Atomic Nuclei at Low Resolution
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- Jessica Foster
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1 Atomic Department of Physics Ohio State University November, 29 Collaborators: E. Anderson, S. Bogner, S. Glazek, E. Jurgenson, R. Perry, S. Ramanan, A. Schwenk + UNEDF collaboration
2 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD Nuclear Theory Group Ulrich Heinz Sabine Jeschonnek Yuri Kovchegov Robert Perry
3 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD What is modern Nuclear Theory? There are 4 known fundamental interactions in nature: STRONG ELECTRIC WEAK GRAVITY
4 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD STRONG ELECTRIC WEAK GRAVITY Standard Model of Particle Physics (2 th century) 21 st Century NUCLEAR PHYSICS 21 st Century Particle Physics (+ beyond standard model) String Theory
5 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD Richard ( Dick ) Furnstahl Research interests: Applications of renormalization group to nuclei Effective field theories for many-body systems Field theory at finite temperature and/or density
6 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD
7 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD
8 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD
9 Overview DOFs EFT UNEDF Resolution SRG Group Forces Interests QCD
10 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Degrees of Freedom: From QCD to Nuclei Density Functional Theory A>1 Coupled Cluster, Shell Model A<1 Exact methods A 12 GFMC, NCSM Lattice QCD Low-mom. interactions Chiral EFT interactions (low-energy theory of QCD) QCD Lagrangian Renormalization Group = focus on relevant dof s
11 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Plan: Use DOF s to Simplify and Make 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!
12 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Plan: Use DOF s to Simplify and Make 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 vaudeville joke about a doctor and patient...
13 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Plan: Use DOF s to Simplify and Make 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 vaudeville joke about a doctor and patient... Patient: Doctor, doctor, it hurts when I do this!
14 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Plan: Use DOF s to Simplify and Make 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 vaudeville joke about a doctor and patient... Patient: Doctor, doctor, it hurts when I do this! Doctor: Then don t do that.
15 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Diffraction 11 (Review)
16 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Diffraction and Resolution
17 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Diffraction and Resolution
18 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Diffraction and Resolution
19 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Diffraction and Resolution
20 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Diffraction and Resolution
21 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Intensity as a Function of Angle Single-slit diffraction = first minimum when sin θ min = λ/a = Pattern widens as λ increases
22 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
23 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
24 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
25 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
26 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
27 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
28 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
29 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
30 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Wavelength and Resolution
31 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Consequences For a slit of width a, θ min = sin 1 (λ/a) so if λ a, you don t learn anything about the details of the slit
32 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Quantum Mechanics and Resolving Power de Broglie relation: λ = h/p λ 1 1 m = probe atoms λ 1 14 m = probe nucleus λ 1 18 m = probe quarks
33 Overview DOFs EFT UNEDF Resolution SRG Big Plan Diffraction QM Rough calculation: θ min 8 8π hc 124 Mev-fm rad and λ = 18 pc 8 MeV = a λ 18 θ min fm 11 fm = radius of 28 Pb 6 fm
34 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Principle of Any Effective Low-Energy Description
35 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Principle of Any Effective Low-Energy Description If system is probed at low energies, fine details not resolved
36 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF 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
37 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF 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 Could be a model or systematic (e.g., effective field theory) physics can change with resolution! Low density low interaction energy low resolution
38 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Quark (QCD) vs. Hadronic NN 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
39 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Quark (QCD) vs. Hadronic NN 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 Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities (resolutions) where phase transitions happen
40 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Making Progress on Complicated Problems General theoretical approach to complicated problems = replace something complicated by something simpler without distorting the physics allow for systematic improvement with estimate of error limited but well-defined regime of applicability Models fall short! Effective Field Theory = use local field theory low energy = low resolution choose appropriate low-energy degrees of freedom = contact interactions + long-range physics powerful renormalization tools
41 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF How Do We Calculate Systematically? Use physics insight to identify small parameters = Expand! Basic mathematical tool = Taylor series expansion in simplest case, expand f (x) for small x: f (x) = a + a 1 x + a 2 x 2 + a 3 x 3 + example: sin(x) = x 1 6 x x 5 + truncation error depends on size of x and size of coefficient Does this work for any function of x? No, it fails for nonanalytic functions (like x or ln x) EFT uses analog of Taylor expansion and dimensional analysis estimate truncation error without calculating next term non-analytic behavior generated explicitly
42 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Simplifying a Problem: A Falling Eraser Conventional Physics 11 approach to dropping an eraser: F = mg = constant acceleration But gravititational force is different between hand and floor! = How do you justify ignoring this? Identify ratio of length scales as small expansion parameter Expand exact result for g for small height r above floor: g GM e ( r + R e ) 2 = GM e R 2 e 2(GM e) r R 3 e + 3(GM e)( r) 2 R 4 e + O[( r) 3 ].
