Nuclear Forces / DFT for Nuclei III

Size: px

Start display at page:

Download "Nuclear Forces / DFT for Nuclei III"

Nicholas Willis
5 years ago
Views:

1 Nuclear Forces / DFT for Nuclei III Department of Physics Ohio State University August, 2008 I. Overview of EFT/RG. II. Chiral effective field theory. III. RG for nuclear forces. EFT for many-body systems. IV. EFT/DFT for dilute Fermi systems. V. Ab initio nuclear DFT.

2 Outline RG Dilute Renormalization Group for Nuclear Forces EFT for Dilute Fermi Systems

3 Outline RG Dilute Coupling Flow Decoupling V low k vs. SRG 3-Body Renormalization Group for Nuclear Forces EFT for Dilute Fermi Systems

4 Sources of Nonperturbative Physics for NN 1 Strong short-range repulsion ( hard core or singular V 2π ) 2 Iterated tensor (S 12 ) interaction 3 Near zero-energy bound states

5 Sources of Nonperturbative Physics for NN 1 Strong short-range repulsion ( hard core or singular V 2π ) 2 Iterated tensor (S 12 ) interaction 3 Near zero-energy bound states

6 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 + 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 Various methods: V low k, SRG, UCOM, phase shift (degrees) AV18 E lab (MeV) 1 S0 AV18 [k max = 2.2 fm -1 ]

Computational Aside: Digital Potentials Although momentum is continuous in principle, in practice represented as discrete (gaussian quadrature) grid: = Calculations become

7 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

8 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 )

9 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 )

10 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 )

11 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 )

12 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 )

13 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 )

14 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 )

15 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 )

16 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 )

17 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 )

18 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 )

19 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 )

20 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 )

21 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 )

22 Unitary Transformation: Bare vs. SRG phase shifts 0 δ [deg] S0 1 3 P1 P0 bare ps vsrg ps δ [deg] P1 F3 G E lab [MeV] E lab [MeV] E lab [MeV]

23 Low-Pass Filters Work! phase shift [deg] RG Dilute Coupling Flow Decoupling V low k vs. SRG 3-Body [nucl-th/ ] Phase shifts with V s (k, k ) = 0 for k, k > k max 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 ]

24 Consequences of a Repulsive Core Revisited S1 deuteron probability density 0.3 uncorrelated correlated 300 ψ(r) 2 [fm 3 ] Argonne v 18 ψ(r) 2 [fm 3 ] 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 100

25 Consequences of a Repulsive Core Revisited S1 deuteron probability density 0.3 uncorrelated correlated 300 ψ(r) 2 [fm -3 ] Argonne v 18 λ = 4.0 fm -1 λ = 3.0 fm -1 λ = 2.0 fm -1 ψ(r) 2 [fm 3 ] 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 100

26 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) 10 0 N 3 LO [550/600] E d - E var [MeV] N max = 6 N max = 10 N max = λ [fm 1 ]

27 bare r d [fm] Long-Distance Physics Preserved Matrix elements dominated by long range run slowly for λ 2 fm 1 Here: examples from deuteron (compressed scales) N 3 LO (550/600 MeV) V srg Deuteron rms radius λ [fm 1 ] bare Q d [fm 2 ] bare <1/r> expt. Deuteron quadrupole N 3 LO (550/600 MeV) V srg λ [fm 1 ] Deuteron <1/r> N 3 LO (550/600 MeV) V srg λ [fm 1 ]

28 Compare to Usual RG Transformation [AV18 3 S 1 ] Conventional RG = Lower a cutoff Λ in relative k, k SRG = Drive the Hamiltonian toward diagonal [λ 1/s 1/4 ] Both transformations decouple high and low momenta

29 Flow Equations and the SRG: History In the early 1970 s, Ken Wilson and Franz Wegner = critical phenomena and renormalization group (RG) Twenty years later, Wilson and Wegner innovate again Unitary RG flow to make many-particle Hamiltonians increasingly energy diagonal Glazek and Wilson, Renormalization of Hamiltonians (1993) = SRG for QCD on the light front Wegner, Flow Equations for Hamiltonians (1994) = condensed matter problems S. Kehrein, Flow-Equation Approach to Many-Particle Systems Dissipative quantum systems to correlated electron physics to non-equilibrium problems to... Particularly well suited for low-energy nuclear physics! Only applied since one year ago [nucl-th/ ] See ntg/srg/

30 Band Diagonalizing with SRG RG Dilute Coupling Flow Decoupling V low k vs. SRG 3-Body Transform an initial hamiltonian, H = T + V: H s = U(s)HU (s) T + V s, [nucl-th/ ] 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...

