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 0νββ decay

M0ν

i

= ⟨0+

F |ˆ

O0ν

i |0+ I ⟩

Decay mechanism: transition

• perator (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] Song, JMY, Ring & Meng, PRC(2017)

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 Hod continuously Unitary transformation H(s) = U(s)H0U†(s) dH(s) ds = [η(s), H(s)], dU(s) ds = η(s)U(s) Formal solution: U(s) = 𝒯 exp[ ∫︂ s ds′η(s′)]

Hergert, Bogner, Morris, Schwenk, & Tsukiyama, PR (2016)

IMSRG: Magnus expansion

The Magnus expansion: rewriting U(s) ≡ eˆ

Ω(s) leads to the following ODE

dˆ Ω(s) ds =

∞

∑︂

k=0

Bk k! adk

ˆ Ωη(s),

Ω(0) = 0 (1) ad0

Ωη(s) = η(s), adk Ωη(s) = [Ω(s), adk−1 Ω ] and Bk are the Bernoulli numbers.

ˆ Ω(s) = ∫︂ s η(s1)ds1 + 1 2 ∫︂ s ds1 ∫︂ s1 ds2[η(s1), η(s2)] + · · · (2) The unitary of U(s) is guaranteed by the anti-hermitian Ω(s). Operator ˆ O(s) (BCH expansion) ˆ O(s) = e

ˆ Ω(s) ˆ

O0e−ˆ

Ω(s) =

∑︂

k=0

1 k!adk

ˆ Ω(s) ˆ

O0 = ˆ O0 + [ˆ Ω(s), ˆ O0] + 1 2![ˆ Ω(s), [ˆ Ω(s), ˆ O0]] + · · · (3)

Blanes, Casas, Oteo, Ros, PR(2009); Morries, Parzuchowski, Bogner, PRC (2015).

IMSRG: Brillouin Generator

One-body term: ηk

l (s)

≡ ⟨Φ|[ˆ H(s), ˜ Ak

l ]|Φ⟩ ∼ λ1B, λ2B

(4) Two-body term: ηkl

mn(s)

≡ ⟨Φ|[ˆ H(s), ˜ Akl

mn]|Φ⟩ ∼ λ1B, λ2B, λ3B

(5) λ1B, λ2B, λ3B: irreducible 1B, 2B, and 3B density matrices λi

j

= ρi

j,

(6a) λij

kl

= ρij

kl − ˆ

𝒝(λi

kλj l),

(6b) λijk

lmn

= ρijk

lmn − ˆ

𝒝(λi

lλjk mn + λi lλj mλk n).

(6c) Hierarchy in irreducible DME: λ1B >> λ2B >> λ3B >> · · · Convergence: η(∞) = 0 and dΩ(s) ds |s=∞ = 0.

Brief summary of last talk at MSU

λi

j, λij kl from GCM (β2, γ, φpn)

NN Interaction: KB3G Minimum: E(48Ti) = −23.88 MeV (SM: 23.66 MeV). MR-IMSRG(2): does not work for 48Ti? NME for 0νββ M0ν = ⟨0+

F |e ˆ ΩF(s) ˆ

O0νe−ˆ

ΩI(s)|0+ I ⟩

SM GCM ImSRG(2)(ΩI) ImSRG(2)(ΩF) 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 λ3B terms in η(2B)?

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/ λ3B 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/ λ3B in η(s)

MR-IMSRG(2): Benchmark Calc. from a Deformed State

E(s) = E(0) + 1 1![ˆ Ω(s), ˆ H0]0B + 1 2![ˆ Ω(s), [ˆ Ω(s), ˆ H0]]0B + 1 3![ˆ Ω(s), [ˆ Ω(s), [ˆ Ω(s), ˆ H0]]]0B + . . .

