Bound states in a box
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1 Bound states in a box Sebastian König in collaboration with D. Lee, H.-W. Hammer; S. Bour, U.-G. Meißner Dr. Klaus Erkelenz Preis Kolloquium HISKP, Universität Bonn December 19, 2013 Bound states in a box p. 1
2 Past and present future The Bonn potential K. Erkelenz et al. based on field theoretical approach designed to be used in nuclear structure calculations Chiral effective interactions Modern structure calculations Nuclear Lattice No-core shell model Coupled cluster etc. SRG-evolved interactions... Epelbaum, Krebs, Lähde, Lee, Meißner Bound states in a box p. 2
3 Outline effective range expansion modified ERE long-range interactions van der Waals force causality bounds pionless EFT Coulomb force large scattering length p-d system shallow bound states low-energy universality EFT periodic boundary conditions finite-range interactions Coulomb wave functions Gamow factor p-d scattering lengths finite-volume mass shifts topological correction factors radial Schroedinger equation ANC Bound states in a box p. 3
4 Outline effective range expansion modified ERE long-range interactions van der Waals force causality bounds pionless EFT Coulomb force large scattering length p-d system shallow bound states low-energy universality EFT periodic boundary conditions finite-range interactions Coulomb wave functions Gamow factor p-d scattering lengths finite-volume mass shifts topological correction factors radial Schroedinger equation ANC Bound states in a box p. 3
5 Prelude Low-energy universality and finite-range interactions Bound states in a box p. 4
6 Low-energy universality low energy large wavelength high energy short wavelength low resolution high resolution Bound states in a box p. 5
7 Low-energy universality low energy large wavelength high energy short wavelength low resolution high resolution Resolution and diffraction Bound states in a box p. 5
8 Low-energy universality low energy large wavelength high energy short wavelength low resolution high resolution Resolution and diffraction Bound states in a box p. 5
9 Low-energy universality V (r) r AlphaZeta, Wikimedia Commons Bound states in a box p. 6
10 Low-energy universality V (r) R r Bound states in a box p. 6
11 Bound states in a box The box periodic finite volume cube of size L 3 Bound states in a box p. 7
12 Bound states in a box The box periodic finite volume cube of size L 3 The bound states 2-body bound states wavefunction ψ Bound states in a box p. 7
13 Bound states in a box The box periodic finite volume cube of size L 3 The bound states 2-body bound states wavefunction ψ The finite volume changes the properties of the system! Bound states in a box p. 7
14 Bound states in a box The box periodic finite volume cube of size L 3 The bound states 2-body bound states wavefunction ψ The finite volume changes the properties of the system! Important for numerical calculations Lattice Bound states in a box p. 7
15 Lattice calculations Solve a physical theory by putting it on a spacetime-lattice! Lattice QCD QCD observables from first principles quarks and gluons as degrees of freedom D. Lee Nuclear Lattice Calculations nuclei from first principles nucleons and pions as d.o.f. based on chiral effective theory Bound states in a box p. 8
