Effective Field Theory for Density Functional Theory I

Transcription

1 Effective Field Theory for Density Functional Theory I Department of Physics Ohio State University February, 26 I. Overview of EFT, RG, DFT for fermion many-body systems II. EFT/DFT for dilute Fermi systems III. Refinements: Toward EFT/DFT for nuclei IV. Loose ends and challenges, Cold atoms, RG/DFT

2 Outline Intro DFT Nuclei Summary Appendix Outline Overview of Fermion Many-Body Systems Density Functional Theory for Coulomb Systems DFT for Nuclei? = EFT and RG Summary I

3 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF Outline Overview of Fermion Many-Body Systems Density Functional Theory for Coulomb Systems DFT for Nuclei? = EFT and RG Summary I

4 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF Examples of Fermion Many-Body Systems Collections of fundamental fermions (electrons, quarks,... ) or of composites of odd number of fermions (e.g., protons)

5 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF Examples of Fermion Many-Body Systems Collections of fundamental fermions (electrons, quarks,... ) or of composites of odd number of fermions (e.g., protons) Isolated atoms or molecules electrons interacting via long-range (screened) Coulomb find charge distribution, binding energy, bond lengths,... Bulk solid-state materials metals, insulators, semiconductors, superconductors,... Liquid 3 He (superfluid!) Cold fermionic atoms in (optical) traps [ 6 Li or 7 Li?] Atomic nuclei Neutron stars neutron matter color superconducting quark matter

6 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF Examples of Fermion Many-Body Systems Collections of fundamental fermions (electrons, quarks,... ) or of composites of odd number of fermions (e.g., protons) Isolated atoms or molecules electrons interacting via long-range (screened) Coulomb find charge distribution, binding energy, bond lengths,... Bulk solid-state materials metals, insulators, semiconductors, superconductors,... Liquid 3 He (superfluid!) Cold fermionic atoms in (optical) traps [ 6 Li or 7 Li?] Atomic nuclei Neutron stars neutron matter color superconducting quark matter

7 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF Nuclear and Cold Atom Many-Body Problems Lennard-Jones and nucleon-nucleon potentials O -O potential (mev) n-n potential (MeV) n-n system O -O system n-n distance (fm) O 2-O 2 distance (nm) [figure borrowed from J. Dobaczewski] Are there universal features of such many-body systems? How can we deal with hard cores in many-body systems?

8 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF The Big Picture (adapted from

9 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF 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

10 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF 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!

11 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF 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!

12 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF Hartree-Fock Wave Function Ψ HF Ĥ Ψ HF = 1 2 Best single Slater determinant in variational sense Ψ HF = det{φ i (x), i = 1 A}, x = (r, σ, τ) Hartree-Fock energy: A i=1 + = 2 dx φ i φ i + 1 2M 2 A i,j=1 A dx dy φ i (x)φ i(y)v(x, y)φ j (y)φ j(x) + i,j=1 dx dy φ i (x) 2 v(x, y) φ j (y) 2 A dy v ext (y) φ j (y) 2 Determine the φ i by varying with fixed normalization: δ ( A δφ i (x) Ψ HF Ĥ Ψ HF ɛ j dy φ j (y) 2) = j=1 i=1

13 Outline Intro DFT Nuclei Summary Appendix Systems Big Picture WF Hartree-Fock Wave Function Best single Slater determinant in variational sense Ψ HF = det{φ i (x), i = 1 A}, x = (r, σ, τ) The φ i (x) satisfy non-local Schrödinger equations: ( ) 2 2M φ i(x) + V H (x) + v ext (x) φ i (x) + dy V E (x, y)φ i (y) = ɛ i φ i (x) with V H (x) = dy A φ j (y) 2 v(x, y), V E (x, y) = v(x, y) j=1 + = A φ j (x)φ j (y) j=1 Solve self-consistently; non-trivial because non-local

14 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Outline Overview of Fermion Many-Body Systems Density Functional Theory for Coulomb Systems DFT for Nuclei? = EFT and RG Summary I

15 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Density Functional Theory (DFT) Dominant application: inhomogeneous electron gas Interacting point electrons in static potential of atomic nuclei number of retrieved records per year 1 1 Hartree Fock Density Functional Theory Ab initio calculations of atoms, molecules, crystals, surfaces, year

16 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Density Functional Theory (DFT) Hohenberg-Kohn: There exists an energy functional E vext [ρ]... E vext [ρ] = F HK [ρ] + d 3 x v ext (x)ρ(x) F HK is universal (same for any external v ext ) = H 2 to DNA! Useful if you can approximate the energy functional Introduce orbitals and minimize energy functional = E gs, ρ gs

