Hoyle state nuclei almost always decay right back into beryllium and an alpha particle. But once out of every 2,500 times, these swollen carbons relax into their stable, ground-state configuration, giving off the extra energy as a burst of gamma rays. The brand new carbon-12 nuclei that are created subsequently populate the periodic table: Some stay as they are, while others fuse with another alpha particle to become oxygen. A fraction of the oxygen nuclei are stripped of a proton, transforming into nitrogen; others fuse with yet another alpha to become neon, and so on. If the star ends in a cataclysmic explosion called a supernova, it disperses these newly minted elements into space, where they eventually become the building blocks of future solar systems.
Hoyle, who died in 2001, knew that without the Hoyle state as their starting point, these elements would not arise. The Hoyle state is a “resonance” of carbon formed by a beryllium atom and alpha particle, meaning it has almost exactly the same energy as their combined masses. Ground-state carbon-12 has a lower energy, and so it does not form through the fusion of an alpha particle and beryllium, just as two plus two does not equal three. “For all these stable states to happen, there must be a resonance,” Hjorth-Jensen said.
But Hoyle only predicted the energy of the resonant state of carbon; he could say nothing about the forces and interactions that enable it to form, nor anything about its physical properties. Because carbon contains six protons and six neutrons, each of which contains three quarks, the Hoyle state amounts to a highly complex 36-body problem. Despite decades of work by nuclear physicists, even with modern computing an exact calculation of the state remains beyond reach.
Now an approach called chiral effective field theory, developed by the Nobel laureate Steven Weinberg, has enabled Lee and his colleagues to closely approximate the structure of the Hoyle state. The trick takes advantage of the fact that protons and neutrons tend to keep their distance from one another inside atomic nuclei, so they “see” one another not as three-quark structures but as single, albeit slightly complicated particles.
Forgetting about quarks turns a 36-body problem into a 12-body problem, but with the strong nuclear force, electromagnetism and “higher-order” chiral forces acting between every particle, even this problem resists an exact solution. “Pinning down where all twelve protons and neutrons are is just a horribly complicated thing,” said Lee.
To make the calculation possible, chiral effective field theory employs a math trick sometimes used in high school calculus. In the same way that a mathematical function, such as a curve on a graph, can be approximated by calculating the first few terms of a “Taylor series expansion” — an infinite sum of progressively smaller terms — around a point on the curve, the researchers approximated the forces that shape the Hoyle state by considering only the first few terms of a Taylor series expansion of those forces.
“I like to compare it to shooting a par-3 in golf,” Lee said. The first stroke, analogous to the leading terms in the Taylor series equation, “drives the ball as close to the hole as possible.” The second stroke, representing the terms with middling influence on the motion of the particles, gets you even closer. The third is a subtle correction. After three strokes, you have a very good approximation of the structure and energy of the Hoyle state.
When a supercomputer applies this calculation to a simulation of six protons and six neutrons distributed on a three-dimensional lattice, the particles can arrange themselves in response to it in infinitely many ways. However, only the lowest-energy configurations of the particles are common in nature. Of these, the researchers found that one of the solutions was the ground-state carbon nucleus. Another was the Hoyle state, with its 7.65 MeV of extra energy.