Shell Model Description of the ${}^{14}\mathrm{C}$ Dating $\beta$ Decay with Brown-Rho-Scaled $NN$ Interactions

We present shell model calculations for the beta decay of ${}^{14}\mathrm{C}$ to the ${}^{14}\mathrm{N}$ ground state, treating the states of the $A=14$ multiplet as two $0p$ holes in an ${}^{16}\mathrm{O}$ core. We employ low-momentum nucleon-nucleon ($NN$) interactions derived from the realistic Bonn-B potential and find that the Gamow-Teller (GT) matrix element is too large to describe the known lifetime. By using a modified version of this potential that incorporates the effects of Brown-Rho scaling medium modifications, we find that the GT matrix element vanishes for a nuclear density around 85% that of nuclear matter. We find that the splitting between the $(J^{\pi },T)=(1^{+},0)$ and $(J^{\pi },T)=(0^{+},1)$ states in ${}^{14}\mathrm{N}$ is improved using the medium-modified Bonn-B potential and that the transition strengths from excited states of ${}^{14}\mathrm{C}$ to the ${}^{14}\mathrm{N}$ ground state are compatible with recent experiments.

Figure 1

The radial part of the nuclear tensor force given in Eq. (2) from $\pi$ and $\rho$ meson exchange at zero density and nuclear matter density under the assumption of BRS.

Figure 2

The $B(\mathrm{GT})$ values for transitions from the states of ${}^{14}\mathrm{C}$ to the ${}^{14}\mathrm{N}$ ground state as a function of the nuclear density and the experimental values from 36. Note that there are three experimental low-lying $2^{+}$ states compared to two theoretical $2^{+}$ states in the $p^{-2}$ configuration.

Figure 3

The half-life of ${}^{14}\mathrm{C}$, as a function of the nuclear density, calculated from the MM Bonn-B potential.

Figure 4

Twice the charge distribution of ${}^{14}\mathrm{N}$ taken from 37, 38 and the fourth power of the $p$-shell wave functions.

Figure 5

The splitting between the $1_1^{+}$ and $0_1^{+}$ levels in ${}^{14}\mathrm{N}$ for different values of the nuclear density. Also included is the experimental value.