Similarity RG and Many-Body Operators

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1 Similarity RG and Many-Body Operators Department of Physics Ohio State University February, 2010 Collaborators: E. Anderson, S. Bogner, S. Glazek, E. Jurgenson, P. Navratil, R. Perry, L. Platter, A. Schwenk, K. Wendt

2 Outline Prelude Context 3NF Questions Prelude: Many-body operators from SRG Context: Nuclear physics and three-body forces Explicit running of three-body (and higher) interactions Open questions and issues

3 Outline Prelude Context 3NF Questions Flow Run 3NF Prelude: Many-body operators from SRG Context: Nuclear physics and three-body forces Explicit running of three-body (and higher) interactions Open questions and issues

4 Prelude Context 3NF Questions Flow Run 3NF Recap: SRG flow equations Transform an initial hamiltonian, H = T + V: H s = U s HU s T + V s, [arxiv: ] where s is the flow parameter. Differentiating wrt s: dh s ds = [η s, H s ] with η s du s ds U s = η s. η s is specified by the commutator with generator G s : η s = [G s, H s ], which yields the flow equation (T held fixed), dh s ds = dv s ds = [[G s, H s ], H s ]. G s determines flow = many choices (T, H D, H BD... )

5 Prelude Context 3NF Questions Flow Run 3NF Recap: SRG flow equations Implementation issues choosing G s choosing a basis Features evident from two-body system decoupling of low-energy from high-energy [arxiv: ] evolves toward universal low-k interactions in free space consistent evolution of operators What about applications to few- or many-body systems?

Prelude Context 3NF Questions Flow Run 3NF Flow in momentum basis with

dv s ds = [[T rel, V s ], H s ] with T rel k = ɛ k and λ 2 = 1/ s dv λ

(q, k ) V λ=2.5 (k, k ) 1st term 2nd term V λ=2.

6 Prelude Context 3NF Questions Flow Run 3NF Flow in momentum basis with η(s) = [T, H s ] For NN only, project onto relative momentum states k dv s ds = [[T rel, V s ], H s ] with T rel k = ɛ k and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k )+ q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k ) V λ=2.5 (k, k ) 1st term 2nd term V λ=2.0 (k, k ) First term drives V λ toward diagonal: V λ (k, k ) = V λ= (k, k ) e [(ɛ k ɛ k )/λ 2 ] 2 +

7 Prelude Context 3NF Questions Flow Run 3NF Variational Calculations in Three-Nucleon Systems Triton ground-state energy vs. size of harmonic oscillator basis ( ω excitations) Rapid convergence as λ decreases Different binding energies = 3-body contribution E t [MeV] MeV initial λ = 4 fm 1 λ = 3 fm 1 λ = 2 fm 1 λ = 1 fm 1 550/600 MeV

8 Prelude Context 3NF Questions Flow Run 3NF Running of E gs with two-body interaction (NN) only E gs [MeV] Not unitary for A 3 = ground-state energy depends on λ 3 H SRG NCSM N 3 LO (500 MeV) λ [fm 1 ] E gs [MeV] SRG NCSM 4 He N 3 LO (500 MeV) λ [fm 1 ] E gs [MeV] Li N 3 LO (500 MeV) SRG NCSM λ [fm 1 ] Same qualitative behavior (from NN repulsion/attraction) Gives running of net three body contribution Error bars are from extrapolated results (not converged)

9 Prelude Context 3NF Questions Flow Run 3NF Flow equations lead to many-body forces Schematically: dv s [[ ds = a a, ] a } a {{ aa }, ] a } a {{ aa } = + a } a {{ a aaa } + 2-body 2-body 3-body! so there will be A-body forces generated Is this a problem? Normal ordering and truncation at NN may be sufficient [Achim] Ok if induced many-body forces are same size as natural ones Nuclear 3-body forces already needed in unevolved potential In fact, there are A-body forces initially Natural hierarchy from chiral EFT = stop flow equations before unnatural 3-body size Many-body methods must deal with them! SRG is a tractable method to evolve many-body operators