16 Lattice artifacts lattice spacing a lattice size L a 0 : continuum limit L : infinite-volume limit Bound states in a box p. 9
17 Lüscher s famous formula Lüscher s idea Use the volume dependence as a tool! p cot δ 0 (p) = 1 ( ) Lp 2 πl S(η), η = 2π p = p ( E(L) ) measure energy levels in finite volume extract physical scattering phase shift Bound states in a box p. 10
18 Outline Overview Part I Mass shift of bound states with angular momentum arxiv: , Part II Topological factors in scattering systems arxiv: Summary Bound states in a box p. 11
19 Outline Overview Part I Mass shift of bound states with angular momentum arxiv: , Part II Topological factors in scattering systems arxiv: Summary Bound states in a box p. 11
20 Part I Mass shift of bound states with angular momentum Lüscher s result for S-waves Bound states in a finite volume General result for arbitrary partial waves Sign of the mass shift Numerical tests Bound states in a box p. 12
21 Starting point S-wave bound state Lüscher (1986) m B = 24π A 2 e κl ml + O( e 2κL ) Bound states in a box p. 13
22 Starting point S-wave bound state Lüscher (1986) m B = 24π A 2 e κl ml + O( e 2κL ) What s the the result for states with angular momentum? Bound states in a box p. 13
23 Why care about higher partial waves? Bound states in a box p. 14
24 Halo nuclei single nucleon weakly bound to a tight core nucleus by Cam-Ann, Wikimedia Commons Bound states in a box p. 15
25 Halo nuclei single nucleon weakly bound to a tight core can be described as an effective 2-body state nucleus by Cam-Ann, Wikimedia Commons Bound states in a box p. 15
26 Halo nuclei single nucleon weakly bound to a tight core can be described as an effective 2-body state nucleus by Cam-Ann, Wikimedia Commons Halo-EFT expansion in R core /R halo effective field theory Bound states in a box p. 15
27 Halo nuclei single nucleon weakly bound to a tight core can be described as an effective 2-body state nucleus by Cam-Ann, Wikimedia Commons Halo-EFT expansion in R core /R halo effective field theory Example P-wave state just below 10 Be + n threshold in 11 Be TUNL Nuclear Data Bound states in a box p. 15
28 Schrödinger equation Ĥ = 1 2µ r + V (r) R Ĥ ψ B = κ2 2µ ψ B finite-range interaction: V (r) = 0 for r > R Bound states in a box p. 16
29 Schrödinger equation Ĥ = 1 2µ r + V (r) R Ĥ ψ B = κ2 2µ ψ B finite-range interaction: V (r) = 0 for r > R Radial Schrödinger equation ψ B (r) = u l(r) r ( d Yl m 2 ) l(l + 1) (θ, φ) dr2 r 2 2µV (r) κ 2 u l (r) = 0 Bound states in a box p. 16
30 Asymptotic wavefunction Radial Schrödinger equation ( d 2 + 1) l(l dr2 r 2 2µV (r) κ )u 2 l (r) = 0 for r > R R R Bound states in a box p. 17
31 Asymptotic wavefunction Radial Schrödinger equation ( d 2 + 1) l(l dr2 r 2 2µV (r) κ )u 2 l (r) = 0 for r > R u l (r) = i l γ ĥ+ l (iκr) Riccati Hankel functions ĥ + 0 (z) = eiz ( ĥ + 1 (z) = 1 + i ) e i(z π/2) z ĥ + 2 (z) = R R Bound states in a box p. 17
32 Finite volume Periodic boundary conditions infinitely many copies of the potential V L (r) = n Z 3 V (r + nl), L R Bound states in a box p. 18
