Nuclear Observables at Low Resolution

Transcription

1 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, P. Navrátil, R.J. Perry, & K.A. Wendt Work supported by NSF and UNEDF/SciDAC (DOE)

2 Outline 1 Overview Resolution SRG 2 Operator Evolution Properties Many-Body 3 Factorization Short-Range Correlations Principles 4 Conclusions Overview Operators Factorization Conclusions

3 Outline 1 Overview Resolution SRG 2 Operator Evolution Properties Many-Body 3 Factorization Short-Range Correlations Principles 4 Conclusions Overview Operators Factorization Conclusions Resolution SRG

4 Resolution Scales of Nuclear Physics Overview Operators Factorization Conclusions Resolution SRG

5 Low resolution makes physics easier + efficient Weinberg s Third Law of Progress in Theoretical Physics: You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you ll be sorry! Overview Operators Factorization Conclusions Resolution SRG

6 Low resolution makes physics easier + efficient Weinberg s Third Law of Progress in Theoretical Physics: You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you ll be sorry! There s an old joke about a doctor and patient... Patient: Doctor, doctor, it hurts when I do this! Overview Operators Factorization Conclusions Resolution SRG

7 Low resolution makes physics easier + efficient Weinberg s Third Law of Progress in Theoretical Physics: You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you ll be sorry! There s an old joke about a doctor and patient... Patient: Doctor, doctor, it hurts when I do this! Doctor: Then don t do that. Overview Operators Factorization Conclusions Resolution SRG

8 Resolution Scale of Theory Overview Operators Factorization Conclusions Resolution SRG

9 Resolution Scale of Theory Overview Operators Factorization Conclusions Resolution SRG

10 Resolution Scale of Theory Overview Operators Factorization Conclusions Resolution SRG

11 Resolution Scale of Theory Claim: Nuclear physics with textbook V (r) is like using beer coasters! Overview Operators Factorization Conclusions Resolution SRG

12 Textbook Nuclear Interaction Hard core makes calculations difficult Overview Operators Factorization Conclusions Resolution SRG

13 Less painful to use a low-resolution version! High resolution Low resolution Can greatly reduce storage without distorting message Lower resolution here by block spinning Alternative: apply a low-pass filter Overview Operators Factorization Conclusions Resolution SRG

14 Low-pass filter on an image Much less information needed Long-wavelength info is preserved Overview Operators Factorization Conclusions Resolution SRG

15 The Nuclear Interaction High momentum matrix elements lead to computationally infeasible many-body problem for all but lightest nuclei Overview Operators Factorization Conclusions Resolution SRG

16 The Nuclear Interaction High momentum matrix elements lead to computationally infeasible many-body problem for all but lightest nuclei Overview Operators Factorization Conclusions Resolution SRG

17 Growth of Configuration interaction matrices Overview Operators Factorization Conclusions Resolution SRG

18 Effect of low-pass filter on observables Try setting high momentum (k 2 fm 1 ) matrix elements to zero and see the effect on low-energy observables 60 1 S0 k = 2 fm 1 phase shift (degrees) AV18 phase shifts E lab (MeV) Overview Operators Factorization Conclusions Resolution SRG

19 Effect of low-pass filter on observables Try setting high momentum (k 2 fm 1 ) matrix elements to zero and see the effect on low-energy observables 60 1 S0 k = 2 fm 1 phase shift (degrees) AV18 phase shifts after low-pass filter E lab (MeV) Overview Operators Factorization Conclusions Resolution SRG

20 Effect of low-pass filter on observables Try setting high momentum (k 2 fm 1 ) matrix elements to zero and see the effect on low-energy observables 60 1 S0 k = 2 fm 1 phase shift (degrees) Overview Operators Factorization Conclusions Resolution SRG AV18 phase shifts after low-pass filter E lab (MeV) Problem: Coupling of high and low momentum

21 Introduction to the SRG: Potentials The Similarity Renormalization Group (SRG) provides a means to systematically evolve computationally difficult Hamiltonians toward diagonal or decoupled form simplifies calculations with nuclear potentials Based on unitary transformations as shown here: H s = U s H s=0 U s T rel + V s Differentiating with respect to s gives the flow equation: dh s ds = [η s, H s ] where η s = du s ds U s = η s The flow can be specified in η s by a flow operator G s : Typically: η s = [G s, H s ] G s = T rel = dh s ds = [[T rel, H s ], H s ] Overview Operators Factorization Conclusions Resolution SRG

