Multi-Reference In-medium Similarity Renormalization Group for the - PowerPoint PPT Presentation

Multi-Reference In-medium Similarity Renormalization Group for the Nuclear Matrix Elements of Neutrinoless Double Beta Decay Jiangming Yao Department of Physics and Astronomy, University of North Carolina at Chapel Hill, North Carolina, 27599-3255, USA DOE topical collaboration “Nuclear Theory for Double-Beta Decay and Fundamental Symmetries”, UMass Amherst, Feb.3, 2017

Nuclear Matrix Elements of 0 νββ decay = ⟨ 0 + F | ˆ i | 0 + M 0 ν O 0 ν I ⟩ i Song, JMY, Ring & Meng, PRC(2017) Decay mechanism: transition operator (limited to 1B current mostly, L/H ν ) Nuclear structure: wave functions of initial and final nuclei Model-dependence ( factor of 2-3) Engel & Menendez, arXiv:1610.06548v1 [nucl-th]

ab initio approaches for heavy deformed nuclei Rapid developments in ab initio approaches (CC, SCGF, IMSRG) for nuclei Limitation: spherical or light nuclei Multi-Reference IMSRG: a promising approach for heavy deformed nuclei IMSRG: suppression H od continuously Unitary transformation H ( s ) = U ( s ) H 0 U † ( s ) dH ( s ) = [ η ( s ) , H ( s )] , ds dU ( s ) = η ( s ) U ( s ) ds Formal solution: Hergert, Bogner, Morris, Schwenk, & Tsukiyama, PR ∫︂ s ds ′ η ( s ′ )] U ( s ) = 𝒯 exp [ (2016) 0

IMSRG: Magnus expansion The Magnus expansion: rewriting U ( s ) ≡ e ˆ Ω( s ) leads to the following ODE ∞ d ˆ Ω( s ) B k ∑︂ k ! ad k = Ω η ( s ) , Ω( 0 ) = 0 (1) ˆ ds k = 0 ad 0 Ω η ( s ) = η ( s ) , ad k Ω η ( s ) = [Ω( s ) , ad k − 1 Ω ] and B k are the Bernoulli numbers. ∫︂ s ∫︂ s ∫︂ s 1 η ( s 1 ) ds 1 + 1 ˆ Ω( s ) = ds 2 [ η ( s 1 ) , η ( s 2 )] + · · · ds 1 (2) 2 0 0 0 The unitary of U ( s ) is guaranteed by the anti-hermitian Ω( s ) . Operator ˆ O ( s ) (BCH expansion) 1 Ω( s ) ˆ ˆ O 0 e − ˆ Ω( s ) = ˆ ∑︂ k ! ad k Ω( s ) ˆ O ( s ) = e O 0 ˆ k = 0 O 0 ] + 1 O 0 + [ˆ ˆ Ω( s ) , ˆ 2 ![ˆ Ω( s ) , [ˆ Ω( s ) , ˆ = O 0 ]] + · · · (3) Blanes, Casas, Oteo, Ros, PR(2009); Morries, Parzuchowski, Bogner, PRC (2015).

IMSRG: Brillouin Generator One-body term: η k ⟨ Φ | [ˆ H ( s ) , ˜ A k l ] | Φ ⟩ ∼ λ 1 B , λ 2 B l ( s ) ≡ (4) Two-body term: ⟨ Φ | [ˆ H ( s ) , ˜ η kl A kl mn ] | Φ ⟩ ∼ λ 1 B , λ 2 B , λ 3 B mn ( s ) ≡ (5) λ 1 B , λ 2 B , λ 3 B : irreducible 1B, 2B, and 3B density matrices λ i ρ i = j , (6a) j λ ij ρ ij kl − ˆ k λ j 𝒝 ( λ i = l ) , (6b) kl λ ijk ρ ijk lmn − ˆ 𝒝 ( λ i l λ jk mn + λ i l λ j m λ k = n ) . (6c) lmn Hierarchy in irreducible DME: λ 1 B >> λ 2 B >> λ 3 B >> · · · Convergence: η ( ∞ ) = 0 and d Ω( s ) | s = ∞ = 0. ds

