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

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3 Mathematical basics thogonal transformation of the degenerate eigenstates will build an eigenbasis, too. Only the total number of eigenstates |EJTi〉 which is given by the trace of the antisymmetrizer stays the same. Consequently, different numerical eigenvalue- problem solvers will produce different orderings and different eigenstates of the totally antisymmetrized basis states. Once we fixed these eigenstates and their or- dering in our calculations, we have to stick to it for the rest of the calculation. 38
SECTION 4 Three-body Jacobi matrix element transformation into the m-scheme As already mentioned at the end of section 2, the starting point of our investi- gation of the N2LO three-nucleon interaction are matrix elements with respect to antisymmetrized three-particle Jacobi states |EJMJTMTi〉. Since m-scheme matrix elements are needed in our many-body calculations, we have to transform the given matrix elements into the m-scheme . To investigate the transformation, we assume a charge invariant and rotational invariant interaction, meaning that the interaction does not depend on the MT or MJ quantum numbers. Moreover, the interaction does not change the total rel- ative angular momentum J and the total isospin T of the three nucleons. This summarizes to 〈EJMJTMTi|V NNN |E ′ J ′ M ′ J T ′ M ′ T i′ 〉 = 〈EJTi|V NNN |E ′ JTi ′ 〉 δJ,J ′ δMJ,M ′ J δT,T ′ δMT ,M ′ T . (4.1) Furthermore, we denote the interaction operator throughout this section as V NNN , indicating a three-nucleon interaction. Nevertheless, all formulas are valid also for two-plus-three-body interactions as long as their matrix elements are given in the basis |EJTi〉. In subsection 4.1 we deal with the transformation into m-scheme matrix elements. During this transformation we encounter a complicated overlap that is calculated separately in subsection 4.2. In subsection 4.3 we discuss a serious limita- tion with the direct transformation into m-scheme matrix elements, since storing them as well as the overlap calculated in subsection 4.2 is very memory demand- J. Langhammer 39
Page 1: Consistent chiral three-nucleon int
Page 4 and 5: Contents 5.2.1 Evolution in three-b
Page 6 and 7: 1 Introduction state of 10 B which
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: to arrive at the final result 〈[(
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
Page 65 and 66: 4.4 J , T -coupling of the m-scheme
Page 67 and 68: plug eq. (4.72) into eq. (4.70) and
Page 69 and 70: 4.4 J , T -coupling of the m-scheme
Page 71: 4.4 J , T -coupling of the m-scheme
Page 74 and 75: 5 Similarity renormalization group
Page 80 and 81: . . . 5 Similarity renormalization
Page 89 and 90: SECTION 6 Chiral three-body force i
Page 91 and 92: 6.1 Studies of the N2LO three-body
Page 93 and 94:
6.1 Studies of the N2LO three-body
Page 95 and 96:
energy [MeV] . -18 -20 -22 -24 -26
Page 97 and 98:
energy [MeV] . energy [MeV] . -22 -
Page 99 and 100:
6.1 Studies of the N2LO three-body
Page 101 and 102:
6.2 Three-body interactions and the
Page 103 and 104:
E−Eexp A [MeV] . 5 2.5 0 -2.5 -5
Page 105:
6.2 Three-body interactions and the
Page 108 and 109:
7 Summary and outlook flow paramete
Page 111:
APPENDICES
Page 114 and 115:
A Implementation dition (−1) (l12
Page 116 and 117:
A Implementation We want to briefly
Page 118 and 119:
A Implementation CGC ,SixJ ,NineJ I
Page 120 and 121:
A Implementation • L12 Lower Bou
Page 122 and 123:
A Implementation which can be found
Page 124 and 125:
B Two-body Talmi Moshinsky transfor
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
References [13] P. Navratil, G. P.
Page 134 and 135:
References [41] N. Imai, H. J. Ong,
Page 137:
ERKLÄRUNG ZUR EIGENSTÄNDIGKEIT Hi
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