22 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

23 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

24 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

25 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

26 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

27 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

28 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

29 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

30 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

31 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

32 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

33 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

34 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

35 Flow equations in action: NN only In each partial wave with ɛ k = 2 k 2 /M and λ = 1/s 1/4 fm 1 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 ) Overview Operators Factorization Conclusions Resolution SRG

36 Low-pass filters work! [Jurgenson et al. (2008)] Phase shifts with V s (k, k ) = 0 for k, k > k max phase shift [deg] S0 3 S P0 3 F E lab [MeV] D AV18 AV18 [k max = 2.2 fm -1 ] V s [k max = 2.2 fm -1 ] Overview Operators Factorization Conclusions Resolution SRG

37 NCSM Calculations for Light Nuclei Ground State Energy [MeV] Harmonic oscillator basis with N max shells for excitations Graphs show convergence for soft chiral EFT potential and evolved SRG potentials (including NNN) Softened with SRG Helium-4 ground-state energy V NN = N 3 LO (500 MeV) V NNN = N 2 LO Original expt Matrix Size [N max ] Ground-State Energy [MeV] Original Lithium-6 Softened with SRG λ = 2.0 fm 1 [E. Jurgenson, P. Navrátil, R.J. Furnstahl (PRL, 2011)] ground-state energy V NN = N 3 LO (500 MeV) V NNN = N 2 LO h- Ω = 20 MeV λ = 1.5 fm 1 expt Matrix Size [N max ] Better convergence, but rapid growth of basis still a problem (solution: importance sampling of matrix elements [R. Roth]) Overview Operators Factorization Conclusions Resolution SRG

38 Evolution in coordinate space Non-diagonal ( nonlocal ) potential, so how to plot? Calculate integral over off-diagonal LP(r) = dr V (r, r ) If local, then equal to V (r) Core meltdown! AV18 1 S 0 Potential Overview Operators Factorization Conclusions Resolution SRG

39 Three-body Forces, etc. Classically: Effective Field Theory: Three-body forces arise from eliminating/decoupling dof s excited states of nucleon relativistic effects high-momentum intermediate states π, ρ, ω Δ, N π, ρ, ω N π, ρ, ω π, ρ, ω Force between earth and moon depends on position of sun Nucleons are composite objects too... Overview Operators Factorization Conclusions Resolution SRG π π π c1, c3, c4 cd ce How does SRG affect many-body forces?

40 Hierarchy of Many-Body Effects with T rel SRG Evolution Consider a s and a s wrt s.p. basis and reference state: dv s [[ ] = a a ds }{{}, } a a {{ aa }, ] } a a {{ aa } = + } a a {{ a aaa } + G s 2-body 2-body 3-body! so there will be A-body forces (and operators) generated Compare 2-body only to full body evolution: [E. Jurgenson, P. Navrátil, R.J. Furnstahl (PRL, 2011)] 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 ] Overview Operators Factorization Conclusions Resolution SRG λ [fm 1 ]

41 Hierarchy of Many-Body Effects with T rel SRG Evolution Consider a s and a s wrt s.p. basis and reference state: dv s [[ ] = a a ds }{{}, } a a {{ aa }, ] } a a {{ aa } = + } a a {{ a aaa } + G s 2-body 2-body 3-body! so there will be A-body forces (and operators) generated Compare 2-body only to full body evolution: [E. Jurgenson, P. Navrátil, R.J. Furnstahl (PRL, 2011)] 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 ] Overview Operators Factorization Conclusions Resolution SRG λ [fm 1 ]

42 Hierarchy of Many-Body Effects with T rel SRG Evolution Consider a s and a s wrt s.p. basis and reference state: dv s [[ ] = a a ds }{{}, } a a {{ aa }, ] } a a {{ aa } = + } a a {{ a aaa } + G s 2-body 2-body 3-body! so there will be A-body forces (and operators) generated Compare 2-body only to full body evolution: [E. Jurgenson, P. Navrátil, R.J. Furnstahl (PRL, 2011)] 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 ] Overview Operators Factorization Conclusions Resolution SRG λ [fm 1 ]

