Ab initio electromagnetic observables with the in-medium similarity renormalization group

We present the formalism for consistently transforming transition operators within the in-medium similarity renormalization group framework. We implement the operator transformation in both the equations-of-motion and valence-space variants, and present first results for electromagnetic transitions and moments in medium-mass nuclei using consistently evolved operators, including the induced two-body parts. These results are compared to experimental values, and—where possible—the results of no-core shell-model calculations using the same input chiral interaction. We find good agreement between the equations-of-motion and valence-space approaches. Magnetic dipole observables are generally in reasonable agreement with experiment, while the more collective electric quadrupole and octupole observables are significantly underpredicted, often by over an order of magnitude, indicating missing physics at the present level of truncation.

Figure 1

The quantity ${\xi }_{\mathrm{c}.\mathrm{m}.}$, which gives an indication of the degree to which the c.m. wave function is a Gaussian (see text), calculated for the ground state of ${}^{14}\mathrm{C}$. Also plotted are two methods of estimating the trapping frequency $\hslash \tilde{\omega }$ (MeV). Column (a) is without a c.m. trap, while column (b) is with $\beta =3$ at a frequency $\hslash \tilde{\omega }=16$ MeV.

Figure 2

Ground state, $2_1{}^{+}$ excitation energy, and $B\left(E2\right)$ values (in $e^2{\mathrm{fm}}^4$) calculated at several values of the Lawson-Gloeckner scaling parameter $\beta$, for ${}^{14}\mathrm{C}$ with the EOM-IMSRG.

Figure 3

Ground-state properties of the deuteron calculated with a full diagonalization (labeled FCI), compared to the same properties calculated in the $0s$ valence space using operators transformed with the IMSRG. Also shown is the result obtained by diagonalizing in a Jacobi basis with $e_{\max }=20$ and 200, in order to gauge convergence. Note that VS-IMSRG and FCI values are nearly the same.

Figure 4

Convergence of the ground-state energy, first $3^{+}$ excitation energy, quadrupole moments (in $e{\mathrm{fm}}^2$), $B(E2;3_1{}^{+}\to 1_1{}^{+})$ ($e^2{\mathrm{fm}}^4$), and $B(M1;0_1{}^{+}\to 1_1{}^{+})$ (${\mu }_n^2$) of ${}^6\mathrm{Li}$. The VS-IMSRG method [column (b)] is compared with NCSM results [column (a)] and experiment [76, 77, 78].

Figure 5

Convergence of the ground-state energy, first $2^{+}$ excitation energy, and $B\left(E2\right)$ (in $e^2{\mathrm{fm}}^4$) to the ground state of ${}^6\mathrm{He}$. Again, VS-IMSRG [column (b)] is compared with NCSM [column (a)] and experiment [78].

Figure 6

Convergence of the first $2^{+}$ excitation energy and $B\left(E2\right)$ (in $e^2{\mathrm{fm}}^4$) to ground state of ${}^{14}\mathrm{C}$. VS- and EOM-IMSRG methods [columns (b) and (c) respectively] are compared with NCSM [column (a)] and experiment [78].

Figure 7

Convergence of $0_1{}^{+}$ excitation energy, $B\left(M1\right)$ (in ${\mu }_N^2$) to ground state, and magnetic dipole moment of ${}^{14}\mathrm{N}$. VS- and EOM-IMSRG methods [columns (b) and (c) respectively] are compared with NCSM [column (a)] and experiment [77, 83].

Figure 8

Results of EOM-IMSRG(2,2) and VS-IMSRG(2) calculations of the $2_1{}^{+}$ excitation energy (a), and the $B(E2;2_1{}^{+}\to 0_1{}^{+})$ value (b) for several closed-shell nuclei in the $sd$ and $pf$ shells. Due to experimental values that vary by several orders of magnitude, the $B\left(E2\right)$ values are scaled such that experiment is unity. Computations are performed at $\hslash \omega =20$ MeV and $e_{\max }=12$. Experimental results are taken from [78].

Figure 9

Convergence of the $1_1{}^{+}$ excitation energy, $B(M1;1_1{}^{+}\to 0_1{}^{+})$ (in ${\mu }_N^2$) and $2_1{}^{+}$ magnetic dipole moment (${\mu }_N$) of ${}^{32}\mathrm{S}$. VS-IMSRG [column (a)] and EOM-IMSRG [column (b)] methods are compared with experiment [77, 88].

Figure 10

Energy and magnetic dipole moment (in ${\mu }_N$) of the $1_1{}^{+}$ state, and $B\left(M1\right)$ to the $2_1{}^{+}$ state (${\mu }_N^2$) of ${}^{32}\mathrm{Cl}$. VS-IMSRG [column (a)] and EOM-IMSRG [column (b)] methods are compared with experiment [77], where available.

Figure 11

Excitation energies of the first $3^{-}$ state of ${}^{16}\mathrm{O}$, along with corresponding $B\left(E3\right)$ strength (in $e^2{\mathrm{fm}}^6$) for the transition to ground state. These values are computed using the EOM-IMSRG(2,2) with ${\mathrm{N}}^2{\mathrm{LO}}_{\mathrm{sat}}$ [column (b)] and the Entem and Machleidt NN$\left(500\right)-3N\left(400\right)$ interaction [column (a)], and are compared with experiment [91].

Figure 12

Excitation energies of the first $3^{-}$ state of ${}^{40}\mathrm{Ca}$, along with corresponding $B\left(E3\right)$ strength (in ${10}^2{e}^2{\mathrm{fm}}^6$) for the transition to ground state. These values are computed using the EOM-IMSRG(2,2) with ${\mathrm{N}}^2{\mathrm{LO}}_{\mathrm{sat}}$ [column (b)] and the Entem and Machleidt NN$\left(500\right)-3N\left(400\right)$ interaction [column (a)], and are compared with experiment [91].

Figure 13

Transition matrix elements $\langle 0_1{}^{+}\parallel E2\parallel 2_1{}^{+}\rangle$ and $\langle 0_1{}^{+}\parallel M1\parallel 1_1{}^{+}\rangle$ computed for select nuclei using EOM-IMSRG(2,2). Calculations are performed for bare operators (orange bars) and operators dressed by consistent IMSRG evolution, both the one-body part (green single-hashed bars) and the one-body plus two-body part (blue double-hashed bars). Values are expressed in $e{\mathrm{fm}}^2$ and ${\mu }_N$ for $E2$ and $M1$ operators, respectively.