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Calculation of Three-Body Density within GCM Method Long-Jun Wang Department of Physics and Astronomy, UNC-Chapel Hill Feb. 03, 2017 Outline Definition and Motivation 1 Calculation of Three-Body (3) and (3) 2 Numerical Check 3 L.-J.

  1. Calculation of Three-Body Density within GCM Method Long-Jun Wang Department of Physics and Astronomy, UNC-Chapel Hill Feb. 03, 2017

  2. Outline Definition and Motivation 1 Calculation of Three-Body ρ (3) and λ (3) 2 Numerical Check 3 L.-J. Wang (UNC) Three-Body Density Feb. 03, 2017 2 / 15

  3. Definition and Motivation Definition ( J -scheme and M -scheme) of ρ (3) � J � 0 � ��� ρ (3) J ≡ � J � � J 12 ˆ � J 45 ˆ � �� � c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) 0 + � 0 + ˆ ˆ ˆ ˆ (1) � � i f j 1 j 2 j 3 ˜ ˜ ˜ � j 4 j 5 j 6 � ρ (3) M ≡ c † c † c † 0 + � 0 + � � � � � ˆ τ 1 j 1 m 1 ˆ τ 2 j 2 m 2 ˆ τ 3 j 3 m 3 ˆ c τ 4 j 4 m 4 ˆ c τ 5 j 5 m 5 ˆ c τ 6 j 6 m 6 (2) i f From M -scheme to J -scheme ′ ρ (3) J = Coeff (123456 , J 12 , J 45 , J, Sig ) � ρ (3) M � (3) ( m 1 m 2 m 3 m 4 m 5 m 6 ) Important for NME of 0 νββ Menendez: PRL (2011); Engel: PRC (2014) From Wendt’s notes From PRL 107, 062501 L.-J. Wang (UNC) Three-Body Density Feb. 03, 2017 3 / 15

  4. Definition and Motivation Definition ( M -scheme) of λ (3) a · · · c † A a ··· k l ··· q = c † k c q · · · c l (4) ρ k r = � Ψ | A k r | Ψ � = ρ rk , (5) rs | Ψ � = ρ (2) ρ kl rs = � Ψ | A kl rs,kl , (6) rst | Ψ � = ρ (3) ρ klm rst = � Ψ | A klm rst,klm . (7) ρ k r ≡ λ k r , (8) ρ kl rs ≡ λ kl rs + A ( λ k r λ l s ) , (9) ρ klm rst ≡ λ klm rst + A ( λ k r λ l s λ m t + λ k r λ lm st ) . (10) The antisymmetrizer A generates all unique permutations of the indices of the product of tensors it is applied to. H. Hergert: In-Medium SRG Notes (2015) L.-J. Wang (UNC) Three-Body Density Feb. 03, 2017 4 / 15

  5. Calculation of Three-Body ρ (3) and λ (3) ρ (3) : from M - to J -scheme � J � 0 ��� � J � J 12 ˆ � J 45 ˆ �� c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) ˆ ˆ ˆ ˆ j 1 j 2 j 3 � ˜ ˜ ˜ j 4 j 5 j 6 � JM � JM ′ � J 12 ˆ � J 45 ˆ �� �� c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) � C 00 = ˆ ˆ ˆ ˆ JMJM ′ j 1 j 2 j 3 ˜ ˜ ˜ � j 4 j 5 j 6 MM ′ � J 12 M 12 ˆ � J 45 M 45 ˆ � � c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) � � � C 00 JMJM ′ C JM J 12 M 12 j 3 m 3 C JM ′ = ˆ ˆ ˆ ˆ J 45 M 45 j 6 m 6 j 1 j 2 j 3 m 3 � ˜ ˜ ˜ j 4 j 5 j 6 m 6 MM ′ M 12 m 3 M 45 m 6 � � � � � C 00 JMJM ′ C JM J 12 M 12 j 3 m 3 C JM ′ J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 = j 4 m 4 j 5 m 5 MM ′ M 12 m 3 M 45 m 6 m 1 m 2 m 4 m 5 c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) × ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 m 4 ˆ j 5 m 5 ˆ ˜ ˜ ˜ j 6 m 6 1 � � � � � ( − ) J − M 2 J + 1 C JM J 12 M 12 j 3 m 3 C J − M J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 √ = j 4 m 4 j 5 m 5 m 1 m 2 m 4 m 5 M M 12 m 3 M 45 m 6 c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) × ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 m 4 ˆ j 5 m 5 ˆ (11) ˜ ˜ ˜ j 6 m 6 L.-J. Wang (UNC) Three-Body Density Feb. 03, 2017 5 / 15

