SRG and Valence-Space Renormalization of the 0 Decay Operator - - PowerPoint PPT Presentation

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SRG and Valence-Space Renormalization of the 0 Decay Operator - - PowerPoint PPT Presentation

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Canadas national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nuclaire et en physique des particules SRG and Valence-Space Renormalization of the 0 Decay Operator TRIUMF

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SLIDE 1

Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada

Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules

Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne

SRG and Valence-Space Renormalization of the 0νββ Decay Operator

TRIUMF Workshop on “Interfacing theory and experiment for reliable double-beta decay matrix element calculations” Vancouver, Canada, May 11-13, 2016.

Petr Navratil | TRIUMF

A=8

  • 2

2 4

A=7 A=10

  • 0.6

0.6 1.2 C(r) [fm

  • 1]

5 10 r [fm]

  • 0.5

0.5 1 1.5 full Obare + Heff Oeff + Heff

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SLIDE 2

Outline

§ Motivation § Ab initio in nuclear physics § No-core shell model § GT transitions in 6He quenching § SRG evolution of operators § Okubo-Lee-Suzuki renormalization of operators in the valence space § Neutrinoless double beta decay toy model § Outlook

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SLIDE 3
  • (i) Renormalization due to missing short-range correlations

– Applies to many ab initio techniques

  • NCSM, CCM, IM-SRG…

– Applies also to phenomenological approaches using effective interactions – SRG is the tool to do the renormalization (surely if SRG evolved interactions are used)

  • (ii) Renormalization due to the valence space truncation

– This is typically on top of the short-range renormalization (i) – Ab initio: Valence space IM-SRG, CCEI, NCSM with core, MBPT – Phenomenology (SM, IBM): effective charges, quenching, MBPT…

M0νββ (or any other) operator renormalization

3

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SLIDE 4

Ab initio calculations in nuclear physics

4

² All nucleons are active ² Exact Pauli principle ² Realistic inter-nucleon interactions

² Accurate description of NN (and 3N) data

² Controllable approximations

INPUT: Realistic inter-nucleon interactions from chiral perturbation theory (N3LO) NN+ (N2LO) 3N

Hα =Uα HUα

+ ⇒ dHα

dα = T, Hα

[ ], Hα

" # $ % α = 1

λ 4

( )

Softening of chiral NN+3N interactions by similarity renormalization group (SRG) unitary transformations: Induce significant 3N interactions Induced 4N and higher much less important

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SLIDE 5
  • No-core shell model (NCSM)

– A-nucleon wave function expansion in the harmonic-oscillator (HO) basis – short- and medium range correlations – Bound-states, narrow resonances

No-core shell model with continuum

5

1

max +

= N N

A

Ψ A = cNi ΦNi

A i

∑

N=0 Nmax

∑

Ψ(A) = cλ

λ

∑

,λ + d r γv( r )

∫

ˆ Aν

ν

∑

,ν

Unknowns

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SLIDE 6

Calculations with chiral 3N: SRG renormalization needed

  • Chiral N3LO NN plus N2LO NNN

potential

– Bare interaction (black line)

  • Strong short-range

correlations § Large basis needed – SRG evolved effective interaction (red line)

  • Unitary transformation
  • Two- plus three-body

components, four-body

  • mitted
  • Softens the interaction

§ Smaller basis sufficient

2 4 6 8 10 12 14 16 18 20 22

Nmax

−29 −28 −27 −26 −25 −24

E [MeV]

bare (36) SRG (2.0/28)

4He

NN + NNN N

3LO (500 MeV)

Hα =Uα HUα

+ ⇒ dHα

dα = T, Hα

[ ], Hα

" # $ % α = 1

λ 4

( )

A=3 binding energy and half life constraint cD=-0.2, cE=-0.205, Λ=500 MeV

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SLIDE 7
  • Continuous transformation driving Hamiltonian to band-diagonal form

with respect to a chosen basis

  • Unitary transformation
  • Setting with Hermitian
  • Customary choice in nuclear physics …kinetic energy operator