43 Hierarchy of Many-Body Effects with T rel SRG Evolution Consider a s and a s wrt s.p. basis and reference state: dv s [[ ] = a a ds }{{}, } a a {{ aa }, ] } a a {{ aa } = + } a a {{ a aaa } + G s 2-body 2-body 3-body! so there will be A-body forces (and operators) generated Compare 2-body only to full body evolution: [E. Jurgenson, P. Navrátil, R.J. Furnstahl (PRL, 2011)] g.s. Expectation Value (MeV) H N max = 18 N max -A3 = NN+NNN h- ω = 28 <Trel> <V NN > <V 3N > λ g.s. Expectation Value (MeV) He N max = 18 N max -A3 = NN+NNN h- ω = 28 <T rel > <V NN > <V 3N > <V 4N > λ λ 100 Overview Operators Factorization Conclusions Resolution SRG λ

44 Outline 1 Overview Resolution SRG 2 Operator Evolution Properties Many-Body 3 Factorization Short-Range Correlations Principles 4 Conclusions Overview Operators Factorization Conclusions Properties Many-Body

45 Introduction to the SRG: Operator Evolution Same unitary transformation evolves potentials and operators! O s = U s O s=0 U s do s ds = [η s, O s ] = [[T rel, H s ], O s ] Can construct transformation directly... U s = ψ α(s) ψ α(0) α U s(k i, k j ) = k i ψ α(s) ψ α(0) k j α or, e.g., evolve from DE do s(k, k ) = ds 2 q 2 dq [(k 2 q 2 )V s(k, q)o s(q, k ) π 0 + (k 2 q 2 )O s(k, q)v s(q, k )] Evolve operators consistent with N3LO 500MeV potential Number Operator, RMS radius, Quadrupole moment, 1 r, etc. Observables in deuteron [ERA, S.K. Bogner, R.J. Furnstahl, R.J. Perry, (PRC, 2010)] Overview Operators Factorization Conclusions Properties Many-Body

46 Deuteron Momentum Distribution π [u(q) 2 + w(q) 2 ] [fm 3 ] a q a qdeuteron N 3 LO unevolved V s at λ = 2.0 fm 1 V s at λ = 1.5 fm q [fm 1 ] Overview Operators Factorization Conclusions Properties Many-Body

47 Number Operator - High Momentum In partial wave momentum basis (for q = 3.02 fm 1 ): Operator: k U λ a qa q U λ k Integrand: ψ λ d Uλ a qa q U λ ψ λ d Integrand on log scale High momentum components suppressed Integrated value does not change, but nature of operator does Overview Operators Factorization Conclusions Properties Many-Body

48 Deuteron Momentum Distribution π [u(q) 2 + w(q) 2 ] [fm 3 ] a q a qdeuteron N 3 LO unevolved V s at λ = 2.0 fm 1 V s at λ = 1.5 fm q [fm 1 ] Overview Operators Factorization Conclusions Properties Many-Body

49 Number Operator - Low Momentum In partial wave momentum basis (for q = 0.34 fm 1 ): Operator: k U λ a qa q U λ k Integrand: ψ λ d Uλ a qa q U λ ψ λ d Integrand on log scale Strength remains at low momentum Similar for other long distance operators: r 2, Q d, & 1 r Overview Operators Factorization Conclusions Properties Many-Body

50 RMS Radius r 2 In partial wave momentum basis: Operator: r 2 Integrand Value: r fm Amplified view of integrand Overview Operators Factorization Conclusions Properties Many-Body

51 Quadrupole Moment r 2 Y 20 In partial wave momentum basis: Integrand with Deuteron wave function Integrated value: fm 2 If, for example, the bare operator is used with the evolved wave function At λ = 6.0 fm 1 λ = 3.0 fm 1 λ = 2.0 fm 1 λ = 1.5 fm 1 r 2 Y fm fm fm fm 2 Note: Q expt fm 2 Overview Operators Factorization Conclusions Properties Many-Body