• 72
• 70
• 68
• 66
• 64

10

• 4

10

• 3

10

• 2

10

• 1

10 10

1

10

2

• 8
• 6
• 4
• 2

2

2 4 6 8 10

• 0.10
• 0.08
• 0.06
• 0.04
• 0.02

0.00

• 69.7 MeV

MR-ImSRG (β=0.0) MR-ImSRG (β=0.05) MR-ImSRG (β=0.10)

E (MeV)

GCN2850

76Ge

SM: -70.1 MeV

dE/ds (MeV) Flow parameter s

||η|| (MeV)

λ3B to energy (β = 0.00 case): ∼ +1 × 10−3 MeV

1 2 3 4 5 6 7 8 9 10

• 8
• 6
• 4
• 2

2 4

• 65.11
• 0.23

1.09 2.95

E (MeV)

• 8.31

E(conv.)=-69.64 MeV

76Ge (β=0.00)

MR-IMSRG(2): Benchmark Calc. from a Deformed State

E(s) = E(0) + 1 1![ˆ Ω(s), ˆ H0]0B + 1 2![ˆ Ω(s), [ˆ Ω(s), ˆ H0]]0B + 1 3![ˆ Ω(s), [ˆ Ω(s), [ˆ Ω(s), ˆ H0]]]0B + . . .

• 72
• 70
• 68
• 66
• 64

10

• 4

10

• 3

10

• 2

10

• 1

10 10

1

10

2

• 8
• 6
• 4
• 2

2

2 4 6 8 10

• 0.10
• 0.08
• 0.06
• 0.04
• 0.02

0.00

• 69.7 MeV

MR-ImSRG (β=0.0) MR-ImSRG (β=0.05) MR-ImSRG (β=0.10)

E (MeV)

GCN2850

76Ge

SM: -70.1 MeV

dE/ds (MeV) Flow parameter s

||η|| (MeV)

λ3B to energy (β = 0.05 case): ∼ +4 × 10−3 MeV

1 2 3 4 5 6 7 8 9 10

• 8
• 6
• 4
• 2

2 4

• 65.75
• 0.23

1.44 2.21

E (MeV)

• 7.74
• 0.09

E(conv.)=-70.13 MeV

76Ge (β=0.05)

MR-IMSRG(2): Benchmark Calc. from a Deformed State

E(s) = E(0) + 1 1![ˆ Ω(s), ˆ H0]0B + 1 2![ˆ Ω(s), [ˆ Ω(s), ˆ H0]]0B + 1 3![ˆ Ω(s), [ˆ Ω(s), [ˆ Ω(s), ˆ H0]]]0B + . . .

• 72
• 70
• 68
• 66
• 64

10

• 4

10

• 3

10

• 2

10

• 1

10 10

1

10

2

• 8
• 6
• 4
• 2

2

2 4 6 8 10

• 0.10
• 0.08
• 0.06
• 0.04
• 0.02

0.00

• 71.5 MeV
• 69.7 MeV

MR-ImSRG (β=0.0) MR-ImSRG (β=0.05) MR-ImSRG (β=0.10)

E (MeV)

GCN2850

76Ge

SM: -70.1 MeV

dE/ds (MeV) Flow parameter s

||η|| (MeV)

λ3B to energy (β = 0.10 case): ∼ +0.15 MeV. [ˆ Ω(s), ˆ H0]3B?

1 2 3 4 5 6 7 8 9 10

• 8
• 6
• 4
• 2

2 4

• 66.73

0.22 1.89 0.45

E (MeV)

• 6.97
• 0.21

E(conv.)=-71.35 MeV

76Ge (β=0.10)

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 (λ2B, λ3B, · · · ). 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 0νββ 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 λ3B λpqr

stu ≃ −[npqr stu ]−1 1

4 ∑︂

a

ˆ 𝒝(λpa

st λqr au)

(7) with npqr

stu =

∑︂

a=p,q,r,s,t,u

λa

a − 3

(8)

• 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: λ2B, λ3B 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/ λ3B in η(s)

MR-IMSRG(2): Benchmark Calc. from a Deformed State

• 72
• 70
• 68
• 66
• 64

10

• 4

10

• 3

10

• 2

10

• 1

10 10

1

10

2

• 8
• 6
• 4
• 2

2

2 4 6 8 10

• 0.10
• 0.08
• 0.06
• 0.04
• 0.02

0.00

• 71.5 MeV
• 69.7 MeV

MR-ImSRG (β=0.0) MR-ImSRG (β=0.05) MR-ImSRG (β=0.07) MR-ImSRG (β=0.09) MR-ImSRG (β=0.10)

E (MeV)

GCN2850

76Ge

SM: -70.1 MeV

dE/ds (MeV) Flow parameter s

||η|| (MeV)