– band-diagonal in momentum space plane-wave basis

  • Initial condition

Similarity Renormalization Group (SRG) evolution

7

Hα =Uα HUα

+

Uα Uα

+ =Uα +Uα =1

dHα dα = dUα dα HUα

+ +UαH dUα +

dα = dUα dα Uα

+UαHUα + +UαHUα +Uα

dUα

+

dα = dUα dα Uα

+Hα + HαUα

dUα

+

dα = ηα, Hα

[ ]

anti-Hermitian generator

ηα ≡ dUα dα Uα

+ = −ηα +

ηα = Gα, Hα

[ ] Gα dHα dα = Gα, Hα

[ ], Hα

! " # $ Gα = T Hα=0 = Hλ=∞ = H λ 2 =1/ α

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SLIDE 8

Light nuclei with SRG evolved interactions

  • 29
  • 28
  • 27
  • 26
  • 25
  • 24
  • 23

. Egs [MeV] (a)

NN only

4He Ω = 20 MeV (b)

NN+3N-induced

exp. (c)

NN+3N-full

2 4 6 8 10 12 14 ∞ Nmax

  • 34
  • 32
  • 30
  • 28
  • 26
  • 24
  • 22

. Egs [MeV] (d) 6Li Ω = 20 MeV 2 4 6 8 10 12 14 ∞ Nmax (e) exp. 2 4 6 8 10 12 14 ∞ Nmax (f)

  • Fast convergence
  • Significant 3N induced

interaction

  • No 4N induced

interaction

  • ★

α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4 Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1

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SLIDE 9

6He half-life

9 PHYSICAL REVIEW C 86, 035506 (2012)

Precision measurement of the 6He half-life and the weak axial current in nuclei

  • A. Knecht,1,* R. Hong,1 D. W. Zumwalt,1 B. G. Delbridge,1 A. Garc´

ıa,1 P. M¨ uller,2 H. E. Swanson,1 I. S. Towner,3 S. Utsuno,1

  • W. Williams,2,† and C. Wrede1,‡

1Department of Physics and Center for Experimental Nuclear Physics and Astrophysics, University of Washington,

Precision measurement of 6He beta decay … challenge and test

  • f ab initio calculations,

nuclear forces and currents

Improvement with the NNN interaction MEC must be included Also: Operator renormalization & continuum

2 4 6 8 10

Nmax

2 2.2 2.4 2.6 2.8 |M(GT; 0

+ 1 -> 1 + 0)| N

3LO NN + N 2LO NNN(500)

N

3LO NN

Expt

SRG Λ=1.7 fm

  • 1

hΩ=16 MeV

6He-> 6Li

NCSM

slide-10
SLIDE 10

6He half-life

10 PHYSICAL REVIEW C 86, 035506 (2012)

Precision measurement of the 6He half-life and the weak axial current in nuclei

  • A. Knecht,1,* R. Hong,1 D. W. Zumwalt,1 B. G. Delbridge,1 A. Garc´

ıa,1 P. M¨ uller,2 H. E. Swanson,1 I. S. Towner,3 S. Utsuno,1

  • W. Williams,2,† and C. Wrede1,‡

1Department of Physics and Center for Experimental Nuclear Physics and Astrophysics, University of Washington,

Precision measurement of 6He beta decay … challenge and test

  • f ab initio calculations,

nuclear forces and currents

Improvement with the NNN interaction Improvement with MEC Also: Operator renormalization & continuum

2 4 6 8 10

Nmax

2 2.2 2.4 2.6 2.8 |M(GT; 0

+ 1 -> 1 + 0)| N

3LO NN - 1b

N

3LO NN + N 2LO 3N(500) - 1b

N

3LO NN + N 2LO 3N(500) - 1b+2b

Expt

SRG Λ=1.7 fm

  • 1

hΩ=16 MeV

6He-> 6Li

NCSM

slide-11
SLIDE 11

6He half-life

11 PHYSICAL REVIEW C 86, 035506 (2012)

Precision measurement of the 6He half-life and the weak axial current in nuclei

  • A. Knecht,1,* R. Hong,1 D. W. Zumwalt,1 B. G. Delbridge,1 A. Garc´

ıa,1 P. M¨ uller,2 H. E. Swanson,1 I. S. Towner,3 S. Utsuno,1

  • W. Williams,2,† and C. Wrede1,‡

1Department of Physics and Center for Experimental Nuclear Physics and Astrophysics, University of Washington,