17 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Kohn-Sham DFT for v ext = V HO Harmonic Trap V HO = V KS

18 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Kohn-Sham DFT for v ext = V HO Harmonic Trap V HO = V KS Interacting density in V HO Non-interacting density in V KS Orbitals {ψ i (x)} in local potential V KS ([ρ], x) [ 2 /2m + V KS (x)]ψ i = ε i ψ i = ρ(x) = A ψ i (x) 2 i=1 Find Kohn-Sham potential V KS ([ρ], x) from δe vext [ρ]/δρ(x) Solve self-consistently

19 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham DFT for Solid-State or Molecular Systems HK free energy for an inhomogeneous electron gas F HK [ρ(x)] = T KS [ρ(x)] + e2 d 3 x d 3 x ρ(x)ρ(x ) 2 x x + E xc [ρ(x)] Then V KS = v ext eφ + v xc with v xc (x) = δe xc /δρ(x) Kohn-Sham T KS [ρ(x)]: find normalized {ψ i, ɛ i } from ) ( 2 2m 2 + V KS (x) ψ i (x) = ɛ i ψ i (x) so that ρ(x) = A i=1 ψ i(x) 2 and T KS [ρ(x)] = A i=1 ψ i 2 2m 2 i ψ i = A i=1 ɛ i d 3 x ρ(x)v KS (x)

20 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham ε Local density approximation: E xc [ρ(x)] d 3 x E xc (ρ(x)) fit E xc (ρ) to Monte Carlo calculation of uniform electron gas ε xc ε tot r s n r 2 2 1s 1 2s+p Ar Z=18 3s+p DFT T F r in practice, use parametric formulas for energy density, e.g., E xc (ρ)/ρ =.458/r s.666g(r s /11.4) with G(x) = 1 { (1 + x) 3 log(1 + x 1 ) x x 1 } 3 just like naive Hartree approach with additional potential: v xc (x) = d[e xc(ρ)] dρ ρ=ρ(x)

21 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Density Functional Theory (DFT) Dominant application: inhomogeneous electron gas Interacting point electrons in static potential of atomic nuclei Ab initio calculations of atoms, molecules, crystals, surfaces,... HF is good starting point, DFT/LDA is better, DFT/GGA is best % deviation from experiment Atomization Energies of Hydrocarbon Molecules Hartree-Fock DFT Local Spin Density Approximation DFT Generalized Gradient Approximation H 2 C 2 C 2 H 2 CH 4 C 2 H 4 C 2 H 6 C 6 H 6 molecule e.g., van Leeuwen Baerends GGA v xc (r) = βρ 1/3 x 2 (r) (r) 1 + 3βx(r) sinh 1 (x(r)) with x = ρ /ρ 4/3

22 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Quotes From the DFT Literature

23 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Quotes From the DFT Literature A Chemist s Guide to DFT (Koch & Holthausen, 2) To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations.

24 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Quotes From the DFT Literature A Chemist s Guide to DFT (Koch & Holthausen, 2) To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. Density Functional Theory (AJP, Argaman & Makov, 2) It is important to stress that all practical applications of DFT rest on essentially uncontrolled approximations, such as the LDA...

25 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Quotes From the DFT Literature A Chemist s Guide to DFT (Koch & Holthausen, 2) To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. Density Functional Theory (AJP, Argaman & Makov, 2) It is important to stress that all practical applications of DFT rest on essentially uncontrolled approximations, such as the LDA... Meta-Generalized Gradient Approximation (Perdew et al., 1999) Some say that there is no systematic way to construct density functional approximations. But there are more or less systematic ways, and the approach taken... here is one of the former.

26 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Preview of DFT as Effective Action Recall ordinary thermodynamics with N particles at T = Use a chemical potential µ as source to change N ( ) Ω Ω(µ) = kt ln Z (µ) and N = µ Invert to find µ(n), Legendre transform to F (N) F (N) = Ω(µ(N)) + µ(n)n = This is our energy function! TV

27 Outline Intro DFT Nuclei Summary Appendix HK Kohn-Sham Preview of DFT as Effective Action Recall ordinary thermodynamics with N particles at T = Use a chemical potential µ as source to change N ( ) Ω Ω(µ) = kt ln Z (µ) and N = µ Invert to find µ(n), Legendre transform to F (N) F (N) = Ω(µ(N)) + µ(n)n = This is our energy function! Generalize to spatially dependent chemical potential J(x) Z (µ) Z [J(x)] and µn = µ ψ ψ J(x)ψ ψ(x) LT from ln Z [J(x)] to Γ[ρ(x)], where ρ = ψ ψ J = DFT! TV

