Few Body Methods in Nuclear Physics - Lecture I

Transcription

1 Few Body Methods in Nuclear Physics - Lecture I Nir Barnea The Hebrew University, Jerusalem, Israel Sept. 2010

2 Course Outline 1 Introduction - Few-Body Nuclear Physics 2 Gaussian Expansion - The Stochastic Variational Method 3 The Harmonic Oscillator Basis - The No Core Shell Model 4 Hyperspherical coordinates and Hyperspherical Harmonics 5 Reactions - The Lorentz Integral Transform Method

3 Outline 1 Few-Body systems in Nuclear Physics 2 The Nuclear Theory 3 The Argonne AV18 interaction 4 Effective Field Theory 5 Nuclear currents 6 Bound state methods in few-body nuclear physics

4 Few-Body systems in Nuclear Physics Few-Body systems in Nuclear Physics Goals For small systems it possible to obtain an accurate and controlled solution of the underlying equations.

5 Few-Body systems in Nuclear Physics Few-Body systems in Nuclear Physics Goals For small systems it possible to obtain an accurate and controlled solution of the underlying equations. Few-body systems are the test ground for nuclear physics.

6 Few-Body systems in Nuclear Physics Few-Body systems in Nuclear Physics Goals For small systems it possible to obtain an accurate and controlled solution of the underlying equations. Few-body systems are the test ground for nuclear physics. The aim it to nail down the nuclear interaction.

7 Few-Body systems in Nuclear Physics Few-Body systems in Nuclear Physics Goals For small systems it possible to obtain an accurate and controlled solution of the underlying equations. Few-body systems are the test ground for nuclear physics. The aim it to nail down the nuclear interaction. Derive and verify the nuclear currents (vector and axial vector currents).

8 Few-Body systems in Nuclear Physics Few-Body systems in Nuclear Physics Goals For small systems it possible to obtain an accurate and controlled solution of the underlying equations. Few-body systems are the test ground for nuclear physics. The aim it to nail down the nuclear interaction. Derive and verify the nuclear currents (vector and axial vector currents). Understand generic properties of the systems at hand.

9 Few-Body systems in Nuclear Physics Few-Body systems in Nuclear Physics Goals For small systems it possible to obtain an accurate and controlled solution of the underlying equations. Few-body systems are the test ground for nuclear physics. The aim it to nail down the nuclear interaction. Derive and verify the nuclear currents (vector and axial vector currents). Understand generic properties of the systems at hand. Astrophysical reaction rates.

10 Few-Body systems in Nuclear Physics Few-Body systems in Nuclear Physics Goals For small systems it possible to obtain an accurate and controlled solution of the underlying equations. Few-body systems are the test ground for nuclear physics. The aim it to nail down the nuclear interaction. Derive and verify the nuclear currents (vector and axial vector currents). Understand generic properties of the systems at hand. Astrophysical reaction rates. These goals are achieved through a combined experimental and theoretical effort to measure and calculate nuclear properties, reaction rates and cross sections to very high precision.

11 The Nuclear Theory Nuclear Theory Background The underlying theory for nuclear physics is QCD.

12 The Nuclear Theory Nuclear Theory Background The underlying theory for nuclear physics is QCD. At low energy QCD is non-perturbative lattice (see Introduction to lattice QCD by C. Alexandro)

13 The Nuclear Theory Nuclear Theory Background The underlying theory for nuclear physics is QCD. At low energy QCD is non-perturbative lattice (see Introduction to lattice QCD by C. Alexandro) Currently no reliable NN interactions can be derived from lattice calculations (this might change in the near future).

14 The Nuclear Theory Nuclear Theory Background The underlying theory for nuclear physics is QCD. At low energy QCD is non-perturbative lattice (see Introduction to lattice QCD by C. Alexandro) Currently no reliable NN interactions can be derived from lattice calculations (this might change in the near future). Therefore the Nuclear theory is based on phenomenology.