Precision measurement of 6He beta decay … challenge and test

  • f ab initio calculations,

nuclear forces and currents

Improvement with the NNN interaction Improvement with MEC Still to be done: Operator renormalization & continuum

2 4 6 8 10

Nmax

2 2.2 2.4 2.6 2.8 |M(GT; 0

+ 1 -> 1 + 0)| N

3LO NN - 1b

N

3LO NN + N 2LO 3N(500) - 1b

N

3LO NN + N 2LO 3N(500) - 1b+2b

Expt

SRG Λ=1.7 fm

  • 1

hΩ=16 MeV

6He-> 6Li

NCSM

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SLIDE 12

SRG evolution of transition operators

ˆ Oλ

JT = ˆ

Uλ

f ˆ

Oλ=∞

JT ˆ

Uλ

i*;

ˆ Uλ = ψα(λ) ψα(λ = ∞)

α

∑

Final/initial unitary transformations Eigenstates after & before evolution Bare λ = 2 fm-1 E1 E1

3S1à3P2 3S1à3P2

Induces 2-body (& higher-body)

  • perators

Bare

  • perator

3-body evolved

  • perator

PHYSICAL REVIEW C 90, 011301(R) (2014)

Operator evolution for ab initio theory of light nuclei

Micah D. Schuster,1,2 Sofia Quaglioni,2 Calvin W. Johnson,1 Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

PHYSICAL REVIEW C 92, 014320 (2015)

Operator evolution for ab initio electric dipole transitions of 4He

Micah D. Schuster,1,* Sofia Quaglioni,2,† Calvin W. Johnson,1,‡ Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

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SLIDE 13

SRG evolution of transition operators

ˆ Oλ

JT = ˆ

Uλ

f ˆ

Oλ=∞

JT ˆ

Uλ

i*;

ˆ Uλ = ψα(λ) ψα(λ = ∞)

α

∑

Final/initial unitary transformations Eigenstates after & before evolution Bare λ = 2 fm-1 E1 E1

3S1à3P2 3S1à3P2

Induces 2-body (& higher-body)

  • perators

PHYSICAL REVIEW C 90, 011301(R) (2014)

Operator evolution for ab initio theory of light nuclei

Micah D. Schuster,1,2 Sofia Quaglioni,2 Calvin W. Johnson,1 Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

PHYSICAL REVIEW C 92, 014320 (2015)

Operator evolution for ab initio electric dipole transitions of 4He

Micah D. Schuster,1,* Sofia Quaglioni,2,† Calvin W. Johnson,1,‡ Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

2 4 6 8 10 12 14 16 18 0.8 0.82 0.84 0.86 0.88 0.9 0.92

Bare operator

(a)

M0 [fm2] Ω = 28 MeV NA2max = 300 NA3max = 40 Nmax

4He

λ = 1.8 fm−1 λ = 2.0 fm−1 λ = 2.2 fm−1 λ = 2.5 fm−1 λ = 3.0 fm−1

2 4 6 8 10 12 14 16 18 0.8 0.82 0.84 0.86 0.88 0.9 0.92 (b)

2B evolved operator M0 [fm2] Ω = 28 MeV NA2max = 300 NA3max = 40 Nmax

4He

λ = 1.8 fm−1 λ = 2.0 fm−1 λ = 2.2 fm−1 λ = 2.5 fm−1 λ = 3.0 fm−1

M0 = Ψ0 ˆ D+ ˆ D Ψ0 = ˆ D Ψ0

2

Total E1 dipole strength

2B evolved E1 operator ˆ

D

slide-14
SLIDE 14

SRG evolution of transition operators

ˆ Oλ

JT = ˆ

Uλ

f ˆ

Oλ=∞

JT ˆ

Uλ

i*;

ˆ Uλ = ψα(λ) ψα(λ = ∞)

α

∑

Final/initial unitary transformations Eigenstates after & before evolution

In 4He, the inclusion of up to three-body induced terms all but completely restores the invariance of transitions under SRG

Bare λ = 2 fm-1 E1 E1

3S1à3P2 3S1à3P2

Induces 2-body (& higher-body)