28 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Outline Overview of Fermion Many-Body Systems Density Functional Theory for Coulomb Systems DFT for Nuclei? = EFT and RG Summary I

29 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Table of the Nuclides

30 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Nuclear and Cold Atom Many-Body Problems Lennard-Jones and nucleon-nucleon potentials O -O potential (mev) n-n potential (MeV) n-n system O -O system n-n distance (fm) O 2-O 2 distance (nm) [figure borrowed from J. Dobaczewski] Are there universal features of such many-body systems? How can we deal with hard cores in many-body systems?

31 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Nuclear Matter in Low-Order Perturbation Theory Standard Argonne v 18 potential Brueckner ladders order-by-order 1st order is Hartree-Fock = unbound! Repulsive core = series diverges E/A [MeV] st order 2nd order pp ladder 3rd order pp ladder Argonne v k f [fm -1 ]

32 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Hartree-Fock Wave Function Best single Slater determinant in variational sense Ψ HF = det{φ i (x), i = 1 A}, x = (r, σ, τ) The φ i (x) satisfy non-local Schrödinger equations: with V H (x) = dy 2 2M φ i(x) + V H (x)φ i (x) + dy V ex (x, y)φ i (y) = ɛ i φ i (x) A φ j (y) 2 v(x, y), V ex (x, y) = v(x, y) j=1 + = A φ j (x)φ j (y) j=1 Solve self-consistently; somewhat tricky because non-local = much simpler if v(x, y) δ(x y)

33 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Skyrme Hartree-Fock Energy Functionals Skyrme: Do Hartree-Fock with V Skyrme 2 + V Skyrme 3, where k V Skyrme 2 k = t t 1(k 2 +k 2 )+t 2 k k +iw (σ 1 +σ 2 ) k k

34 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Skyrme Hartree-Fock Energy Functionals Skyrme: Do Hartree-Fock with V Skyrme 2 + V Skyrme 3, where k V Skyrme 2 k = t t 1(k 2 +k 2 )+t 2 k k +iw (σ 1 +σ 2 ) k k Motivates Skyrme energy density functional (for N = Z ): E[ρ, τ, J] = 1 2M τ t ρ t 3ρ 2+α (3t 1 + 5t 2 )ρτ (9t 1 5t 2 )( ρ) W ρ J (t 1 t 2 )J 2 where ρ(x) = i φ i(x) 2 and τ(x) = i φ i(x) 2 (and J)

35 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Skyrme Hartree-Fock Energy Functionals Skyrme: Do Hartree-Fock with V Skyrme 2 + V Skyrme 3, where k V Skyrme 2 k = t t 1(k 2 +k 2 )+t 2 k k +iw (σ 1 +σ 2 ) k k Motivates Skyrme energy density functional (for N = Z ): E[ρ, τ, J] = 1 2M τ t ρ t 3ρ 2+α (3t 1 + 5t 2 )ρτ (9t 1 5t 2 )( ρ) W ρ J (t 1 t 2 )J 2 where ρ(x) = i φ i(x) 2 and τ(x) = i φ i(x) 2 (and J) Minimize E = dx E[ρ, τ, J] by varying the (normalized) φ i s ( 1 2M (x) + U(x) W ρ 1 ) i σ φ i (x) = ɛ i φ i (x), U = 3 4 t ρ + ( 3 16 t t 2)τ + and 1 2M (x) = 1 2M + ( 3 16 t t 2)ρ Iterate until φ i s and ɛ i s are self-consistent

36 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Problems with Extrapolations Mass formulas and energy functionals do well where there is data, but elsewhere... two-neutron separation energies S 2n (MeV) Sn data exist Experiment HFB-SLy4 HFB-SkP HFB-D1S SkX RHB-NL3 LEDF Mass Formulae Neutron Number data do not exist Neutron Number Neutron Number exp FRDM CKZ JM MJ T+ Figure 6: Predicted two-neutron separation energies EFT for the for even-even DFT I Sn isotopes using several

37 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Questions and Criticisms of Skyrme HF Typical [e.g., SkIII] model parameters (in units of MeV-fm n ): t = 1129 t 1 = 395 t 2 = 95 t 3 = 14 W = 12 These seem large; is there an expansion parameter? Where does ρ 2+α come from? Why not ρ 2+β? Parameter Fitting [von Neumann via Fermi via Dyson]: With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

