Breakthrough calculations of collisions between two helium nuclei pave the way to a quantitative understanding of how the elements carbon and oxygen were made in stars — and to improved models of stellar evolution. **See Letter ** p.111

The life-enabling elements carbon and oxygen were mainly made in red giant and supergiant stars, through a sequence of fusion reactions known as helium burning. In this process, helium nuclei (^{4}He) — aggregates of two protons and two neutrons, named alpha particles by Ernest Rutherford^{1} — are progressively converted into carbon and oxygen nuclei (^{12}C and ^{16}O, respectively). But exactly how each of these reactions happens at a fundamental level remains unexplained. On page 111 of this issue, Elhatisari *et al*.^{2} take a crucial step towards addressing this question by describing the collision of two alpha particles (alpha–alpha scattering) from first principles. The theoretical–computational methods described in this work could also be used to characterize collisions between certain other composite quantum particles.

Helium burning starts when two alpha particles collide with each other and attempt to fuse. But the product of that fusion is an unstable beryllium isotope, ^{8}Be, which decays almost instantly back into two helium nuclei. A third alpha particle therefore has to be captured nearly simultaneously with the collision of the original pair for ^{12}C to be formed. This process is known as the triple-alpha reaction^{3} and was first proposed in 1952. Oxygen is then created when ^{12}C captures a fourth alpha particle^{4} (Fig. 1). The ratio of the amount of carbon to that of oxygen produced through these processes has profound repercussions on the later evolutionary phases of magnificent massive stars such as Orion's Betelgeuse, and their ultimate fate once they explode as supernovae^{5}.

The odds of any of these reactions taking place are small at the low energies encountered in stellar environments, largely because the positively charged colliding nuclei electrically repel each other. The low reaction rates make helium-burning reactions difficult to replicate and measure in a laboratory. This prevents the carbon-to-oxygen ratio produced in stars from being accurately estimated, and introduces large uncertainties in stellar-evolution models and simulations of the processes that create nuclei (nucleosynthetic processes). A computational approach capable of describing helium burning from first principles could come to the rescue by providing 'measurements' from simulations. Of course, before such theoretical predictions can be trusted, their accuracy in reproducing experimentally determined data must be ascertained. The scattering of two alpha particles has been characterized experimentally, and is a good starting point in this case.

Why did we have to wait until 2015 to obtain a first-principles description of alpha–alpha scattering? The simple answer is that developing a fundamental understanding of nuclei and their interactions is one of the most complicated problems in science. It involves unravelling the properties of an ensemble of nucleons (protons or neutrons), exerting forces on each other, that emerge from the underlying theory of strong interactions, while also accounting for all the quantum-mechanical laws that govern microscopic objects. The complexity of the numerical calculations needed explodes as the number of nucleons increases. Explaining the dynamic interactions of clusters of nucleons is especially hard. Even the scattering of deuterium (a two-nucleon system) from alpha particles has only recently been described from first principles^{6}.

Elhatisari and co-authors report a clever way to break through this computational ceiling. They start with a supercomputer-friendly formulation — on a four-dimensional space-time lattice — of chiral effective field theory. This theory^{7,8} links the interactions of nucleons to quantum chromodynamics (the underlying theory of the strong force) by describing them as the sum of an infinite number of terms, systematically organized in order of importance so that all but the first few terms can be neglected.

The authors use their formulation to follow the evolution of the wavefunction of mutually interacting nucleons in a pair of alpha particles, although this is easier said than done. It requires a trick^{9,10} developed in the 1950s and an aptitude for 'gambling': the problem is reformulated as a much simpler system of independent nucleons interacting with a background of auxiliary particles; the most-likely backgrounds are drawn from a probability distribution. This technique, known as auxiliary-field Monte Carlo, is then used to 'cool' the two alpha particles, which are initially placed at a distance from each other, to their correct physical state — that is, to a low-energy quantum-mechanical solution of two dynamically interacting nucleon clusters. This process is repeated over and over, each time with a new relative position for the initial pair of alpha particles, again drawn from an appropriate probability distribution, and the resulting physical states are then used to compute the effective interaction experienced by the two particles as a whole. *Voilà*! The ferocious eight-body problem is transformed into a docile two-cluster problem.

After 2 million hours of parallel computations, the alpha–alpha scattering properties obtained by the authors — including the first three interaction terms of the sequence provided by chiral effective field theory — show promising agreement with experimentally obtained values. The calculation is admittedly somewhat shy of the accuracy required to make quantitative predictions for nucleosynthesis and stellar evolution. Improvements should be made by including the next term in the sequence, investigating the dependence of the results on the spacing of the space-time lattice, and doing other precision tests. Extensions to enable the treatment of three-cluster dynamics^{11} are also required before the method can be applied to the triple-alpha process.

Most impressively, Elhatisari *et al*. have devised a first-principles method for simulating scattering and reactions in which the number of computing operations is proportional to the square of the number of nucleons, and therefore grows relatively slowly. Scattering between alpha particles and ^{12}C, and the conversion of these particles to ^{16}O — a problem that is only four times as difficult as alpha–alpha scattering using the authors' approach — are now within reach of simulations. Furthermore, analogous methods could be used to solve other puzzles. For example, predictive calculations of hyperon–neutron scattering could help to settle whether or not 'strange' particles can exist in the cores of neutron stars^{12}, thus providing insight into the phases of dense nuclear matter that can exist. But that is another story.

## Notes

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### Sofia Quaglioni is in the Nuclear and Chemical Sciences Division, Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, California 94551-0808, USA.

- Sofia Quaglioni

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