10 Outline Prelude Context 3NF Questions GFMC 3NF χeft NM Prelude: Many-body operators from SRG Context: Nuclear physics and three-body forces Explicit running of three-body (and higher) interactions Open questions and issues

Prelude Context 3NF Questions GFMC 3NF χeft NM Degrees of

A>100 Coupled Cluster, Shell Model A<100 Exact methods A

interactions Chiral EFT interactions (low-energy theory

11 Prelude Context 3NF Questions GFMC 3NF χeft NM Degrees of Freedom: From QCD to Nuclei Density Functional Theory A>100 Coupled Cluster, Shell Model A<100 Exact methods A 12 GFMC, NCSM Lattice QCD Low-mom. interactions Chiral EFT interactions (low-energy theory of QCD) QCD Lagrangian RG = make nuclear structure look more like quantum chemistry

12 Prelude Context 3NF Questions GFMC 3NF χeft NM Light nuclei: Pieper/Wiringa (Bonner Prize!) Three-body forces needed for energies, splittings,...

13 Prelude Context 3NF Questions GFMC 3NF χeft NM Atomic 3-body forces: Axilrod-Teller term (1943) Three-body potential for atoms/molecules from triple-dipole mutual polarization (3rd-order perturbation correction) V (i, j, k) = ν(1 + 3 cos θ i cos θ j cos θ k ) (r ij r ik r jk ) 3 Usually negligible in metals and semiconductors Can be important for ground-state energy of solids bound by van der Waals potentials Bell and Zuker (1976): 10% of energy in solid xenon

14 Prelude Context 3NF Questions GFMC 3NF χeft NM Observations on three-body forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on Λ/λ e.g., Tjon line E b ( 4 He) [MeV] π, ρ, ω, N π, ρ, ω π, ρ, ω A=3,4 binding energies SRG NN only, λ in fm 1 λ=1.25 N λ=3.0 π, ρ, ω λ=1.75 λ=1.5 λ=2.0 λ=2.25 Expt. λ= λ=1.0 N 3 LO NN potentials SRG N 3 LO (500 MeV) E b ( 3 H) [MeV]

15 Prelude Context 3NF Questions GFMC 3NF χeft NM Observations on three-body forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on Λ/λ e.g., Tjon line E gs [MeV] π π π c 1, c 3, c 4 c D c E SRG NCSM 6 Li N 3 LO (500 MeV) NNN at different Λ/λ can be fit to χeft or evolved how large is 4-body? λ [fm 1 ]

16 Prelude Context 3NF Questions GFMC 3NF χeft NM Observations on three-body forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on Λ/λ e.g., Tjon line NNN at different Λ/λ can be fit to χeft or evolved how large is 4-body? saturation of nuclear matter cutoff dependence as tool Energy/nucleon [MeV] π π π c 1, c 3, c 4 c D c E V low k NN from N 3 LO (500 MeV) 3NF fit to E 3 H and r 4He Λ = 2.0 fm 1 3NF pp ladders Λ = 1.8 fm 1 Λ = 2.8 fm 1 Λ = 1.8 fm 1 NN only Λ = 2.8 fm 1 NN only NN + 3N NN only k F [fm 1 ]

17 Prelude Context 3NF Questions GFMC 3NF χeft NM Chiral EFT hierarchy Power counting still unsettled But many-body hierarchy consistent with calculations in few-body systems 3-body at N 2 LO without and NLO with 4-body at N 3 LO

18 Prelude Context 3NF Questions GFMC 3NF χeft NM Nuclear matter with RG-evolved NN plus fit NNN π, ρ, ω, N π, ρ, ω π, ρ, ω N π, ρ, ω = π π π Energy/nucleon [MeV] Hartree-Fock Empirical saturation point k F [fm 1 ] V low k NN from N 3 LO (500 MeV) 3NF fit to E 3H and r 4He 2nd order k F [fm 1 ] c 1, c 3, c 4 c D c E pp ladders Λ = 1.8 fm 1 Λ = 2.0 fm 1 Λ = 2.2 fm 1 Λ = 2.8 fm < Λ 3NF < 2.5 fm k F [fm 1 ] At low resolution, nuclear saturation driven by NNN Can we validate use of the chiral EFT operator basis?

