A stringent upper limit on the direct 3$α$ decay of the Hoyle State in $^{12}$C
We investigate an implication of the most recent observation of a second $J^π= 2^+$ state in $^{12}$C, which was measured using the $^{12}$C($γ$,$α$)$^8$Be$_{\textrm{(g.s.)}}$ reaction. In addition to the dissociation of $^{12}$C to an $α$-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$α$ partial decay width to be determined as $Γ_{3α} < 32(4)$ keV. Since this 2$^+$ state is predicted by all theoretical models to be a collective excitation of the Hoyle state, the 3$α$ partial width of the Hoyle state is calculable from the ratio of 3$α$ 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 ${Γ_{3α} \over Γ} < 5.7 \times 10^{-6}$, over an order of magnitude lower than previously reported. This result places the direct measurement of this rare decay mode beyond current experimental capabilities.