  • perators
  • 28
  • 27
  • 26
  • 25

NN-only NN+3N-induced NN+3N

1.50 1.55 1.60

Bare operator 2B evolved operator 3B evolved operator

1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 0.90 0.95 1.00

(a)

(b) (c)

ˆ r2 1/2 [fm] Ψ0| ˆ D2|Ψ0 [fm2] Egs [MeV] λ [fm−1]

Ω = 28 MeV NA2max = 300 NA3max = 40 Nmax = 18 PHYSICAL REVIEW C 90, 011301(R) (2014)

Operator evolution for ab initio theory of light nuclei

Micah D. Schuster,1,2 Sofia Quaglioni,2 Calvin W. Johnson,1 Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

PHYSICAL REVIEW C 92, 014320 (2015)

Operator evolution for ab initio electric dipole transitions of 4He

Micah D. Schuster,1,* Sofia Quaglioni,2,† Calvin W. Johnson,1,‡ Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

slide-15
SLIDE 15

SRG evolution of transition operators

ˆ Oλ

JT = ˆ

Uλ

f ˆ

Oλ=∞

JT ˆ

Uλ

i*;

ˆ Uλ = ψα(λ) ψα(λ = ∞)

α

∑

Final/initial unitary transformations Eigenstates after & before evolution

1) The shorter the range the more renormalization 2) The 3B contribution relatively more important for the longer range Lesson for the neutrinoless double β decay: SRG evolve the M0νββ operator

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 2 4 6 8 10 12 14 16 18 20

λ=2.5 fm

  • 1 2B evolved

λ=2.0 fm

  • 1 2B evolved

λ=1.5 fm

  • 1 2B evolved

λ=2.5 fm

  • 1 3B evolved

λ=2.0 fm

  • 1 3B evolved

λ=1.5 fm

  • 1 3B evolved

Renormalization (%) a0 [fm]

ˆ O(⃗ r1,⃗ r2) = A exp

  • − (⃗

r1 − ⃗ r2)2 a2

  • A
  • exp
  • − r2

a2

  • d⃗

r = 1

PHYSICAL REVIEW C 90, 011301(R) (2014)

Operator evolution for ab initio theory of light nuclei

Micah D. Schuster,1,2 Sofia Quaglioni,2 Calvin W. Johnson,1 Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

PHYSICAL REVIEW C 92, 014320 (2015)

Operator evolution for ab initio electric dipole transitions of 4He

Micah D. Schuster,1,* Sofia Quaglioni,2,† Calvin W. Johnson,1,‡ Eric D. Jurgenson,2 and Petr Navr´ atil3

1San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA

slide-16
SLIDE 16
  • Applications in SM calculations presented by Mihai Horoi
  • n Thursday

– “0νββ Decay: To Quench or Not to Quench”

  • 2B SRG evolution of the light neutrino 0νββ

– chiral N3LO NN, SRG λ=2 fm-1

  • ~5% renormalization in 76Ge
  • 2B SRG evolution of the heavy neutrino 0νββ

– chiral N3LO NN, SRG λ=2 fm-1

  • ~25% renormalization in 76Ge

SRG evolution of the M0νββ operator

16

n n′ 20 40 60 80 100 20 40 60 80 100 n 20 40 60 80 100 n 20 40 60 80 100 n 20 40 60 80 100 20 40 60 80 100

Bare λ=2.5fm−1 λ=2.0fm−1 λ=1.5fm−1

3P0

slide-17
SLIDE 17

Operator renormalization in the valence space

17

Microscopic origins of effective charges in the shell model

Petr Navra ´til,* Michael Thoresen, and Bruce R. Barrett

Department of Physics, University of Arizona, Tucson, Arizona 85721 PHYSICAL REVIEW C FEBRUARY 1997 VOLUME 55, NUMBER 2

PHYSICAL REVIEW C 84, 044316 (2011)

Nonperturbative renormalization of the neutrinoless double-β operator in p-shell nuclei

Deepshikha Shukla and Jonathan Engel

Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina, 27516-3255, USA

Petr Navratil

TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3 Canada and Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, California 94551, USA (Received 15 August 2011; published 19 October 2011) PHYSICAL REVIEW C 80, 024315 (2009)

