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

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