52 Demonstration of Decoupling In Expectation Values Evolve Hamiltonian & operators to λ in full space TRUNCATE at Λ: Unevolved & SRG Evolved λ= Unevolved & SRG Evolved λ= a + a a + a SRG Evolved & Truncated at Λ= SRG Evolved & Truncated at Λ= q (fm -1 ) Momentum distribution Calculated with AV18 potential. Λ = 2.5 fm 1 λ = 6.0 fm 1, 3.0 fm 1, and 2.0 fm 1 Decoupling for all q is successful when λ < Λ Expectation values reproduced in q (fm -1 ) 10 truncated basis a + a SRG Evolved & Truncated at Λ=2.5 Overview Operators Factorization Conclusions Properties Many-Body Unevolved & SRG Evolved q (fm -1 ) λ=2.0

53 Variational calculations in the deuteron u(k) = Test whether operators are fine-tuned Try a simple 4-parameter variational ansatz ( ) 1 k 2 2 ak 2 ( ) (k 2 + γ 2 )(k 2 + μ 2 ) e λ 2 k 2 2 w(k) = (k 2 + γ 2 )(k 2 + ν 2 ) 2 e λ Error in energy for different starting potentials small λ works great! No fine-tuning (E var E d ) [MeV] AV18 N 3 LO 500 MeV NNLO 550/600 MeV λ (fm 1 ) Overview Operators Factorization Conclusions Properties Many-Body

54 Variational calculations in the deuteron u(k) = Test whether operators are fine-tuned Try a simple 4-parameter variational ansatz ( ) 1 k 2 2 ak 2 ( ) (k 2 + γ 2 )(k 2 + μ 2 ) e λ 2 k 2 2 w(k) = (k 2 + γ 2 )(k 2 + ν 2 ) 2 e λ 2 Momentum distribution for AV18 at several λ s small λ works great! No fine-tuning (a + q a q ) deuteron AV18 Direct Calculation Variational λ=6.0 fm 1 Variational λ=4.0 fm 1 Variational λ=1.5 fm q (fm 1 ) Overview Operators Factorization Conclusions Properties Many-Body

55 Variational calculations in the deuteron u(k) = Test whether operators are fine-tuned Try a simple 4-parameter variational ansatz ( ) 1 k 2 2 ak 2 ( ) (k 2 + γ 2 )(k 2 + μ 2 ) e λ 2 k 2 2 w(k) = (k 2 + γ 2 )(k 2 + ν 2 ) 2 e λ NNLO 550/600 MeV Form factor G M for NNLO potential at two λ s vs. direct small λ works great! No fine-tuning MG M /M d Direct Calculation Variational λ=5.0 fm 1 Variational λ=1.2 fm 1 Overview Operators Factorization Conclusions Properties Many-Body q (fm 1 )

56 Many-Body evolution of Operators Many-body evolution with operators normal ordered in the vacuum: dôs = [[T rel, H s ], Ôs ] = T ij a a i j, T ds i j a ij i j i a j + 1 V pqkl s a p 2 a q a l a k +, Ôs pqkl Only one non-vanishing contraction in the vacuum: a i a j = δ ij A general operator Ô for an A-body system can be written as Ô = Ô(1) + Ô(2) + Ô(3) + + Ô(A) where the Ô(i) label the i = 1, 2, 3,..., A-body components SRG operator Ôs will have contributions for all n so that Ô(n) Ô(n) s. Expanding commutators and making contractions, one finds: Evolution of an operator is fixed in each n-particle subspace How do we deal with this in practice? Overview Operators Factorization Conclusions Properties Many-Body

57 Operator Evolution & Extraction Process Overview Operators Factorization Conclusions Properties Many-Body

58 Operator Evolution & Extraction Process Overview Operators Factorization Conclusions Properties Many-Body

59 Operator Evolution & Extraction Process Overview Operators Factorization Conclusions Properties Many-Body

60 Operator Evolution & Extraction Process Overview Operators Factorization Conclusions Properties Many-Body

61 Operator Evolution & Extraction Process Overview Operators Factorization Conclusions Properties Many-Body

62 Operator Evolution & Extraction Process Overview Operators Factorization Conclusions Properties Many-Body

63 Electromagnetic Form Factors - 1-body Initial Current Elastic electron-deuteron scattering Calculation of G C, G Q, G M operators at LO 1-body versus 2-body evolution? bare versus evolved G C λ=1.5 Unevolved SRG Evolved SRG Evolved wf with Bare Operator q (fm -1 ) Unevolved SRG Evolved SRG Evolved wf with Bare Operator Unevolved SRG Evolved SRG Evolved wf with Bare Operator G Q 10-2 λ=1.5 MG M /M d 10-2 λ= q (fm -1 ) q (fm -1 ) Wave function is derived from the NNLO 550/600 MeV Epelbaum et al. potential and the evolution is run to λ = 1.5 fm 1. Overview Operators Factorization Conclusions Properties Many-Body