Effective operators from exact many-body renormalization

  • A. F. Lisetskiy,1,2,* M. K. G. Kruse,1 B. R. Barrett,1 P. Navratil,3 I. Stetcu,4 and J. P. Vary5

1Department of Physics, University of Arizona, Tucson, Arizona 85721, USA

PHYSICAL REVIEW C 78, 044302 (2008)

Ab-initio shell model with a core

  • A. F. Lisetskiy,1,* B. R. Barrett,1 M. K. G. Kruse,1 P. Navratil,2 I. Stetcu,3 and J. P. Vary4

1Department of Physics, University of Arizona, Tucson, Arizona 85721, USA

Ab Initio Coupled-Cluster Effective Interactions for the Shell Model: Application to Neutron-Rich Oxygen and Carbon Isotopes

  • G. R. Jansen,1,2 J. Engel,3 G. Hagen,1,2 P. Navratil,4 and A. Signoracci1,2

1Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

PRL 113, 142502 (2014) P H Y S I C A L R E V I E W L E T T E R S

week ending 3 OCTOBER 2014

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SLIDE 18

H

P Q P Q

Nmax Nmax

Effective Hamiltonian & operators from Okubo-Lee-Suzuki transformation

Heff

QXHX-1 Q

H : E1, E2, E3,…EdP,…E∞

Heff : E1, E2, E3,…Ed P

QXHX −1P = 0

Heff = PXHX −1P

model space dimension

unitary X=exp[-arctanh(ω+-ω)]

⟨q| ω |p⟩ =

d−1

  • k=0

⟨q|k⟩ ⟨k|p⟩ M = P + ω†ω = P (1 + ω†ω)P. Oeff = M− 1

2 (P + ω†)O(P + ω)M− 1 2

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SLIDE 19

Valence-space renormalization of the M0νββ operator

19

Toy problem: 6He 6Be transition NCSM calculations in Nmax=6-10 space projected to Nmax=0 Effective operator used for transitions in A=7,8,10 systems

Mf i ≡ ⟨f |

  • ab

MGT

ab + MF ab + MT ab |i⟩

MGT

ab = HGT (rab) σ a · σ b,

MF

ab = HF(rab) ,

HK(r) = 2R πr ∞ hK(q) sin qr q + ¯ ω dq , K = GT, F. ∞ C(r) dr = Mf i

slide-20
SLIDE 20

Valence-space renormalization of the M0νββ operator

20

Toy problem: 6He 6Be transition NCSM calculations in Nmax=6-10 space projected to Nmax=0 Effective operator used for transitions in A=7,8,10 systems:

7,8,10He 7,8,10Be

∞ C(r) dr = Mf i

A=8

  • 2

2 4

A=7 A=10

  • 0.6

0.6 1.2 C(r) [fm

  • 1]

5 10 r [fm]

  • 0.5

0.5 1 1.5 full Obare + Heff Oeff + Heff

C(r) in Fig. 1. 7 8 10 full 1.76 0.48 0.79 bare 1.49 0.18 0.91 effective 1.90 0.59 1.23

Non-perturbative renormalization

  • f the transition operator improves the shell model

ability to reproduce ab initio results

slide-21
SLIDE 21

Conclusions and Outlook

  • Possible contribution to the neutrinoless double beta decay:

– Renormalization of the transition operator to account for the short-range correlations using the SRG evolution – Renormalization to account for the valence-space truncation using the Okubo- Lee-Suzuki transformation and/or valence space IM-SRG – Benchmark calculations in 48Ca and beyond

  • Ab initio calculations of nuclear structure and reactions is a dynamic field

with significant advances

slide-22
SLIDE 22

Micah Shuster (ORNL) Sofia Quaglioni, Eric Jurgenson (LLNL) Calvin Johnson (SDSU) Mihai Horoi (CMU) Michael Desrochers (UBC), Doron Gazit (Hebrew U) Jon Engel, D. Shukla (UNC)

  • A. Calci (TRIUMF), R. Roth (TU Darmstadt), D. Furnstahl (OSU)

Collaborators contributing to presented results