  6. Calculation of Three-Body ρ (3) and λ (3) ρ (3) : from M - to J -scheme In the signature basis 1 k ≡ 1 ˆ d † ˆ c † c † d k ≡ √ 2(ˆ c ˜ k + ˆ c k ) , √ 2(ˆ k − ˆ k ) , (12) ¯ ˜ 1 k ≡ 1 ˆ d † ˆ c † c † d ¯ k ≡ √ 2(ˆ c ˜ k − ˆ c k ) , √ 2(ˆ k + ˆ k ) . (13) ˜ � ˆ � ˆ d † d † � � e − iπ ˆ e iπ ˆ J x = ± i k k J x (14) ˆ ˆ d † d † ¯ ¯ k k Where +( − ) for m = 1 2 , − 3 2 , 5 2 · · · ( − 1 2 , 3 2 , − 5 2 · · · ) , i.e., m = even + 1 2 for positive signature. Rotated matrix elements in signature basis � ˆ d † τ 1 j 1 m 1 ˆ d † τ 2 j 2 m 2 ˆ d † τ 3 j 3 m 3 ˆ d τ 4 j 4 m 4 ˆ d τ 5 j 5 m 5 ˆ � ˜ � � � � φ f d τ 6 j 6 m 6 φ i (15) N. Hinohara: Notes (2015) L.-J. Wang (UNC) Three-Body Density Feb. 03, 2017 6 / 15

  7. Calculation of Three-Body ρ (3) and λ (3) ρ (3) : from M - to J -scheme ′ 1 � � ( − ) J − M √ = 2 J + 1 × ( m 1 m 2 m 3 m 4 m 5 m 6 ) ( M 12 M 45 M )  c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 )  J 12 M 12 j 3 m 3 C J − M J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 C JM ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 m 4 ˆ j 5 m 5 ˆ j 4 m 4 j 5 m 5 ˜ ˜ ˜   j 6 m 6     c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) J 12 M 12 j 3 m 3 C J − M J 45 M 45 j 6 − m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45  + C JM  ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 m 4 ˆ j 5 m 5 ˆ    j 4 m 4 j 5 m 5 ˜ ˜ ˜  j 6 − m 6     c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 )  J 12 M 12 j 3 m 3 C J − M J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45  + C JM ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 m 4 ˆ j 5 − m 5 ˆ     j 4 m 4 j 5 − m 5 ˜ ˜ ˜  j 6 m 6      c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) J 12 M 12 j 3 m 3 C J − M J 45 M 45 j 6 − m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 + C JM   ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 m 4 ˆ j 5 − m 5 ˆ   j 4 m 4 j 5 − m 5 ˜ ˜ ˜   j 6 − m 6     c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) J 12 M 12 j 3 m 3 C J − M J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45  + C JM  ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 − m 4 ˆ j 5 m 5 ˆ    j 4 − m 4 j 5 m 5 ˜ ˜ ˜  j 6 m 6     c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 )  J 45 M 45 j 6 − m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45  J 12 M 12 j 3 m 3 C J − M + C JM ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 − m 4 ˆ j 5 m 5 ˆ     j 4 − m 4 j 5 m 5 ˜ ˜ ˜   j 6 − m 6     c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 J 12 M 12 j 3 m 3 C J − M  + C JM  ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 − m 4 ˆ j 5 − m 5 ˆ (16) j 4 − m 4 j 5 − m 5 ˜ ˜ ˜ j 6 m 6 c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) J 12 M 12 j 3 m 3 C J − M J 45 M 45 j 6 − m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 + C JM j 4 − m 4 j 5 − m 5 ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 m 3 � ˆ j 4 − m 4 ˆ j 5 − m 5 ˆ     ˜ ˜ ˜ j 6 − m 6      c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 )  J 12 M 12 j 3 − m 3 C J − M J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 + C JM ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 − m 3 � ˆ j 4 m 4 ˆ j 5 m 5 ˆ     j 4 m 4 j 5 m 5 ˜ ˜ ˜   j 6 m 6     c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) J 12 M 12 j 3 − m 3 C J − M J 45 M 45 j 6 − m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45  + C JM  ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 − m 3 � ˆ j 4 m 4 ˆ j 5 m 5 ˆ    j 4 m 4 j 5 m 5 ˜ ˜ ˜  j 6 − m 6     c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 )  J 12 M 12 j 3 − m 3 C J − M J 45 M 45 j 6 m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45  + C JM ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 − m 3 � ˆ j 4 m 4 ˆ j 5 − m 5 ˆ     j 4 m 4 j 5 − m 5 ˜ ˜ ˜  j 6 m 6      c ( τ 1 ) † c ( τ 2 ) † c ( τ 3 ) † c ( τ 4 ) c ( τ 5 ) c ( τ 6 ) J 12 M 12 j 3 − m 3 C J − M J 45 M 45 j 6 − m 6 C J 12 M 12 j 1 m 1 j 2 m 2 C J 45 M 45 + C JM  j 4 m 4 j 5 − m 5 ˆ j 1 m 1 ˆ j 2 m 2 ˆ j 3 − m 3 � ˆ j 4 m 4 ˆ j 5 − m 5 ˆ    ˜ ˜ ˜   j 6 − m 6     + · · · · · ·         + · · · · · ·   L.-J. Wang (UNC) Three-Body Density Feb. 03, 2017 7 / 15

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