31 MATLAB Code for SRG is Easy to Play With! % V_s is a vector of the current potential; convert to square matrix V_s_matrix = reshape(v_s, tot_pts, tot_pts); H_s_matrix = T_matrix + V_s_matrix; % form the Hamiltontian % Matrix for the right side of the SRG differential equation if (strcmp(evolution, T )) rhs_matrix = my_commutator( my_commutator(t_matrix, H_s_matrix),... H_s_matrix ); elseif (strcmp(evolution, Wegner )) rhs_matrix = my_commutator( my_commutator(diag(diag(h_s_matrix)),... H_s_matrix), H_s_matrix ); elseif ( strcmp(evolution, sharp_bd ) strcmp(evolution, smooth_bd )... strcmp(evolution, weird_bd ) ) % P_matrix is a matrix that builds the block diagonal constraint % or the matrix made from the f regulator. rhs_matrix = my_commutator(... my_commutator(p_matrix.* H_s_matrix.* P_matrix +... (1-P_matrix).* H_s_matrix.* (1-P_matrix ),... H_s_matrix), H_s_matrix );

Flow in NN Momentum Basis with η(s) = [T, H s ] For NN only,

32 Flow in NN Momentum Basis with η(s) = [T, H s ] For NN only, project onto partial-wave rel. momentum k dv s ds = [[T rel, V s ], H s ] 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 ) V λ=2.5 (k, k ) 1st term 2nd term V λ=2.0 (k, k ) First term drives V λ toward diagonal: V λ (k, k ) = V λ= (k, k ) e [(ɛ k ɛ k )/λ 2 ] 2 +

33 Flow of N 3 LO Chiral EFT Potentials 1 S 0 from N 3 LO (500 MeV) of Entem/Machleidt 1 S 0 from N 3 LO (550/600 MeV) of Epelbaum et al. See ntg/srg/ for more!

34 Flow of N 3 LO Potentials RG Dilute Coupling Flow Decoupling V low k vs. SRG 3-Body 3 S 1 from N 3 LO (500 MeV) of Entem/Machleidt 3 S 1 from N 3 LO (550/600 MeV) of Epelbaum et al. See srg/ for more!

35 Run to Lower λ via SRG = Universality λ = 5.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 5.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

36 Run to Lower λ via SRG = Universality λ = 4.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 4.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

37 Run to Lower λ via SRG = Universality λ = 3.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 3.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

38 Run to Lower λ via SRG = Universality λ = 2.5 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 2.5 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

39 Run to Lower λ via SRG = Universality λ = 2.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 2.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

40 Run to Lower λ via SRG = Universality λ = 1.5 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 1.5 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

41 Low-Momentum Interactions from RG [AV18 3 S 1 ] V low k = Lower a cutoff Λ in relative k, k [sharp] SRG = Drive the Hamiltonian toward diagonal [λ 1/s 1/4 ] Other transformations also decouple (e.g., UCOM) Isn t chiral EFT already soft? Or why not use a lower cutoff? [e.g., E/G/M: 450 MeV, E/M: N3LOW (400 MeV)]

42 Low-Momentum Interactions from RG [AV18 3 S 1 ] V low k = Lower a cutoff Λ in relative k, k [e (k 2 /Λ 2 ) 8 ] SRG = Drive the Hamiltonian toward diagonal [λ 1/s 1/4 ] Other transformations also decouple (e.g., UCOM) Isn t chiral EFT already soft? Or why not use a lower cutoff? [e.g., E/G/M: 450 MeV, E/M: N3LOW (400 MeV)]

43 Low-Momentum Interactions from RG [AV18 3 S 1 ] V low k = Lower a cutoff Λ in k, k [e (k 2 /Λ 2 ) 4 ] SRG = Drive the Hamiltonian toward diagonal [λ 1/s 1/4 ] Other transformations also decouple (e.g., UCOM) Isn t chiral EFT already soft? Or why not use a lower cutoff? [e.g., E/G/M: 450 MeV, E/M: N3LOW (400 MeV)]

44 Lower Cutoff via RG = Universality T Bogner, Kuo, Schwenk k +k k +k k +k = V Λ k +k T + q Λ V Λ Require dt dλ = 0 = RG equation for V Λ Integrated = Lee-Suzuki Run cutoff to Λ 2 fm 1 Same long distance physics = coalesce to V low k V low k (k,k) [fm] [nucl-th/ ] Λ = 3.0 fm 1 1 S /500 [E/G/M] 550/600 [E/G/M] /700 [E/G/M] 500 [E/M] 600 [E/M] k/λ

45 Lower Cutoff via RG = Universality T Bogner, Kuo, Schwenk k +k k +k k +k = V Λ k +k T + q Λ V Λ Require dt dλ = 0 = RG equation for V Λ Integrated = Lee-Suzuki Run cutoff to Λ 2 fm 1 Same long distance physics = coalesce to V low k V low k (k,k) [fm] [nucl-th/ ] Λ = 2.5 fm 1 1 S /500 [E/G/M] 550/600 [E/G/M] /700 [E/G/M] 500 [E/M] 600 [E/M] k/λ