38 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Questions and Criticisms of Skyrme HF Typical [e.g., SkIII] model parameters (in units of MeV-fm n ): t = 1129 t 1 = 395 t 2 = 95 t 3 = 14 W = 12 These seem large; is there an expansion parameter? Where does ρ 2+α come from? Why not ρ 2+β? Parameter Fitting [von Neumann via Fermi via Dyson]: With four parameters I can fit an elephant, and with five I can make him wiggle his trunk. Skyrme HF is only mean-field; too simple for NN correlations Law of the Conservation of Difficulty Difficulty in a solution to a problem is always conserved regardless of the technique used in the solution. How do we improve the approach? Is pairing treated correctly?

39 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Questions and Criticisms of Skyrme HF Typical [e.g., SkIII] model parameters (in units of MeV-fm n ): t = 1129 t 1 = 395 t 2 = 95 t 3 = 14 W = 12 These seem large; is there an expansion parameter? Where does ρ 2+α come from? Why not ρ 2+β? Parameter Fitting [von Neumann via Fermi via Dyson]: With four parameters I can fit an elephant, and with five I can make him wiggle his trunk. Skyrme HF is only mean-field; too simple for NN correlations Law of the Conservation of Difficulty Difficulty in a solution to a problem is always conserved regardless of the technique used in the solution. How do we improve the approach? Is pairing treated correctly? How does Skyrme HF relate to NN (and NNN) forces?

40 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Nuclear) Many-Body Physics: Old Approach

41 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Nuclear) Many-Body Physics: Old Approach One Hamiltonian for all problems and energy/length scales (not QCD!) For nuclear structure, protons and neutrons with a local potential fit to NN data

42 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Nuclear) Many-Body Physics: Old Approach One Hamiltonian for all problems and energy/length scales (not QCD!) Find the best potential For nuclear structure, protons and neutrons with a local potential fit to NN data NN potential with χ 2 /dof 1 up to 3 MeV energy

43 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Nuclear) Many-Body Physics: Old Approach One Hamiltonian for all problems and energy/length scales (not QCD!) Find the best potential Two-body data may be sufficient; many-body forces as last resort For nuclear structure, protons and neutrons with a local potential fit to NN data NN potential with χ 2 /dof 1 up to 3 MeV energy Let phenomenology dictate whether three-body forces are needed (answer: yes!)

44 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Nuclear) Many-Body Physics: Old Approach One Hamiltonian for all problems and energy/length scales (not QCD!) Find the best potential Two-body data may be sufficient; many-body forces as last resort Avoid (hide) divergences For nuclear structure, protons and neutrons with a local potential fit to NN data NN potential with χ 2 /dof 1 up to 3 MeV energy Let phenomenology dictate whether three-body forces are needed (answer: yes!) Add form factor to suppress high-energy intermediate states; don t consider cutoff dependence

45 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Nuclear) Many-Body Physics: Old Approach One Hamiltonian for all problems and energy/length scales (not QCD!) Find the best potential Two-body data may be sufficient; many-body forces as last resort Avoid (hide) divergences Choose approximations by art For nuclear structure, protons and neutrons with a local potential fit to NN data NN potential with χ 2 /dof 1 up to 3 MeV energy Let phenomenology dictate whether three-body forces are needed (answer: yes!) Add form factor to suppress high-energy intermediate states; don t consider cutoff dependence Use physical arguments (often handwaving) to justify the subset of diagrams used

46 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Why Use EFT For Many-Body Physics? Systematic calculations with error estimates Reliable, model independent extrapolation Analogy between EFT and basic numerical analysis naive error analysis: pick a method and reduce the mesh size (e.g., increase grid points) until the error is acceptable sophisticated error analysis: understand scaling and sources of error (e.g., algorithm vs. round-off errors) = Does it work as well as it should? representation dependence = not all are equally effective! extrapolation: completeness of an expansion basis Quantum mechanics makes EFT trickier!

47 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Error Plots in Numerical Analysis relative error h n to h n Numerical Integration trapezoid rule O(h 2 ) mesh size h

48 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Error Plots in Numerical Analysis Numerical Integration relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) mesh size h

49 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Error Plots in Numerical Analysis Numerical Integration relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) mesh size h

50 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx

51 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration mesh size h

52 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

53 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

54 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

55 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Principles of Effective Low-Energy Theories

56 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Principles of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved

57 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Principles 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

58 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

59 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

60 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

61 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

62 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

63 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

64 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

65 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

66 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Wavelength and Resolution

67 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Sources of Nonperturbative Physics for NN 1 Strong short-range repulsion ( hard core ) 2 Iterated tensor (S 12 ) interaction 3 Near zero-energy bound states

68 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Sources of Nonperturbative Physics for NN 1 Strong short-range repulsion ( hard core ) 2 Iterated tensor (S 12 ) interaction 3 Near zero-energy bound states Consequences: In Coulomb DFT, Hartree-Fock gives dominate contribution = correlations are small corrections = DFT works! cf. NN interactions = correlations HF = DFT fails??