19 Outline Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Prelude: Many-body operators from SRG Context: Nuclear physics and three-body forces Explicit running of three-body (and higher) interactions Open questions and issues

20 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis SRG with normal-ordering in the vacuum SRG flow equation dhs ds = [[G s, H s ], H s ], e.g., G s = T rel Right side evaluated w/o solving bound-state or scattering eqs. Can be applied directly in three-particle space A-body operators completely fixed in A-particle subspace What about spectator nucleons? Decoupling of 3N part in momentum space dv s ds = dv 12 ds + dv 13 ds + dv 23 ds + dv 123 = [[T rel, V s], H s], ds = dv 123 ds = [[T 12, V 12 ], (T 3 + V 13 + V 23 + V 123 )] + { } + { } + [[T rel, V 123 ], H s] No multi-valued two-body interactions (dependence on excitation energy of unlinked spectators) Or, direct solution in discrete harmonic oscillator basis

21 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Diagrams for SRG = Disconnected cancels V (2) s = [T, V (2) s ] = [[T, V (2) s ], T ] = V (3) s = [T, V (3) s ] = [[T, V (3) s ], T ] = dv s (2) (a, b) = a b + a c b a c b ds (ɛ a ɛ b ) 2 V (2) s (a, b) c [(ɛ a ɛ c ) (ɛ c ɛ b )] V (2) s (a, c) V (2) s (c, b) dv (3) s ds =

2 ) V (N 3, N 3 ) Diagonalize symmetrizer N A N A 1 ; n A 1 ; use recursively

22 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Embedding: Initial potential Symmetrized Jacobi oscillator basis (here: 1D bosons) V (p, p ) V (N 2, N 2 ) V (N 3, N 3 ) Diagonalize symmetrizer N A N A 1 ; n A 1 ; use recursively Embedding is everything, SRG coding is trivial 3D: Use Navratil et al. technology for NCSM

Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Embedding: SRG evolved potential at λ = 2 Symmetrized Jacobi oscillator basis (here: bosons) V (p, p ) V (N

23 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Embedding: SRG evolved potential at λ = 2 Symmetrized Jacobi oscillator basis (here: bosons) V (p, p ) V (N 2, N 2 ) V (N 3, N 3 ) Diagonalize symmetrizer N A N A 1 ; n A 1 ; use recursively Embedding is everything, SRG coding is trivial 3D: Use Navratil et al. technology for NCSM

24 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Few-Body Embedding Legend: Embedding, Evolving, BE calculation, Initial 3NF A=3 (2N only): V (2) SRG osc = V (2) embed λ,osc = V (2) diag λ,3nosc = BE (2Nonly) 3 A=4 (2N only): V (2) SRG osc = V (2) embed λ,osc = V (2) embed λ,3nosc = V (2) diag λ,4nosc = BE (2Nonly) 4 A=4 (2N+3N only): V (2) embed osc = V (2) SRG 3Nosc = V (2+3) embed λ,3nosc = V (2+3) diag λ,4nosc = BE (2N+3Nonly) 4 3NF = +V (3init) 3Nosc...