64 Number Operator - High Momentum In partial wave momentum basis (for q = 3.02 fm 1 ): Operator: k a qa q k Integrand: ψ d a qa q ψ d Integrand on log scale High momentum components suppressed Integrated value does not change, but nature of operator does Overview Operators Factorization Conclusions Properties Many-Body

65 Outline 1 Overview Resolution SRG 2 Operator Evolution Properties Many-Body 3 Factorization Short-Range Correlations Principles 4 Conclusions Overview Operators Factorization Conclusions Short-Range Correlations Principles

66 Correlations in Nuclear Systems Subedi et al., Science 320,1476 (2008) E.g.: Detection of knocked out pairs with large relative momenta How is vertex modified? How Understand in Context of SRG and low-momentum interactions? = Overview Operators Factorization Conclusions Short-Range Correlations Principles

67 Nucleon Momentum Distributions From Pieper, Wiringa, and Pandharipande (1992). Scaling behavior of momentum distribution function at large q n A (k) = a 2(A, d) n d (k) for k > k Fermi explained by dominance of NN potential & short-range correlations Dominance of np pairs over pp pairs: explained by tensor forces Hard interactions used (high resolution)... difficult calculations Alternative: Calculation of pair density at low resolution... Start with calculation of nuclear matter in MBPT: Overview Operators Factorization Conclusions Short-Range Correlations Principles

68 Evolved Operators in Many-Body Perturbation Theory Rewrite unitary transformation in 2 nd quantization: Û λ = Example: [ ( k1 k Uλ 2, k 3 k 4 ) ( 2 2 δ k1 k 2, k 3 k 4 )] 2 2 δ (k1 + k 2, k 3 + k 4) a k 1 a k 2 a k4 a k3 + Apply to 1-body operator for momentum distribution Ô = a qa q: Û λ ÔÛ λ a qa q a qa i a ( k1 k ia q+u 2 ) λ, i q 2 2 a k 1 a k 2 a k4 a k3 U ( i q λ, k 3 k 4 ) Apply to 2-body pair density operator Ô = a P/2+q a P/2 q a P/2 qa P/2+q : Û λ Ô Û λ = U λ ( k1 k 2 2, 2q ) a k 1 a k 2 a k4 a k3 U λ ( 2q, k 3 k 4 ) 2 + Implement perturbative expansion at small λ = fm 1 Work done with K. Hebeler Overview Operators Factorization Conclusions Short-Range Correlations Principles

69 SRG Evolution of Operators in Nuclear Matter Using AV18 potential Pair-densities are approximately resolution independent Enhancement of np over nn pairs due to tensor force To do: Compare to finite nuclei results with LDA Work done with K. Hebeler Overview Operators Factorization Conclusions Short-Range Correlations Principles

70 SRG Evolution of Operators in Nuclear Matter Using AV18 potential Pair-densities are approximately resolution independent Enhancement of np over nn pairs due to tensor force To do: Compare to finite nuclei results with LDA Work done with K. Hebeler Overview Operators Factorization Conclusions Short-Range Correlations Principles

