Ab Initio Calculation of the Hoyle State

The Hoyle state plays a crucial role in the helium burning of stars heavier than our Sun and in the production of carbon and other elements necessary for life. This excited state of the carbon-12 nucleus was postulated by Hoyle as a necessary ingredient for the fusion of three alpha particles to produce carbon at stellar temperatures. Although the Hoyle state was seen experimentally more than a half century ago nuclear theorists have not yet uncovered the nature of this state from first principles. In this Letter we report the first ab initio calculation of the low-lying states of carbon-12 using supercomputer lattice simulations and a theoretical framework known as effective field theory. In addition to the ground state and excited spin-2 state, we find a resonance at $-85(3)\mathrm{MeV}$ with all of the properties of the Hoyle state and in agreement with the experimentally observed energy.

Figure 1

Extraction of the excited states of ${}^{12}\mathrm{C}$ from the time dependence of the projection amplitude at LO. The slope of the logarithm of $Z(t)/Z_{0_1^{+}}(t)$ at large $t$ determines the energy relative to the ground state.

Figure 2

Summary of lattice results for the carbon-12 spectrum and comparison with the experimental values. For each order in chiral EFT labeled on the left, results are shown for the ground state (blue circles), Hoyle state (red squares), and the $J_z=0$ (open black circles) and $J_z=2$ (filled black circles) projections of the spin-2 state.

Figure 3

The radial distribution function $f_{\mathrm{pp}}(r)$ for the ground state (A), Hoyle state (B), and in the $J_z=0$ (C) and $J_z=2$ (D) projections of the spin-2 state. The yellow bands denote error bars.