69 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Sources of Nonperturbative Physics for NN 1 Strong short-range repulsion ( hard core ) 2 Iterated tensor (S 12 ) interaction 3 Near zero-energy bound states Consequences: In Coulomb DFT, Hartree-Fock gives dominate contribution = correlations are small corrections = DFT works! cf. NN interactions = correlations HF = DFT fails?? However... the first two depend on the resolution = different cutoffs third one is affected by Pauli blocking

70 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Deuteron (Bound np) at High Resolution ψ(r) 2 [fm -3 ] S1 deuteron probability density Argonne v 18 ψ (k) 2 [fm 3 ] S1 deuteron w.f. momentum probability density Argonne v r [fm] k [fm -1 ] Repulsive core = short-distance suppression = important high-momentum (small λ) components Makes the many-body problem complicated!

71 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Deuteron at Lower Resolutions S1 deuteron probability density S1 deuteron w.f. momentum probability density ψ(r) 2 [fm -3 ].15.1 Argonne v 18 Λ = 4. fm -1 Λ = 3. fm -1 Λ = 2. fm -1 ψ (k) 2 [fm 3 ] Argonne v 18 Λ = 4 fm -1 Λ = 3 fm -1 Λ = 2 fm r [fm] Repulsive core = short-distance suppression / high-momentum components k [fm -1 ] Low-momentum potential = much simpler wave function!

72 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Deuteron at Different Resolutions more Integrand of ψ d V Λ ψ d for Λ =6. fm k (fm 1 ) k (fm 1 ) 1

73 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Deuteron at Different Resolutions more Integrand of ψ d V Λ ψ d for Λ =5. fm k (fm 1 ) k (fm 1 ) 1

74 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Deuteron at Different Resolutions more Integrand of ψ d V Λ ψ d for Λ =4. fm k (fm 1 ) k (fm 1 ) 1

75 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Deuteron at Different Resolutions more Integrand of ψ d V Λ ψ d for Λ =3. fm k (fm 1 ) k (fm 1 ) 1

76 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM The Deuteron at Different Resolutions more Integrand of ψ d V Λ ψ d for Λ =2. fm k (fm 1 ) k (fm 1 ) 1

77 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Consequence for Basis Expansions [nucl-th/6217] Error in Deuteron Binding vs. Cutoff N max = 4 hω (Ω optimized at each Λ) (E var - E d ) [MeV] sharp cutoff exponential cutoff (n=4) Λ [fm -1 ]

78 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Consequence for Basis Expansions [nucl-th/6217] 1 Error in Triton Binding vs. Cutoff N max = 6 N max = 1 N max = 2 (E t - E var )/E t Λ [fm -1 ]

79 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Consequence for Basis Expansions [nucl-th/6217] 8. Triton Binding Energy vs. HO Basis Size 8.1 b osc = 1.5 fm solid --> smooth cutoff Λ = 2. fm -1 Λ = 1.6 fm -1 E t [MeV] N max

80 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Two-Body Correlations in Nuclear Matter?.2 χ(r) S defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = ) Argonne v r [fm]

81 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Two-Body Correlations in Nuclear Matter?.2 χ(r) S defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = ) Argonne v 18 Λ = 4.5 fm r [fm]

82 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Two-Body Correlations in Nuclear Matter?.2 χ(r) S defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = ) Argonne v 18 Λ = 4.5 fm -1 Λ = 3.5 fm r [fm]

83 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Two-Body Correlations in Nuclear Matter?.2 χ(r) S defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = ) Argonne v 18 Λ = 4.5 fm -1 Λ = 3.5 fm -1 Λ = 2.5 fm r [fm]

84 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Two-Body Correlations in Nuclear Matter?.2 χ(r) S defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = ) Argonne v 18 Λ = 4.5 fm -1 Λ = 3.5 fm -1 Λ = 2.5 fm -1 N 3 LO r [fm]