43 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF We identify a small, dimensionless parameter x r/r e : g = GM e Re 2 (1 2x + 3x 2 + O[x 3 ]) we verify that this is the correct choice of scales because coefficients are natural (order unity) = estimate truncation error without exact solution! there may be multiple scales and/or combinatoric factors But we can also apply this approach without knowing the answer! postulate a Taylor expansion in x with unknown coefficients
44 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF EFT Analogy: Relativistic Kinetic Energy K (v 2 ) If we couldn t calculate K (v 2 ) of an e at (low) speed v? Can do the experiment: [W. Bertozzi, Am. J. Phys. 32 (1964) 551]
45 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF EFT Analogy: Symmetries and Naturalness Analyzing K at low energies without knowing the answer... Use known symmetries to constrain K : What could K depend on ( free particle )? x?, t?, v? Hint: space is homogenous and isotropic, time is homogenous Symmetries = K depends only on v 2 v 2 Subtlety: K is analytic (otherwise v could appear) EFT form: K eft (v 2 ) = 1 2 mv 2 [a + a 2 v 2 + a 4 v 4 + ] All terms in [ ] must be dimensionless = introduce scale of velocity (call it c ) Physics assumption: only one velocity scale
46 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Final form: K eft (v 2 ) = 1 2 mv 2 [b + b 2 (v/c) 2 + b 4 (v/c) 4 + ] If correct c = O(1) dimensionless constants b i (naturalness) Power counting : each order = two powers of (v/c) = truncation error estimate find coefficients b i by matching to experiment or theory match at lowest values of v, where EFT is most accurate K error v v
47 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Matching to Theoretical Calculation Suppose square roots were expensive to calculate = for small (v/c), it s cheaper to evaluate effective theory! Start with EFT: K eft (v 2 ) = 1 2 mv 2 [b + b 2 (v/c) 2 + b 4 (v/c) 4 + ] Match to underlying theory = Taylor expand K exact : K exact = mc 2 [1/ 1 (v/c) 2 1] = mv mv 4 8c 2 + 5mv 6 16c mv 8 128c 6 + O(v 1 ). read off b i : b = 1, b 2 = 3 4, b 4 = 5 8, b 6 = 35 64, could even match at unphysical speed (e.g., imaginary v) Now use K eft for any particle with a small velocity Coefficients of (v/c) 2 are natural = predict truncation error!