15 The Nuclear Theory Nuclear Theory Background The underlying theory for nuclear physics is QCD. At low energy QCD is non-perturbative lattice (see Introduction to lattice QCD by C. Alexandro) Currently no reliable NN interactions can be derived from lattice calculations (this might change in the near future). Therefore the Nuclear theory is based on phenomenology. However at low energy the quarks and glueons degrees of freedom are replaced by baryons and mesons. L QCD (q, G) L Nucl (N, π,...)

16 The Nuclear Theory Low energy Nuclear Physics Background The relevant low-energy degrees of freedom are neutrons, protons and pions m p = MeV m n = MeV m π± = MeV m π0 = MeV

17 The Nuclear Theory Low energy Nuclear Physics Background The relevant low-energy degrees of freedom are neutrons, protons and pions m p = MeV m n = MeV m π± = MeV m π0 = MeV Neutrons and protons are assume to be identical particles with internal degree of freedom - isospin. Same goes for the pions.

18 The Nuclear Theory Low energy Nuclear Physics Background The relevant low-energy degrees of freedom are neutrons, protons and pions m p = MeV m n = MeV m π± = MeV m π0 = MeV Neutrons and protons are assume to be identical particles with internal degree of freedom - isospin. Same goes for the pions. It is assumed that the nuclei are non-relativistic objects ψγ µ µ ψ ψ 2 2 ψ

19 The Nuclear Theory Low energy Nuclear Physics Background The relevant low-energy degrees of freedom are neutrons, protons and pions m p = MeV m n = MeV m π± = MeV m π0 = MeV Neutrons and protons are assume to be identical particles with internal degree of freedom - isospin. Same goes for the pions. It is assumed that the nuclei are non-relativistic objects ψγ µ µ ψ ψ 2 2 ψ We seek an Hamiltonian of the form H Nucl = i 2 i 2 + V Nucl(x i ) + V EM (x i ) x = r, σ, τ

20 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties

21 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties Hermiticity V NN = V NN

22 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties Hermiticity Permutational symmetry V NN = V NN [(1, 2), V NN ] = 0

23 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties Hermiticity Permutational symmetry Invariance under Rotation V NN = V NN [(1, 2), V NN ] = 0 [J a, V NN ] = 0

24 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties Hermiticity Permutational symmetry Invariance under Rotation Parity P : r r P is an unitary operator V NN = V NN [(1, 2), V NN ] = 0 [J a, V NN ] = 0 PV NN P 1 = V NN PiP 1 = i

25 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties Hermiticity Permutational symmetry Invariance under Rotation Parity P : r r P is an unitary operator Time reversal T : t t T is an anti-unitary operator V NN = V NN [(1, 2), V NN ] = 0 [J a, V NN ] = 0 PV NN P 1 = V NN PiP 1 = i TV NN T 1 = V NN TiT 1 = i

26 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties Hermiticity Permutational symmetry Invariance under Rotation Parity P : r r P is an unitary operator Time reversal T : t t T is an anti-unitary operator Isoscalar (only at leading orders) V NN = V NN [(1, 2), V NN ] = 0 [J a, V NN ] = 0 PV NN P 1 = V NN PiP 1 = i TV NN T 1 = V NN TiT 1 = i [τ a, V NN ] = 0

27 The Nuclear Theory Nuclear Theory The nuclear force fulfills the following properties Hermiticity Permutational symmetry Invariance under Rotation Parity P : r r P is an unitary operator Time reversal T : t t T is an anti-unitary operator Isoscalar (only at leading orders) Invariance under Galilean transformation V NN = V NN [(1, 2), V NN ] = 0 [J a, V NN ] = 0 PV NN P 1 = V NN PiP 1 = i TV NN T 1 = V NN TiT 1 = i [τ a, V NN ] = 0 V NN (r 1 +c, r 2 +c,...) = V NN (r 1, r 2,...)

