Electric dipole polarizability from first principles calculations

The electric dipole polarizability quantifies the low-energy behavior of the dipole strength and is related to critical observables such as the radii of the proton and neutron distributions. Its computation is challenging because most of the dipole strength lies in the scattering continuum. In this paper we combine integral transforms with the coupled-cluster method and compute the dipole polarizability using bound-state techniques. Employing different interactions from chiral effective field theory, we confirm the strong correlation between the dipole polarizability and the charge radius, and study its dependence on three-nucleon forces. We find good agreement with data for the ${}^4\mathrm{He},{}^{40}\mathrm{Ca}$, and ${}^{16}\mathrm{O}$ nuclei, and predict the dipole polarizability for the rare nucleus ${}^{22}\mathrm{O}$.

Figure 1

The Stieltjes integral transform $\mathcal{I}\left(\sigma \right)$ as a function of $\sigma$ in the case of ${}^4\mathrm{He}$.

Figure 2

The electric dipole polarizability in ${}^4\mathrm{He}$ as a function of the model space size $N_{\max }$. Curves for different values of $\hslash \mathrm{\Omega }$, the underlying harmonic oscillator frequency, are shown.

Figure 3

The electric dipole polarizability ${\alpha }_D\left(\varepsilon \right)$ in ${}^4\mathrm{He}$ as a function of the integration energy $\varepsilon$: (i) using the LIT and Eq. (1) in blue (band); (ii) using Eq. (16) in red (solid); (iii) using the continued fraction of Eq. (35) in black (dashed). Calculations are performed for $\hslash \mathrm{\Omega }=22$ MeV and $N_{\max }=14$.

Figure 4

Electric dipole polarizability in ${}^{16}\mathrm{O}$ as a function of the model space size $N_{\max }$ for different values of $\hslash \mathrm{\Omega }$.

Figure 5

Charge radius in ${}^{16}\mathrm{O}$ as a function of the model space size $N_{\max }$ for different values of $\hslash \mathrm{\Omega }$.

Figure 6

The electric dipole polarizability ${\alpha }_D\left(\varepsilon \right)$ in ${}^{16}\mathrm{O}$ as a function of the integration limit $\varepsilon$. The blue band (i) is obtained integrating the weighted response function as in Eq. (1); the red solid curve (ii) is calculated integrating the weighted LIT at small $\mathrm{\Gamma }$ as in Eq. (16); the black dashed line (iii) is obtained from the continued fraction of Eq. (35). Calculations are performed with $N_{\max }=14$ and $\hslash \mathrm{\Omega }=22$ MeV.

Figure 7

The electric dipole polarizability ${\alpha }_D$ in ${}^{22}\mathrm{O}$ as a function of the model space size $N_{\max }$. Different curves for different values of the underlying harmonic oscillator basis frequency $\hslash \mathrm{\Omega }$ are shown.

Figure 8

The electric dipole polarizability ${\alpha }_D\left(\varepsilon \right)$ in ${}^{22}\mathrm{O}$ as a function of the integration limit $\varepsilon$. The red solid curve (ii) is calculated integrating the weighted LIT at small $\mathrm{\Gamma }$ as in Eq. (16); the black dashed line (iii) is obtained from the continued fraction of Eq. (35). Calculations are performed with $N_{\max }=14$ and $\hslash \mathrm{\Omega }=18$ MeV.

Figure 9

${}^4\mathrm{He}$ photoabsorption response function calculated with different methods and interactions (see text for details) compared with experimental data from by Nakayama et al. [68] (blue circles), Arkatov et al. [58, 59] (white squares), Nilsson et al. [69] (yellow squares), Shima et al. [70, 71] (magenta circles), and Tornow et al. [72] (green squares).

Figure 10

${}^{16}\mathrm{O}$ photoabsorption response function calculated with coupled cluster with singles-and-doubles using a NN interaction only [35, 52] (dark band) and ${\mathrm{NNLO}}_{\mathrm{sat}}$ [26] (light band). The red circles are the experimental data from Ishkhanov et al. [73] while the white triangles with error bars are the experimental results by Ahrens et al. [64].

Figure 11

${\alpha }_D$ vs $r_{\mathrm{ch}}$ in ${}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}$. Empty symbols refer to calculations with NN potentials only: (a) SRG evolved Entem-Machleidt interaction [52] with $\mathrm{\Lambda }=500$ MeV/c and $\lambda =\infty ,3.5,3.0,2.5,$ and $2.0{\mathrm{fm}}^{-1}$, (b) SRG evolved Entem-Machleidt interaction [52] with $\mathrm{\Lambda }=600$ MeV/c and $\lambda =3.5,3.0$ and $2.5{\mathrm{fm}}^{-1}$, (c) SRG evolved CD-BONN [54] interaction with $\lambda =4.0$ and $3.5{\mathrm{fm}}^{-1}$, (d) ${\mathrm{V}}_{\mathrm{low}-k}$ evolved CD-BONN potentials with $\lambda =3.0,2.5$, and $2.0{\mathrm{fm}}^{-1}$, and (e) ${\mathrm{V}}_{\mathrm{low}-k}$-evolved AV18 [57] interaction and $\lambda =3.0$ and $2.5{\mathrm{fm}}^{-1}$. The red diamonds (f) refer to calculations that include 3NF: The large one is from ${\mathrm{NNLO}}_{\mathrm{sat}}$ [26] and the others are from chiral interactions as in Ref. [53]. The green bands (exp) show the experimental data [64, 82].