48 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Chiral EFT for Two Nucleons as Prototype Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy 1π 2π contact LO 1S 3S P 1D D3 3G
49 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Chiral EFT for Two Nucleons as Prototype Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy 1π 2π contact LO NLO 1S 3S P 1D D3 3G
50 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Chiral EFT for Two Nucleons as Prototype Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy 1π 2π contact LO NLO N 2 LO 1S 3S P 1D D3 3G
51 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Chiral EFT for Two Nucleons as Prototype Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy 1π 2π contact LO NLO N 2 LO N 3 LO 4 (15) 1S 3S P 1D D3 3G
52 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF State of the Art: N 3 LO (Epelbaum, nucl-th/5932) Phase Shift [deg] Phase Shift [deg] Phase Shift [deg] Phase Shift [deg] ε 1 3 D2 1 S 3 P Lab. Energy [MeV] D2 3 D3 3 S1 3 P Lab. Energy [MeV] P1 3 P2 3 D1 ε Lab. Energy [MeV]
53 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF State of the Art: N 3 LO (Epelbaum, nucl-th/5932) dσ/dω A y A y dσ/dω dσ/dω θ [deg] -.4 A y θ [deg]
54 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Many-Body Forces are Inevitable in EFT! What if we have three nucleons interacting? Successive two-body scatterings with short-lived high-energy intermediate states unresolved = must be absorbed into three-body force 2N forces 3N forces 4N forces A feature, not a bug! How do we organize (3, 4, ) body forces? EFT! [nucl-th/31263]
55 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF Tidal Analog to Nuclear 3-Body Forces Microscopic calculations of nuclear properties are complicated by the importance of three-body forces between protons and neutrons. These are analogous to tidal forces: the gravitational force on the Earth is not just the pairwise sum of Earth-Moon and Earth-Sun forces. The computational cost of three-body nuclear forces is high; e.g., a factor of ten for matrix diagonalization methods.
56 Overview DOFs EFT UNEDF Resolution SRG Principle Taylor Analogy Chiral EFT 3NF 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): 1% of energy in solid xenon
57 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star Given an Interaction, Why Not Just Solve?
58 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star The Many-Body Schrödinger Wave Function [adapted from Joe Carlson] How to represent the wave function for an A-body nucleus? Consider 8 Be (Z = 4 protons, N = 4 neutrons) Ψ = σ,τ χ σ χ τ φ(r) where R are the 3A spatial coordinates χ σ = 1 2 A (2 A terms) = 256 for A = 8 χ τ = n 1 n 2 p A ( A! N!Z! terms) = 7 for 8 Be
59 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star The Many-Body Schrödinger Wave Function [adapted from Joe Carlson] How to represent the wave function for an A-body nucleus? Consider 8 Be (Z = 4 protons, N = 4 neutrons) Ψ = σ,τ χ σ χ τ φ(r) where R are the 3A spatial coordinates χ σ = 1 2 A (2 A terms) = 256 for A = 8 χ τ = n 1 n 2 p A ( A! N!Z! terms) = 7 for 8 Be So for 8 Be there are 17,92 complex functions in 3A 3 = 21 dimensions!
60 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star The Many-Body Schrödinger Wave Function [adapted from Joe Carlson] How to represent the wave function for an A-body nucleus? Consider 8 Be (Z = 4 protons, N = 4 neutrons) Ψ = σ,τ χ σ χ τ φ(r) where R are the 3A spatial coordinates χ σ = 1 2 A (2 A terms) = 256 for A = 8 χ τ = n 1 n 2 p A ( A! N!Z! terms) = 7 for 8 Be So for 8 Be there are 17,92 complex functions in 3A 3 = 21 dimensions! Suppose you represent this for a nucleus of size 1 fm with a mesh spacing of.5 fm. You would need 1 27 grid points! Obviously this is not the way it is done!
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62 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star Universal Nuclear Energy Density Functional
63 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star Universal Nuclear Energy Density Functional
64 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star Goals of SciDAC 2 Project: Building a Universal Nuclear Energy Density Functional Understand nuclear properties for element formation, for properties of stars, and for present and future energy and defense applications Scope is all nuclei (there are more than 5!), with particular interest in reliable calculations of unstable nuclei and in reactions Order of magnitude improvement over present capabilities = precision calculations Connected to the best microscopic physics Maximum predictive power with well-quantified uncertainties [See by Bertsch, Dean, and Nazarewicz]
65 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star The Big Picture: From QCD to Nuclei Density Functional Theory A>1 Coupled Cluster, Shell Model A<1 Exact methods A 12 GFMC, NCSM Lattice QCD Low-mom. interactions Chiral EFT interactions (low-energy theory of QCD) QCD Lagrangian
66 Overview DOFs EFT UNEDF Resolution SRG Solving UNEDF N-Star Neutron Star Anatomy (from D. Page) Dilute neutrons at surface = neutron superfluid (cf. cold atoms?) Outer core: bulk EOS calibrated by nuclei? High density inner core: color superconductor from quark Cooper pairs?