25 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis 3D SRG Evolution with T rel in a Jacobi HO Basis Evolve in any basis [momentum space in progress by L. Platter] Here: use anti-symmetric Jacobi HO basis from NCSM directly obtain SRG matrix elements in HO basis separate 3-body evolution not needed Compare 2-body only to full body evolution: Ground-State Energy [MeV] H NN-only N 3 LO (500 MeV) Ground-State Energy [MeV] He N 3 LO (500 MeV) NN-only Expt. 8.6 Expt λ [fm 1 ] λ [fm 1 ]

26 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis 3D SRG Evolution with T rel in a Jacobi HO Basis Evolve in any basis [momentum space in progress by L. Platter] Here: use anti-symmetric Jacobi HO basis from NCSM directly obtain SRG matrix elements in HO basis separate 3-body evolution not needed Compare 2-body only to full body evolution: Ground-State Energy [MeV] H NN-only NN + NNN-induced N 3 LO (500 MeV) Ground-State Energy [MeV] He N 3 LO (500 MeV) NN-only NN+NNN-induced Expt. 8.6 Expt λ [fm 1 ] λ [fm 1 ]

27 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis 3D SRG Evolution with T rel in a Jacobi HO Basis Evolve in any basis [momentum space in progress by L. Platter] Here: use anti-symmetric Jacobi HO basis from NCSM directly obtain SRG matrix elements in HO basis separate 3-body evolution not needed Compare 2-body only to full body evolution: Ground-State Energy [MeV] H NN-only NN + NNN-induced NN + NNN Ground-State Energy [MeV] He N 3 LO (500 MeV) NN-only NN+NNN-induced +NNN-initial Expt. 8.6 Expt λ [fm 1 ] λ [fm 1 ]

28 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis 3D SRG Evolution with T rel in a Jacobi HO Basis Good convergence properties independent of 3-body: E [MeV] N 3 LO (500 MeV) H bare (28) SRG (2.0/20) NN (+ NNNinduced) E [MeV] He bare (36) SRG (2.0/36) SRG (2.0/28) N 3 LO (500 MeV) NN + NNN NN + NNN HO matrix elements (to be) available for NCFC, CC,... Challenge: efficient (on-the-fly) conversion to m-scheme

29 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis 3D SRG Evolution with T rel in a Jacobi HO Basis Good convergence properties independent of 3-body: E [MeV] N 3 LO (500 MeV) H bare (28) L-S (28) SRG (2.0/20) NN (+ NNNinduced) E [MeV] He bare (36) LS (36) LS (28) SRG (2.0/36) SRG (2.0/28) N 3 LO (500 MeV) NN + NNN NN + NNN HO matrix elements (to be) available for NCFC, CC,... Challenge: efficient (on-the-fly) conversion to m-scheme

30 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Tjon line revisited E b ( 4 He) [MeV] Tjon line for NN-only potentials SRG NN-only SRG NN+NNN (λ >1.7 fm 1 ) λ=1.2 N 3 LO (500 MeV) λ=1.5 λ=3.0 λ=1.8 λ= λ=2.0 Expt E b ( 3 H) [MeV]

C 39 1076 (1989)]: V (2) (x) = V 1 e x 2 /σ 2 V 1 + 2 e x 2 /σ 2 2 σ 1 π σ 2 π λ =

31 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Explore Using a One-Dimensional Model 1-D model [Negele et al.: Phys.Rev.C (1989)]: V (2) (x) = V 1 e x 2 /σ 2 V e x 2 /σ 2 2 σ 1 π σ 2 π λ = λ = 5 λ = 3 λ = 2 Same features as in 3D, but much easier! See E. Jurgenson, rjf, arxiv: for details

32 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Induced Many-Body Forces: A = 3 Ground-State Energy V (2) (x) = V 1 σ 1 π e x 2 /σ V 2 σ 2 π e x 2 /σ 2 2 A = 3 = 28 2-body only V (2) (λ= ) = V α c E = λ Ground-State Energy A = 3 = 28 V α σ 1 =.05 c E = body only V 2 = λ Basis independent: same evolution in k or HO basis Black: Same evolution pattern for 2-body-only as 3D NN-only

33 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Induced Many-Body Forces: A = 3 Ground-State Energy V (2) (x) = V 1 σ 1 π e x 2 /σ V 2 σ 2 π e x 2 /σ 2 2 A = 3 = 28 2-body only 2+3-body V (2) (λ= ) = V α c E = λ Ground-State Energy A = 3 = 28 V α σ 1 =.05 c E = body only 2+3-body V 2 = λ Basis independent: same evolution in k or HO basis Black: Same evolution pattern for 2-body-only as 3D NN-only Red: Includes induced 3NF - Unitary!

