Consistent chiral three-nucleon interactions in ... - Theory Center

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1 Introduction state of 10 B which has, as a result of a calculation with the Argonne-V18 or CD- Bonn potential, the wrong angular momentum J = 1 instead of the experimental value J = 3 [1]. To overcome these discrepancies, the two-nucleon interactions were supplemented by various three-nucleon potentials. Spectra obtained with these potentials combined with the two-nucleon potentials match experimental data much better. But it is not obvious which two-nucleon interaction should be combined with which three-nucleon interaction. Furthermore, one should aim at a more fundamental derivation of the nuclear interaction starting from Quantum Chromodynamics (QCD) as the underlying theory. A direct derivation of a nuclear potential from QCD is problematic because of its nonperturbative character in the low-energy regime. Moreover, describing nu- clei as systems of interacting quarks and gluons, is not feasible nowadays. Fortu- nately, this detailed description is not necessary, because we investigate an energy regime, where quarks and gluons are not resolved and, thus, we can adopt an effective theory valid for the low-energy regime of nuclear physics. A modern approach is the so-called chiral effective field theory (χEFT) which treats nucleons and pi- ons as degrees of freedom and takes all relevant symmetries of QCD, especially the chiral symmetry, into account. In this framework a systematic expansion of the nuclear potential exists, as Weinberg showed in [2]. More details will be discussed in section 2. It turns out that the aforementioned problems can be remedied by this pro- cedure. Firstly, we have a more fundamental derivation of the potential, which has the nice feature that only two new parameters occur when we include three- body interactions at next-to-next-to leading order (N2LO). Secondly, it is now clear which three-nucleon interaction should be used together with the two-nucleon interaction, namely the one corresponding to the same expansion order. The first time the three-nucleon interaction shows up is at N2LO, and three-nucleon forces of higher order are not available yet. Therefore, this order is considered in this thesis, especially the three-nucleon part is investigated. Results of first no-core shell model calculations with Lee-Suzuki-transformed chiral two-body interaction (NN) and two-plus-three-body interaction (NN+NNN) in a small model space Nmax = 6 are compared to experimental results for 10 B in fig. 1 [3]. Obviously, the quantum numbers of the ground state are correct if one in- cludes the three-body interaction. The chiral two-body interaction alone, like the CD-Bonn or Argonne-V18 two-body potentials predict an incorrect ground state of 10 B. Moreover, the overall agreement with experiment is significantly improved with the use of the three-body interaction. 2 We start our investigations with the matrix elements of the chiral interaction
Figure 1 – Spectrum of 10 B in MeV as result of Lee-Suzuki no-core shell model calculations in a Nmax = 6 model space. The used harmonic-oscillator frequency is Ω = 14 MeV. Results including the chiral two-body interaction (right), the chiral two-plus-three-body interaction (left) are compared to experiment (middle) [3]. up to N3LO for two-nucleon interactions and N2LO for three-nucleon forces and extract the three-nucleon interaction matrix elements. These three-nucleon matrix elements are given with respect to fully antisymmetrized three-particle Jacobi states [4]. For nuclear structure calculations it is more convenient to work with matrix elements with respect to a Slater-determinant basis consisting of single- particle harmonic oscillator states based on Cartesian coordinates. This basis is also known as m-scheme basis. Consequently, we have to derive and implement the transformation from Jacobi to m-scheme matrix elements, which will include two harmonic oscillator brackets. It turns out that this is a non-trivial task, because it is computational demanding. Generally, we follow the strategy outlined in [5] and precompute a certain overlap coefficient to speed up the transformation code. Having the m-scheme matrix elements we are in principle able to attack the eigenvalue problem of the Hamiltonian. But the storage of m-scheme matrix elements becomes problematic even for modest model space sizes. Therefore, our strategy is to compute matrix elements with respect to states that carry a total coupled angular momentum J and isospin T . These matrix elements are then decoupled to m-scheme matrix elements in the many-body calculation. With this strategy it is possible for the first time to calculate matrix elements of the N2LO three-nucleon interaction up to a total three-particle harmonic oscillator energies J. Langhammer 3
Page 1: Consistent chiral three-nucleon int
Page 4 and 5: Contents 5.2.1 Evolution in three-b
Page 8 and 9: 1 Introduction of at least E3max =
Page 11 and 12: SECTION 2 Chiral effective field th
Page 13 and 14: comparable to the high-precision Ar
Page 15 and 16: Figure 3 - Trajectories in the cD-c
Page 17 and 18: SECTION 3 Mathematical basics In th
Page 19 and 20: 3.1 Angular momentum coupling are f
Page 21 and 22: • |[j1, (j2, j3)J23]J ′ M ′
Page 23 and 24: and r2, respectively, these are def
Page 25 and 26: 3.3 Harmonic oscillator brackets (H
Page 27 and 28: in coordinate basis representation
Page 29 and 30: matrix. Moreover, we can use the re
Page 31 and 32: write down the definition 〈〈n1l
Page 33 and 34: 3.3 Harmonic oscillator brackets (H
Page 35 and 36: 3.4 Antisymmetrizer in basis repres
Page 37 and 38: 3.4 Antisymmetrizer in basis repres
Page 39 and 40: Thus the isospin matrix element is
Page 41 and 42: to arrive at the final result 〈[(
Page 43 and 44: SECTION 4 Three-body Jacobi matrix
Page 45 and 46: 4.1 Matrix elements of the three-nu
Page 51 and 52: mations. 4.2 Calculation of the T -
Page 53 and 54: ◮ (nala, nblb)Lab −→ (N12L12(
Page 55 and 56: 4.2 Calculation of the T -coefficie
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4.2 Calculation of the T -coefficie
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4.4 J , T -coupling of the m-scheme
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plug eq. (4.72) into eq. (4.70) and
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5 Similarity renormalization group
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. . . 5 Similarity renormalization
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SECTION 6 Chiral three-body force i
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6.1 Studies of the N2LO three-body
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energy [MeV] . -18 -20 -22 -24 -26
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energy [MeV] . energy [MeV] . -22 -
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6.2 Three-body interactions and the
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E−Eexp A [MeV] . 5 2.5 0 -2.5 -5
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6.2 Three-body interactions and the
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7 Summary and outlook flow paramete
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APPENDICES
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A Implementation dition (−1) (l12
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A Implementation We want to briefly
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A Implementation CGC ,SixJ ,NineJ I
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A Implementation • L12 Lower Bou
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A Implementation which can be found
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B Two-body Talmi Moshinsky transfor
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References [13] P. Navratil, G. P.
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References [41] N. Imai, H. J. Ong,
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ERKLÄRUNG ZUR EIGENSTÄNDIGKEIT Hi
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