71 Factorization Idea: If k < λ and q λ = factorization: U λ (k, q) K λ (k)q λ (q) Motivation: The Operator Product Expansion (OPE) of Nonrelativistic Wave Function (Lepage) 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 λ Ψ α (q) γλ (q) p 2 dp Z(λ)Ψ λ α (p) 0 λ +η λ (q) p 2 dp p 2 Z(λ) Ψ λ α (p) 0 By projecting the nuclear potential in momentum subspace, recover OPE via: 1 γ λ (q) λ q 2 dq q Q λ H q V (q, 0) Q λ η λ (q) q 2 dq 1 q λ Q λ H q 2 Q λ p 2 V (q, p) p 2 =0 Questions: ψ d U λ a qa qu λ ψ d is independent of λ. What is the nature of U λ a qa qu λ? If k < λ and q λ = factorization? U λ K λ (k)q λ (q) Can construct transformation directly: U λ (k, q) = k ψα λ ψ α q α ] λ k ψα λ p 2 dp Z(λ)Ψ λ α (p) γ λ (q) α 0 [ αlow U λ (k, q) K λ (k)q λ (q) Overview Operators Factorization Conclusions Short-Range Correlations Principles

72 Numerical Factorization U(k i,q) / U(k 0,q) A preliminary test of factorization in U can be made by assuming 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 ), if k < λ. K λ (k 0 ) As shown below, one can infer this behavior from the plateaus for q 2fm 1 when k i < λ S0 k 0 = 0.1 fm 1 λ = 2.0 fm 1 k 1 = 0.5 fm 1 k 2 = 1.0 fm 1 k 3 = 1.5 fm 1 k 4 = 3.0 fm q [fm 1 ] Singular Value Decomposition (SVD) tool to quantitatively analyze the extent to which U factorizes The SVD can be expressed as an outer product expansion G = r i d i u i v t i where r is the rank and the d i are the singular values (in order of decreasing value). Evidence: Shown below at 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 Overview Operators Factorization Conclusions Short-Range Correlations Principles

73 Practical Use From Decoupling: write ψ λ U λ Ô U λ ψ λ = λ λ dk dq dq dk ψ λ (k )U λ (k, q )Ô(q, q)u λ (q, k)ψ λ (k) Using Factorization: set U λ (k, q) K λ (k)q λ (q), where k < λ and q λ. λ [ λ λ ] λ = ψ λ (k ) U λ (k, q )Ô(q, q)u λ (q, k) }{{} +I QOQ K λ (k )K λ (k) ψ λ (k) }{{} Low Momentum Structure where I QOQ = ] λ dq λ [Q dq λ (q )Ô(q, q)q λ (q) Universal Valid when initial operators weakly couple high and low momentum, e.g., r 2 1 r G C (q = 3.02 fm 1 ) a qa q(q = 3.02 fm 1 ) Overview Operators Factorization Conclusions Short-Range Correlations Principles

74 Factorization in Few-Body Nuclei Variational Monte Carlo Calculation Using AV14 NN potential 1D few-body HO space calculation System of A bosons interacting via a model potential body NN-only 3-body NN-only 4-body NN-only 10-2 N(k) / A N max = From Pieper, Wiringa, and Pandharipande (1992). Possible explanation of scaling behavior Results from dominance of NN potential and short-range correlations (Frankfurt, et al.) k A Test Bed for 3D NCSM calculations: Alternative explanation of scaling behavior Results from factorization λ λ 0 0 ψ λ (k ) [I QOQ K λ (k )K λ (k)] ψ λ (k) Overview Operators Factorization Conclusions Short-Range Correlations Principles

75 Factorization in Few-Body Nuclei Variational Monte Carlo Calculation Using AV14 NN potential 1D few-body HO space calculation System of A bosons interacting via a model potential body NN-only 3-body NN-only 4-body NN-only PHQ 2-body NN-only λ=2 PHQ 3-body NN-only λ=2 PHQ 4-body NN-only λ=2 N(k) / A N max = From Pieper, Wiringa, and Pandharipande (1992). Possible explanation of scaling behavior Results from dominance of NN potential and short-range correlations (Frankfurt, et al.) k A Test Bed for 3D NCSM calculations: Alternative explanation of scaling behavior Results from factorization λ λ 0 0 ψ λ (k ) [I QOQ K λ (k )K λ (k)] ψ λ (k) Overview Operators Factorization Conclusions Short-Range Correlations Principles