34 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Induced Many-Body Forces: A = 3 Ground-State Energy V (2) (x) = V 1 σ 1 π e x 2 /σ 2 1 A = 3 = 28 2-body only V (2) (λ= ) = V β c E = λ Ground-State Energy attractive only A = 3 = 28 V 2 = 1 σ 2 = 1.2 V β c E = 0 2-body only λ Basis independent: same evolution in k or HO basis Black: Evolution pattern for attractive 2-body-only

35 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Induced Many-Body Forces: A = 3 Ground-State Energy V (2) (x) = V 1 σ 1 π e x 2 /σ 2 1 A = 3 = 28 2-body only 2+3-body V (2) (λ= ) = V β c E = λ Ground-State Energy attractive only A = 3 = 28 V 2 = 1 σ 2 = 1.2 V β c E = 0 2-body only 2+3-body λ Basis independent: same evolution in k or HO basis Black: Evolution pattern for attractive 2-body-only Red: Includes induced 3NF - Unitary!

36 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Induced Many-Body Forces: A = V (3) (p, q, p, q ) = c E e ((p 2 +q 2 )/Λ 2 ) n e ((p2 +q 2 )/Λ 2 ) n (Λ = 2 n = 4) 4 A = 4 = 28 2-body only Ground-State Energy V (2) (λ= ) = V α λ c E = 0.05 c E = λ c E = λ

37 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Induced Many-Body Forces: A = V (3) (p, q, p, q ) = c E e ((p 2 +q 2 )/Λ 2 ) n e ((p2 +q 2 )/Λ 2 ) n (Λ = 2 n = 4) Ground-State Energy A = 4 = 28 V (2) (λ= ) = V α body 2+3-body only 2-body only λ c E = 0.05 c E = λ c E = λ

38 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis Induced Many-Body Forces: A = 5 Ground-State Energy A = 3 = 28 V α c E = 0 2-body only 2+3-body λ Ground-State Energy A = 4 = 28 V α c E = body 2+3-body only 2-body only λ Ground-State Energy A = 5 = 28 V α c E = 0 2-body only 2+3-body only body only body λ Five-body force is negligible Hierarchy of induced many-body forces

39 V (3) analysis Ground-State Energy Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis d dλ ψ(3) (3) λ V λ ψ(3) λ λ = ψ(3) (2) λ [V λ, V (2) λ ] c [V (3) A = 3 = 28 2-body only 2+3-body V (2) (λ= ) = V α c E = 0 Ground-State Expectation Value λ, V (3) λ ] ψ(3) λ d V (3) /dλ [V (2),V (2) ] c [V (3),V (3) ] A = 3 = 28 V α c E = λ Majority evolution dominated by [V (2), V (2) ], (V [T, V ]) Hierarchy of contributions >

40 Prelude Context 3NF Questions Diagrams Embed 3D Tjon 1D Analysis V (4) analysis in A = 4 d dλ ψ(4) (4) λ V λ ψ(4) λ = ψ(4) λ [V (2) λ, V (3) λ ] c + [V (3) +[V (3) λ, V (3) λ ] c [V (4) λ, V (2) λ λ, V (4) ] ψ(4) λ λ ] c Ground-State Energy A = 4 = 28 V α c E = body 2+3-body only 2-body only Ground-State Expectation Value dv (4) /dλ - [η,v (4) ] d V (4) /dλ [V (2),V (3) ] c [V (3),V (2) ] c [V (3),V (3) ] c [V (4),V (4) ] c A = 4 = 28 V c α E = λ λ No [V (2), V (2) ] = Induced 4-body is small! Initial hierarchy of few-body forces is maintained

41 Outline Prelude Context 3NF Questions Universality Ops Factorization HO Summary Prelude: Many-body operators from SRG Context: Nuclear physics and three-body forces Explicit running of three-body (and higher) interactions Open questions and issues

42 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Run to Lower λ via SRG = Universal λ = 5.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 5.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] Will evolved NNN interactions be universal?