28 The Nuclear Theory Expanding the 2-body potential The 2-body interaction is a function of the operators r = (r 2 r 1 ), p = 1 2 (p 2 p 1 ), σ 1, σ 2, τ 1, τ 2

29 The Nuclear Theory Expanding the 2-body potential The 2-body interaction is a function of the operators r = (r 2 r 1 ), p = 1 2 (p 2 p 1 ), σ 1, σ 2, τ 1, τ 2 Starting with the spin operators we have tensors of ranks 0,1,2 σ 1 σ 2, S = σ 1 + σ 2, [σ 1 σ 2 ] 2, [σ 1 σ 2 ] 1, σ 1 σ 2 2

30 The Nuclear Theory Expanding the 2-body potential The 2-body interaction is a function of the operators r = (r 2 r 1 ), p = 1 2 (p 2 p 1 ), σ 1, σ 2, τ 1, τ 2 Starting with the spin operators we have tensors of ranks 0,1,2 σ 1 σ 2, S = σ 1 + σ 2, [σ 1 σ 2 ] 2, [σ 1 σ 2 ] 1, σ 1 σ 2 2 Due to permutational symmetry the last two terms must be multiplied with r or p and appear with even powers = becoming a function of the other terms.

31 The Nuclear Theory Expanding the 2-body potential The 2-body interaction is a function of the operators r = (r 2 r 1 ), p = 1 2 (p 2 p 1 ), σ 1, σ 2, τ 1, τ 2 Starting with the spin operators we have tensors of ranks 0,1,2 σ 1 σ 2, S = σ 1 + σ 2, [σ 1 σ 2 ] 2, [σ 1 σ 2 ] 1, σ 1 σ 2 2 Due to permutational symmetry the last two terms must be multiplied with r or p and appear with even powers = becoming a function of the other terms. The simplest scalar spatial operators are r 2, p 2, L 2 = (r p) 2, r p.

32 The Nuclear Theory Expanding the 2-body potential The 2-body interaction is a function of the operators r = (r 2 r 1 ), p = 1 2 (p 2 p 1 ), σ 1, σ 2, τ 1, τ 2 Starting with the spin operators we have tensors of ranks 0,1,2 σ 1 σ 2, S = σ 1 + σ 2, [σ 1 σ 2 ] 2, [σ 1 σ 2 ] 1, σ 1 σ 2 2 Due to permutational symmetry the last two terms must be multiplied with r or p and appear with even powers = becoming a function of the other terms. The simplest scalar spatial operators are r 2, p 2, L 2 = (r p) 2, r p. Due to time reversal the last term must appear with even powers and becomes a function r 2, p 2, L 2.

33 The Nuclear Theory Expanding the 2-body potential The 2-body interaction is a function of the operators r = (r 2 r 1 ), p = 1 2 (p 2 p 1 ), σ 1, σ 2, τ 1, τ 2 Starting with the spin operators we have tensors of ranks 0,1,2 σ 1 σ 2, S = σ 1 + σ 2, [σ 1 σ 2 ] 2, [σ 1 σ 2 ] 1, σ 1 σ 2 2 Due to permutational symmetry the last two terms must be multiplied with r or p and appear with even powers = becoming a function of the other terms. The simplest scalar spatial operators are r 2, p 2, L 2 = (r p) 2, r p. Due to time reversal the last term must appear with even powers and becomes a function r 2, p 2, L 2. There is only one rank 1 operator L S

34 The Nuclear Theory The 2-body potential There are three rank 2 operators S 12 = (σ 1 ˆr)(σ 2 ˆr) 1 3 (σ 1 σ 2 ) S 12 = (σ 1 ˆp)(σ 2 ˆp) 1 3 (σ 1 σ 2 ) Q 12 = 1 2 {(σ 1 L)(σ 2 L) + (σ 2 L)(σ 1 L)}