76 Short Range Correlations and the EMC effect Deep inelastic scattering ratio at Q 2 2 GeV 2 and 0.35 x B 0.7 and inelastic scattering at Q GeV 2 and 1.5 x B 2.0 Strong linear correlation between slope of ratio of DIS cross sections (nucleus A vs. deuterium) and nuclear scaling ratio L.B. Weinstein, et al., Phys. Rev. Lett. 106, (2011) Can nuclear scaling and EMC effect be explained via factorization of operators and low momentum structure of the nuclei? We can calculate a 2 in MBPT Same dependence on nuclear structure for high momentum operators EMC effect? Overview Operators Factorization Conclusions Short-Range Correlations Principles

77 Outline 1 Overview Resolution SRG 2 Operator Evolution Properties Many-Body 3 Factorization Short-Range Correlations Principles 4 Conclusions Overview Operators Factorization Conclusions

78 Conclusions Summary: Outlook: The SRG provides a means to lower the resolution needed in nuclear interactions, thereby reducing the computational difficulty of the nuclear many-body problem Nuclear operators can be consistently evolved and calculated with the SRG Many-body contributions can be extracted and embedded at each order Factorization: U λ (k, q) K λ (k)q λ (q). Offers method to explore high resolution probes in low momentum systems. EMC effect? Work in progress on perturbative calculations of nuclear matter and few-body nuclei using SRG interactions Higher order electromagnetic interactions in few-body nuclei Establish bounds on growth of many-body operators Overview Operators Factorization Conclusions

79 Overview Operators Factorization Conclusions The End

80 R. Roth et al. N 3 LO(500) E/A and R ch in MBPT 16 O 7.7 MeV 2.3 fm 1 Coupled cluster NN-only results: 40 Ca 8.6 MeV 2.8 fm 1 48 Ca 8.3 MeV 2.9 fm 1 Overview Operators Factorization Conclusions

81 Ab initio approach to light-ion reactions NCSM/RGM using low momentum SRG NN interactions See, e.g., Navrátil, Roth, Quaglioni, PRC 82, (2010) for nucleon-nucleus scattering Overview Operators Factorization Conclusions

82 Ab initio approach to light-ion reactions Applications to fusion energy systems and stellar evolution Still to do: including SRG-evolved NNN interactions Overview Operators Factorization Conclusions

83 1/r Expectation Value Expand into partial wave momentum basis: Integrand with Deuteron wave function Integrated value: fm 1 Overview Operators Factorization Conclusions

84 Momentum Distribution in Few-Body Model Space, I A Test Bed for 3D NCSM calculations: Embed in 1D few-body HO space (E.D. Jurgenson) Effectively N A a k a k N A N A = Symmetric Jacobi states of A bosons Proof of principle using SRG evolved operators in few-body system Same embedding procedure and unitary transformation as for potentials is used in these calculations Future: E/M Operators in 3D n(k)/a A=2, 3, 4 Boson System N max = 40 A = 2 A = 3 A = k Overview Operators Factorization Conclusions

85 Momentum Distribution in Few-Body Model Space, II n(k)/a A=2, 3, & 4 Boson System with bare operator A = 2 A = 3 λ = λ = A = 4 λ = 5.00 λ = 3.00 N max = k k k High momentum components suppressed Wave functions are simplified... what about the operators? Overview Operators Factorization Conclusions

86 Embedding Other Operators Consider the following momentum distributions in 1D model... Problem Corrected N(p) / A A=2, 2-body only A=3, 2-body only A=4, 2-body only A=2, 2-body only λ=2 A=3, 2-body only λ=2 A=4, 2-body only λ=2 N(p) / A A=2, 2-body only A=3, 2-body only A=4, 2-body only A=2, 2-body only λ=2 A=3, 2-body only λ=2 A=4, 2-body only λ= N max = N max = p p Solution: Operators must be boosted to embed into A-particle space Revised Embedding Process... Overview Operators Factorization Conclusions

87 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

88 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

89 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

90 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

91 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

92 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

93 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

94 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

95 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

96 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

97 Custom Potentials Can we tailor the potential to other shapes with the SRG? Consider dh s ds = [[G s, H s ], H s ] in the 1 P 1 partial wave with a strange choice for G s Overview Operators Factorization Conclusions

98 An Example: Block Diagonalization Alternative SRG Evolution The flow equation is still given by dh s = [η s, H s ] ds but with the generator η s = [G s, H s ] where ( ) PHs P 0 G s = 0 QH s Q Note: λ = 1 s 1/4 fm 1 Overview Operators Factorization Conclusions

