Decay Modes of the Hoyle State in $^{12}C$

Abstract: Recent experimental results give an upper limit less than 0.043\% (95\% C.L.) to the direct decay of the Hoyle state into 3$\alpha$ respect to the sequential decay into $^8${Be}+$\alpha$. We performed one and two-dimensional tunneling calculations to estimate such a ratio and found it to be more than one order of magnitude smaller than experiment depending on the range of the nuclear force. This is within high statistics experimental capabilities. Our results can also be tested by measuring the decay modes of high excitation energy states of $^{12}$C where the ratio of direct to sequential decay might reach 10\% at $E^*$($^{12}$C)=10.3 MeV. The link between a Bose Einstein Condensate (BEC) and the direct decay of the Hoyle state is also addressed. We discuss a hypothetical `Efimov state' at $E^*$($^{12}$C)=7.458 MeV, which would mainly {\it sequentially} decay with 3$\alpha$ of {\it equal energies}: a counterintuitive result of tunneling. Such a state, if it would exist, is at least 8 orders of magnitude less probable than the Hoyle's, thus below the sensitivity of recent and past experiments.