46 Lower Cutoff via RG = Universality T Bogner, Kuo, Schwenk k +k k +k k +k = V Λ k +k T + q Λ V Λ Require dt dλ = 0 = RG equation for V Λ Integrated = Lee-Suzuki Run cutoff to Λ 2 fm 1 Same long distance physics = coalesce to V low k V low k (k,k) [fm] [nucl-th/ ] Λ = 2.0 fm 1 1 S /500 [E/G/M] 550/600 [E/G/M] /700 [E/G/M] 500 [E/M] 600 [E/M] k/λ

47 Lower Cutoff via RG = Universality T Bogner, Kuo, Schwenk k +k k +k k +k = V Λ k +k T + q Λ V Λ Require dt dλ = 0 = RG equation for V Λ Integrated = Lee-Suzuki Run cutoff to Λ 2 fm 1 Same long distance physics = coalesce to V low k V low k (k,k) [fm] [nucl-th/ ] Λ = 1.5 fm 1 1 S /500 [E/G/M] 550/600 [E/G/M] /700 [E/G/M] 500 [E/M] 600 [E/M] k/λ

Block Diagonalization Via SRG [arxiv:0801.

48 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

49 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

50 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

51 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

52 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

53 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

54 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

55 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

56 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

57 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

58 Block Diagonalization Via SRG [arxiv: ] Can we get a Λ = 2 fm 1 V low k -like potential with SRG? ( Yes! Use dhs ds = [[G PHs P 0 s, H s ], H s ] with G s = 0 QH s Q )

59 Compare SRG Block Diagonal and V low k Decoupling means that they are functionally equivalent Can we directly relate Lee-Suzuki V low k and this SRG? Can we use this SRG to evolve in the 3N system?

60 Compare SRG Block Diagonal and V low k Decoupling means that they are functionally equivalent Can we directly relate Lee-Suzuki V low k and this SRG? Can we use this SRG to evolve in the 3N system?

61 Weinberg Eigenvalue Analysis of Convergence Born Series: T (E) = V + V 1 E H 0 V + V 1 E H 0 V 1 E H 0 V + For fixed E, find (complex) eigenvalues η ν (E) [Weinberg] 1 E H 0 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 [nucl-th/ ] Im η Im η 1 S0 3 S1 3 D Re η Re η AV18 CD-Bonn N 3 LO (500 MeV) 1 2 AV18 CD-Bonn N 3 LO

62 Lowering the Cutoff Increases Perturbativeness Weinberg eigenvalue analysis (repulsive) [nucl-th/ ] Im η 1 S S Re η Λ = 10 fm -1 Λ = 7 fm -1 Λ = 5 fm -1 Λ = 4 fm -1 Λ = 3 fm -1 Λ = 2 fm -1 N 3 LO η ν (E=0) Argonne v 18 N 2 LO-550/600 [19] N 3 LO-550/600 [14] N 3 LO [13] Λ (fm -1 )

63 Lowering the Cutoff Increases Perturbativeness Weinberg eigenvalue analysis (repulsive) [nucl-th/ ] Im η 3 S1 3 D S1 3 D Re η Λ = 10 fm -1 Λ = 7 fm -1 Λ = 5 fm -1 Λ = 4 fm -1 Λ = 3 fm -1 Λ = 2 fm -1 N 3 LO η ν (E=0) Argonne v 18 N 2 LO-550/600 N 3 LO-550/600 N 3 LO [Entem] Λ (fm -1 )

64 Lowering the Cutoff Increases Perturbativeness Weinberg eigenvalue analysis (η ν at 2.22 MeV vs. density) 1 3 S1 with Pauli blocking η ν (B d ) Λ = 4.0 fm -1 Λ = 3.0 fm -1 Λ = 2.0 fm k F [fm -1 ] Pauli blocking in nuclear matter increases it even more! at Fermi surface, pairing revealed by η ν > 1

65 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 1 550/600 MeV N max

66 Flow Equations Lead to Many-Body Forces Schematically: dv s [[ ds = a a, ] a } a {{ aa }, ] a } a {{ aa } = + a } a {{ a aaa } + 2-body 2-body 3-body! so there will be A-body forces generated (!). Is this a problem? Normal ordering and truncation may be sufficient Ok if induced many-body forces are same size as natural ones Nuclear 3-body forces already needed in unevolved potential Natural hierarchy from chiral EFT = stop flow equations if unnatural Many-body methods must deal with them!

