A stringent upper limit on the direct 3$\alpha$ decay of the Hoyle State in $^{12}$C

We investigate an implication of the most recent observation of a second $J^\pi = 2^+$ state in $^{12}$C, which was measured using the $^{12}$C($\gamma$,$\alpha$)$^8$Be$_{\textrm{(g.s.)}}$ reaction. In addition to the dissociation of $^{12}$C to an $\alpha$-particle and $^8$Be in its ground state, a small fraction of events (2%) were identified as direct decays and decays to excited states in $^8$Be. This allowed a limit on the direct 3$\alpha$ partial decay width to be determined as $\Gamma_{3\alpha} < 32(4)$ keV. Since this 2$^+$ state is predicted by all theoretical models to be a collective excitation of the Hoyle state, the 3$\alpha$ partial width of the Hoyle state is calculable from the ratio of 3$\alpha$ decay penetrabilities of the Hoyle and 2$^+$ states. This was calculated using the semi-classical WKB approach and we deduce a stringent upper limit for the direct decay branching ratio of the Hoyle state of ${\Gamma_{3\alpha} \over \Gamma} < 5.7 \times 10^{-6}$, over an order of magnitude lower than previously reported. This result places the direct measurement of this rare decay mode beyond current experimental capabilities.