Stringent upper limit on the direct $3\alpha$ decay of the Hoyle state in ${}^{12}\mathrm{C}$

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

Figure 1

Different types of events photographed by the O-TPC for $E\gamma =9.6$ MeV. (upper panel) A ${}^{12}\mathrm{C}(\gamma ,{\alpha }_0){}^8\mathrm{Be}_{(\mathrm{g}.\mathrm{s}.)}$ sequential decay. (lower panel) A ${}^{12}\mathrm{C}(\gamma ,{\alpha }_1){}^8\mathrm{Be}_{2^{+}}$ sequential decay or ${}^{12}\mathrm{C}(\gamma ,3\alpha )$ direct decay, which are indistinguishable.

Figure 2

The Coulomb potential as a function of the hyperradius for various direct $3\alpha$ decay configurations of the Hoyle state. The horizontal and vertical dashed lines show the energy above the $3\alpha$ threshold and the ${\rho }_0$ value, respectively. Inset shows the decay configurations corresponding to the upper, center, and lower lines in the plot.

Figure 3

The ratio of the $3\alpha$ penetrability factors as a function of the $R_0$ parameter. The vertical dashed line marks the $R_0$ value that best reproduces the experimental $\alpha$-particle radius.