Operator evolution for ab initio electric dipole transitions of ${}^4\mathrm{He}$

A goal of nuclear theory is to make quantitative predictions of low-energy nuclear observables starting from accurate microscopic internucleon forces. A major element of such an effort is applying unitary transformations to soften the nuclear Hamiltonian and hence accelerate the convergence of ab initio calculations as a function of the model space size. The consistent simultaneous transformation of external operators, however, has been overlooked in applications of the theory, particularly for nonscalar transitions. We study the evolution of the electric dipole operator in the framework of the similarity renormalization group method and apply the renormalized matrix elements to the calculation of the ${}^4\mathrm{He}$ total photoabsorption cross section and electric dipole polarizability. All observables are calculated within the ab initio no-core shell model. We find that, although seemingly small, the effects of evolved operators on the photoabsorption cross section are comparable in magnitude to the correction produced by including the chiral three-nucleon force and cannot be neglected.

Figure 1

SRG evolution of the 2B dipole operator in HO space for the ${}^3{S}_1$ to ${}^3{P}_2$ transition. The color bar represents the value of the dipole matrix elements and is truncated to highlight the off-diagonal behavior as a function of evolution, from bare $\left(\lambda =\infty \right)$ to $\lambda =1.5 {\mathrm{fm}}^{-1}$. The matrix elements have units of femtometers.

Figure 2

Convergence of the total dipole strength $M_0$ of ${}^4\mathrm{He}$ as a function of $N_{\text{max}}$ using the bare operator and evolved wave functions from the $NN+3N$ Hamiltonian with $\lambda =2.5 {\mathrm{fm}}^{-1}$ at $\hslash \mathrm{\Omega }=22$, 28, 34, and 40 MeV.

Figure 3

Convergence of the total dipole strength $M_0$ of ${}^4\mathrm{He}$ as a function of $N_{\text{max}}$ at $\hslash \mathrm{\Omega }=28$ MeV using (a) the bare and (b) the 2B evolved $\widehat{D}$ operator and wave functions from the $NN+3N$ Hamiltonian with $\lambda =1.8,2.2,2.5$, and $3.0 {\mathrm{fm}}^{-1}$.

Figure 4

Convergence as a function of $N_{\text{max}}$ of the 2B evolved total dipole strength, $M_0$, of ${}^4\mathrm{He}$ computed as $\left|\right|\widehat{D}|{\mathrm{\Psi }}_0{\rangle \left|\right|}^2$ (squares) and ${\widehat{D}}^{†}\widehat{D}$ (circles) for $\lambda =1.8 {\mathrm{fm}}^{-1}$ and $\hslash \mathrm{\Omega }=22$ MeV (dashed lines) and 28 MeV (solid lines). The arrow shows the converged value of $M_0$ computed with the bare operator. Results were obtained using the wave function from the $NN+3N$ Hamiltonian.

Figure 5

Convergence of the bare (circles) and 2B SRG evolved (squares) electric polarizability of ${}^4\mathrm{He}$ as a function of $N_{\text{max}}$ for (a) $\lambda =1.8 {\mathrm{fm}}^{-1}$ with $\hslash \mathrm{\Omega }=22$ MeV (dashed line) and 28 MeV (solid line) and (b) with fixed $\hslash \mathrm{\Omega }=28$ MeV at $\lambda =1.8$ (dashed line) and $2.5 {\mathrm{fm}}^{-1}$ (solid line). Results were obtained using the wave function from the $NN+3N$ Hamiltonian.

Figure 6

Dependence of the ${}^4\mathrm{He}$ total photoabsorption cross section computed with the $NN+3N$-induced (region delimited by dashed [blue] lines) and $NN+3N$ (region delimited by solid [red] lines) Hamiltonians and a 2B evolved dipole operator on (a) the model-space size $N_{\max }$ at $\hslash \mathrm{\Omega }=28$ MeV and $\lambda =1.8 {\mathrm{fm}}^{-1}$ and (b) the HO frequency $\hslash \mathrm{\Omega }$ at $N_{\max }=18/19$ and $\lambda =2.5 {\mathrm{fm}}^{-1}$. Also shown (dotted [black] line) is the result of the LS calculation of Ref. [33] using the ${\mathrm{N}}^3\mathrm{LO} NN$ interaction.

Figure 7

Dependence of (a) total strength of the dipole transition and (b) electric dipole polarizability on variations of the SRG flow parameter, $\lambda$, for $N_{\text{max}}=18$ and $\hslash \mathrm{\Omega }=28$ MeV, obtained using wave functions from the $NN+3N$-induced (dashed lines) and $NN+3N$ (solid lines) Hamiltonians along with four types of operators: bare (circles), 2B evolved $\widehat{D}$ (squares), 2B evolved ${\widehat{D}}^{†}\widehat{D}$ (diamonds), and 3B evolved ${\widehat{D}}^{†}\widehat{D}$ (triangles). The dotted line in panel (b) indicates the evaluation of Ref. [34] based on a LS renormalization of the ${\mathrm{N}}^3\mathrm{LO} NN$ plus ${\mathrm{N}}^2\mathrm{LO} 3N$ interactions and bare dipole operator. See the text for more details.

Figure 8

Dependence (represented as the width of the bands) on the variation of $\lambda$ between 1.8 and $3.0 {\mathrm{fm}}^{-1}$ of the ${}^4\mathrm{He}$ photoabsorption cross section, ${\sigma }_{\gamma }\left(\omega \right)$, as a function of the photon energy, $\omega$, at $N_{\text{max}}=18/19$ and $\hslash \mathrm{\Omega }=28$ MeV, using the $NN+3N$-induced (dashed contours) and $NN+3N$ (solid contours) wave functions. Calculations were obtained (a) with the bare dipole operator; (b) with the 2B evolved dipole operator; and (c) by rescaling the 2B evolved results by the ratio $\langle {\mathrm{\Psi }}_0|{\widehat{D}}^{†}\widehat{D}|{\mathrm{\Psi }}_0\rangle /\left|\right|\widehat{D}|{\mathrm{\Psi }}_0{\rangle \left|\right|}^2$, with the ${\widehat{D}}^{†}\widehat{D}$ operator evolved in the $3N$ space (3B rescaled operator, see text for details).

Figure 9

The ${}^4\mathrm{He}$ photoabsorption cross section as a function of excitation energy, $\omega$, for $NN+3N$-induced (blue dashed line) and $NN+3N$ (red solid line) interactions with a model-space truncation of $N_{\text{max}}=18/19$ and oscillator frequency of $\hslash \mathrm{\Omega }=28$ MeV. The total cross section is compared to experimental results from Wells et al. [60], Shima et al. [61], Nilsson et al. [62], Nakayama et al. [63], and Raut et al. [64]. See the text for more details.