85 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Why is In-Medium T Perturbative for V low k? Phase space in pp-channel strongly suppressed: q 2 dq V NN(k, q)v NN(q, k) k F k 2 q 2 Λ vs. q 2 dq V low k(k, q)v low k(q, k) k F k 2 q 2 Tames hard core, tensor, and bound state Λ: P/2 ± k > k F k P/2 F: P/2 ± k < k F and k < Λ Λ k F

86 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Nuclear Matter with NN Ladders Only st order 2nd order pp ladder 3rd order pp ladder Brueckner ladders order-by-order Repulsive core = series diverges E/A [MeV] 5 Argonne v k f [fm -1 ]

87 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Nuclear Matter with NN Ladders Only st order 2nd order pp ladder 3rd order pp ladder Brueckner ladders order-by-order Repulsive core = series diverges E/A [MeV] 5 Argonne v 18 V low k converges -5 V low k ( Λ=1.9 fm -1 ) k f [fm -1 ]

88 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Nuclear Matter with NN Ladders Only st order 2nd order pp ladder 3rd order pp ladder Brueckner ladders order-by-order Repulsive core = series diverges E/A [MeV] 5 Argonne v 18 V low k converges No saturation in sight! -5-1 V low k ( Λ=1.9 fm -1 ) k f [fm -1 ]

89 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Deja Vu All Over Again? There were active attempts to transform away hard cores and soften the tensor interaction in the late sixties and early seventies. But the requiem for soft potentials was given by Bethe (1971): Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required. Next 3+ years trying to solve accurately with hard potential

90 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Deja Vu All Over Again? There were active attempts to transform away hard cores and soften the tensor interaction in the late sixties and early seventies. But the requiem for soft potentials was given by Bethe (1971): Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required. Next 3+ years trying to solve accurately with hard potential But the story is not complete: three-nucleon forces (3NF)!

91 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM V low k with Chiral 3NF Ideal: Start with chiral NN + 3NF EFT and run Λ Possible now: Run NN and fit 3NF EFT at each Λ Bogner, Nogga, Schwenk, Phys. Rev. C 7 (24) 612 two-pion-exchange c i s from NN PSA fit two free parameters fit to 3 H and 4 He binding energies ratio 2NF/3NF consistent with chiral counting

92 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Approximate) Nuclear Matter with NN and NNN Hartree-Fock 5 Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 Λ = 2.1 fm -1 [no V 3N ] E/A [MeV] k F [fm -1 ]

93 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM (Approximate) Nuclear Matter with NN and NNN Hartree-Fock 2nd Order 5 Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 Λ = 2.1 fm -1 [no V 3N ] 5 Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 Λ = 2.1 fm -1 [no V 3N ] E/A [MeV] -5 E/A [MeV] k F [fm -1 ] k F [fm -1 ] 3-body out of control?

94 Outline Intro DFT Nuclei Summary Appendix Skyrme Analogs V lowk NM Low-Momentum Potential for Neutron Matter Removing hard core = simpler many-body starting point for neutron matter [Schwenk, Friman, Brown] 25 2 Akmal et al. AV18 (1998) BHF: Bao et al. (1994) V low k Hartree-Fock Akmal et al. AV18+UIX+boost (1998) E/A [MeV] ρ [fm -3 ] Simple Hartree-Fock (circles) matches best calculations!

95 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow Outline Overview of Fermion Many-Body Systems Density Functional Theory for Coulomb Systems DFT for Nuclei? = EFT and RG Summary I

96 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow Bethe and Calculating Nuclear Matter Hans Bethe in review of nuclear matter (1971): The theory must be such that it can deal with any nucleon-nucleon (NN) force, including hard or soft core, tensor forces, and other complications. It ought not to be necessary to tailor the NN force for the sake of making the computation of nuclear matter (or finite nuclei) easier, but the force should be chosen on the basis of NN experiments (and possibly subsidiary experimental evidence, like the binding energy of H 3 ).

97 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow EFT and RG Make Many-Body Physics Easier There s an old vaudeville joke about a doctor and patient...

98 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow EFT and RG Make Many-Body Physics Easier There s an old vaudeville joke about a doctor and patient... Patient: Doctor, doctor, it hurts when I do this!

99 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow EFT and RG Make Many-Body Physics Easier 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.

100 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow EFT and RG Make Many-Body Physics Easier 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. 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!

101 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow (Nuclear) Many-Body Physics: Old vs. New

102 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians

103 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one!

