Chiral three-nucleon interaction and the ${}^{14}\mathrm{C}$-dating $\beta$ decay

We present a shell-model calculation for the $\beta$ decay of ${}^{14}\mathrm{C}$ to the ${}^{14}\mathrm{N}$ ground state, treating the relevant nuclear states as two $0p$ holes in an ${}^{16}\mathrm{O}$ core. Employing the universal low-momentum nucleon-nucleon potential $V_{\mathrm{low}\hspace{0.5em}k}$ only, one finds that the Gamow-Teller matrix element is too large to describe the known (very long) lifetime of ${}^{14}\mathrm{C}$. As a novel approach to this problem, we invoke the chiral three-nucleon force (3NF) at leading order and derive from it a density-dependent in-medium $\mathit{NN}$ interaction. By including this effective in-medium $\mathit{NN}$ interaction, the Gamow-Teller matrix element vanishes for a nuclear density close to that of saturated nuclear matter, ${\rho }_0=0.16\hspace{0.5em}{\mathrm{fm}}^{-3}$. The genuine short-range part of the three-nucleon interaction plays a particularly important role in this context, since the medium modifications to the pion propagator and pion-nucleon vertex (owing to the long-range 3NF) tend to cancel out in the relevant observable. We discuss also uncertainties related to the off-shell extrapolation of the in-medium $\mathit{NN}$ interaction. Using the off-shell behavior of $V_{\mathrm{low}\hspace{0.5em}k}$ as a guide, we find that these uncertainties are rather small.

Figure 1

In-medium $\mathit{NN}$ interaction generated by the two-pion exchange component ($~c_{1,3,4}$) of the chiral three-nucleon interaction. The short double line symbolizes the filled Fermi sea of nucleons [i.e., the medium insertion $-2\pi \delta (l_0)\theta (k_f-|\overrightarrow{l}\hspace{0.5em}|)$ in the in-medium nucleon propagator]. Reflected diagrams are not shown.

Figure 2

In-medium $\mathit{NN}$ interaction generated by the one-pion exchange ($~c_D$) and short-range component ($~c_E$) of the chiral three-nucleon interaction.

Figure 3

Two different off-shell extrapolations for the density-dependent $V_{\mathit{NN}}^{\mathrm{med},2}$ at $\rho ={\rho }_0$ compared to the off-shell dependence of the low-momentum one-pion exchange in the ${}^1{S}_0$ partial wave. Matrix elements are shifted and scaled for comparison purposes (see text for details).

Figure 4

Diagrams contributing to the effective interaction $V_{\mathrm{eff}}$ in our present calculation. Wavy lines represent the density-dependent in-medium nuclear interaction calculated in Sec. 2.

Figure 5

The average strength of the three $\langle \alpha \beta ;0^{+}1\left|V_{\mathit{NN}}^{\mathrm{med},\mathrm{i}}\right|\gamma \delta ;0^{+}1\rangle$ matrix elements for the six density-dependent contributions to the in-medium $\mathit{NN}$ interaction relative to $V_{\mathrm{low}\hspace{0.5em}k}$. The low-momentum interaction is constructed from the Idaho N${}^3\mathrm{LO}$ potential for ${\Lambda }_{\mathrm{low}\hspace{0.5em}k}=2.1$ ${\mathrm{fm}}^{-1}$.

Figure 6

The average strength of the six $\langle \alpha \beta ;1^{+}0\left|V_{\mathit{NN}}^{\mathrm{med},\mathrm{i}}\right|\gamma \delta ;1^{+}0\rangle$ matrix elements for the six density-dependent contributions to the in-medium $\mathit{NN}$ interaction relative to $V_{\mathrm{low}\hspace{0.5em}k}$. The low-momentum interaction is constructed from the Idaho N${}^3\mathrm{LO}$ potential for ${\Lambda }_{\mathrm{low}\hspace{0.5em}k}=2.1$ ${\mathrm{fm}}^{-1}$.

Figure 7

The effects of the various density-dependent contributions to the in-medium $\mathit{NN}$ interaction on the ground state to ground state $B(\mathrm{GT})$ value. The legend denotes which of the $V_{\mathit{NN}}^{\mathrm{med},i}$ are included together with $V_{\mathrm{low}\hspace{0.5em}k}$.

Figure 8

Dependence of the ground state to ground state $B(\mathrm{GT})$ value on the off-shell extrapolation of the density-dependent $V_{\mathit{NN}}^{\mathrm{med}}$ interaction. Both the symmetric and asymmetric off-shell extrapolations are shown.

Figure 9

The $B(\mathrm{GT})$ values for transitions from the low-lying states of ${}^{14}\mathrm{C}$ to the ground state of ${}^{14}\mathrm{N}$ as a function of the nuclear density. The experimental values are from Ref. [13]. Note that there are three experimental low-lying $2^{+}$ states compared to two theoretical $2^{+}$ states in the $0p^{-2}$ configuration.

Figure 10

The uncertainty in the calculated value of $B(\mathrm{GT})$ obtained by varying the low-momentum cutoff ${\Lambda }_{\mathrm{low}\hspace{0.5em}k}$ between 2.1 and 2.3 ${\mathrm{fm}}^{-1}$. The shaded region corresponds to nuclear densities close to that experienced by valence $0p$-shell nucleons in ${}^{14}\mathrm{C}$.

Figure 11

Twice the charge distribution of ${}^{14}\mathrm{N}$ taken from Refs. [39, 40] together with the square of the radial $0p$-shell wave functions.