67 Observations on Three-Body Forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on λ e.g., Tjon line 3-body contributions increase with density saturates nuclear matter E b ( 4 He) [MeV] how large is 4-body? E b ( 3 H) [MeV] A=3,4 binding energies SRG NN only, λ in fm 1 λ=1.0 N 3 LO λ=1.25 λ=1.75 λ=1.5 λ=2.0 λ=2.25 Expt. λ=2.5 λ=3.0 NN potentials SRG N 3 LO (500 MeV)

68 Observations on Three-Body Forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on λ e.g., Tjon line 3-body contributions increase with density saturates nuclear matter E gs [MeV] Li N 3 SRG NCSM LO (500 MeV) how large is 4-body? λ [fm 1 ]

69 Diagrams for SRG = Three-body Decouples V (2) s = [T, V (2) s ] = [[T, V (2) s ], T ] = V (3) s = [T, V (3) s ] = [[T, V (3) s ], T ] = dv s (2) (a, b) = a b + a c b a c b ds (ɛ a ɛ b ) 2 V (2) s (a, b) c [(ɛ a ɛ c ) (ɛ c ɛ b )] V (2) s (a, c) V (2) s (c, b) dv (3) s ds =

70 1D Laboratory for SRG Few-Body Physics E. Jurgenson et al. [OSU] Use a simple sum-of-gaussians potential, e.g., V (x) = 34 e (x/0.2)2 8.5 e (x/0.8)2 SRG results in two-particle system qualitatively like NN: ψ(x) λ = infinity V(x) x Bound-state energies: 0.92 (A Nuclear = Forces/DFT 2), 2.57 (A = 3),...

71 1D Laboratory for SRG Few-Body Physics E. Jurgenson et al. [OSU] Use a simple sum-of-gaussians potential, e.g., V (x) = 34 e (x/0.2)2 8.5 e (x/0.8)2 SRG results in two-particle system qualitatively like NN: ψ(x) λ = infinity λ = V(x) x Bound-state energies: 0.92 (A Nuclear = Forces/DFT 2), 2.57 (A = 3),...

72 1D Laboratory for SRG Few-Body Physics E. Jurgenson et al. [OSU] Use a simple sum-of-gaussians potential, e.g., V (x) = 34 e (x/0.2)2 8.5 e (x/0.8)2 SRG results in two-particle system qualitatively like NN: ψ(x) λ = infinity λ = 5 λ = V(x) x Bound-state energies: 0.92 (A Nuclear = Forces/DFT 2), 2.57 (A = 3),...

73 1D Laboratory for SRG Few-Body Physics E. Jurgenson et al. [OSU] Use a simple sum-of-gaussians potential, e.g., V (x) = 34 e (x/0.2)2 8.5 e (x/0.8)2 SRG results in two-particle system qualitatively like NN: ψ(x) λ = infinity λ = 5 λ = 3 λ = V(x) x Bound-state energies: 0.92 (A Nuclear = Forces/DFT 2), 2.57 (A = 3),...

74 1D SRG Evolution with T rel in a Jacobi HO Basis Rather than evolving in momentum space use recursive symmetrization formalism developed for NCSM directly obtain SRG matrix elements in HO basis separate 3-body evolution not needed Compare A = 3 2-body only to full body evolution: A = 3 bosons SRG running in h.o. basis A = 3 bosons N max = 40 Ground-State Energy Binding Energy body only λ = infinity λ = 4 λ = body only λ N max cut

75 1D SRG Evolution with T rel in a Jacobi HO Basis Rather than evolving in momentum space use recursive symmetrization formalism developed for NCSM directly obtain SRG matrix elements in HO basis separate 3-body evolution not needed Compare A = 3 2-body only to full body evolution: A = 3 bosons SRG running in h.o. basis A = 3 bosons N max = 40 Ground-State Energy Binding Energy body λ = infinity λ = 4 λ = body only body λ N max cut

76 1D SRG Evolution with T rel in a Jacobi HO Basis A 4 follows by embedding A 1 in A (Navratil et al.) Compare A = 4 2-body only to full body evolution 4.5 Ground-State Energy A = 4 bosons SRG running in h.o. basis 2-body only λ

77 1D SRG Evolution with T rel in a Jacobi HO Basis A 4 follows by embedding A 1 in A (Navratil et al.) Compare A = 4 2-body only to full body evolution 4.5 Ground-State Energy A = 4 bosons SRG running in h.o. basis 2-body only body λ

78 1D SRG Evolution with T rel in a Jacobi HO Basis A 4 follows by embedding A 1 in A (Navratil et al.) Compare A = 4 2-body only to full body evolution Now extract 3NF from A = 3 and look at body only 4.5 Ground-State Energy A = 4 bosons SRG running in h.o. basis 2-body only body body only λ Year 2 3: other generators/schemes, scaling analysis, 3D

79 What are the measurable quantities? True observables do not change under field redefinitions or unitary transformations in low-energy effective theories Examples: cross sections and conserved quantities like charge Many useful quantities are extracted from measurements via a convolution (e.g., using some type of factorization) But these will vary with the convention used E.g., parton distributions Conventions are renormalization prescriptions, cutoffs,... Different potentials reflect different conventions The convention for the long-range part of NN N potentials is agreed to be (local) pion exchange, but differs widely for the short-range part. (Note: V low k preserves long-distance part.)