35 The Nuclear Theory The 2-body potential There are three rank 2 operators S 12 = (σ 1 ˆr)(σ 2 ˆr) 1 3 (σ 1 σ 2 ) S 12 = (σ 1 ˆp)(σ 2 ˆp) 1 3 (σ 1 σ 2 ) Q 12 = 1 2 {(σ 1 L)(σ 2 L) + (σ 2 L)(σ 1 L)} Summing up we have the following operators Ô p = { 1, σ 1 σ 2, p 2, r 2, L 2, L S, S 12, Q 12, S 12 } {1, τ1 τ 2 }

36 The Nuclear Theory The 2-body potential There are three rank 2 operators S 12 = (σ 1 ˆr)(σ 2 ˆr) 1 3 (σ 1 σ 2 ) S 12 = (σ 1 ˆp)(σ 2 ˆp) 1 3 (σ 1 σ 2 ) Q 12 = 1 2 {(σ 1 L)(σ 2 L) + (σ 2 L)(σ 1 L)} Summing up we have the following operators Ô p = { 1, σ 1 σ 2, p 2, r 2, L 2, L S, S 12, Q 12, S 12 } {1, τ1 τ 2 } The most general 2-body potential can be written as an expansion in these operators or powers of them.

37 The Nuclear Theory The 2-body potential There are three rank 2 operators S 12 = (σ 1 ˆr)(σ 2 ˆr) 1 3 (σ 1 σ 2 ) S 12 = (σ 1 ˆp)(σ 2 ˆp) 1 3 (σ 1 σ 2 ) Q 12 = 1 2 {(σ 1 L)(σ 2 L) + (σ 2 L)(σ 1 L)} Summing up we have the following operators Ô p = { 1, σ 1 σ 2, p 2, r 2, L 2, L S, S 12, Q 12, S 12 } {1, τ1 τ 2 } The most general 2-body potential can be written as an expansion in these operators or powers of them. In practice we make the choice either working in configuration or momentum space. Thus the practical expansion might take somewhat different form.

38 The Nuclear Theory Nuclear Potentials Examples for NN interaction Argonne V18 potential - AV18 Effective Field Theory - EFT potential

39 The Argonne AV18 interaction Phenomenology - AV18 The nuclear force can be expanded into a series of operators

40 The Argonne AV18 interaction Phenomenology - AV18 The nuclear force can be expanded into a series of operators The terms in the expansion are dictated by the symmetries

41 The Argonne AV18 interaction Phenomenology - AV18 The nuclear force can be expanded into a series of operators The terms in the expansion are dictated by the symmetries The nuclear Hamiltonian H = K i + V ij + K i = K CI i i i<j i<j<k V ijk + K CSB i = [ ( ) + τ ( z i 1 1 )] 2 i 2 2 m p m n 2 m p m n

42 The Argonne AV18 interaction Phenomenology - AV18 The nuclear force can be expanded into a series of operators The terms in the expansion are dictated by the symmetries The nuclear Hamiltonian H = K i + V ij + i i<j i<j<k K i = K CI i + K CSB i = [ ( ) + τ ( z i 1 1 )] 2 i 2 2 m p m n 2 m p m n The Potential is composed of EM and NUCLEAR short, intermediate and 1-pion exchange terms V ij = v EM ij V ijk + v π ij + vi ij + vs ij

43 The Argonne AV18 interaction Phenomenology - AV18 The nuclear force can be expanded into a series of operators The terms in the expansion are dictated by the symmetries The nuclear Hamiltonian H = K i + V ij + i i<j i<j<k K i = K CI i + K CSB i = [ ( ) + τ ( z i 1 1 )] 2 i 2 2 m p m n 2 m p m n The Potential is composed of EM and NUCLEAR short, intermediate and 1-pion exchange terms v EM ij V ij = v EM ij + v π ij + vi ij + vs ij - includes pp, pn, & nn electromagnetic terms. V ijk

44 The Argonne AV18 interaction Phenomenology - AV18 The nuclear force can be expanded into a series of operators The terms in the expansion are dictated by the symmetries The nuclear Hamiltonian H = K i + V ij + i i<j i<j<k K i = K CI i + K CSB i = [ ( ) + τ ( z i 1 1 )] 2 i 2 2 m p m n 2 m p m n The Potential is composed of EM and NUCLEAR short, intermediate and 1-pion exchange terms V ij = v EM ij + v π ij + vi ij + vs ij v EM ij - includes pp, pn, & nn electromagnetic terms. The pion exchange term V ijk v π ij = [Y π(r ij )σ i σ j + T π (r ij )S ij ]τ i τ j here Y π is the Yukawa function with cutoff T π is the corresponding function to the tensor term.

