Unified Description of ${}^6\mathrm{Li}$ Structure and Deuterium-${}^4\mathrm{He}$ Dynamics with Chiral Two- and Three-Nucleon Forces

We provide a unified ab initio description of the ${}^6\mathrm{Li}$ ground state and elastic scattering of deuterium ($d$) on ${}^4\mathrm{He}$ ($\alpha$) using two- and three-nucleon forces from chiral effective field theory. We analyze the influence of the three-nucleon force and reveal the role of continuum degrees of freedom in shaping the low-lying spectrum of ${}^6\mathrm{Li}$. The calculation reproduces the empirical binding energy of ${}^6\mathrm{Li}$, yielding an asymptotic $D$- to $S$-state ratio of the ${}^6\mathrm{Li}$ wave function in the $d+\alpha$ configuration of $-0.027$, in agreement with a determination from ${}^6\mathrm{Li}\text{−}{}^4\mathrm{He}$ elastic scattering, but overestimates the excitation energy of the $3^{+}$ state by 350 keV. The bulk of the computed differential cross section is in good agreement with data. These results endorse the application of the present approach to the evaluation of the ${}^2\mathrm{H}(\alpha ,\gamma ){}^6\mathrm{Li}$ radiative capture, responsible for the big-bang nucleosynthesis of ${}^6\mathrm{Li}$.

Figure 1

Computed $d\text{−}{}^4\mathrm{He}$ $S$- and $D$-wave phase shifts at $N_{\max }=8$ and $\hslash \mathrm{\Omega }=20\mathrm{MeV}$, obtained with fifteen square-integrable ${}^6\mathrm{Li}$ eigenstates and up to seven ${}^2\mathrm{H}$ pseudostates in each of the ${{}^{3}S}_1-{{}^{3}D}_1$, ${{}^{3}D}_2$, and ${{}^{3}D}_3-{{}^{3}G}_3$ channels. The two-body part of the similarity renormalization group (SRG) evolved $\mathrm{N}{}^3\mathrm{LO}NN$ potential [25] ($NN$-only) with $\mathrm{\Lambda }=2.0{\mathrm{fm}}^{-1}$ was used.

Figure 2

$S$-, ${{}^{3}P}_0\text{−}$, and $D$-wave $d\text{−}{}^4\mathrm{He}$ phase shifts computed with the $NN$-only, $NN+3N\text{−}\mathrm{ind}$, and $NN+3N$ Hamiltonians (lines) compared to those of the $R$-matrix analyses of [34, 35] (symbols). More details in the text.

Figure 3

Ground-state energy and low-lying ${}^6\mathrm{Li}$ positive-parity $T=0$ resonance parameters extracted [36] from the phase shifts of Fig. 2 (NCSMC) compared to the evaluated centroids and widths (indicated by $\mathrm{\Gamma }$) of Ref. [1] (Expt.). Also shown on the left-hand side are the best ($N_{\max }=12$) and extrapolated [37] NCSM energy levels. The zero energy is set to the respective computed (experimental) $d+{}^4\mathrm{He}$ breakup thresholds. Absolute g.s. energies can be found in Table 1.

Figure 4

Computed (a) ${}^2\mathrm{H}(\alpha ,d){}^4\mathrm{He}$ laboratory-frame and (b) ${}^4\mathrm{He}(d,d){}^4\mathrm{He}$ center-of-mass frame angular differential cross sections (lines) using the $NN+3N$ Hamiltonian at the deuteron laboratory and c.m. angles of, respectively, ${\phi }_d=30°$ and ${\theta }_d=164°$ as a function of the laboratory helium ($E_{\alpha }$) and deuteron ($E_d$) incident energies, compared with data (symbols) from Refs. [7, 8, 9, 42, 43, 44, 45]. (c) Calculated (lines) and measured (symbols) center-of-mass angular distributions at $E_d=2.93,6.96,8.97$ [46], and 12 MeV [47] are scaled by a factor of 20,5,2, and 1, respectively. All positive- and negative-parity partial waves up to $J=3$ were included in the calculations.