Applications of Renormalization Group Methods in Nuclear Physics 1

Transcription

1 Applications of Renormalization Group Methods in Nuclear Physics 1 Dick Furnstahl Department of Physics Ohio State University HUGS 2014

2 Outline: Lecture 1 Lecture 1: Overview Preview: Running couplings/potentials Goals of low-energy nuclear physics Breakthroughs in low-energy nuclear theory Nuclear scales and resolution

3 Outline: Lecture 1 Lecture 1: Overview Preview: Running couplings/potentials Goals of low-energy nuclear physics Breakthroughs in low-energy nuclear theory Nuclear scales and resolution

4 Renormalization group and QCD coupling: Running α s (Q 2 ) 0.5 α s (Q) July 2009 Deep Inelastic Scattering e + e Annihilation Heavy Quarkonia Extractions from experiment can be compared (here at M Z ): τ-decays (N3LO) Quarkonia (lattice) Υ decays (NLO) DIS F 2 (N3LO) DIS jets (NLO) 0.1 QCD α s(μ Z) = ± Q [GeV] The QCD coupling is scale dependent ( running ): α s (Q 2 ) [β 0 ln(q 2 /Λ 2 QCD )] 1 From µ 2 (dα s /dµ 2 ) = β(α s ) and β(α s ) = α 2 s(β 0 + β 1 α s + ) (this is the beta function ) e + e jets & shps (NNLO) electroweak fits (N3LO) e + e jets & shapes (NNLO) α s (Μ Z) cf. QED, where α em (Q 2 ) is effectively constant for soft Q 2 : α em (Q 2 = 0) 1/137 fixed H for quantum chemistry

5 Running QCD α s (Q 2 ) vs. running nuclear V λ (k, k ) 0.5 α s (Q) July 2009 Deep Inelastic Scattering e + e Annihilation Heavy Quarkonia Vary scale ( resolution ) with RG = diff. eq. for potential V Scale dependence: RG running of initial potential with scale λ (decoupling or separation scale) QCD α s(μ Z) = ± Q [GeV] The QCD coupling is scale dependent (cf. low-e QED): α s (Q 2 ) [β 0 ln(q 2 /Λ 2 QCD )] 1 Different Hamiltonians? Do you get different answers from the same Feynman diagrams with α s (µ 2 1 ) and α s(µ 2 2 )? But all are (NN) phase equivalent! (predict the same NN cross sections) Shift contributions between interaction and sums over intermediate states

6 Outline: Lecture 1 Lecture 1: Overview Preview: Running couplings/potentials Goals of low-energy nuclear physics Breakthroughs in low-energy nuclear theory Nuclear scales and resolution

7 Low-energy playground: Table of the nuclides

8 Isotope science: XXI science New frontiers from rare isotope facilities (worldwide) Heavy Ion Research Facility Lanzhou VECC Kolkata Sao Paulo Pelletron Asymptotic freedom? FRIB$ DFT$ current$ from B. Sherrill Challenge of open quantum systems! (continuum channels,... )

9 fusion Validated Nuclear Interac/ons Chiral EFT Ab- ini/o Op/miza/on Model valida/on Uncertainty Quan/fica/on Stellar burning Structure and Reac/ons: Light and Medium Nuclei Ab- ini/o RGM CI Neutron drops EOS Correla/ons Load balancing Eigensolvers Nonlinear solvers Model valida/on Uncertainty Quan/fica/on Structure and Reac/ons: Heavy Nuclei DFT TDDFT Load balancing Op/miza/on Model valida/on Uncertainty Quan/fica/on Eigensolvers Nonlinear solvers Mul/resolu/on analysis Neutrinos and Fundamental Symmetries Neutron Stars Fission

