the Doorway to carbon-
Life on earth is based on carbon, more precisely on the stable isotope 12C. Carbon is generated in stars in the so-called triple alpha process, in which two 4He nuclei (also called alpha particles) fuse to an instable but long-lived 8Be. A third alpha then fuses with this beryllium nucleus to generate carbon, see Figure 1. However, calculations in the early 1950's showed that if the reaction would proceed through the ground state of 12C, orders of magnitude too little carbon is produced to account for the abundances measured
on earth and in the universe. This obstacle was overcome by Fred Hoyle in 1954, who predicted an excited state of 12C just 350 keV above the 4He+8Be threshold . Due to this resonance condition, the production rate of carbon was increased by many orders of magnitude. This state with its quantum numbers spin zero and parity plus was experimentally confirmed at Caltech in 1957. However, the so-called Hoyle state has been an enigma for nuclear structure theory ever since - no one was able to calculate it ab initio. This means combining precise
models for the nucleon-nucleon interactions with the most sophisticated many-body approaches such as the no-core shell model or the fixed node Green's function Monte Carlo method which have been very successful in describing the spectra of light nuclei. Given its role in the formation of life-essential elements, the Hoyle state is commonly mentioned in anthropic arguments explaining the fine-tuning of fundamental parameters of the universe.
|Figure 1: Illustration of the triple-alpha process
The first ab initio calculation of the low-lying spectrum of 12C using the framework of chiral effective field theory and Monte Carlo lattice simulations was reported in Ref. . This combination of the modern approach to the nuclear force problem 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 (see the review ). The main advantages of this scheme are: i) it can be improved systematically by including higher orders in a systematic power counting, ii) it generates consistent
two and three nucleons forces and iii) it is firmly rooted in the symmetries underlying the strong interactions, described by a non-abelian gauge theory,
Quantum Chromodynamics. The nuclear
interactions consist of long-ranged pion exchanges and shorter-ranged multi-nucleon contact terms. The latter come with undetermined parameters. These can be fixed by a fit to the cornucopia of data in two-nucleon scattering and a few three-nucleon observables such as the triton binding energy. The so determined forces can be used in exact continuum calculations for three and four nucleon systems and lead to a very precise descriptions of thousands of data to a very high accuracy.
|Figure 2: Euclidean lattice with point-like nucleons
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 Ls and temporal length Lt, and nucleons are placed on the lattice sites (see left panel of Fig. 2). The minimal length on the lattice, the so-called lattice
spacing a (see the right panel of Fig. 2) entails a maximum momentum, pmax = /a. Such a lattice representation is ideally suited for parallel computing and the corresponding Monte Carlo codes to calculate the nuclear energies scale ideally with several thousand processors
on JUGENE, the world's largest Blue Gene/P installation located at the Jülich Supercomputing Centre. Furthermore, the nuclear interactions share an SU(4) spin-isospin symmetry - already observed by Wigner in 1937 - that strongly sup
press the malicious fermion sign problem that haunts any Monte Carlo simulation at finite density. Nuclear ground state energies can simply be obtained by letting the Euclidean time tend to infinity, as in that limit the lowest energy state is filtered out. To investigate the excited states, one has to start with a large number of initial states - combined to give the appropriate quantum numbers - and diagonalize the resulting transition matrix.
|Figure 3: Binding energy difference between the triton and 3He nuclei as a function of the lattice volume V = L3. The three lines correspond to fitting various subsets of the data so as to determine the systematical error.
||Figure 4: Left: Ground state energy of 12C as a function of Euclidean time. Right: Spectrum of 12C obtained from nuclear lattice simulations compared to the experimental values.
As a first check, we show a typical result for the three-nucleon systems that serve as benchmarks of our calculational scheme. In Figure 3, the binding energy difference between the triton and 3He is shown for various lattice lengths and fitted with a Lüscher type formula, that allows to take the infinite volume limit. The resulting energy difference of 0.78(5) MeV is in good agreement with the experimental value of 0.76 MeV. Given that this notoriously difficult observable could be calculated with good accuracy gives us the confidence to apply the method to larger nuclei, including 4He, 8Be and 12C. The resulting ground state energies are -28.4(3.0) MeV, -58(2) MeV and -91(3) MeV, in good agreement with the experimental values -28.3 MeV, -56.5 MeV and -92.2 MeV, respectively. The leading order ground state energy of 12C as a function of Euclidean time is shown in the left panel of Figure 4.
The more complicated calculation of the excited states is described in Ref. . The resulting spectrum is shown in the right panel of Figure 4, apart from the lowest-lying spin-2, even parity excitation we also find the Hoyle state at the
correct energy. Also, the 4He+8Be threshold comes out at -86(2) MeV, i.e. within the accuracy of the calculation it is degenerate with the Hoyle state.
At present, we are analyzing the structure of the Hoyle state - it appears
to be a chain of alpha particles. This
scenario can be tested as we predict a
2+ resonance on top of the Hoyle state.
Such a state is not experimentally
confirmed. Furthermore, we can now
address the anthropic principle by
varying the fundamental parameters of
the underlying theory. Performing lattice
simulations for such unphysical para-
meters will answer the question whether
or not the Hoyle is close to the 4He+8Be
threshold solely for the physical values
of these parameters or in a much wider class of theories. In the latter case, the Hoyle state could no longer serve as one of the prime examples of this principle.
I thank my collaborators Evgeny Epelbaum,
Hermann Krebs and Dean Lee for a superb collaboration and the JSC for providing the computational resources.
 Hoyle, F.
Astrophys. J. Suppl. Ser. 1 (1954) 121
 Epelbaum, E., Krebs, H., Lee, D.,
Phys. Rev. Lett. 106 (2011) 192501 [arXiv:1101.2547 [nucl-th]]
 Epelbaum, E., Hammer, H. W.,
Rev. Mod. Phys. 81 (2009) 1773
• Ulf-G. Meißner
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