104 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Two-body data may be sufficient; many-body forces as last resort Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one! Many-body data needed and many-body forces inevitable

105 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Two-body data may be sufficient; many-body forces as last resort Avoid divergences Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one! Many-body data needed and many-body forces inevitable Exploit divergences (cutoff dependence as tool)

106 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow (Nuclear) Many-Body Physics: Old vs. New One Hamiltonian for all problems and energy/length scales Find the best potential Two-body data may be sufficient; many-body forces as last resort Avoid divergences Choose diagrams by art Infinite # of low-energy potentials; different resolutions = different dof s and Hamiltonians There is no best potential = use a convenient one! Many-body data needed and many-body forces inevitable Exploit divergences (cutoff dependence as tool) Power counting determines diagrams and truncation error

107 Outline Intro DFT Nuclei Summary Appendix Bethe EFT/RG Tomorrow Simple Many-Body Problem: Hard Spheres Infinite potential at radius R Scattering solutions are simple: u (r) = sin[k(r-r)] E k = k 2 /M R r R >>R What is the energy / particle and density profile of the trapped many-body system?

108 Outline Renormalization deuteron Weinberg Renormalization Old vs. New More Deuteron Variational Weinberg Eigenvalues

109 Outline Renormalization deuteron Weinberg Renormalization: Old vs. New

110 Outline Renormalization deuteron Weinberg Renormalization: Old vs. New Renormalization is technical device to get rid of divergences in perturbation theory. Renormalization is an expression of the variation of the structure of physical interactions with changes in the scale of the phenomena being probed.

111 Outline Renormalization deuteron Weinberg Renormalization: Old vs. New Renormalization is technical device to get rid of divergences in perturbation theory. Focus on the high-energy behavior and on ways of circumventing divergences Renormalization is an expression of the variation of the structure of physical interactions with changes in the scale of the phenomena being probed. Focus on finite variation of physical interactions with finite changes of energy

112 Outline Renormalization deuteron Weinberg Renormalization: Old vs. New Renormalization is technical device to get rid of divergences in perturbation theory. Focus on the high-energy behavior and on ways of circumventing divergences Cutoff Λ is artificial variable; Λ at end Renormalization is an expression of the variation of the structure of physical interactions with changes in the scale of the phenomena being probed. Focus on finite variation of physical interactions with finite changes of energy Λ is boundary of unknown/unresolved physics; keep Λ finite. Remove Λ dependence systematically

113 Outline Renormalization deuteron Weinberg Renormalization: Old vs. New Renormalization is technical device to get rid of divergences in perturbation theory. Focus on the high-energy behavior and on ways of circumventing divergences Cutoff Λ is artificial variable; Λ at end Non-renormalizable means no predictive power = renormalizable theories Renormalization is an expression of the variation of the structure of physical interactions with changes in the scale of the phenomena being probed. Focus on finite variation of physical interactions with finite changes of energy Λ is boundary of unknown/unresolved physics; keep Λ finite. Remove Λ dependence systematically Non-renormalizable = systematic expansion! Effective field theories

114 Outline Renormalization deuteron Weinberg More Deuteron Variational Calculations (E var - E d ) [MeV] Eq. (1) Eq. (11), j max = 3 Eq. (11), j max = Λ (fm -1 )

115 Outline Renormalization deuteron Weinberg Convergence of the Born Series for Scattering Consider whether the Born series converges for given E T (E) = V + V 1 E H V + V 1 E H V 1 E H V + For fixed E, find (complex) eigenvalues η ν (E) [Weinberg] 1 E H V Γ ν = η ν Γ ν = T (E) Γ ν = V Γ ν (1 + η ν + η 2 ν + ) = T diverges if any η ν (E) 1

116 Outline Renormalization deuteron Weinberg Convergence of the Born Series for Scattering Consider whether the Born series converges for given E T (E) = V + V 1 E H V + V 1 E H V 1 E H V + For fixed E, find (complex) eigenvalues η ν (E) [Weinberg] 1 E H V Γ ν = η ν Γ ν = T (E) Γ ν = V Γ ν (1 + η ν + η 2 ν + ) = T diverges if any η ν (E) 1 For E <, same as finding η ν where V /η ν has bound state (H + V /η ν ) b = E b with η ν > or η ν <

117 Outline Renormalization deuteron Weinberg Weinberg Eigenvalues as Function of Cutoff 3 S1 (E cm = MeV) Deuteron = attractive eigenvalue η ν Λ = unchanged Hard core = repulsive eigenvalue η ν Λ = reduced η ν free space, η > free space, η < Λ [fm -1 ]