10 JLab: Understanding short-range correlations in nuclei Subedi et al., Science 320, 1476 (2008) e A e q What is this vertex? k k q = k k p 1 p 2 N N A!2 Higinbotham, arxiv: ν = E k E k Q 2 = q 2 x B = Q2 2m N ν r( 56 Fe/ 3 He) r( 12 C/ 3 He) r( 4 He/ 3 He) a) b) c) x B 1.4 <Q 2 < 2.6 GeV 2 Egiyan et al. PRL 96, (2006) p 1 q p 1 p 2 p 2 SRC interpretation: NN interaction can scatter states with p 1,p 2 k F to intermediate states with p 1,p 2 k F which are knocked out by the photon How to explain cross sections in terms of low-momentum interactions? Vertex depends on the resolution!

11 Outline: Lecture 1 Lecture 1: Overview Preview: Running couplings/potentials Goals of low-energy nuclear physics Breakthroughs in low-energy nuclear theory Nuclear scales and resolution

12 Historical perspective: JLab experimentalist [L. Weinstein (2012)] Comprehensive Theory Overview Nuclear Theory - circa 2000 Nuclear Theory - today: 1, 2, 3, 12, many

13 Historical perspective: Ab initio structure years ago Figure from the RIA (now FRIB) white paper (2000) Limits of nuclear existence r-process protons rp-process A=10 A=12 50 neutrons A~60 82 Density Functional Theory Selfconsistent Mean Field Ab initio few-body calculations 0Ñω Shell Model No-Core Shell Model G-matrix Many-body approaches for ordinary nuclei Ab initio: Only up to 12 C (GFMC and NCSM with NN+3N)

14 Historical perspective: Ab initio structure years ago From the start of the SciDAC UNEDF project (2007) dimension of the problem Interfaces provide crucial clues Ab initio: Selected nuclei up to 40 Ca (with CC, but 3N?)

15 ain s key to explain 24O as heaviest oxygen isotope ki, What Holt, Schwenk, Akaishi, Phys. 105,structure (2010).today? is feasible for Rev. ab Lett. initio 0 A NN + 3N-full (400) O E 14 d increased binding for neutron-rich 25 3 Max 1.9 fm 50 1 Hergert&et&al.,&PRL&110,&242501&(2013)& Nature'498,'346'(2013)' E MeV d in precision Penning trap exp. 51,52Ca from AME! deviation in 75 Theory'preceded' accurate'experiment' 100 boration + Holt, Menendez, Schwenk, submitted n global predictions? E/A [MeV] -10 IM SRG 14 (a) CCSD! N3 LO 10 12! N2 LOopt NN+3N-induced IT NCSM exp A 0.5 Energy/particle. 110, (2013) -0.5 (b) (in MeV) from oxygen to tin (A = 132!) ected -7 Hartree-Fock-Bogoliubov. (examples) Two-neutron separation energies calcium for Calcium isotope chain NN+3N-full (c) Binder'et'al.,'arXiv: '(12/2013)' -8 exp -9 ent -10many-body methods! Λ3N = 400 MeV/c! Λ3N = 350 MeV/c Reactions: 0.5 Experimental and Ab Initio Theoretical Perspectives, TRIUMF, 02/19/ O 24 O 36 Ca 40 Ca 48 Ca 52 Ca 54 Ca 48 Ni 56 Ni 60 Ni 62 Ni 66 Ni (d) 68 Ni 78 Ni 88 Sr 90 Zr 100 Sn 106 Sn 108 Sn 114 Sn 116 Sn 118 Sn 120 Sn 132 Sn

16 Outline: Lecture 1 Lecture 1: Overview Preview: Running couplings/potentials Goals of low-energy nuclear physics Breakthroughs in low-energy nuclear theory Nuclear scales and resolution

17 Standing on the shoulders of giants... Steven Weinberg (1933 ) Nobel Prize 1979 electroweak theory (GWS),... effective field theory (EFT) applied to nuclear physics Kenneth G. Wilson ( ) Nobel Prize 1982 renormalization group (RG) and critical phenomena,... similarity RG = Conceptual basis for changing resolution and tools to do it!