Brief summary of last talk at MSU j , λ ij λ i kl from GCM ( β 2 , γ, φ pn ) NN Interaction: KB3G Minimum: E ( 48 Ti ) = − 23 . 88 MeV (SM: 23.66 MeV). MR-IMSRG(2): does not work for 48 Ti? NME for 0 νββ M 0 ν = ⟨ 0 + Ω F ( s ) ˆ ˆ O 0 ν e − ˆ Ω I ( s ) | 0 + F | e I ⟩ GCM ImSRG(2)( Ω I ) ImSRG(2)( Ω F ) SM GT 0.848 0.883 1.058 0.941 Fermi 0.146 0.207 0.233 0.172 Tensor -0.058 -0.071 -0.067 -0.060 Total 0.936 1.019 1.224 1.053 Missing the irreducible λ 3 B terms in η ( 2 B ) ?

MR-IMSRG(2): Benchmark Calc. from a Spherical State Numerical Details Reference state: spherical HFB NN Interaction: shell-model KB3G Techniques: PNP-ImSRG and MR-ImSRG Better convergence w / λ 3 B in η ( s )

MR-IMSRG(2): Benchmark Calc. from a Spherical State Numerical Details Reference state: spherical HFB NN Interaction: shell-model GCN2850 Techniques: PNP-ImSRG and MR-ImSRG Better convergence w / λ 3 B in η ( s )

MR-IMSRG(2): Benchmark Calc. from a Deformed State E ( 0 ) + 1 H 0 ] 0 B + 1 H 0 ]] 0 B + 1 H 0 ]]] 0 B + . . . 1 ![ˆ 2 ![ˆ Ω( s ) , [ˆ 3 ![ˆ Ω( s ) , [ˆ Ω( s ) , [ˆ Ω( s ) , ˆ Ω( s ) , ˆ Ω( s ) , ˆ E ( s ) = λ 3 B to energy ( β = 0 . 00 case): 10 MR-ImSRG ( β =0.0) || η || (MeV) ∼ + 1 × 10 − 3 MeV 8 MR-ImSRG ( β =0.05) MR-ImSRG ( β =0.10) 6 4 4 2 GCN2850 2.95 0 2 1.09 -64 -0.23 E (MeV) -66 0 0 1 2 3 4 5 6 7 8 9 10 E (MeV) -68 -2 -69.7 MeV SM: -70.1 MeV -70 -4 -72 76Ge ( β =0.00 ) 2 -6 dE/ds (MeV) 0 0.00 E(conv.)=-69.64 MeV -0.02 -2 -0.04 -8 -0.06 -8.31 -4 -0.08 -65.11 -0.10 76 Ge -6 -8 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 Flow parameter s

MR-IMSRG(2): Benchmark Calc. from a Deformed State E ( 0 ) + 1 H 0 ] 0 B + 1 H 0 ]] 0 B + 1 H 0 ]]] 0 B + . . . 1 ![ˆ Ω( s ) , ˆ 2 ![ˆ Ω( s ) , [ˆ Ω( s ) , ˆ 3 ![ˆ Ω( s ) , [ˆ Ω( s ) , [ˆ Ω( s ) , ˆ E ( s ) = λ 3 B to energy ( β = 0 . 05 case): 10 MR-ImSRG ( β =0.0) || η || (MeV) ∼ + 4 × 10 − 3 MeV 8 MR-ImSRG ( β =0.05) MR-ImSRG ( β =0.10) 6 4 4 2 GCN2850 2.21 0 1.44 2 -64 -0.23 -0.09 E (MeV) 0 -66 0 1 2 3 4 5 6 7 8 9 10 E (MeV) -68 -2 -69.7 MeV SM: -70.1 MeV -70 -4 -72 76Ge ( β =0.05 ) 2 -6 dE/ds (MeV) 0 0.00 E(conv.)=-70.13 MeV -0.02 -2 -8 -0.04 -7.74 -0.06 -4 -0.08 -65.75 -0.10 76 Ge -6 -8 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 Flow parameter s

