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# Ab initio alpha–alpha scattering

## Abstract

Processes such as the scattering of alpha particles (4He), the triple-alpha reaction, and alpha capture play a major role in stellar nucleosynthesis. In particular, alpha capture on carbon determines the ratio of carbon to oxygen during helium burning, and affects subsequent carbon, neon, oxygen, and silicon burning stages. It also substantially affects models of thermonuclear type Ia supernovae, owing to carbon detonation in accreting carbon–oxygen white-dwarf stars1,2,3. In these reactions, the accurate calculation of the elastic scattering of alpha particles and alpha-like nuclei—nuclei with even and equal numbers of protons and neutrons—is important for understanding background and resonant scattering contributions. First-principles calculations of processes involving alpha particles and alpha-like nuclei have so far been impractical, owing to the exponential growth of the number of computational operations with the number of particles. Here we describe an ab initio calculation of alpha–alpha scattering that uses lattice Monte Carlo simulations. We use lattice effective field theory to describe the low-energy interactions of protons and neutrons, and apply a technique called the ‘adiabatic projection method’ to reduce the eight-body system to a two-cluster system. We take advantage of the computational efficiency and the more favourable scaling with system size of auxiliary-field Monte Carlo simulations to compute an ab initio effective Hamiltonian for the two clusters. We find promising agreement between lattice results and experimental phase shifts for s-wave and d-wave scattering. The approximately quadratic scaling of computational operations with particle number suggests that it should be possible to compute alpha scattering and capture on carbon and oxygen in the near future. The methods described here can be applied to ultracold atomic few-body systems as well as to hadronic systems using lattice quantum chromodynamics to describe the interactions of quarks and gluons.

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## Acknowledgements

We acknowledge discussions with G. Hale and partial financial support from the Deutsche Forschungsgemeinschaft (Sino-German CRC 110), the Helmholtz Association (contract no. VH-VI-417), BMBF (grant no. 05P12PDFTE), the US Department of Energy (DE-FG02-03ER41260), and US National Science Foundation grant no. PHY-1307453. Further support was provided by the EU HadronPhysics3 project, the ERC project no. 259218 NUCLEAREFT, and the Magnus Ehrnrooth Foundation of the Finnish Society of Sciences and Letters. The computational resources were provided by the Jülich Supercomputing Centre at Forschungszentrum Jülich and by RWTH Aachen.

## Author information

Authors

### Contributions

S.E. performed the analysis of the scattering data. S.E., E.E., H.K., D.L., and G.R. were involved in conceptual development of the adiabatic projection method. S.E. and T.L. produced the figures. T.A.L., D.L., T.L., and U.-G.M. obtained supercomputing time and developed the code. All authors were involved in writing, editing, and reviewing the manuscript.

### Corresponding author

Correspondence to Dean Lee.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

## Extended data figures and tables

### Extended Data Figure 1 Schematic overview of our method.

We start with an eight-body system of protons and neutrons. Each alpha-particle wave packet consists of four nucleons. The protons are red, the neutrons are blue, and the spins are represented by arrows. Next, we perfom ab initio lattice Monte Carlo simulations to construct the adiabatic Hamiltonian for two alpha clusters (grey spheres). Finally, we use the adiabatic Hamiltonian to compute alpha–alpha scattering phase shifts.

### Extended Data Figure 2 s-wave extrapolations at NNLO.

ag, NNLO results (circles) for the s-wave phase shift δ0 versus Lt at laboratory energy ELab = 1.00 MeV, 2.00 MeV, 3.00 MeV, 6.96 MeV, 8.87 MeV, 10.88 MeV, and 12.30 MeV, respectively, as labelled. The theoretical error bars indicate 1 s.d. uncertainty due to Monte Carlo errors. The dot-dashed lines are fits to the data, used to extrapolate the Lt → ∞ limits. The red hatched regions indicate the 1 s.d. error estimate of the extrapolation.

### Extended Data Figure 3 d-wave extrapolations at NNLO.

ag, NNLO results (circles) for the d-wave phase shift δ2 versus Lt at laboratory energy ELab = 2.00 MeV, 3.00 MeV, 5.26 MeV, 6.96 MeV, 8.87 MeV, 9.88 MeV, and 10.88 MeV, respectively, as labelled. The theoretical error bars indicate 1 s.d. uncertainty due to Monte Carlo errors. The dot-dashed lines are fits to the data, used to extrapolate the Lt → ∞ limits. The red hatched regions indicate the 1 s.d. error estimate of the extrapolation.

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Elhatisari, S., Lee, D., Rupak, G. et al. Ab initio alpha–alpha scattering. Nature 528, 111–114 (2015). https://doi.org/10.1038/nature16067

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• DOI: https://doi.org/10.1038/nature16067

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