80 Quantities that vary with convention = not observables deuteron D-state probability [e.g., Friar, PRC 20 (1979)] off-shell effects [Fearing/Scherer] occupation numbers [Hammer/rjf] wound integrals short-range part of wave functions short-range potentials; e.g., contribution of short-range 3-body forces P D N 3 LO (500 MeV) with e (p2 /Λ 2 ) Binding energy (MeV) E d D-state probability 2.23 Asymptotic D-S ratio Λ (fm -1 ) η d

81 Every Operator Flows RG Dilute Coupling Flow Decoupling V low k vs. SRG 3-Body Evolution with s of any operator O is given by: O s = U(s)OU (s) so O s evolves via do s ds = [[T rel, V s ], O s ] Matrix elements of evolved operators are unchanged 4π [u(q) 2 + w(q) 2 ] [fm 3 ] AV18 V s at λ = 2 fm 1 V s at λ = 1.5 fm 1 V low k at Λ = 2.0 fm 1 CD-Bonn Consider momentum distribution < ψ d a qa q ψ d > at q = 4.5 fm 1 q [fm 1 ]

82 Integrand of < ψ d Ua qa q U ψ d > at q = 4.5 fm 1 Flow of deuteron matrix element integrand is toward low k Simple variational ansatz works well = No fine-tuning Factorization: U(k, q) K (k)q(q) for k λ, q λ

83 Summary: Atomic Nuclei at Low Resolution Strategy: Lower the resolution as much as possible High resolution = high momenta can be painful! ( It hurts when I do this. Then don t do that. ) Correlations in wave functions reduced dramatically Non-local potentials and many-body forces generated

84 Summary: Atomic Nuclei at Low Resolution Strategy: Lower the resolution as much as possible High resolution = high momenta can be painful! ( It hurts when I do this. Then don t do that. ) Correlations in wave functions reduced dramatically Non-local potentials and many-body forces generated Flow equations (SRG) achieve low resolution by decoupling Band (or block) diagonalizing Hamiltonian matrix Unitary transformations: observables don t change! Nuclear case: evolve until few-body forces start to explode

85 Summary: Atomic Nuclei at Low Resolution Strategy: Lower the resolution as much as possible High resolution = high momenta can be painful! ( It hurts when I do this. Then don t do that. ) Correlations in wave functions reduced dramatically Non-local potentials and many-body forces generated Flow equations (SRG) achieve low resolution by decoupling Band (or block) diagonalizing Hamiltonian matrix Unitary transformations: observables don t change! Nuclear case: evolve until few-body forces start to explode Applications to nuclei and beyond Coupled cluster and FCI for nuclei converges much faster MBPT works = constructive nuclear density functional theory Flow equations already applied to QCD and condensed matter Systems of cold atoms?

86 Outline RG Dilute Fermi 3-Body Renormalization Group for Nuclear Forces EFT for Dilute Fermi Systems

87 RG Dilute Fermi 3-Body Simple Many-Body Problem: Hard Spheres Infinite potential at radius R sin(kr+δ) R 0 R r Scattering length a 0 = R Dilute nr 3 1 = k F a 0 1 What is the energy / particle of the uniform system? [nucl-th/ ] DFT: Add a trap ~ 1/ k F

88 RG Dilute Fermi 3-Body At Low Energies: Effective Range Expansion As first shown by Schwinger, k l+1 cot δ l (k) has a power series expansion. For l = 0: k cot δ 0 = 1 a r 0k 2 Pr 3 0 k 4 + defines the scattering length a 0 and the effective range r 0 While r 0 R, the range of the potential, a 0 can be anything if a 0 R, it is called natural a 0 R (unnatural) is particularly interesting = cold atoms The effective range expansion for hard sphere scattering is: k cot( kr) = 1 R Rk 2 + = a 0 = R r 0 = 2R/3 so the low-energy effective theory is natural

89 RG Dilute Fermi 3-Body EFT for Natural Short-Range Interaction A simple, general interaction is a sum of delta functions and derivatives of delta functions. In momentum space, k V eft k = C C 2(k 2 + k 2 ) + C 2 k k + Or, L eft has most general local (contact) interactions: L eft = ψ [ i t + 2 2M ] ψ C 0 2 (ψ ψ) 2 + C 2 [ (ψψ) (ψ 2 ψ) + h.c. ] 16 + C 2 8 (ψ ψ) (ψ ψ) D 0 6 (ψ ψ) Dimensional analysis = C 2i 4π M R2i+1, D 2i 4π M R2i+4

