Connecting ground-state properties of ${}^6$Li to each other and to scattering data

TL;DR

This work analyzes how the ${}^6$Li s-wave ANC $C_0$ correlates with the deuteron separation energy $E_b$ in the ${}^6$Li–αd system using twelve ab initio NCSMC calculations and simple two-body models. It demonstrates that the strong $E_b$–$C_0^2$ correlation arises because the depth of the $oldsymbol{ ext{α–d}}$ central potential changes only slightly with $E_b$, a result supported by both R-matrix and interior-norm analyses, and it provides a perturbative interpretation based on the linearity of the interior norm. The paper then investigates how reliably $C_0^2$ can be extracted from phase-shift data via CM-ERE and $R$-matrix extrapolations, finding that $R$-matrix converges faster and is more robust to data choices, while CM-ERE requires many higher-order terms and shows sensitivity to the energy range and pole constraints. The results emphasize the need for careful uncertainty quantification and suggest that, for precise astrophysical rates, multi-channel effects and rigorous error analyses are essential when inferring ANCs from scattering data.

Abstract

We examine the relationship between the Asymptotic Normalization Coefficient (ANC) of $^6$Li and other low-energy observables in the $α$-deuteron system. Our analysis uses a set of calculations carried out within the {\it ab initio} No Core Shell Model with Continuum (NCSMC) using a variety of inter-nucleon interactions and basis sizes, and yielding ${}^6$Li deuteron separation energies between 1.3 and 1.8 MeV [Phys. Rev. Lett. 129, 042503 (2022)]. These NCSMC calculations show that the square of the ANC is strongly correlated with the separation energy over this range. In this work, we investigate the origin of this correlation using the phenomenological $R$-matrix, a single-channel potential and a perturbative approach. We show that this correlation occurs because the depth of the $α$-deuteron central potential changes by only a small relative amount as the separation energy varies. We then investigate if the ANC can be accurately extracted from $α$-deuteron phase shifts in an ideal case in which low-energy data are available and there are no experimental errors. We find that both $R$-matrix and Coulomb-modified effective-range theory (CM-ERE) yield extracted ANCs close to, although not exactly equal to, the true value, provided the extrapolation is constrained by the known position of the bound-state pole and at least three terms are included in the fit function. The $R$-matrix approach converges faster than the CM-ERE as the number of parameters increases and is also more robust against the inclusion of low-energy and high-energy phase shift data. Finally, our study also shows that a naive quantification of uncertainties by comparing different truncations used in both theories is not accurate, and suggests the accuracy of ANCs extracted from phase shift data needs further investigation.

Figures (7)

• Figure 1: ANC$^2$s versus the $\alpha$-$d$ separation energy. A strong correlation is indicated by the NCSMC calculations (black points). The correlation predicted by the $R$-matrix approach \ref{['eq:anc']} for three different values of the channel radius $a$ is shown as the solid curves. The dot-dashed curve shows the correlation predicted using a Woods-Saxon (WS) potential.
• Figure 2: Ratio $\mathcal{R}(\tilde{R};\kappa)$\ref{['eq:calR']} obtained with various NCSMC calculations. The inset shows the asymptotic values $\mathcal{R}_{as}(\kappa)$ as a function of the binding momentum $\kappa$. The red band corresponds to the $1\sigma$ uncertainty obtained from the linear regression of the NCSMC calculations for $\mathcal{R}_{as}(\kappa)$, evaluated at 10 fm.
• Figure 3: Comparison of the correlation between ANC$^2$s and the binding energy obtained using the perturbation expansion \ref{['eq:prediction']} (red band) and CM-ERE. The red band represents the $1\sigma$ uncertainty associated to the fit of $R_{\rm as}(\kappa)$.
• Figure 4: The $s$-wave phase shift effective range function (top panel), the absolute error of the CM-ERE (middle panel) and the inverse amplitude (bottom panel) from NCSMC calculation 9 and the CM-ERE for fits at various orders. The point indicates the location of the bound state in the calculation. The left (right) plots correspond to the results without (with) the bound state pole position imposed.
• Figure 5: The sensitivity of the ANC extracted from three-free-parameter $R$-matrix fits to the channel radius. The $y$-axis shows the ANC$^2$ compared to the NCSMC value. The phase-shift data used for the extrapolation to the bound-state pole were from NCSMC calculation 9.