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

Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules SRG and Valence-Space Renormalization of the 0 νββ Decay Operator TRIUMF Workshop on “Interfacing theory and experiment for reliable double-beta decay matrix element calculations” 4 A=7 Vancouver, Canada, May 11-13, 2016. 2 0 Petr Navratil | TRIUMF -2 A=8 full 1.2 O bare + H eff O eff + H eff -1 ] 0.6 C(r) [fm 0 -0.6 1.5 A=10 1 0.5 0 Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne -0.5 0 5 10 Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada r [fm] Propriété d’un consortium d’universités canadiennes, géré en co-entreprise à partir d’une contribution administrée par le Conseil national de recherches Canada

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

M 0 νββ (or any other) operator renormalization • (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 … 3

Ab initio calculations in nuclear physics INPUT: Realistic inter-nucleon interactions ² All nucleons are active from chiral perturbation theory ² Exact Pauli principle (N 3 LO) NN+ (N 2 LO) 3N ² Realistic inter-nucleon interactions ² Accurate description of NN (and 3N) data ² Controllable approximations Softening of chiral NN+3N interactions by similarity renormalization group (SRG) unitary transformations: Induce significant 3N interactions Induced 4N and higher much less important + ⇒ dH α ( ) " $ % α = 1 [ ] , H α H α = U α HU α T , H α d α = # λ 4 4

No-core shell model with continuum • No-core shell model (NCSM) N = N 1 max + – A -nucleon wave function expansion in the harmonic-oscillator (HO) basis – short- and medium range correlations A N max – Bound-states, narrow resonances Ψ A = ∑ ∑ A c Ni Φ Ni N = 0 i d  r γ v (  Ψ ( A ) = ˆ ∑ ∑ ∫ c λ , λ + r ) A ν , ν λ ν 5 Unknowns

Calculations with chiral 3N: SRG renormalization needed − 24 • Chiral N 3 LO NN plus N 2 LO NNN bare (36) 4 He potential SRG (2.0/28) − 25 – Bare interaction (black line) • Strong short-range correlations 3 LO (500 MeV) − 26 E [MeV] N § Large basis needed NN + NNN – SRG evolved effective − 27 interaction (red line) • Unitary transformation − 28 + ⇒ dH α ( ) " % α = 1 $ H α = U α HU α [ T , H α ] , H α d α = # λ 4 • Two- plus three -body − 29 2 4 6 8 10 12 14 16 18 20 22 components, four -body N max omitted • Softens the interaction § Smaller basis sufficient A =3 binding energy and half life constraint c D =-0.2, c E =-0.205, Λ =500 MeV

Similarity Renormalization Group (SRG) evolution • Continuous transformation driving Hamiltonian to band-diagonal form with respect to a chosen basis + = U α + U α = 1 + U α U α • Unitary transformation H α = U α HU α + + dH α d α = dU α + + U α H dU α d α = dU α dU α + + U α HU α + U α HU α + U α d α HU α d α U α d α + = dU α dU α η α ≡ dU α + = − η α + H α + H α U α [ ] d α U α d α = η α , H α + d α U α anti-Hermitian generator • Setting with Hermitian [ ] G α η α = G α , H α dH α ! # [ ] , H α G α , H α d α = " $ • Customary choice in nuclear physics … kinetic energy operator G α = T – band-diagonal in momentum space plane-wave basis λ 2 = 1/ α • Initial condition H α = 0 = H λ = ∞ = H 7

Light nuclei with SRG evolved interactions NN only NN + 3N-induced NN + 3N-full (a) (b) (c) -23 4 He -24 � Ω = 20 MeV E gs [MeV] -25 -26 -27 • Fast convergence -28 exp. • Significant 3N induced -29 . interaction -22 (d) (e) (f) • No 4N induced 6 Li interaction -24 � Ω = 20 MeV E gs [MeV] -26 -28 -30 -32 exp. . -34 2 4 6 8 10 12 14 ∞ 2 4 6 8 10 12 14 ∞ 2 4 6 8 10 12 14 ∞ N max N max N max � ● ★ � � α = 0 . 04 fm 4 α = 0 . 05 fm 4 α = 0 . 0625 fm 4 α = 0 . 08 fm 4 α = 0 . 16 fm 4 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