45 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The Yukawa functions Y π (r) = e µr (1 e cr2) ; T π (r) = µr ( µr + 3 ) e µr (1 (µr) 2 e cr2) 2 µr

46 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The Yukawa functions Y π (r) = e µr (1 e cr2) ; T π (r) = µr The tensor operator S ij = 3(σ i ˆr ij )(σ j ˆr ij ) σ i σ j ( µr + 3 ) e µr (1 (µr) 2 e cr2) 2 µr

47 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The Yukawa functions Y π (r) = e µr (1 e cr2) ; T π (r) = µr ( µr + 3 ) e µr (1 (µr) 2 e cr2) 2 µr The tensor operator S ij = 3(σ i ˆr ij )(σ j ˆr ij ) σ i σ j The long range part is governed by the pion mass µ = m π c 2 / c 0.7[fm 1 ]

48 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The Yukawa functions Y π (r) = e µr (1 e cr2) ; T π (r) = µr ( µr + 3 ) e µr (1 (µr) 2 e cr2) 2 µr The tensor operator S ij = 3(σ i ˆr ij )(σ j ˆr ij ) σ i σ j The long range part is governed by the pion mass µ = m π c 2 / c 0.7[fm 1 ] The intermediate term has 2-pion exchange range v I ij = c I ptπ(r 2 ij )Ô p ij p

49 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The Yukawa functions Y π (r) = e µr (1 e cr2) ; T π (r) = µr ( µr + 3 ) e µr (1 (µr) 2 e cr2) 2 µr The tensor operator S ij = 3(σ i ˆr ij )(σ j ˆr ij ) σ i σ j The long range part is governed by the pion mass µ = m π c 2 / c 0.7[fm 1 ] The intermediate term has 2-pion exchange range v I ij = c I ptπ(r 2 ij )Ô p ij The Short range term p v S ij = [c s 0,p + cs 1,p r ij + c s 2,p r2 ij]w(r ij )Ô p ij p

50 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The Yukawa functions Y π (r) = e µr (1 e cr2) ; T π (r) = µr ( µr + 3 ) e µr (1 (µr) 2 e cr2) 2 µr The tensor operator S ij = 3(σ i ˆr ij )(σ j ˆr ij ) σ i σ j The long range part is governed by the pion mass µ = m π c 2 / c 0.7[fm 1 ] The intermediate term has 2-pion exchange range v I ij = c I ptπ(r 2 ij )Ô p ij The Short range term p v S ij = [c s 0,p + cs 1,p r ij + c s 2,p r2 ij]w(r ij )Ô p ij p The function W(r) is a Woods-Saxon function which provides the hard core W(r) = [ 1 + e (r r 0)/a ] 1 r 0 = 0.5fm, a = 0.2fm

51 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The Yukawa functions Y π (r) = e µr (1 e cr2) ; T π (r) = µr ( µr + 3 ) e µr (1 (µr) 2 e cr2) 2 µr The tensor operator S ij = 3(σ i ˆr ij )(σ j ˆr ij ) σ i σ j The long range part is governed by the pion mass µ = m π c 2 / c 0.7[fm 1 ] The intermediate term has 2-pion exchange range v I ij = c I ptπ(r 2 ij )Ô p ij The Short range term p v S ij = [c s 0,p + cs 1,p r ij + c s 2,p r2 ij]w(r ij )Ô p ij p The function W(r) is a Woods-Saxon function which provides the hard core W(r) = [ 1 + e (r r 0)/a ] 1 There are about 30 adjustable parameters!!! r 0 = 0.5fm, a = 0.2fm

