**Principal Investigator:**

Ulf-G. Meißner

**Affiliation:**

Universität Bonn and Forschungszentrum Jülich

**Local Project ID:**

hfz02

**HPC Platform used:**

JUQUEEN of JSC

**Date published:**

**Lattice simulations of nuclear reactions that are relevant to nucleosynthesis in stars have recently become possible. As a first step, researchers from the Nuclear Lattice Effective Field Theory Collaboration have performed an ab initiocalculation of the low-energy scattering of two alpha particles. This paves the way for a deeper understanding of the element generation and the limits of nuclear stability.**

The elements that make up the Earth and life on it are generated in hot, old stars. Of particular importance are processes involving alpha particles (Helium-4 nuclei) and alpha-like nuclei (like e.g. Carbon-12 or Oxygen-16) as these comprise a major part of stellar nucleosynthesis. While the underlying processes have been understood from nuclear modeling since long, a first principles calculation has not been possible until recently. A first step towards such *ab initio*calculations has been described in Ref. [1], where a parameter-free computation of alpha-alpha scattering showed an amazingly good agreement with the existing data, as discussed in more detail below.

It is important to note that the fusion processes in the stars take place at energies that are not available in the laboratory, contrary to common thinking, these are much lower than can be measured in laboratory scattering processes. The thermonuclear reaction in stars take place in a relatively narow energy window, the so-called Gamov peak, which is located at an energy of a mere 300 keV for the “holy grail of nuclear astrophysics” [2], the reaction α +^{12}C→^{16}O+γ, while the lowest energies reached in the laboratory to measure this reaction have been around 1000 keV. Therefore, one needs a controled method to bridge this gap – nuclear models used so far that describe well the existing data cannot provide the required cross sections at such low energies without an uncontrolled systematic uncertainty. This is where our new method based on lattice simulations enters the game.

Why is such a calculation difficult? Reaction dynamics has been considered in nuclear theory for many decades, but all *ab initio* approaches used so far show a factorial or even exponentially scaling with the number of particles involved. Here, *ab initio* means that the nuclear many-body problem is solved numerically exactly, with nuclear forces determined before in systems with two or three nucleons (protons and/or neutrons). Contrary to the conventional approaches, the new method described below exhibits a mild scaling, that is the computing time increases only quadratically with the number of nucleons.

The framework we have developed and which was used to calculate alpha-alpha scattering is the method of nuclear lattice simulations, that had its breakthrough by allowing for the first *ab initio* calculation of the so-called Hoyle state in the spectrum of ^{12}C [3]. This resonant state is mandatory for a sufficent generation of Carbon-12 and thus without it, life on Earth would not be possible. This combination of the modern approach to the nuclear force problem based on an effective field theory with high-performance computing methods defines a completely new method to exactly solve the nuclear *A*-body problem (with *A* the number of nucleons, that is protons and neutrons, in a nucleus). The first ingredient of this method is a systematic and precise effective field theory description of the forces between two and three nucleons, that has been worked out in the last decade by various groups worldwide. To go beyond atomic number four, one has to devise a method to exactly solve the *A*-body problem. Such a method is given by *nuclear lattice simulations*. Space-time is discretized with spatial length L_{s} and temporal length L_{t} , and nucleons are placed on the lattice sites. The minimal length on the lattice, the so-called lattice spacing a, entails a maximum momentum, p_{max} = π / a. Such a lattice representation is ideally suited for parallel computing. Given this framework, the structure and spectrum of ^{12}C [4] and ^{16}O [5] as well as the ground state energies of all alpha-type nuclei up to ^{28}Si have been calulated within a 1% accuracy [6], based on the same microscopic Hamiltonian.

To calculate scattering processes on such a space-time lattice, our calculation proceeds in two steps. First, using exactly the same microscopic Hamiltonian as in the earlier nuclear structure calculations, allows one to construct an *ab initio* cluster Hamiltonian. This step is depicted in Fig. 1, where two clusters at large separation *R* are shown. Within the clusters, the full microscopic dynamics (strong and electromagnetic interactions) is included, such as polarisation and deformation effects or the Pauli exclusion principle. As the separation *R*becomes very large, we can describe the system in terms of an effective cluster Hamiltonian (the free lattice Hamiltonian for two clusters) plus infinite-range interactions (like the Coulomb interaction). In the second step, we can then compute the two-cluster scattering phase shifts or reaction amplitudes using this adiabatic Hamiltonian. Here, one has to account for the strong and short-range Coulomb interactions between the protons and the neutrons in the clusters as well the long-range Coulomb interactions between the protons.