118 Outline Renormalization deuteron Weinberg Weinberg Eigenvalues as Function of Cutoff 3 S1 (E cm = MeV) Deuteron = attractive eigenvalue η ν Λ = unchanged Hard core = repulsive eigenvalue η ν Λ = reduced In medium: both reduced η ν 1 for Λ 2 fm 1 = perturbative (in pp) η ν free space, η > free space, η < k f = 1.35 fm -1, η > k f = 1.35 fm -1, η < Λ [fm -1 ]

119 Outline Renormalization deuteron Weinberg Weinberg Eigenvalues as Function of Density 1 3 S1 with Pauli blocking η ν (B d ) -1 Λ = 8. fm k f [fm -1 ]

120 Outline Renormalization deuteron Weinberg Weinberg Eigenvalues as Function of Density 1 3 S1 with Pauli blocking η ν (B d ) -1-2 Λ = 8. fm -1 Λ = 4. fm k f [fm -1 ]

121 Outline Renormalization deuteron Weinberg Weinberg Eigenvalues as Function of Density 1 3 S1 with Pauli blocking η ν (B d ) -1-2 Λ = 8. fm -1 Λ = 4. fm -1 Λ = 3. fm k f [fm -1 ]

122 Outline Renormalization deuteron Weinberg Weinberg Eigenvalues as Function of Density 1 3 S1 with Pauli blocking η ν (B d ) -1-2 Λ = 8. fm -1 Λ = 4. fm -1 Λ = 3. fm -1 Λ = 2. fm k f [fm -1 ]

123 Outline Renormalization deuteron Weinberg Collapse of Weinberg Eigenvalues.2 1 S η ν (E=) Argonne v 18 N 2 LO-55/6 [19] N 3 LO-45/5 [14] N 3 LO-55/6 [14] N 3 LO-6/7 [14] N 3 LO [13] Λ (fm -1 )

124 Outline Renormalization deuteron Weinberg Collapse of Weinberg Eigenvalues.5 3 S1 3 D Λ (fm -1 )

125 O 2 -O 2 potential (mev) n-n potential (MeV) n-n system O -O system n-n distance (fm) O 2-O 2 distance (nm) [figure borrowed from J. Dobaczewski]

126 Atomization Energies of Hydrocarbon Molecules 2 % deviation from experiment Hartree-Fock DFT Local Spin Density Approximation DFT Generalized Gradient Approximation -1 H 2 C 2 C 2 H 2 CH 4 C 2 H 4 C 2 H 6 C 6 H 6 molecule

127 Local density approximation fit E xc (ρ) to Monte Carlo cal ε.1 ε tot.2.3 ε xc r s

128 culation of uniform electron gas 2 1s 2s+p Ar Z=18 DFT T F n r 2 1 3s+p r

129 Outline Renormalization deuteron Weinberg The Deuteron at Different Resolutions back Integrand of ψ d V Λ ψ d for Λ =2. fm k (fm 1 ) k (fm 1 ).5 4 2

130 Outline Renormalization deuteron Weinberg The Deuteron at Different Resolutions back Integrand of ψ d V Λ ψ d for Λ =1.8 fm k (fm 1 ) k (fm 1 ).5 4 2

131 Outline Renormalization deuteron Weinberg The Deuteron at Different Resolutions back Integrand of ψ d V Λ ψ d for Λ =1.6 fm k (fm 1 ) k (fm 1 ).5 4 2

132 Outline Renormalization deuteron Weinberg The Deuteron at Different Resolutions back Integrand of ψ d V Λ ψ d for Λ =1.4 fm k (fm 1 ) k (fm 1 ).5 4 2

133 Outline Renormalization deuteron Weinberg The Deuteron at Different Resolutions back Integrand of ψ d V Λ ψ d for Λ =1.2 fm k (fm 1 ) k (fm 1 ).5 4 2

134 Outline Renormalization deuteron Weinberg The Deuteron at Different Resolutions back Integrand of ψ d V Λ ψ d for Λ =1. fm k (fm 1 ) k (fm 1 ).5 4 2

135 Outline Renormalization deuteron Weinberg Is Three-Body Contribution Out of Control? back Check ratios: 1 Saturation driven by 3NF Unnaturally large? Chiral: V 3N (Q/Λ) 3 V NN = consistent V 3N / V low k.1 Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm H 4 He k F [fm -1 ]

136 Outline Renormalization deuteron Weinberg Is Three-Body Contribution Out of Control? back Check ratios: Saturation driven by 3NF Unnaturally large? Chiral: V 3N (Q/Λ) 3 V NN = consistent Power counting still needed with NN + 3N HF at LO Four-body contributions? V 3N / V low k Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 3 H 4 He k F [fm -1 ]