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

We present a shell-model calculation for the β decay of ¹⁴C to the ¹⁴N ground state, treating the relevant nuclear states as two 0p holes in an ¹⁶O core. Employing the universal low-momentum nucleon-nucleon potential V_low k only, one finds that the Gamow-Teller matrix element is too large to describe the known (very long) lifetime of ¹⁴C. 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 NN interaction. By including this effective in-medium NN interaction, the Gamow-Teller matrix element vanishes for a nuclear density close to that of saturated nuclear matter, ρ₀=0.16 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 NN interaction. Using the off-shell behavior of V_low k as a guide, we find that these uncertainties are rather small.