43 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Run to Lower λ via SRG = Universal λ = 4.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 4.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] Will evolved NNN interactions be universal?

44 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Run to Lower λ via SRG = Universal λ = 3.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 3.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] Will evolved NNN interactions be universal?

45 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Run to Lower λ via SRG = Universal λ = 2.5 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 2.5 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] Will evolved NNN interactions be universal?

46 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Run to Lower λ via SRG = Universal λ = 2.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 2.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] Will evolved NNN interactions be universal?

47 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Run to Lower λ via SRG = Universal λ = 1.5 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 1.5 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] Will evolved NNN interactions be universal?

48 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Every operator flows Evolution with s of any operator O is given by: O s = U s OU s so O s evolves via do s ds = [[G s, H s ], O s ] U s = i ψ(0) i ψ(s) i Matrix elements of evolved operators are unchanged Consider momentum distribution < ψ d a qa q ψ d > at q = 4.5 fm 1 4π [u(q) 2 + w(q) 2 ] [fm 3 ] q [fm 1 ] AV18 V s at λ = 2 fm 1 V s at λ = 1.5 fm 1 CD-Bonn

49 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Integrand of < ψ d Ua qa q U ψ d > at q = 4.5 fm 1 Flow of deuteron matrix element integrand is toward low k Simple variational ansatz works well = No fine-tuning Factorization: U(k, q) K (k)q(q) for k λ, q λ

Prelude Context 3NF Questions Universality Ops Factorization HO Summary High and low momentum operators in deuteron Integrand of ψ d (Ua qa q U ) ψ d for q = 0.

50 Prelude Context 3NF Questions Universality Ops Factorization HO Summary High and low momentum operators in deuteron Integrand of ψ d (Ua qa q U ) ψ d for q = 0.34 fm 1 Momentum distribution a q a qdeuteron Integrand for q = 3.02 fm 1 4π [u(q) 2 + w(q) 2 ] [fm 3 ] N 3 LO unevolved V s at λ = 2.0 fm 1 V s at λ = 1.5 fm q [fm 1 ] Decoupling = High momentum components suppressed Integrated value does not change, but nature of operator does Similar for other operators: r 2, Q d, 1/r

51 Prelude Context 3NF Questions Universality Ops Factorization HO Summary High and low momentum operators in deuteron Integrand of ψ d (Ua qa q U ) ψ d for q = 0.34 fm 1 Momentum distribution a q a qdeuteron Integrand for q = 3.02 fm 1 4π [u(q) 2 + w(q) 2 ] [fm 3 ] N 3 LO unevolved V s at λ = 2.0 fm 1 V s at λ = 1.5 fm q [fm 1 ] Decoupling = High momentum components suppressed Integrated value does not change, but nature of operator does Similar for other operators: r 2, Q d, 1/r

52 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Decoupling in operator expectation values Evolve to λ in full space TRUNCATE at Λ = 2.5 fm 1 : a + a 10 0 Evolved to λ=6.0, Truncated at Λ=2.5 Unevolved SRG Evolved λ = 6 fm q (fm 1 ) Here: momentum distribution a qa q Decoupling for all q works when λ < Λ

53 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Decoupling in operator expectation values Evolve to λ in full space TRUNCATE at Λ = 2.5 fm 1 : a + a 10 0 Evolved to λ=3.0, Truncated at Λ=2.5 Unevolved SRG Evolved λ = 3 fm q (fm 1 ) Here: momentum distribution a qa q Decoupling for all q works when λ < Λ

