Anomaly in first-forbidden transitions of ${}^{176}$Lu

${}^{176}$Lu is the key nucleus for understanding the evolution of planetary bodies in the solar system, and the temperature and nucleosynthesis of the $s$ process, due to its very long terrestrial half-life $T_{1\mathrm{/}2}\approx 3.7\times {10}^{10}$ years. The very long half-life is caused by two anomalous first-forbidden transitions from the $7^{-}$ state with extremely large comparative half-life log$ft\approx 19$ which have never been appeared in other cases of the existing nuclear databases. We analyze the underlying mechanism and reason for the anomaly in first-forbidden transitions of ${}^{176}$Lu for the first time, which is based on the projected shell model. It is found that the possible $K$-forbidden nature is indispensable for describing the two extremely weak first-forbidden transitions, and the transition strengths are very sensitive to the detailed configuration mixing and $K$ mixing in the nuclear wave functions. The half-life of the $7^{-}$ state is calculated to be $T_{1\mathrm{/}2}\approx 1.95\times {10}^{10}$ years.