma$)${}^7$Be.

Effective Field Theory analysis of ${}^3$He-$\alpha$ scattering data