67 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter What Does the Nuclear Potential Look Like? Textbook answer (cf. force between atoms): Momentum units ( = c = 1): typical relative momentum in large nucleus 1 fm 1 2 MeV (?)
68 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Consequences of a Repulsive Core S1 deuteron probability density.4.3 uncorrelated correlated 4 3 ψ(r) 2 [fm 3 ].1 Argonne v 18 ψ(r) 2 [fm 3 ].2.1 V(r) [MeV] r [fm] r [fm] 1 Probability at short separations suppressed = correlations Greatly complicates expansion of many-body wave functions Short-distance structure high-momentum components
69 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Recasting a Problem in Fourier Space Amplitude as a function of time for plucked guitar: Same information is contained in amplitude of frequencies
70 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter
71 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Energy by Adding Up Frequencies/Wavelengths Describe energy by where particle is and how fast it is moving or by adding up energy in each wavelength given the distribution of wavelengths
72 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter 2D Fourier Transforms of Images
73 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter S-Wave (L = ) NN Potential in Momentum Space Fourier transform in partial waves (Bessel transform) V L= (k, k ) = d 3 r j (kr) V (r) j (k r) = k V L= k Repulsive core = big high-k ( 2 fm 1 ) components
74 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter S-Wave (L = ) NN Potential in Momentum Space Fourier transform in partial waves (Bessel transform) V L= (k, k ) = d 3 r j (kr) V (r) j (k r) = k V L= k Repulsive core = big high-k ( 2 fm 1 ) components
75 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Many Short Wavelengths = Large Matrices
76 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Digital Resolution: Higher Resolution is Better (?) Computer screens, printers, digital cameras,... Higher resolution = more pixels Pixel size characteristic scale = greater detail Greater resolution = more $$$
77 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Principle of Effective Low-Energy Theories
78 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Principle of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved
79 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Principle of Effective Low-Energy Theories 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 systematically achieved by effective field theory
80 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter What if your theory has too much resolution?
81 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter What if your theory has too much resolution?
82 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter What if your theory has too much resolution?
83 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter What if your theory has too much resolution? Claim: Nuclear physics with usual V (r) is like using beer coasters
84 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Less Painful to use a Low-Resolution Version! High resolution Low resolution Lower resolution by block spinning or low-pass filter Can greatly reduce storage without distorting message
85 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Low-Pass Filter on an Image
86 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Try a Low-Pass Filter on V (r) = Set to zero high momentum (k 2 fm 1 ) matrix elements and see the effect on low-energy observables
87 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter Use Phase Shifts to Test sin(kr+δ) 6 1 S R r phase shift (degrees) 4 2 AV E lab (MeV) Here: 1 S (spin-singlet, L =, J = ) neutron-proton scattering Different phase shifts in each partial wave channel
88 Overview DOFs EFT UNEDF Resolution SRG Potential Fourier Digital Low-Energy OSU Filter A Low-Pass Filter on V (r) Fails At Low Energy 6 1 S R sin(kr+δ) r phase shift (degrees) 4 2 AV18 AV18 [k max = 2.2 fm -1 ] E lab (MeV)
89 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Why Did Our Low-Pass Filter Fail? Basic problem: low k and high k are coupled E.g., perturbation theory for (tangent of) phase shift: k V k + k k V k k V k (k 2 k 2 )/m S Solution: Unitary transformation of the H matrix = decouple! E n = Ψ n H Ψ n = ( Ψ n U )UHU (U Ψ n ) = Ψ n H Ψ n phase shift (degrees) 4 2 AV18 AV18 [k max = 2.2 fm -1 ] Here: Use flow equations E lab (MeV)