33 Finite volume Periodic boundary conditions infinitely many copies of the potential V L (r) = n Z 3 V (r + nl), L R 3 L 2 L 2 L 2 3 L 2 V (r) = V 0 θ(r r) Bound states in a box p. 18
34 Finite volume Periodic boundary conditions infinitely many copies of the potential V L (r) = n Z 3 V (r + nl), L R 3 L 2 L 2 L 2 3 L 2 V (r) = V 0 exp( r 2 /R 2 ) Bound states in a box p. 18
35 Finite volume Periodic boundary conditions infinitely many copies of the potential V L (r) = n Z 3 V (r + nl), L R 3 L 2 L 2 L 2 3 L 2 V (r) = V 0 exp( r/r) r Bound states in a box p. 18
36 Finite volume Ĥ L ψ = E B (L) ψ Ĥ ψ B = E B ( ) ψ B Mass shift m B E B ( ) E B (L) m B = M E B Bound states in a box p. 19
37 Finite volume Ĥ L ψ = E B (L) ψ Ĥ ψ B = E B ( ) ψ B Mass shift m B E B ( ) E B (L) m B = M E B The wavefunction ψ(r) has to be periodic, too! ψ(r + nl) = ψ(r) Ansatz: ψ 0 (r) = n Z 3 ψ B (r+nl) Bound states in a box p. 19
38 Finite volume Ĥ L ψ = E B (L) ψ Ĥ ψ B = E B ( ) ψ B Mass shift m B E B ( ) E B (L) m B = M E B The wavefunction ψ(r) has to be periodic, too! ψ(r + nl) = ψ(r) Ansatz: ψ 0 (r) = n Z 3 ψ B (r+nl) ĤL ψ 0 = E B ( ) ψ 0 + η η(r) = n n V (r + nl) ψ B (r + n L) ψ ĤL ψ 0 = E B (L) ψ ψ 0 = E B ( ) ψ ψ 0 + ψ η Bound states in a box p. 19
39 Finite volume Ĥ L ψ = E B (L) ψ Ĥ ψ B = E B ( ) ψ B Mass shift m B E B ( ) E B (L) m B = M E B The wavefunction ψ(r) has to be periodic, too! ψ(r + nl) = ψ(r) Ansatz: ψ 0 (r) = n Z 3 ψ B (r+nl) ĤL ψ 0 = E B ( ) ψ 0 + η η(r) = n n V (r + nl) ψ B (r + n L) Result ψ ĤL ψ 0 = E B (L) ψ ψ 0 = E B ( ) ψ ψ 0 + ψ η m B = ψ η ψ 0 ψ 0 = d 3 r ψb(r) V (r) ψ B (r + nl) + O ( e 2κL ) n =1 Bound states in a box p. 19
40 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) Bound states in a box p. 20
41 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) Bound states in a box p. 20
42 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) Bound states in a box p. 20
43 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) Bound states in a box p. 20
44 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) Bound states in a box p. 20
45 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) Bound states in a box p. 20
46 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L Bound states in a box p. 20
47 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L Bound states in a box p. 20
48 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L Bound states in a box p. 20
49 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) L Bound states in a box p. 20
50 Finite volume mass shift m B = n =1 d 3 r ψ B(r) V (r) ψ B (r + nl) + O ( e 2κL ) It s all determined by the tail! L Bound states in a box p. 20
51 Finite volume mass shift m B = n =1 d 3 r ψb(r nl) V (r nl) Yl m (θ, φ) ilγĥ+ l (iκr) + r Bound states in a box p. 21
52 Finite volume mass shift m B = n =1 d 3 r ψb(r nl) V (r nl) Yl m (θ, φ) ilγĥ+ l (iκr) + r Bound states in a box p. 21
53 Finite volume mass shift m B = n =1 d 3 r 1 [ r κ 2] ψ 2µ B(r nl) Yl m (θ, φ) ilγĥ+ l (iκr) + r Bound states in a box p. 21
54 Finite volume mass shift m B = n =1 d 3 r 1 [ r κ 2] ψ 2µ B(r nl) Yl m (θ, φ) ilγĥ+ l (iκr) + r S-waves Y0 0 (θ, φ) = 1, ĥ + 0 (iκr) = e κr 4π Bound states in a box p. 21