MR-IMSRG(2): Benchmark Calc. from a Deformed State E ( 0 ) + 1 H 0 ] 0 B + 1 H 0 ]] 0 B + 1 H 0 ]]] 0 B + . . . 1 ![ˆ Ω( s ) , ˆ 2 ![ˆ Ω( s ) , [ˆ Ω( s ) , ˆ 3 ![ˆ Ω( s ) , [ˆ Ω( s ) , [ˆ Ω( s ) , ˆ E ( s ) = λ 3 B to energy ( β = 0 . 10 case): 10 MR-ImSRG ( β =0.0) || η || (MeV) ∼ + 0 . 15 MeV. [ˆ Ω( s ) , ˆ H 0 ] 3 B ? 8 MR-ImSRG ( β =0.05) MR-ImSRG ( β =0.10) 6 4 4 2 GCN2850 1.89 0 2 -64 0.45 0.22 -0.21 E (MeV) -66 0 0 1 2 3 4 5 6 7 8 9 10 E (MeV) -68 -69.7 MeV -2 SM: -70.1 MeV -70 -71.5 MeV -4 -72 76Ge ( β =0.10 ) 2 -6 dE/ds (MeV) 0 0.00 E(conv.)=-71.35 MeV -0.02 -6.97 -2 -0.04 -8 -0.06 -4 -0.08 -66.73 -0.10 76 Ge -6 -8 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 Flow parameter s

Summary and outlook Summary The MR-IMSRG(2) based on shell-model interactions works well for near-spherical Ref. states, but not for large deformed ones. The more correlation is included in the Ref. state, the more important is the high-rank irreducible density ( λ 2 B , λ 3 B , · · · ). Extension of the MR-IMSRG(2) to MR-IMSRG(2*) or MR-IMSRG(3) is needed for deformed nuclei. Outlook MR-IMSRG based on chiral NN interaction with the MR-IMSRG(2*). Calculation of the NME for 0 νββ with the MR-IMSRG(2*).

Acknowledgement Jonathan Engel, Longjun Wang University of North Carolina, Chapel Hill Changfeng Jiao Central Michigan University Heiko Hergert NSCL, Michigan State University Thanks for your attention

MR-IMSRG: Benchmark Calculation from a Spherical State Mazziotti’s prescription for λ 3 B stu ] − 1 1 λ pqr stu ≃ − [ n pqr 𝒝 ( λ pa ˆ ∑︂ st λ qr au ) (7) 4 a with n pqr ∑︂ λ a stu = a − 3 (8) a = p , q , r , s , t , u D. A. Mazziotti, PRA (1999) Numerical Details Reference state: spherical HFB NN Interaction: shell-model GCN2850 Techniques: MR-ImSRG

MR-IMSRG(2): Benchmark Calc. from a Spherical State Numerical Details Reference state: spherical HF state: λ 2 B , λ 3 B are zero. NN Interaction: shell-model KB3G Techniques: PNP-ImSRG and MR-ImSRG MR-IMSRG(2) works for this case

MR-IMSRG(2): Benchmark Calc. from a Spherical State Numerical Details Reference state: spherical HFB NN Interaction: shell-model GCN2850 Techniques: PNP-ImSRG and MR-ImSRG Better convergence w / λ 3 B in η ( s )

MR-IMSRG(2): Benchmark Calc. from a Deformed State 10 MR-ImSRG ( β =0.0) || η || (MeV) 8 MR-ImSRG ( β =0.05) MR-ImSRG ( β =0.07) 6 MR-ImSRG ( β =0.09) MR-ImSRG ( β =0.10) 4 2 GCN2850 0 -64 E (MeV) -66 -68 -69.7 MeV SM: -70.1 MeV -70 -71.5 MeV -72 2 dE/ds (MeV) 0 0.00 -0.02 -2 -0.04 -0.06 -4 -0.08 76 Ge -0.10 -6 -8 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 Flow parameter s