6 He half-life Precision measurement of 6 He beta decay PHYSICAL REVIEW C 86 , 035506 (2012) Precision measurement of the 6 He 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. Wrede 1, ‡ 1 Department of Physics and Center for Experimental Nuclear Physics and Astrophysics, University of Washington, 2.8 … challenge and test 6 He-> 6 Li 3 LO NN + N 2 LO NNN(500) N of ab initio calculations, 3 LO NN N + 0)| Expt nuclear forces 2.6 NCSM and currents -1 + 1 -> 1 SRG Λ =1.7 fm h Ω =16 MeV 2.4 |M(GT; 0 Improvement with the NNN interaction 2.2 MEC must be included Also: Operator renormalization 2 2 4 8 10 6 & continuum N max 9

6 He half-life Precision measurement of 6 He beta decay PHYSICAL REVIEW C 86 , 035506 (2012) Precision measurement of the 6 He 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. Wrede 1, ‡ 1 Department of Physics and Center for Experimental Nuclear Physics and Astrophysics, University of Washington, 2.8 … challenge and test 3 LO NN - 1b 6 He-> 6 Li N of ab initio calculations, 3 LO NN + N 2 LO 3N(500) - 1b N nuclear forces + 0)| 3 LO NN + N 2 LO 3N(500) - 1b+2b 2.6 N NCSM Expt and currents + 1 -> 1 -1 SRG Λ =1.7 fm h Ω =16 MeV 2.4 |M(GT; 0 Improvement with the NNN interaction 2.2 Improvement with MEC Also: Operator renormalization 2 2 4 6 8 10 & continuum N max 10

6 He half-life Precision measurement of 6 He beta decay PHYSICAL REVIEW C 86 , 035506 (2012) Precision measurement of the 6 He 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. Wrede 1, ‡ 1 Department of Physics and Center for Experimental Nuclear Physics and Astrophysics, University of Washington, 2.8 … challenge and test 3 LO NN - 1b 6 He-> 6 Li N of ab initio calculations, 3 LO NN + N 2 LO 3N(500) - 1b N nuclear forces + 0)| 3 LO NN + N 2 LO 3N(500) - 1b+2b 2.6 N NCSM Expt and currents + 1 -> 1 -1 SRG Λ =1.7 fm h Ω =16 MeV 2.4 |M(GT; 0 Improvement with the NNN interaction 2.2 Improvement with MEC Still to be done: Operator renormalization 2 2 4 6 8 10 & continuum N max 11

SRG evolution of transition operators PHYSICAL REVIEW C 92 , 014320 (2015) PHYSICAL REVIEW C 90 , 011301(R) (2014) Operator evolution for ab initio electric dipole transitions of 4 He Operator evolution for ab initio theory of light nuclei Micah D. Schuster, 1,* Sofia Quaglioni, 2, † Calvin W. Johnson, 1, ‡ Eric D. Jurgenson, 2 and Petr Navr´ Micah D. Schuster, 1,2 Sofia Quaglioni, 2 Calvin W. Johnson, 1 Eric D. Jurgenson, 2 and Petr Navr´ atil 3 atil 3 1 San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA 1 San Diego State University, 5500 Campanile Drive, San Diego, California 92182, USA f ˆ JT ˆ JT = ˆ ˆ ˆ i * ; ∑ O λ U λ O λ = ∞ U λ U λ = ψ α ( λ ) ψ α ( λ = ∞ ) α Final/initial unitary Eigenstates after transformations & before evolution Bare operator E1 E1 3 S 1 à 3 P 2 3 S 1 à 3 P 2 Induces 2-body (& higher-body) operators 3-body evolved Bare λ = 2 fm -1 operator