55 Finite volume mass shift m B = n =1 d 3 r γ [ r κ 2] ψb(r nl) e κr + 16πµ r S-waves Y0 0 (θ, φ) = 1, ĥ + 0 (iκr) = e κr 4π Bound states in a box p. 21
56 Finite volume mass shift m B = n =1 d 3 r γ [ r κ 2] ψb(r nl) e κr + 16πµ r S-waves Y0 0 (θ, φ) = 1, ĥ + 0 (iκr) = e κr 4π [ r κ 2] e κr r = 4πδ (3) (r) Green s function Bound states in a box p. 21
57 Finite volume mass shift m B = n =1 d 3 r γ [ r κ 2] ψb(r nl) e κr + 16πµ r S-waves Y0 0 (θ, φ) = 1, ĥ + 0 (iκr) = e κr 4π [ r κ 2] e κr r = 4πδ (3) (r) Green s function S-waves sum just yields a factor six... m (0,0) B = 3 γ 2 e κl µl + O( e 2κL ) Bound states in a box p. 21
58 Higher partial waves m B = n =1 d 3 r 1 [ r κ 2] ψ 2µ B(r nl) Yl m (θ, φ) ilγĥ+ l (iκr) + r Lemma Yl m (θ, φ) ĥ+ l (iκr) = ( i) l Rl m r ( 1 ) [ κ e κr r r ] Rl m(r) = rl Yl m (ˆr) Bound states in a box p. 22
59 Higher partial waves m B = n =1 d 3 r 1 [ r κ 2] ψ 2µ B(r nl) Yl m (θ, φ) ilγĥ+ l (iκr) + r Lemma Yl m (θ, φ) ĥ+ l (iκr) = ( i) l Rl m r ( 1 ) [ κ e κr r r ] R m l (r) = rl Y m l (ˆr) Lüscher 1991 Bound states in a box p. 22
60 Higher partial waves m B = n =1 d 3 r 1 [ r κ 2] ψ 2µ B(r nl) Yl m (θ, φ) ilγĥ+ l (iκr) + r Lemma Yl m (θ, φ) ĥ+ l (iκr) = ( i) l Rl m r ( 1 ) [ κ e κr r r ] R m l (r) = rl Y m l (ˆr) Lüscher 1991 P-waves m (1,0) B = m (1,±1) B = 3 γ 2 e κl µl + O( e 2κL ) Mass shift for P-wave states exactly reversed compared to S-waves! Bound states in a box p. 22
61 Sign of the mass shift ψ even m B < 0 x Bound states in a box p. 23
62 Sign of the mass shift ψ even m B < 0 x Bound states in a box p. 23
63 Sign of the mass shift ψ even m B < 0 x even parity WF profile relaxed less curvature more deeply bound Bound states in a box p. 23
64 Sign of the mass shift ψ odd m B > 0 ψ even m B < 0 x even parity WF profile relaxed less curvature more deeply bound Bound states in a box p. 23
65 Sign of the mass shift odd parity WF profile compressed more curvature less bound ψ odd m B > 0 ψ even m B < 0 x even parity WF profile relaxed less curvature more deeply bound Bound states in a box p. 23
66 Final result General formula m (l,γ) B ( ) = α 1 κl γ 2 e κl µl + O( e 2κL ) l Γ α(x) 0 A T T x + 135x x x 4 2 E + ( 1 / x + 405x x x 4) Bound states in a box p. 24
67 Final result General formula m (l,γ) B ( ) = α 1 κl γ 2 e κl µl + O( e 2κL ) l Γ α(x) 0 A T T x + 135x x x 4 2 E + ( 1 / x + 405x x x 4) broken symmetry! rotation group SO(3) cubic group O Bound states in a box p. 24
68 Numerical checks Results can be checked with a very simple calculation... Lattice Hamiltonian Ĥ 0 = ˆn 3ˆµ a (ˆn)a(ˆn) 1 2ˆµ Ê(ˆq) = 1ˆµ l=1,2,3 l=1,2,3 (1 cos ˆq i ) = 1 2ˆµ ( a (ˆn)a(ˆn + ê l ) + a (ˆn)a(ˆn ê l )) ˆq l 2 l=1,2,3 [ ] 1 + O(ˆq l 2 ) lattice units: ˆL = L/a, Ê = E a, etc., a = lattice spacing Bound states in a box p. 25
69 Numerical checks Interaction: V step (r) = V 0 θ(r r) Approximate infinite volume with L = 40 Methods to calculate mass shift 1 m B = E B (L ) E B (L) (direct difference) 2 m B = n =1 ( ) 3 m B = α 1 κl γ 2 e κl µl d 3 r ψ B (r) V (r) ψ B(r + nl) (overlap integral) (Green s function) Bound states in a box p. 26
