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

CONTENTS
1 Introduction 1
2 Chiral effective field theory (χEFT) 7
3 Mathematical basics 13
3.1 Angular momentum coupling . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . 13
3.1.2 6j-symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.3 9j-symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.4 3nj-symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Jacobi coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Harmonic oscillator brackets (HOBs) . . . . . . . . . . . . . . . . . . . 21
3.3.1 Definition of harmonic oscillator brackets – our version . . . 22
3.3.2 Symmetry relations . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.3 Definition of harmonic oscillator brackets – alternative ver-
sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.4 Symmetry relations of the alternative HOBs . . . . . . . . . . . 26
3.3.5 Connection between our and the alternative definition of the
HOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.6 Explicit formula for calculation of the HOBs . . . . . . . . . . . 30
3.4 Antisymmetrizer in basis representation . . . . . . . . . . . . . . . . . 31
4 Three-body Jacobi matrix element transformation into the m-scheme 39
4.1 Matrix elements of the three-nucleon interaction at N2LO in the m-
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Calculation of the T -coefficient . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Computational challenges . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 J , T -coupling of the m-scheme matrix elements . . . . . . . . . . . . 61
5 Similarity renormalization group transformation 69
5.1 General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Solution of the flow equation . . . . . . . . . . . . . . . . . . . . . . . 73
J. Langhammer iii