52 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The operators take the form: Central terms p = 1,..., 4 1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 )

53 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The operators take the form: Central terms p = 1,..., 4 1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 ) Tensor force p = 5, 6 S ij, S ij (τ 1 τ 2 )

54 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The operators take the form: Central terms p = 1,..., 4 1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 ) Tensor force p = 5, 6 LS coupling p = 7, 8 S ij, S ij (τ 1 τ 2 ) L S, L S(τ 1 τ 2 )

55 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The operators take the form: Central terms p = 1,..., 4 1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 ) Tensor force p = 5, 6 LS coupling p = 7, 8 L 2 terms p = 9, 10, 11, 12 S ij, S ij (τ 1 τ 2 ) L S, L S(τ 1 τ 2 ) L 2 {1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 )}

56 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The operators take the form: Central terms p = 1,..., 4 1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 ) Tensor force p = 5, 6 LS coupling p = 7, 8 L 2 terms p = 9, 10, 11, 12 S ij, S ij (τ 1 τ 2 ) L S, L S(τ 1 τ 2 ) (LS) 2 coupling p = 13, 14 L 2 {1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 )} (L S) 2, (L S) 2 (τ 1 τ 2 )

57 The Argonne AV18 interaction Phenomenology - AV18 (Continue) The operators take the form: Central terms p = 1,..., 4 1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 ) Tensor force p = 5, 6 LS coupling p = 7, 8 L 2 terms p = 9, 10, 11, 12 S ij, S ij (τ 1 τ 2 ) L S, L S(τ 1 τ 2 ) L 2 {1, (σ 1 σ 2 ), (τ 1 τ 2 ), (σ 1 σ 2 )(τ 1 τ 2 )} (LS) 2 coupling p = 13, 14 (L S) 2, (L S) 2 (τ 1 τ 2 ) Isospin symmetry breaking terms T 12 = 3τ 1z τ 2z (τ 1 τ 2 ), (σ 1 σ 2 )T 12, S 12 T 12, τ 1z + τ 2z

58 The Argonne AV18 interaction The Nuclear force - Argonne V18

59 The Argonne AV18 interaction The Nuclear force - Argonne V18

60 Effective Field Theory Effective Field Theory In Effective field theory the Lagrangian is expanded using the ration between a small momentum Q and cutoff Λ L(N, π) = L n (N, π) The Lagrangian L n (N, π) contain all possible terms of order ( Q Λ ) n compatible with QCD symmetries. n

61 Effective Field Theory Effective Field Theory In Effective field theory the Lagrangian is expanded using the ration between a small momentum Q and cutoff Λ L(N, π) = L n (N, π) The Lagrangian L n (N, π) contain all possible terms of order ( Q Λ ) n compatible with QCD symmetries. The Hamiltonian H = H 0 + H I = H 0 + H 1 + H 2 + H n

62 Effective Field Theory Effective Field Theory In Effective field theory the Lagrangian is expanded using the ration between a small momentum Q and cutoff Λ L(N, π) = L n (N, π) The Lagrangian L n (N, π) contain all possible terms of order ( Q Λ ) n compatible with QCD symmetries. The Hamiltonian H = H 0 + H I = H 0 + H 1 + H 2 + H H N0 = N 2 2m N H π0 = 1 2 π π m2 ππ 2 n

63 Effective Field Theory Effective Field Theory In Effective field theory the Lagrangian is expanded using the ration between a small momentum Q and cutoff Λ L(N, π) = L n (N, π) The Lagrangian L n (N, π) contain all possible terms of order ( Q Λ ) n compatible with QCD symmetries. The Hamiltonian H = H 0 + H I = H 0 + H 1 + H 2 + H H N0 = N 2 2m N H π0 = 1 2 π π m2 ππ 2 H 1 = g A 2f π N τ a σ π a N H 2 = 1 (π 2fπ 2 µ π)(π µ π) + 1 N τ (π π)n 4fπ 2 + C S 2 N NN N + C T 2 N σn N σn n

64 Effective Field Theory EFT potential ( ) 2 ga (σ 1 q)(σ 2 q) V LO = τ 1 τ 2 + C 2f π q 2 + m 2 S + C T σ 1 σ 2 π

65 Effective Field Theory EFT potential

66 Effective Field Theory EFT potential

67 Effective Field Theory EFT potential

68 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential.

