${}^{12}\mathrm{C}$ properties with evolved chiral three-nucleon interactions

We investigate selected static and transition properties of ${}^{12}\mathrm{C}$ using ab initio no-core shell model (NCSM) methods with chiral two- and three-nucleon interactions. We adopt the similarity renormalization group (SRG) to assist convergence including up to three-nucleon ($3N$) contributions. We examine the dependencies of the ${}^{12}$C observables on the SRG evolution scale and on the model-space parameters. We obtain nearly converged low-lying excitation spectra. We compare results of the full NCSM with the importance truncated NCSM in large model spaces for benchmarking purposes. We highlight the effects of the chiral $3N$ interaction on several spectroscopic observables. The agreement of some observables with experiment is improved significantly by the inclusion of $3N$ interactions, e.g., the $B\left(M1\right)$ from the first $J^{\pi }T=1^{+}1$ state to the ground state. However, in some cases the agreement deteriorates, e.g., for the excitation energy of the first $1^{+}0$ state, leaving room for improved next-generation chiral Hamiltonians.

Figure 1

Excitation spectra of ${}^{12}$C with the chiral $NN+3N$ interactions for three different SRG evolution scales as a function of $N_{\max }$ at $\hslash \Omega =20$ MeV compared with experiment. The solid line for each calculated level represents results with $\alpha =0.0625{\text{fm}}^4$. The long-dashed line represents NCSM results with $\alpha =0.08{\text{fm}}^4$ and the short-dashed line represents NCSM results with $\alpha =0.04{\text{fm}}^4$.

Figure 2

Excitation spectra of ${}^{12}$C with the chiral $NN+3N$ interactions, for two different $\hslash \Omega$ values, as a function of $N_{\max }$ at $\alpha =0.0625{\text{fm}}^4$ compared with experiment. The solid line for each calculated state represents results with $\hslash \Omega =20$ MeV. The long-dashed line represents results with $\hslash \Omega =16$ MeV.

Figure 3

Excitation spectra of ${}^{12}$C without and with initial chiral $3N$ interaction for two different $N_{\max }$ values, compared with experiment. These NCSM results are calculated at $\hslash \Omega =20$ MeV for flow parameter $\alpha =0.0625{\text{fm}}^4$.

Figure 4

Reduced magnetic dipole transition matrix element from the $1^{+}1$ to the ground state of ${}^{12}$C (in units of ${\mu }_N^2$) as a function of $N_{\max }$ at three different SRG evolution scales and three different HO basis frequencies. The solid and dashed lines present the results obtained with and without the initial chiral $3N$ interaction, respectively.

Figure 5

Reduced electric quadrupole transition matrix element from the $2^{+}0$ to the ground state of ${}^{12}$C (in units of $e^2$ fm${}^4$) as a function of $N_{\max }$ at three different SRG evolution scales and three different HO basis frequencies. The solid and dashed lines present the results obtained with and without the initial chiral $3N$ interaction, respectively.

Figure 6

Excitation energy of the $3^{-}0$ in ${}^{12}$C as a function of ($N_{\max }$,$N_{\max }$ +1) without (dashed lines) and with (solid lines) initial chiral $3N$ interaction at three different SRG evolution scales and three different HO basis frequencies. The unnatural parity states are computed at $N_{\max }$ +1, while the corresponding excitation energy is calculated with respect to the ground state at $N_{\max }$ so that a pair of basis spaces defines each point in this plot.

Figure 7

Negative-parity excitation spectra of ${}^{12}$C obtained without and with initial chiral $3N$ interaction for two different $N_{\max }$ values, compared with experiment. These NCSM results are calculated at $\hslash \Omega =20$ MeV for flow parameter $\alpha =0.0625{\text{fm}}^4$.

Figure 8

Excitation spectra of ${}^{12}$C with the chiral $NN+3N$ interactions, obtained with the NCSM and the IT-NCSM, as a function of $N_{\max }$, and compared with experiment. The solid lines represent the NCSM result and dashed lines represent the IT-NCSM results, with boxes indicating the typical threshold-extrapolation uncertainties. These results are calculated at $\hslash \Omega =20$ MeV with the SRG evolution scale $\alpha =0.0625{\text{fm}}^4$. For $N_{\max }=10$, only IT-NCSM calculations targeting the lowest four eigenstates are currently available.

Figure 9

Excitation of the negative-parity spectra of ${}^{12}$C with respect to the lowest $3^{-}0$ state using the chiral $NN+3N$ interaction, obtained with the NCSM and the IT-NCSM, as a function of $N_{\max }$, and compared with experiment. The solid lines represent the NCSM result and dashed lines represent the IT-NCSM results. These results are calculated at $\hslash \Omega =20$ MeV with the SRG evolution scale $\alpha =0.0625{\text{fm}}^4$. For $N_{\max }=9$, only IT-NCSM calculated eigenstates are currently available.