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

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4 Three-body Jacobi matrix element transformation into the m-scheme leading to a〈abc|V NNN |a ′ b ′ c ′ 〉a = 3! Jab,J α ncm,lcm MJ mcm i J ′ ab ,J ′ j × ′ a j ′ b m ′ a m′ J b ′ ab M ′ J ab ′ ab j′ c M ′ ab m′ J c ′ M ′ ja jb Jab Jab jc J × ma mb Mab Mab mc M ⎛ ⎞ ⎛ a b c Jab J J ⎜ ⎟ ⎜ × T⎝ ⎠ T⎝ × cα,i c˜α ′ ,i ′ ncm lcm n12 l12 n3 l3 sab j12 j3 tab T MT ×〈EJTi|V NNN |E ′ JTi ′ 〉 . ˜α ′ i ′ lcm J mcm MJ lcm J mcm MJ J ′ M ′ J M a ′ b ′ c ′ J ′ ab ⎞ ′ J J ⎟ ⎠ ncm lcm n ′ 12 l ′ 12 n′ 3 l ′ 3 s ′ ab j ′ 12 j ′ 3 t ′ ab T MT (4.19) Here, α still means a sum over {n12, l12, sab, j12, n3, l3, j3, J, tab, T, MT }, whereas ˜α ′ means a sum over {n ′ 12 , l′ 12 , s′ ab , j′ 12 , n′ 3 , l′ 3 , j3 ′ , t ′ ab }. mcm,MJ Next we use the orthogonality relation (3.11) of the Clebsch-Gordan coefficients lcm J mcm MJ J M lcm J mcm MJ J ′ M ′ = δJ ,J ′ δM,M ′ (4.20) and the three-body m-scheme interaction matrix element finally reads a〈abc|V NNN |a ′ b ′ c ′ 〉a = 3! 44 Jab,J α ncm,lcm i J ′ ab j × ′ a j ′ b m ′ a m′ J b ′ ab M ′ J ab ′ ab j′ c M ′ ab m′ J c M ⎛ ⎞ ⎛ a b c Jab J J ⎜ ⎟ ⎜ × T⎝ ⎠ T⎝ × cα,i c˜α ′ ,i ′ ncm lcm n12 l12 n3 l3 sab j12 j3 tab T MT ×〈EJTi|V NNN |E ′ JTi ′ 〉 δM,M ′ . ˜α ′ i ′ ja jb ma mb Jab Mab Jab jc Mab mc a ′ b ′ c ′ J ′ ab J M J J ncm lcm n ′ 12 l ′ 12 n′ 3 l ′ 3 s ′ ab j ′ 12 j ′ 3 t ′ ab T MT ⎞ ⎟ ⎠ (4.21)
4.1 Matrix elements of the three-nucleon interaction at N2LO in the m-scheme Or with all quantum numbers shown explicitly a〈abc|V NNN |a ′ b ′ c ′ 〉a = 3! with Jab,J ′ ab J ncm,lcm n12,n ′ 12 l12,l ′ 12 sab,s ′ ab j12,j ′ 12 tab,t ′ ab n3,n ′ 3 l3,l ′ 3 j3,j ′ J,T 3 j × ′ a j ′ b m ′ a m′ J b ′ ab M ′ J ab ′ ab j′ c M ′ ab m′ J c M ⎛ ⎞ ⎛ a b c Jab J J ⎜ ⎟ ⎜ × T⎝ ⎠ T⎝ × cα,i ncm lcm n12 l12 n3 l3 ⎡ ⎢ ⎣ sab j12 j3 tab T MT n12 l12 n3 l3 ⎥ sab j12 j3 J⎦c˜α ′ ,i ′ tab T MT i ⎤ ×〈EJTi|V NNN |E ′ JTi ′ 〉 , ⎡ ⎢ ⎣ ja jb ma mb Jab Mab n ′ 12 l ′ 12 n ′ 3 l ′ 3 s ′ ab j ′ 12 j ′ 3 J t ′ ab T MT i ′ Jab jc Mab mc a ′ b ′ c ′ J ′ ab i,i ′ J M J J ncm lcm n ′ 12 l ′ 12 n′ 3 l ′ 3 s ′ ab j ′ 12 j ′ 3 t ′ ab T MT ⎤ ⎥ ⎦ ⎞ ⎟ ⎠ (4.22) Mab = ma + mb , (4.23) M ′ ab = m′ a + m′ b , (4.24) M = ma + mb + mc = m ′ a + m′ b + m′ c MT = mta + mtb + mtc = m ′ ta + m′ tb + m′ tc , (4.25) , (4.26) E = 2n12 + l12 + 2n3 + l3 . (4.27) The large number of summations and the fact that the T -coefficient depends on almost all of them leads to a complicated computational task. Obviously, the explicit execution of the summations would lead to inefficiently large run times. For details on our implementation we refer the interested reader to appendix A. The remaining task is to calculate the T-coefficient defined in eq. (4.9), which will be done in the following subsection. J. Langhammer 45
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 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 47: 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
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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
Page 108 and 109:
7 Summary and outlook flow paramete
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APPENDICES
Page 114 and 115:
A Implementation dition (−1) (l12
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A Implementation We want to briefly
Page 118 and 119:
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
Page 124 and 125:
B Two-body Talmi Moshinsky transfor
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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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