Chiral three-nucleon interaction and the C 14 -dating β decay

We present a shell-model calculation for the $\ensuremath{\beta}$ 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 ${V}_{\mathrm{low} k}$ 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, ${\ensuremath{\rho}}_{0}=0.16 {\mathrm{fm}}^{\ensuremath{-}3}$. 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 ${V}_{\mathrm{low} k}$ as a guide, we find that these uncertainties are rather small.

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