54 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Decoupling in operator expectation values Evolve to λ in full space TRUNCATE at Λ = 2.5 fm 1 : a + a 10 0 Evolved to λ=2.0, Truncated at Λ=2.5 Unevolved SRG Evolved λ = 2 fm q (fm 1 ) Here: momentum distribution a qa q Decoupling for all q works when λ < Λ

55 Factorization Prelude Context 3NF Questions Universality Ops Factorization HO Summary If k < λ and q λ = factorization: U λ (k, q) K λ (k)q λ (q)? Operator product expansion for nonrelativistic wf s (Lepage) Z Z Ψ true(r) = γ(r) dr Ψ eff δ a(r ) + n(r)a 2 dr Ψ eff 2 δ a(r ) + O(a 4 ) Similarly, in momentum space Ψ α (q) γ λ (q) Z λ 0 p 2 dp Z (λ)ψ λ α(p) + η λ (q) Z λ 0 p 2 dp p 2 Z (λ) Ψ λ α(p) + By projecting potential in momentum subspace, recover OPE via: γ λ (q) η λ (q) Z λ Z λ q 2 dq q q 2 dq q 1 bq λ H b Q λ q V (q, 0) 1 bq λ H b Q λ q 2 p 2 V (q, p) p 2 =0 Construct unitary transformation to get U λ (k, q) K λ (k)q λ (q) U λ (k, q) = X hx α low Z λ i k ψα ψ λ α q k ψα λ p 2 dp Z (λ)ψ λ α(p) γ λ (q) + α α 0

56 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Numerical Factorization Test of factorization of U: U λ (k i, q) U λ (k 0, q) K λ(k i )Q λ (q) K λ (k 0 )Q λ (q), so for q λ K λ(k i ) K λ (k 0 ). Look for plateaus for q 2fm 1 Singular value decomposition quantitatively analyze the extent to which U factorizes outer product expansion rx G = d i u i v t i where r is the rank and the d i are decreasing singular values Example: results for λ = 2 fm 1, for q > λ and k < λ 1 S0 Potential d 1 d 2 d 3 AV N3LO 500 MeV N3LO 550/600 MeV S1 3 S 1 AV N3LO 500 MeV N3LO 550/600 MeV i

Prelude Context 3NF Questions Universality Ops Factorization HO Summary Practical use of factorization Decoupling implies ψ λ U λ b O U λ ψ λ = Z λ 0 Z Z dk dq dq 0 0 Z λ 0 dk ψ λ (k )U λ (k, q ) b

57 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Practical use of factorization Decoupling implies ψ λ U λ b O U λ ψ λ = Z λ 0 Z Z dk dq dq 0 0 Z λ 0 dk ψ λ (k )U λ (k, q ) b O(q, q)u λ (q, k)ψ λ(k) Factorization: set U λ (k, q) K λ (k)q λ (q), where k < λ and q λ Z λ Z λ»z λ Z λ = ψ λ (k ) U λ (k, q ) O(q b, q)u λ (q, k) {z } +I QOQ K λ (k )K λ (k) ψ λ (k) {z } Low Momentum Structure where I QOQ ] λ dq [Q λ dq λ (q )Ô(q, q)q λ (q) Universal Valid when initial operators weakly couple high and low momentum: r 2 1/r G C (q = 3.02 fm 1 ) a qa q(q = 3.02 fm 1 )

58 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Factorization in few-body nuclei: n(k) at large k AV14 NN with VMC A bosons in 1D model From Pieper, Wiringa, and Pandharipande (1992). Conventional explanation: Dominance of NN potential and short-range correlations (Frankfurt et al.) Alternative: factorization λ λ 0 0 ψ λ (k ) [I QOQ K λ (k )K λ (k)] ψ λ (k)

59 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Factorization in few-body nuclei: n(k) at large k AV14 NN with VMC A bosons in 1D model From Pieper, Wiringa, and Pandharipande (1992). Conventional explanation: Dominance of NN potential and short-range correlations (Frankfurt et al.) Alternative: factorization λ λ 0 0 ψ λ (k ) [I QOQ K λ (k )K λ (k)] ψ λ (k)

60 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Long-distance observables: radius H r [fm] h- ω = 28 NN-only NN bare NN LS SRG λ = 100 fm 1 SRG λ = 5 fm 1 SRG λ = 3 fm 1 SRG λ = 2 fm Unevolved operator Harmonic oscillator basis is problematic!