18 Connecting degrees of freedom with EFT and RG LQCD constituent quarks scale& separa)on& ab initio CI Resolution DFT collective models

19 Quark (QCD) vs. hadronic NN N interaction!"%/01 %&)-,.2 3A4A5A6'.7&)"')#$AB)8!949B):0;<)=>00?@A 6'.7&"'#$ C3>D %&)-,.2!AEAF%,%*G#)"')#$AB)8!951B)1H)=>00;@A IA)J&K%%B)6A)C.7%B)LA)M#'&+/#B)8KN&A)!"OA)P"''A)99, <::<<>)=:<<Q@ L"*&.,)-.,(")-,.2)PR9ST)K''UTVV#,W%OA.,GVU/-V<0<1A;?0Q Old goal: replace hadronic descriptions at ordinary nuclear densities with quark description (since QCD is the theory)

20 Quark (QCD) vs. hadronic NN N interaction!"#$%&'%()*+($"#,)-.,(" Old goal: replace hadronic descriptions at ordinary nuclear densities with quark description (since QCD is the theory) New goal: use effective hadronic dof s systematically Seek model independence and theory error estimates!"%/01 %&)-,.2 3A4A5A6'.7&)"')#$AB)8!949B):0;<)=>00?@A 6'.7&"'#$ C3>D %&)-,.2!AEAF%,%*G#)"')#$AB)8!951B)1H)=>00;@A Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities (resolutions) where phase IA)J&K%%B)6A)C.7%B)LA)M#'&+/#B)8KN&A)!"OA)P"''A)99, transitions happen or at high momentum<::<<>)=:<<q@ transfers L"*&.,)-.,(")-,.2)PR9ST)K''UTVV#,W%OA.,GVU/-V<0<1A;?0Q

21 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!

22 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!

23 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.

24 Digital resolution: Higher resolution is better (?) Computer screens, printers, digital cameras, TV s... Higher resolution = more pixels Pixel size characteristic scale = greater detail

25 Diffraction and resolution

26 Wavelength and resolution

27 Wavelength and resolution

28 Wavelength and resolution

29 Wavelength and resolution

30 Wavelength and resolution

31 Wavelength and resolution

32 Wavelength and resolution

33 Wavelength and resolution

34 Wavelength and resolution

35 Principle of any effective low-energy description

36 Principle of any effective low-energy description If system is probed at low energies, fine details not resolved

37 Principle of any effective low-energy description If system is probed at low energies, fine details not resolved Use low-energy variables for low-energy processes Short-distance structure can be replaced by something simpler without distorting low-energy observables (cf. multipoles in classical electrodynamics) Could be a model or systematic (e.g., effective field theory)

38 Principle of any effective low-energy description If system is probed at low energies, fine details not resolved Use low-energy variables for low-energy processes Short-distance structure can be replaced by something simpler without distorting low-energy observables (cf. multipoles in classical electrodynamics) Could be a model or systematic (e.g., effective field theory) Density in Pb low momentum low resolution (λ = h/p) but not so fast: the interaction can affect the resolution

39 Why is textbook nuclear physics so hard? V L=0 (k, k ) = d 3 r j 0 (kr) V (r) j 0 (k r) = k V L=0 k = V kk matrix Momentum units ( = c = 1): typical relative momentum in large nucleus 1 fm MeV but... Repulsive core = large high-k ( 2 fm 1 ) components

40 Why is textbook nuclear physics so hard? V L=0 (k, k ) = d 3 r j 0 (kr) V (r) j 0 (k r) = k V L=0 k = V kk matrix Momentum units ( = c = 1): typical relative momentum in large nucleus 1 fm MeV but... Repulsive core = large high-k ( 2 fm 1 ) components