90 RG Dilute Fermi 3-Body Effective Field Theory Ingredients Ref: Hammer, rjf [nucl-th/ ] 1 Use the most general L with low-energy dof s consistent with global and local symmetries of underlying theory L eft = ψ [ ] i t + 2 2M ψ C 0 2 (ψ ψ) 2 D 0 6 (ψ ψ)

91 RG Dilute Fermi 3-Body Effective Field Theory Ingredients Ref: Hammer, rjf [nucl-th/ ] 1 Use the most general L with low-energy dof s consistent with global and local symmetries of underlying theory L eft = ψ [ ] i t + 2 2M ψ C 0 2 (ψ ψ) 2 D 0 6 (ψ ψ) Declaration of regularization and renormalization scheme natural a 0 = dimensional regularization and min. subtraction

92 RG Dilute Fermi 3-Body Effective Field Theory Ingredients Ref: Hammer, rjf [nucl-th/ ] 1 Use the most general L with low-energy dof s consistent with global and local symmetries of underlying theory L eft = ψ [ ] i t + 2 2M ψ C 0 2 (ψ ψ) 2 D 0 6 (ψ ψ) Declaration of regularization and renormalization scheme natural a 0 = dimensional regularization and min. subtraction 3 Well-defined power counting = small expansion parameters use the separation of scales = k F Λ with Λ 1/R = k F a 0, etc. O ( k 6 F) : O ( k 7 F ) : + E = ρ k F 2 [ 3 2M π (k Fa 0 ) + 4 ] 35π 2 (11 2 ln 2)(k Fa 0 ) 2 + cleanly recovers perturbative free-space ERE and in-medium energy density (including logs), plus error estimates

93 RG Dilute Fermi 3-Body Feynman Rules for EFT Vertices L eft = ψ [ i t + 2 2M ] ψ C 0 2 (ψ ψ) 2 + C 2 [ (ψψ) (ψ 2 ψ) + h.c. ] 16 + C 2 8 (ψ ψ) (ψ ψ) D 0 6 (ψ ψ) P/2 + k P/2 + k = P/2 k P/2 k i k k 2 + k 2 V EFT k ic 0 ic 2 2 ic 2 k k = + id 0

94 Renormalization RG Dilute Fermi 3-Body Reproduce f 0 (k) in perturbation theory (Born series): f 0 (k) a 0 ia 2 0k (a 3 0 a2 0r 0 /2)k 2 + O(k 3 a 4 0) Consider the leading potential V (0) EFT (x) = C 0δ(x) or k V (0) eft k = = C 0 Choosing C 0 a 0 gets the first term. Now k VG 0 V k : d 3 q 1 = C 0 M (2π) 3 k 2 q 2 + iɛ C 0! = Linear divergence!

95 Renormalization RG Dilute Fermi 3-Body Reproduce f 0 (k) in perturbation theory (Born series): f 0 (k) a 0 ia 2 0k (a 3 0 a2 0r 0 /2)k 2 + O(k 3 a 4 0) Consider the leading potential V (0) EFT (x) = C 0δ(x) or k V (0) eft k = = C 0 Choosing C 0 a 0 gets the first term. Now k VG 0 V k : = Λc d 3 q (2π) 3 1 k 2 q 2 + iɛ Λ c 2π 2 ik 4π + O(k 2 Λ c ) = If cutoff at Λ c, then can absorb into C 0, but all powers of k 2

96 Renormalization RG Dilute Fermi 3-Body Reproduce f 0 (k) in perturbation theory (Born series): f 0 (k) a 0 ia 2 0k (a 3 0 a2 0r 0 /2)k 2 + O(k 3 a 4 0) Consider the leading potential V (0) EFT (x) = C 0δ(x) or k V (0) eft k = = C 0 Choosing C 0 a 0 gets the first term. Now k VG 0 V k : d D q 1 D 3 = (2π) 3 k 2 q 2 ik + iɛ 4π Dimensional regularization with minimal subtraction = only one power of k!