61 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving NN forces in NCSM A=3 space 1 1 Ground-State Energy [MeV] H NN-only SRG-Trel λ = h- ω = 28 = 30 Ground-State Energy [MeV] H NN-only SRG-Trel λ = h- ω = 20 = cut cut ω = 28 is optimal for the bare interaction ω = 20 is optimal for λ = 2 evolution No improvement in convergence for small λ

62 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving NN forces in NCSM A=3 space 1 1 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 h- ω = 28 = 30 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 h- ω = 20 = cut cut ω = 28 is optimal for the bare interaction ω = 20 is optimal for λ = 2 evolution No improvement in convergence for small λ

63 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving NN forces in NCSM A=3 space 1 1 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 h- ω = 28 = 30 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 h- ω = 20 = cut cut ω = 28 is optimal for the bare interaction ω = 20 is optimal for λ = 2 evolution No improvement in convergence for small λ

64 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving NN forces in NCSM A=3 space 1 1 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 h- ω = 28 = 30 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 h- ω = 20 = cut cut ω = 28 is optimal for the bare interaction ω = 20 is optimal for λ = 2 evolution No improvement in convergence for small λ

65 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving NN forces in NCSM A=3 space 1 1 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 h- ω = 28 = 30 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 h- ω = 20 = cut cut ω = 28 is optimal for the bare interaction ω = 20 is optimal for λ = 2 evolution No improvement in convergence for small λ

66 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving NN forces in NCSM A=3 space 1 1 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 λ = 1.0 h- ω = 28 = 30 Ground-State Energy [MeV] H NN-only SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 λ = 1.0 h- ω = 20 = cut cut ω = 28 is optimal for the bare interaction ω = 20 is optimal for λ = 2 evolution No improvement in convergence for small λ

67 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving NN + NNN in NCSM A=3 space 1 1 Ground-State Energy H NNN SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 λ = 1.0 h- ω = 28 = 30 Ground-State Energy H NNN SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 λ = 1.0 h- ω = 20 = cut cut Same plots but now including an initial 3NF from N2LO No improvement in convergence for small λ

basis: In this basis T rel will not drive H to diagonal form But

68 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Using other SRG Generators 1D matrices T rel and V in NCSM basis: In this basis T rel will not drive H to diagonal form But harmonic oscillator Hamiltonian will! H ho = T rel + V ho is diagonal in this basis

69 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Evolving with H ho in HO Basis Using G = H ho improves convergence dramatically 1 1 Ground-State Energy H NNN SRG-Trel λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 λ = 1.0 h- ω = 28 = 30 Ground-State Energy H NNN SRG-Hho λ = λ = 3.0 λ = 2.0 λ = 1.5 λ = 1.2 λ = 1.0 h- ω = 28 = cut cut Compare T rel on the left with H ho on the right But: 1D study indicates spurious bound states contaminate evolution with H ho with many-body truncation

70 Prelude Context 3NF Questions Universality Ops Factorization HO Summary Summary of open questions and issues Power counting for evolved many-body interactions Do many-body interactions flow to universal form? Operator issues Scaling of many-body operators Factorization for few-body systems Can choices for G s... reduce the many-body forces? improve convergence in HO basis?

Nuclear Observables at Low Resolution

Nuclear Observables at Low Resolution Eric R. Anderson Department of Physics The Ohio State University September 13, 2011 In Collaboration with: S.K Bogner, R.J. Furnstahl, K. Hebeler, E.D. Jurgenson,