We work with the microscopic Hamiltonian at next-to-next-to-leading order (NNLO) in the chiral expansion of the nuclear forces on a coarse lattice with a lattice spacing a ≃ 2 fm. All appearing parameters have been fixed before in the systems with two, three and four nucleons. We thus can make *parameter-free* predictions of the low-energy α-α phase shifts. These are shown for the S- and the D-wave in comparison to the existing data in Fig. 2 at next-to-leading order (NLO) and at NNLO. We find a remarkable agreement, which can futher be improved in the future. These phase shifts provide useful benchmarks to assess systematic errors in calculations of higher-body nuclear systems and our calculation further demonstrates that an ab initio calculation of the holy grail of nuclear astrophysics is in reach.

**Acknowledgements:**

This project was made possible through the HPC system JUQUEEN of the Jülich Supercomputer Centre (JSC).

**Research Team:**

I thank my collaborators Serdar Elhatisari, Dean Lee, Gautam Rupak, Evgeny Epelbaum, Hermann Krebs, Timo A. Lähde and Tom Luu as part of the NLEFT collaboration.

**References**

[1] S. Elhatisari, D. Lee, G. Rupak, E. Epelbaum, H. Krebs, T. A. Lähde, T. Luu and U.- G. Meißner, Nature **528** (2015) 111 [arXiv:1506.03513 [nucl-th]].

[2] W. A. Fowler, Rev. Mod. Phys. **56** (1984) 149.

[3] E. Epelbaum, H. Krebs, D. Lee and U.-G. Meißner, Phys. Rev. Lett. **106** (2011) 192501 [arXiv:1101.2547 [nucl-th]].

[4] E. Epelbaum, H. Krebs, T. A. Lähde, D. Lee and U.-G. Meißner, Phys. Rev. Lett. **109** (2012) 252501 [arXiv:1208.1328 [nucl-th]].

[5] E. Epelbaum, H. Krebs, T. A. L¨ahde, D. Lee, U.-G. Meißner and G. Rupak, Phys. Rev. Lett. **112** (2014) no.10, 102501 [arXiv:1312.7703 [nucl-th]].

[6] T. A. Lähde, E. Epelbaum, H. Krebs, D. Lee, U.-G. Meißner and G. Rupak, Phys. Lett. B **732** (2014) 110 [arXiv:1311.0477 [nucl-th]].

**Scientific Contact:**

Prof. Dr. Ulf-G. Meißner

Universität Bonn und Forschungszentrum Jülich

D-53115 Bonn (Germany)

e-mail: meissner [at] hiskp.uni-bonn.de

**Acknowledgements:**

This project was made possible through the HPC system JUQUEEN of the Jülich Supercomputer Centre (JSC).

**Research Team:**

I thank my collaborators Serdar Elhatisari, Dean Lee, Gautam Rupak, Evgeny Epelbaum, Hermann Krebs, Timo A. Lähde and Tom Luu as part of the NLEFT collaboration.

**References**

[1] S. Elhatisari, D. Lee, G. Rupak, E. Epelbaum, H. Krebs, T. A. Lähde, T. Luu and U.- G. Meißner, Nature **528** (2015) 111 [arXiv:1506.03513 [nucl-th]].

[2] W. A. Fowler, Rev. Mod. Phys. **56** (1984) 149.

[3] E. Epelbaum, H. Krebs, D. Lee and U.-G. Meißner, Phys. Rev. Lett. **106** (2011) 192501 [arXiv:1101.2547 [nucl-th]].

[4] E. Epelbaum, H. Krebs, T. A. Lähde, D. Lee and U.-G. Meißner, Phys. Rev. Lett. **109** (2012) 252501 [arXiv:1208.1328 [nucl-th]].

[5] E. Epelbaum, H. Krebs, T. A. L¨ahde, D. Lee, U.-G. Meißner and G. Rupak, Phys. Rev. Lett. **112** (2014) no.10, 102501 [arXiv:1312.7703 [nucl-th]].

[6] T. A. Lähde, E. Epelbaum, H. Krebs, D. Lee, U.-G. Meißner and G. Rupak, Phys. Lett. B **732** (2014) 110 [arXiv:1311.0477 [nucl-th]].

**Scientific Contact:**

Prof. Dr. Ulf-G. Meißner

Universität Bonn und Forschungszentrum Jülich

D-53115 Bonn (Germany)

e-mail: meissner [at] hiskp.uni-bonn.de