97 RG Dilute Fermi 3-Body Dim. reg. + minimal subtraction = simple power counting: P/2 + k P/2 + k = + P/2 k P/2 k it (k, cos θ) ic 0 M 4π (C0)2 k O(k 3 ) ( M )2 +i (C 0) 3 k 2 ic 2k 2 ic 4π 2k 2 cos θ Matching in free space: C 0 = 4π M a 0 = 4π M R, C 2 = 4π a0 2r 0 M 2 = 4π M R3 3, Recovers effective range expansion order-by-order with perturbative diagrammatic expansion one power of k per diagram estimate truncation error from dimensional analysis

98 RG Dilute Fermi 3-Body Noninteracting Fermi Sea at T = 0 Put system in a large box (V = L 3 ) with periodic bc s spin-isospin degeneracy ν (e.g., for nuclei, ν = 4) fill momentum states up to Fermi momentum k F k F k F N = ν 1, E = ν k k 2 k 2 2M

99 RG Dilute Fermi 3-Body Noninteracting Fermi Sea at T = 0 Put system in a large box (V = L 3 ) with periodic bc s spin-isospin degeneracy ν (e.g., for nuclei, ν = 4) fill momentum states up to Fermi momentum k F k F k F N = ν 1, E = ν k k 2 k 2 2M Use: F(k) dk i F(k i) k i = i F(k i) 2π L n i = 2π L i F (k i)

100 RG Dilute Fermi 3-Body Noninteracting Fermi Sea at T = 0 Put system in a large box (V = L 3 ) with periodic bc s spin-isospin degeneracy ν (e.g., for nuclei, ν = 4) fill momentum states up to Fermi momentum k F k F k F N = ν 1, E = ν k k 2 k 2 2M Use: F(k) dk i F(k i) k i = i F(k i) 2π L n i = 2π L In 1-D: N = ν L 2π In 3-D: N = ν V (2π) 3 +kf k F dk = νk F π L = ρ = N L = νk F π ; kf d 3 k = νk 3 F 6π 2 V = ρ = N V = νk 3 F 6π 2 ; i F (k i) E L = 1 2 kf 2 3 2M ρ E V = 3 2 kf 2 5 2M ρ

101 RG Dilute Fermi 3-Body Noninteracting Fermi Sea at T = 0 Put system in a large box (V = L 3 ) with periodic bc s spin-isospin degeneracy ν (e.g., for nuclei, ν = 4) fill momentum states up to Fermi momentum k F k F k F N = ν 1, E = ν k k 2 k 2 2M Use: F(k) dk i F(k i) k i = i F(k i) 2π L n i = 2π L In 1-D: N = ν L 2π In 3-D: N = ν V (2π) 3 +kf k F dk = νk F π L = ρ = N L = νk F π ; kf d 3 k = νk 3 F 6π 2 V = ρ = N V = νk 3 F 6π 2 ; i F (k i) E L = 1 2 kf 2 3 2M ρ Volume/particle V /N = 1/ρ 1/k 3 F, so spacing 1/k F E V = 3 2 kf 2 5 2M ρ

102 RG Dilute Fermi 3-Body Energy Density From Summing Over Fermi Sea Leading order V (0) EFT (x) = C 0δ(x) = V (0) EFT (k, k ) = C 0 = E LO = C 0 ν(ν 1) 2 ( kf ) 2 1 a 0 k 6 k F

103 RG Dilute Fermi 3-Body Energy Density From Summing Over Fermi Sea Leading order V (0) EFT (x) = C 0δ(x) = V (0) EFT (k, k ) = C 0 = E LO = C 0 ν(ν 1) 2 ( kf At the next order, we get a linear divergence again: = E NLO k F ) 2 1 a 0 k 6 k d 3 q (2π) 3 C 2 0 k 2 q 2 F

104 RG Dilute Fermi 3-Body Energy Density From Summing Over Fermi Sea Leading order V (0) EFT (x) = C 0δ(x) = V (0) EFT (k, k ) = C 0 = E LO = C 0 ν(ν 1) 2 ( kf At the next order, we get a linear divergence again: = E NLO k F ) 2 1 a 0 k 6 k d 3 q (2π) 3 C 2 0 k 2 q 2 Same renormalization fixes it! Particles holes k F 1 k 2 q 2 = 0 1 k 2 q 2 kf 0 1 D 3 k 2 q 2 kf 0 F 1 k 2 q 2 a2 0k 7 F

105 RG Dilute Fermi 3-Body Feynman Rules for Energy Density at T = 0 T = 0 Energy density E is sum of Hugenholtz diagrams same vertices as free space (same renormalization!) Feynman rules: 1 Each line is assigned conserved k (k 0, k) and [ω k k 2 /2M] ( θ(k kf ) ig 0 ( k) αβ = iδ αβ k 0 ω k + iɛ + θ(k ) F k) k 0 ω k iɛ 2 β α δ γ (δ αγ δ βδ + δ αδ δ βγ ) (if spin-independent) 3 After spin summations, δ αα ν in every closed fermion loop. 4 Integrate d 4 k/(2π) 4 with e ik 00 + for tadpoles 5 Symmetry factor i/(s l max l=2 (l!)k ) counts vertex permutations and equivalent l tuples of lines