99 High and Low Momentum Operators in 1D Oscillator Basis Number operator at: q = 15.0 & q = 0.5 n(k)/a 5 x λ=100 λ=3 λ= N cut A=3 System, 1D model boson system Evolve Hamiltonian & operators to λ at large N max Truncate model space at N cut Poor convergence of long range operators Λ UV m ω mn max ω; Λ IR N max Can this be corrected? n(k)/a r λ= λ=3 λ= N cut r 2 Value λ=100 λ=3 λ=2 λ=1.5 λ= N cut Overview Operators Factorization Conclusions

100 An alternative generator - diagonal in the basis The harmonic oscillator Hamiltonian H ho is diagonal in the basis Consider a naive choice (replace T rel with H ho in SRG generator) η s = [H ho, H s ] do s ds = [η s, O s ] r 2 for A=3 Many Contributions to Binding Energy Greatly Improved convergence Generates spurious deep bound states... Overview Operators Factorization Conclusions

101 Controlled IR and UV renormalization I Consider T rel + αr 2, where α is a parameter which can be adjusted to optimize the renormalization (here, α = 1), so that η s = [T rel + αr 2, H s ] r 2 for A=3 Many Contributions to Binding Energy Convergence improves with decreasing λ No spurious deep bound states. Is hierarchy of many-body forces under control? Overview Operators Factorization Conclusions

102 Controlled IR and UV renormalization II Convergence of High (q=15.0) and Low (q=0.5) number operators with η s = [T rel + αr 2, H s] n(k)/a 5 x λ=100 λ=3 λ=2 λ=1 n(k)/a λ=100 λ=3 λ=2 λ= N cut N cut Convergence of binding energy Overview Operators Factorization Conclusions

103 Low Momentum Operators in NCSM Evolve Hamiltonian & operators to λ at large N max Truncate model space at N cut Λ UV m ω mn max ω; Λ IR N max Poor convergence of long range operators... can this be corrected? Radius [fm] N 3 LO 3 H h- ω = 28 λ = fm 1 λ = 5.00fm 1 λ = 3.00fm 1 λ = 2.00fm 1 λ = 1.50fm 1 λ = 1.20fm 1 λ = 1.00fm N max cut SRG evolution of r 2 operator for A=3 Chiral N 3 LO Potential with initial NN+NNN interaction Overview Operators Factorization Conclusions

104 Controlled IR and UV renormalization Consider T rel + αr 2, where α is a parameter which can be adjusted to optimize the renormalization (here, α = 10), so that η s = [T rel + αr 2, H s] N 3 LO h- ω = 28 Radius [fm] H λ = fm 1 λ = 5.00fm 1 λ = 3.00fm 1 λ = 2.00fm 1 λ = 1.50fm 1 λ = 1.20fm 1 λ = 1.00fm N max cut Convergence improves with decreasing λ Preliminary evidence: spurious deep bound states appear in 4 He for small λ when embedding this H s Overview Operators Factorization Conclusions

105 The Similarity Renormalization Group with Novel Generators The Similarity Renormalization Group (SRG) W. Li, era, and R.J. Furnstahl, arxiv: v1 [nucl-th]. dh s ds = [η s, H s] = [[G s, H s], H s] Typically, G s = T rel with λ = 1 s 1/4 fm 1 G inv s = σ 2 /(1 + T /σ 2 ) c + T + AV18 Potential SRG flow can be tailored with alternative choices of G s λ eq redefined to match decoupling Using Gs exp and Gs inv with σ = 2 fm 1, low E part of V still decoupled Much less evolution at high E much faster! G exp s = σ 2 e T /σ2 c + T + Same λ eq = 2 fm 1 : σ controls pattern. Overview Operators Factorization Conclusions

106 The Similarity Renormalization Group with Novel Generators W. Li, era, and R.J. Furnstahl, arxiv: v1 [nucl-th]. Computational performance with novel generators G s = T time [s] G s inv G s exp time [s] 100 G s = T G s exp λ eq [fm 1 ] σ [fm -1 ] Allows evolution to low λ Application to A > 2 evolution Hierarchy of many body forces under control A=2 evolution is a good indication of behavior in larger systems Overview Operators Factorization Conclusions