41 Consequences of a repulsive core uncorrelated correlated S1 deuteron probability density ψ(r) 2 [fm 3 ] V(r) [MeV] ψ(r) 2 [fm 3 ] 0.1 Argonne v r [fm] r [fm] Probability at short separations suppressed = correlations Short-distance structure high-momentum components Greatly complicates expansion of many-body wave functions

42 Expanding wave functions in an HO basis Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: N max ψ Nmax (r) = c α φ α (r) α=0 15 V@rD 10 5 Find c α s by diagonalizing ĤΨ = EΨ E exact' =')1.51' r Out[476]= wf ψ exact (r), ψ 0 (r), 0.5 φ 0 ψ exact (r), ψ 0 (r), 0.5 φ 0 wf r r -0.5 N max = 0, E 0 = 1.30 N max = 0, E 0 = +5.23

43 Expanding wave functions in an HO basis Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: N max ψ Nmax (r) = c α φ α (r) α=0 15 V@rD 10 5 Find c α s by diagonalizing ĤΨ = EΨ E exact' =')1.51' r Out[480]= wf ψ exact (r), ψ 2 (r), 0.5 φ 2 ψ exact (r), ψ 2 (r), 0.5 φ 2 wf r r -0.5 N max = 2, E 2 = 1.46 N max = 2, E 2 = 0.87

44 Expanding wave functions in an HO basis Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: N max ψ Nmax (r) = c α φ α (r) α=0 15 V@rD 10 5 Find c α s by diagonalizing ĤΨ = EΨ E exact' =')1.51' r Out[485]= wf ψ exact (r), ψ 4 (r), 0.5 φ 4 ψ exact (r), ψ 4 (r), 0.5 φ 4 wf r r -0.5 N max = 4, E 4 = 1.46 N max = 4, E 4 = 1.04

45 Expanding wave functions in an HO basis Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: N max ψ Nmax (r) = c α φ α (r) α=0 15 V@rD 10 5 Find c α s by diagonalizing ĤΨ = EΨ E exact' =')1.51' r Out[489]= wf ψ exact (r), ψ 6 (r), 0.5 φ 6 ψ exact (r), ψ 6 (r), 0.5 φ 6 wf r r -0.5 N max = 6, E 6 = 1.50 N max = 6, E 6 = 1.40

46 Expanding wave functions in an HO basis Single-particle radial wf ψ(r) Expand in harmonic oscillator wfs: N max ψ Nmax (r) = c α φ α (r) α=0 Find c α s by diagonalizing ĤΨ = EΨ Extend to many-body system 15 V@rD 10 E exact' =')1.51' r -5 Out[493]= wf ψ exact (r), ψ 8 (r), 0.5 φ 8 ψ exact (r), ψ 8 (r), 0.5 φ 8 wf r r -0.5 N max = 8, E 8 = 1.50 N max = 8, E 8 = 1.43

47 Many-body physics by matrix diagonalization Harmonic oscillator basis with N max shells for excitations Graphs show convergence for soft chiral EFT potential (although not at optimal Ω for 6 Li) Ground State Energy [MeV] 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] Lithium-6 8 ground-state energy 4 V 0 NN = N 3 LO (500 MeV) V 4 NNN = N 2 LO 8 h- Ω = 20 MeV Original expt Matrix Size [N max ] Factorial growth of basis with A = limits calculations Too much resolution from potential = mismatch of scales

48 What if your theory has too much resolution?

49 What if your theory has too much resolution?

50 What if your theory has too much resolution?

51 What if your theory has too much resolution? Claim: Nuclear physics with textbook V (r) is like using beer coasters!

52 Less painful to use a low-resolution version! High resolution Low resolution Can greatly reduce storage without distorting message Resolution was lowered here by block spinning Alternative: apply a low-pass filter

53 Low-pass filter on an image Use 2D Fourier transform Long and short wavelengths separated

54 Low-pass filter on an image Use 2D Fourier transform Long and short wavelengths separated After low-pass filter: Much less information needed Long-wavelength info is preserved