90 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Why Did Our Low-Pass Filter Fail? Can t simply change short-distance structure without changing the low-energy results The reason is that there are virtual intermediate states But the effect of short-distance physics on low-energy physics can be absorbed by adjustments in the basic forces = Renormalization Group α() ; α(m W ) 1 128
91 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Computational Aside: Digital Potentials Although momentum is continuous in principle, in practice represented as discrete (gaussian quadrature) grid: = Calculations become just matrix multiplications! E.g., k V k + k k V k k V k (k 2 k 2 )/m + = V ii+ j V ij V ji 1 (k 2 i k 2 j )/m Resolution is sufficient for many significant figures
92 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
93 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
94 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
95 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
96 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
97 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
98 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
99 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
100 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
101 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
102 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
103 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
104 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
105 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M 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 )
106 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Unitary Transformation: Bare vs. SRG phase shifts δ [deg] S 1 3 P1 P bare ps vsrg ps δ [deg] P1 F3 G E lab [MeV] E lab [MeV] E lab [MeV]
107 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Low-Pass Filters Work! phase shift [deg] [nucl-th/7113] Phase shifts with V s (k, k ) = for k, k > k max S 3 S P 3 F E lab [MeV] D1 5 1 AV18 AV18 [k max = 2.2 fm -1 ] V s [k max = 2.2 fm -1 ]
108 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Consequences of a Repulsive Core Revisited S1 deuteron probability density.3 uncorrelated correlated 3 ψ(r) 2 [fm 3 ].15.1 Argonne v 18 ψ(r) 2 [fm 3 ].2.1 V(r) [MeV] r [fm] r [fm] Probability at short separations suppressed = correlations Greatly complicates expansion of many-body wave functions Short-distance structure high-momentum components 1
109 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Consequences of a Repulsive Core Revisited S1 deuteron probability density.3 uncorrelated correlated 3 ψ(r) 2 [fm -3 ].15.1 Argonne v 18 λ = 4. fm -1 λ = 3. fm -1 λ = 2. fm -1 ψ(r) 2 [fm 3 ].2.1 V(r) [MeV] r [fm] r [fm] Transformed potential = no short-range correlations in wf Potential is now non-local: V (r)ψ(r) d 3 r V (r, r )ψ(r ) Not a problem for many-body methods using ho matrix elements 1
110 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Variational Calculations Converge Rapidly Absolute error vs. λ of the predicted deuteron binding energy from a variational calculation in a fixed-size basis of harmonic oscillators (N max ω excitations) 1 N 3 LO [55/6] E d - E var [MeV] N max = 6 N max = 1 N max = λ [fm 1 ]
111 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Band Diagonalizing with SRG Transform an initial hamiltonian, H = T + V: H s = U(s)HU (s) T + V s, [nucl-th/61145] 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 T diagonal = T D : which yields the flow equation, η(s) = [T D, H s ], dh s ds = dv s ds = [[T D, H s ], H s ]. T D determines flow: T D = T or T 2 or H D ( T + (V s ) D ) or...
112 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body Variational Calculations in Three-Nucleon Systems Triton ground-state energy vs. size of harmonic oscillator basis (N max ω excitations) Rapid convergence as λ decreases See arxiv: for more examples Different binding energies = 3-body contribution E t [MeV] MeV N max initial λ = 4 fm 1 λ = 3 fm 1 λ = 2 fm 1 λ = 1 fm 1 55/6 MeV N max
113 Overview DOFs EFT UNEDF Resolution SRG Coupling Flow Decoupling SRG 3-Body 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): 1% of energy in solid xenon
114
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