106 RG Dilute Fermi 3-Body Power Counting Power counting rules 1 for every propagator (line): M/k 2 F 2 for every loop integration: k 5 F /M 3 for every n body vertex with 2i derivatives: k 2i F /MΛ2i+3n 5 Diagram with V n 2i n body vertices scales as (k F ) β only: β = 5 + n=2 i=0 e.g., O ( k 6 F) : = V 2 0 = 1 (3n + 2i 5)V2i n. = β = 5 + ( ) 1 = 6 = O(k 6 F )

107 RG Dilute Fermi 3-Body Power Counting Power counting rules 1 for every propagator (line): M/k 2 F 2 for every loop integration: k 5 F /M 3 for every n body vertex with 2i derivatives: k 2i F /MΛ2i+3n 5 Diagram with V n 2i n body vertices scales as (k F ) β only: β = 5 + n=2 i=0 (3n + 2i 5)V2i n. e.g., = V 2 0 = 2 = β = 5 + ( ) 2 = 7 = O(k 7 F )

108 RG Dilute Fermi 3-Body T = 0 Energy Density from Hugenholtz Diagrams E V = ρ k F 2 [ 3 2M 5 ]

109 RG Dilute Fermi 3-Body T = 0 Energy Density from Hugenholtz Diagrams E O ( kf 6 ) : V = ρ k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) ]

110 RG Dilute Fermi 3-Body T = 0 Energy Density from Hugenholtz Diagrams O ( k 6 F) : O ( k 7 F) : + E V = ρ k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 ]

111 RG Dilute Fermi 3-Body T = 0 Energy Density from Hugenholtz Diagrams O ( k 6 F) : O ( k 7 F) : + E V = ρ k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 O ( k 8 F) : + ]

112 RG Dilute Fermi 3-Body T = 0 Energy Density from Hugenholtz Diagrams O ( k 6 F) : O ( k 7 F) : + E V = ρ k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 O ( k 8 F) : + + (ν 1) ( (ν 3) ) (k F a 0 ) ]

113 RG Dilute Fermi 3-Body T = 0 Energy Density from Hugenholtz Diagrams O ( k 6 F) : O ( k 7 F) : + E V = ρ k F 2 [ (ν 1) 2M 5 3π (k Fa 0 ) 4 + (ν 1) 35π 2 (11 2 ln 2)(k Fa 0 ) 2 O ( k 8 F) : (ν 1) ( (ν 3) ) (k F a 0 ) 3 + (ν 1) 1 10π (k Fr 0 )(k F a 0 ) 2 + (ν + 1) 1 5π (k Fa p ) 3 + ]

114 RG Dilute Fermi 3-Body Looks Like a Power Series in k F! Is it?

115 RG Dilute Fermi 3-Body Looks Like a Power Series in k F! Is it? New logarithmic divergences in 3 3 scattering + (C 0 ) 4 ln(k/λ c )

116 RG Dilute Fermi 3-Body Looks Like a Power Series in k F! Is it? New logarithmic divergences in 3 3 scattering + (C 0 ) 4 ln(k/λ c ) Changes in Λ c must be absorbed by 3-body coupling D 0 (Λ c ) = D 0 (Λ c ) (C 0 ) 4 ln(a 0 Λ c ) + const. [Braaten & Nieto] [ ] d = 0 = fixes coefficient! dλ c

117 RG Dilute Fermi 3-Body Looks Like a Power Series in k F! Is it? New logarithmic divergences in 3 3 scattering + (C 0 ) 4 ln(k/λ c ) Changes in Λ c must be absorbed by 3-body coupling D 0 (Λ c ) = D 0 (Λ c ) (C 0 ) 4 ln(a 0 Λ c ) + const. [Braaten & Nieto] [ ] d = 0 = fixes coefficient! dλ c What does this imply for the energy density? O ( k 9 F ln(k F ) ) : + + (ν 2)(ν 1) (k Fa 0 ) 4 ln(k F a 0 )

118 RG Dilute Fermi 3-Body Summary: Dilute Fermi System with Natural a 0 The many-body energy density is perturbative in k F a 0 efficiently reproduced by the EFT approach Power counting = error estimate from omitted diagrams Three-body forces are inevitable in a low-energy effective theory and not unique = they depend on the two-body potential The case of a natural scattering length is under control for a uniform system What about a finite # of fermions in a trap? (Lecture IV) What if the scattering length is not natural?

Applications of Renormalization Group Methods in Nuclear Physics 2

Applications of Renormalization Group Methods in Nuclear Physics 2 Dick Furnstahl Department of Physics Ohio State University HUGS 2014 Outline: Lecture 2 Lecture 2: SRG in practice Recap from lecture