55 Try a low-pass filter on nuclear V (r) = Set to zero high momentum (k 2 fm 1 ) matrix elements and see the effect on low-energy observables

56 Use Phase Shifts to Test sin(kr+δ) 60 1 S0 0 R r phase shift (degrees) AV E lab (MeV) Here: 1 S 0 (spin-singlet, L = 0, J = 0) neutron-proton scattering Different phase shifts in each partial wave channel

57 Effect of low-pass filter on observables sin(kr+δ) 0 R r 60 1 S0 k = 2 fm 1 phase shift (degrees) AV18 phase shifts E lab (MeV)

58 Effect of low-pass filter on observables sin(kr+δ) 0 R r 60 1 S0 k = 2 fm 1 phase shift (degrees) AV18 phase shifts after low-pass filter E lab (MeV)

59 Why did our low-pass filter fail? Basic problem: low k and high k are coupled (mismatched dof s!) E.g., perturbation theory for (tangent of) phase shift: k V k + k k V k k V k (k 2 k 2 )/m S0 k = 2 fm 1 Solution: Unitary transformation of the H matrix = decouple! E n = Ψ n H Ψ n U U = 1 = ( Ψ n U )UHU (U Ψ n ) = Ψ n H Ψ n Here: Decouple using RG E lab (MeV) phase shift (degrees) AV18 phase shifts after low-pass filter

60 Preview: Decoupling with the similarity RG

61 Preview: Consequences of a repulsive core revisited uncorrelated correlated S1 deuteron probability density ψ(r) 2 [fm 3 ] V(r) [MeV] ψ(r) 2 [fm 3 ] Argonne v r [fm] r [fm] Probability at short separations suppressed = correlations Short-distance structure high-momentum components Greatly complicates expansion of many-body wave functions

62 Preview: Consequences of a repulsive core revisited uncorrelated correlated S1 deuteron probability density softened ψ(r) 2 [fm 3 ] V(r) [MeV] ψ(r) 2 [fm 3 ] Argonne v 18 λ = 4.0 fm -1 λ = 3.0 fm -1 λ = 2.0 fm r [fm] 100 original r [fm] Transformed potential = no short-range correlations in wf! Can it really be so different in the interior?

63 Preview: Consequences of a repulsive core revisited uncorrelated correlated S1 deuteron probability density r 2 ψ(r) 2 [fm 1 ] V(r) [MeV] r 2 ψ(r) 2 [fm 1 ] Argonne v 18 λ = 4.0 fm -1 λ = 3.0 fm -1 λ = 2.0 fm r [fm] r [fm] Transformed potential = no short-range correlations in wf! What part of the coordinate-space wave function is measurable? What about the high-momentum tail in momentum space?

64 Preview: Revisit the convergence with matrix size (N max ) Harmonic oscillator basis with N max shells for excitations Ground State Energy [MeV] 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] Lithium-6 8 ground-state energy 4 V 0 NN = N 3 LO (500 MeV) V 4 NNN = N 2 LO 8 h- Ω = 20 MeV Original expt Matrix Size [N max ]

65 Preview: Revisit the convergence with matrix size (N max ) Harmonic oscillator basis with N max shells for excitations Ground State Energy [MeV] Softened with SRG Helium-4 ground-state energy Jurgenson et al. (2009) V NN = N 3 LO (500 MeV) V NNN = N 2 LO Original expt Matrix Size [N max ] Ground-State Energy [MeV] Lithium-6 8 ground-state energy 4 V 0 NN = N 3 LO (500 MeV) V 4 NNN = N 2 LO 8 h- Ω = 20 MeV Original Softened with SRG 24 λ = 2.0 fm 1 28 λ = 1.5 fm 1 32 expt Matrix Size [N max ] Graphs show that convergence for soft chiral EFT potential is accelerated for evolved SRG potentials Nuclear structure/reaction calculations more perturbative