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Simultaneous Inference of Effective Range Parameters and EFT Truncation Uncertainty in $^{3}$He-$α$ Scattering

Andrius Burnelis, Daniel R. Phillips

Abstract

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.

Simultaneous Inference of Effective Range Parameters and EFT Truncation Uncertainty in $^{3}$He-$α$ Scattering

Abstract

We extend previous halo effective field theory analyses of low-energy elastic scattering of He-He, including the -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 -wave interactions for data up to MeV, and including -wave interactions for data up to MeV. The inferred breakdown scales in each analysis are consistent with previous work. We find that -wave interactions are needed to describe data for MeV.

Paper Structure

This paper contains 20 sections, 62 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Effective Field Theory
  3. Energy Scales
  4. Halo Effective Field Theory and Effective Range Theory
  5. Power Counting For $s$-Waves and $p$-Waves
  6. Power Counting For $f$-Waves
  7. Data
  8. Scattering Model
  9. Differential Cross Section
  10. S & P Waves
  11. F Waves and Resonances
  12. Uncertainty Model
  13. Covariance Matrix
  14. Simultaneously Sampling Model Parameters and Truncation Uncertainty Parameters
  15. Implementation and Results Without $f$-Waves
  16. ...and 5 more sections

Figures (15)

  • Figure 1: Energy level diagram of the $^{7}$Be system Tilley2002.
  • Figure 2: Graphical representation of the partial wave scattering amplitude $f_{\ell}^{\pm}$. The dashed lines represent the $^{3}$He field $(\psi_{\lambda})$, while the solid lines represent the $^{4}$He field $(\phi)$, and the filled rectangle is the dressed dimer field propagator for that partial wave.
  • Figure 3: Graphical representation of the dressing of the bare dimer propagator. The dashed lines represent the $^{3}$He field $(\psi_{\lambda})$, while the solid lines represent the $^{4}$He field $(\phi)$. The thick solid line represents the dressed dimer propagator.
  • Figure 4: Comparison of sizes of the different terms in the effective range function, relative to the Coulomb characteristic component of the CM-ERE. In each panel, the blue curve corresponds to the scattering length relative to the Coulomb characteristic component, the orange curve corresponds to the effective range relative to the Coulomb characteristic component, and the green curve corresponds to the shape parameter relative to the Coulomb characteristic component.
  • Figure 5: Comparison of the ratios of the amplitudes for the $^{3}$He-$\alpha$ system. The blue curve corresponds to the ratio of $\frac{3}{2}^{-}$ relative to $s$-wave, the orange corresponds to $\frac{1}{2}^{-}$ relative to $s$-wave, the green corresponds to $\frac{7}{2}^{-}$ relative to $s$-wave, and the red corresponds to $\frac{5}{2}^{-}$ relative to $s$-wave. The dashed purple line indicates the nominal $Q^{3}$ curve (with $\Lambda_B=0.90~{\rm fm}^{-1}$) indicating the size of N3LO effects. The short vertical dashed lines indicates the momentum at which we have cross section measurements in the Paneru dataset PaneruPaneru:thesis. The tall vertical dashed-dotted lines indicate the resonance momentum $k_{R}$ and the window of the $\frac{7}{2}^{-}$ resonance.
  • ...and 10 more figures