69 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology.

70 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology. Reproduce NN phase shifts up to pion threshold.

71 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology. Reproduce NN phase shifts up to pion threshold. The 2-body NN force under-binds 3 He, 4 He,...

72 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology. Reproduce NN phase shifts up to pion threshold. The 2-body NN force under-binds 3 He, 4 He,... A 3-body, NNN, force must be supplemented.

73 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology. Reproduce NN phase shifts up to pion threshold. The 2-body NN force under-binds 3 He, 4 He,... A 3-body, NNN, force must be supplemented. The force can be local or non-local.

74 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology. Reproduce NN phase shifts up to pion threshold. The 2-body NN force under-binds 3 He, 4 He,... A 3-body, NNN, force must be supplemented. The force can be local or non-local. Non-central = L, S are not good quantum numbers.

75 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology. Reproduce NN phase shifts up to pion threshold. The 2-body NN force under-binds 3 He, 4 He,... A 3-body, NNN, force must be supplemented. The force can be local or non-local. Non-central = L, S are not good quantum numbers. EFT potentials are naturally formulated in momentum space.

76 Effective Field Theory Nucleon-nucleon interaction Summary Characteristics of realistic NN forces Long range - One Pion Exchange, Yukawa potential. Short range - phenomenology. Reproduce NN phase shifts up to pion threshold. The 2-body NN force under-binds 3 He, 4 He,... A 3-body, NNN, force must be supplemented. The force can be local or non-local. Non-central = L, S are not good quantum numbers. EFT potentials are naturally formulated in momentum space. There are about 30 adjustable parameters.

77 Effective Field Theory Calibrating and Testing the NN interaction The 2 and 3-body observables are used to adjust the nuclear force. Exact calculation of other few-body observables are needed in order test the interaction. Where to test? A > 3 ground states or 3, 4,...-body scattering and reactions

78 Effective Field Theory Calibrating and Testing the NN interaction The 2 and 3-body observables are used to adjust the nuclear force. Exact calculation of other few-body observables are needed in order test the interaction. Where to test? A > 3 ground states or 3, 4,...-body scattering and reactions

79 Nuclear currents A comment about currents In order to calculate reactions we need to know how the nuclei are coupled to external fields. H = H Nucl + dxj µ Nucl (x)a µ(x)

80 Nuclear currents A comment about currents In order to calculate reactions we need to know how the nuclei are coupled to external fields. H = H Nucl + dxj µ Nucl (x)a µ(x) The nuclear current can be vector or axial vector J µ Nucl = Jµ V + Jµ A Even the electric current is not simple in nuclei. For a vector current the charge is conserved thus J = t ρ = i[h, ρ]

81 Nuclear currents A comment about currents In order to calculate reactions we need to know how the nuclei are coupled to external fields. H = H Nucl + dxj µ Nucl (x)a µ(x) The nuclear current can be vector or axial vector J µ Nucl = Jµ V + Jµ A Even the electric current is not simple in nuclei. For a vector current the charge is conserved thus J = t ρ = i[h, ρ] the charge density is ρ(r) = δ(r i r)τ z i i

82 Nuclear currents A comment about currents In order to calculate reactions we need to know how the nuclei are coupled to external fields. H = H Nucl + dxj µ Nucl (x)a µ(x) The nuclear current can be vector or axial vector J µ Nucl = Jµ V + Jµ A Even the electric current is not simple in nuclei. For a vector current the charge is conserved thus J = t ρ = i[h, ρ] the charge density is but ρ(r) = δ(r i r)τ z i i [τ i τ j, τ z i ] = 2i(τ i τ j ) z