70 Numerical checks Interaction: V step (r) = V 0 θ(r r) Approximate infinite volume with L = 40 Methods to calculate mass shift 1 m B = E B (L ) E B (L) (direct difference) 2 m B = n =1 ( ) 3 m B = α 1 κl γ 2 e κl µl d 3 r ψ B (r) V (r) ψ B(r + nl) (overlap integral) (Green s function) Replace e ˆκˆL/ˆL 4πĜˆκ(ˆL, 0, 0) to reduce discretization errors! Lattice Green s function Ĝˆκ (ˆn) = 1 L 3 ˆq e iˆq ˆn Q 2 (ˆq) + ˆκ 2 Bound states in a box p. 26
71 Numerical checks log(l mb ) m B = ±3 γ 2 e κl µl S-wave direct difference overlap integral Green s function P-wave V = V step L Bound states in a box p. 27
72 Numerical checks 0 m B = ( ) 30 κl + γ 2 e κl µl mb direct difference overlap integral Green s function V = V step, D-wave, E rep L Bound states in a box p. 28
73 Numerical checks m B = 1 /2 ( ) κl + γ 2 e κl µl V = V step, D-wave, T2 rep. mb direct difference overlap integral Green s function L Bound states in a box p. 29
74 Part II Topological factors in scattering systems Bound states in a box p. 30
75 Motivation: atom-dimer scattering H 1, τ(η) = 1 H 2, τ(η) = 1 STM equation κ D a AD κ D a latt calculation by S. Bour, D. Lee, U.-G. Meißner Bound states in a box p. 31
76 Motivation: atom-dimer scattering H 1, τ(η) = 1 H 2, τ(η) = 1 STM equation κ D r AD κ D a latt calculation by S. Bour, D. Lee, U.-G. Meißner Bound states in a box p. 32
77 Atom dimer scattering p cot δ 0 (p) = 1 ( ) Lp 2 πl S(η), η =, p = p ( E(L) ) 2π p cot δ 0 (p) = 1 + r AD a AD 2 p2 + O(p 4 ) A part of E(L) is due to the binding of the dimer! Bound states in a box p. 33
78 Part II Topological factors in scattering systems Motivation Moving bound states in a finite volume Mass shift for twisted boundary conditions Corrections for scattering states Conclusion: corrected atom-dimer results Bound states in a box p. 34
79 Bound states in moving frames So far... considered two-particle state directly in relative coordinates wavefunction ψ(r), r = r 1 r 2 r m 2 m 1 Bound states in a box p. 35
80 Bound states in moving frames So far... considered two-particle state directly in relative coordinates wavefunction ψ(r), r = r 1 r 2 Now full wavefunction Ψ(r 1, r 2 ) = e ip R ψ(r) r P = center-of-mass momentum m 1 R = αr 1 + (1 α)r 2, α = m 1 + m 2 m 1 m 2 P Bound states in a box p. 35
81 Bound states in moving frames So far... considered two-particle state directly in relative coordinates wavefunction ψ(r), r = r 1 r 2 Now full wavefunction Ψ(r 1, r 2 ) = e ip R ψ(r) r P = center-of-mass momentum m 1 R = αr 1 + (1 α)r 2, α = m 1 + m 2 m 1 Put system into finite box, impose periodic BC... L m 2 P L Bound states in a box p. 35
82 Twisted boundary conditions m 2 Now Ψ(r 1, r 2 ) has to be periodic! m 1 r P L Ψ(r 1 +nl, r 2 ) = e ip R e iαlp n ψ(r+nl)= Ψ(r 1, r 2 ) ψ(r + nl) = e iαlp n ψ(r) L twisted boundary conditions Bound states in a box p. 36
83 Twisted boundary conditions m 2 Now Ψ(r 1, r 2 ) has to be periodic! m 1 r P L Ψ(r 1 +nl, r 2 ) = e ip R e iαlp n ψ(r+nl)= Ψ(r 1, r 2 ) ψ(r + nl) = e iαlp n ψ(r) L twisted boundary conditions Question What is the finite-volume mass shift in this case? Bound states in a box p. 36
84 Mass shift for twisted boundary conditions boundary condition: ψ(r + nl) = e iθ n ψ(r), θ = αlp new ansatz: ψ 0 (r) = n Z 3 ψ B (r + nl) e iθ n Bound states in a box p. 37
