Simultaneous Inference of Effective Range Parameters and EFT Truncation Uncertainty in $^{3}$He-$\alpha$ Scattering
We extend previous halo effective field theory analyses of low-energy elastic scattering of $^{3}$He-$^{4}$He, including the $\frac{7}{2}^{-}$ $f$-wave resonance as an explicit degree of freedom. The presence of this resonance necessitates a changing power counting scheme depending on the kinematic region. Therefore, we construct a theory uncertainty model at the partial wave amplitude level, allowing us to generate a sophisticated theory covariance matrix that captures the way the theory error structure changes as energy increases. We then perform a Bayesian analysis and simultaneously estimate the joint posterior distributions of the effective range theory parameters and the parameters that characterize the effective field theory truncation uncertainty. We compare two different analyses: no $f$-wave interactions for data up to $E_{\text{max}} = 2.6$ MeV, and including $f$-wave interactions for data up to $E_{\text{max}} = 5.5$ MeV. The inferred breakdown scales in each analysis are consistent with previous work. We find that $f$-wave interactions are needed to describe data for $E_{lab} \gtrapprox 3.6$ MeV.