83 Nuclear currents A comment about currents In order to calculate reactions we need to know how the nuclei are coupled to external fields. H = H Nucl + dxj µ Nucl (x)a µ(x) The nuclear current can be vector or axial vector J µ Nucl = Jµ V + Jµ A Even the electric current is not simple in nuclei. For a vector current the charge is conserved thus J = t ρ = i[h, ρ] the charge density is but ρ(r) = δ(r i r)τ z i i [τ i τ j, τ z i ] = 2i(τ i τ j ) z Conclusion: the nuclear EM current must include 2 and 3 body terms, consistent with the potential. This requirement is naturally fulfilled in EFT where the potentials and currents are calculated from the same starting point.

84 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions.

85 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions. 2 The Hilbert space grows as ( L x ) 3A 3

86 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions. 2 The Hilbert space grows as ( L x ) 3A 3 3 The number of internal degrees of freedom is 4 A.

87 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions. 2 The Hilbert space grows as ( L x ) 3A 3 3 The number of internal degrees of freedom is 4 A. 4 The wave function must be anti-symmetric.

88 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions. 2 The Hilbert space grows as ( L x ) 3A 3 3 The number of internal degrees of freedom is 4 A. 4 The wave function must be anti-symmetric. 5 The wave function should be invariant to Galilean transformation.

89 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions. 2 The Hilbert space grows as ( L x ) 3A 3 3 The number of internal degrees of freedom is 4 A. 4 The wave function must be anti-symmetric. 5 The wave function should be invariant to Galilean transformation. 6 The tensor force is very important, L, S are no good...

90 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions. 2 The Hilbert space grows as ( L x ) 3A 3 3 The number of internal degrees of freedom is 4 A. 4 The wave function must be anti-symmetric. 5 The wave function should be invariant to Galilean transformation. 6 The tensor force is very important, L, S are no good... 7 Continuum wave function is a very complicated object.

91 Bound state methods in few-body nuclear physics Solution of the few-body problem The Problem is... 1 The Schrodinger equation is a partial differential equation in 3A 3 dimensions. 2 The Hilbert space grows as ( L x ) 3A 3 3 The number of internal degrees of freedom is 4 A. 4 The wave function must be anti-symmetric. 5 The wave function should be invariant to Galilean transformation. 6 The tensor force is very important, L, S are no good... 7 Continuum wave function is a very complicated object. 8 Scattering boundary conditions for more then 2 charged particles are problematic.

92 Bound state methods in few-body nuclear physics Solution of the few-body problem

93 Bound state methods in few-body nuclear physics Solution of the few-body problem

94 Bound state methods in few-body nuclear physics BENCHMARK for 4 He ground state with AV8 potential H. Kamada et al., PRC (2001) Method Basis T V E b r2 FY (5) (10) (5) 1.485(3) CRCGV Gaussians SVM Gaussians HH HH (1) GFMC 102.3(1.0) (1.0) (2) 1.490(5) NCSM HO (20) EIHH HH 100.8(9) (9) (10) 1.486

95 Bound state methods in few-body nuclear physics References The Nucleon-Nucleon Interaction E. Epelbaum, PhD Dissertation, Bochum (2000). The Stochastic Variational Method K. Varga and Y. Suzuki, Phys. Rev. C52, 2885 (1995). The No Core Shell Model P. Navrátil and B. R. Barrett, Phys. Rev. C54, 2986 (1996) The HH expansion and results A. Kievsky, S. Rosati, M. Viviani, L. E. Marcucci, and L. Girlanda J. Phys. G (2008) Review of the LIT method and applications V. Efros, W. Leidemann, G. Orlandini, and N. Barnea, J. Phys. G 34 R459 (2007) The Effective Interaction HH method N. Barnea, W. Leidemann, G. Orlandini, Phys. Rev. C (2000).