85 Mass shift for twisted boundary conditions boundary condition: ψ(r + nl) = e iθ n ψ(r), θ = αlp new ansatz: ψ 0 (r) = ψ B (r + nl) e iθ n n Z 3 m B = d 3 r ψb(r) V (r) ψ B (r + nl) e iθ n + O ( e 2κL ) n =1 S-wave result m B = γ 2 e κl µl n=ê x,ê y,ê z cos(θ n) + Bound states in a box p. 37
86 Mass shift for twisted boundary conditions boundary condition: ψ(r + nl) = e iθ n ψ(r), θ = αlp new ansatz: ψ 0 (r) = ψ B (r + nl) e iθ n n Z 3 m B = d 3 r ψb(r) V (r) ψ B (r + nl) e iθ n + O ( e 2κL ) n =1 S-wave result m B = γ 2 e κl µl n=ê x,ê y,ê z cos(θ n) + cos(θ n) = 3 for θ = (0, 0, 0) consistent n Bound states in a box p. 37
87 Mass shift for twisted boundary conditions boundary condition: ψ(r + nl) = e iθ n ψ(r), θ = αlp new ansatz: ψ 0 (r) = ψ B (r + nl) e iθ n n Z 3 m B = d 3 r ψb(r) V (r) ψ B (r + nl) e iθ n + O ( e 2κL ) n =1 S-wave result m B = γ 2 e κl µl n=ê x,ê y,ê z cos(θ n) + cos(θ n) = 3 for θ = (0, 0, 0) consistent n The mass shift vanishes in certain moving frames! Davoudi & Savage, arxiv: Bound states in a box p. 37
88 Scattering states Now consider the scattering of two states A and B... Scattering wavefunction r Ψ p = c k e i 2π k L r ( 2π k/l ) 2 p 2, E AB(p, L) = Ψ p Ĥ Ψ p Ψ p Ψ p E A k (L) E A k (L) E A k ( ) = 0 E B k (L) = γ 2 e κl µl l=1,2,3 cos(2πα B k l )
89 Scattering states Now consider the scattering of two states A and B... Scattering wavefunction r Ψ p = c k e i 2π k L r ( 2π k/l ) 2 p 2, E AB(p, L) = Ψ p Ĥ Ψ p Ψ p Ψ p E A k (L) E A k (L) E A k ( ) = 0 E B k (L) = γ 2 e κl µl l=1,2,3 cos(2πα B k l )
90 Topological correction factors Final result: Topological volume factor τ(η) = 1 N k l=1,2,3 cos(2παk l) 3( k 2 η) 2, η = ( ) Lp 2 2π E AB (p, L) E AB (p, ) = τ A (η) E A 0 (L) + τ B(η) E B 0 (L) Subtract this correction from measured energy levels! Bound states in a box p. 39
91 Corrected atom-dimer results H 1, τ(η) = 1 H 2, τ(η) = 1 STM equation κ D a AD κ D a latt calculation by S. Bour, D. Lee, U.-G. Meißner Bound states in a box p. 40
92 Corrected atom-dimer results κ D a AD H 1, full H 2, full H 1, τ(η) = 1 H 2, τ(η) = 1 STM equation κ D a latt calculation by S. Bour, D. Lee, U.-G. Meißner Bound states in a box p. 40
93 Corrected atom-dimer results H 1, τ(η) = 1 H 2, τ(η) = 1 STM equation κ D r AD κ D a latt calculation by S. Bour, D. Lee, U.-G. Meißner Bound states in a box p. 41
94 Corrected atom-dimer results κ D r AD H 1, full H 2, full H 1, τ(η) = 1 H 2, τ(η) = 1 STM equation κ D a latt calculation by S. Bour, D. Lee, U.-G. Meißner Bound states in a box p. 41
95 The End Summary Mass shift can be calculated for bound states with arbitrary l. Sign of the shift can be related to parity of the states. Predictions can be tested by numerical calculations. Mass shift of composite particles has to be corrected for in scattering calculations. Bound states in a box p. 42
96 The End Summary Mass shift can be calculated for bound states with arbitrary l. Sign of the shift can be related to parity of the states. Predictions can be tested by numerical calculations. Mass shift of composite particles has to be corrected for in scattering calculations. *** Thanks for your attention! Bound states in a box p. 42
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