Nuclear structure I: Introduction and nuclear interactions

Transcription

1 Nuclear structure I: Introduction and nuclear interactions Stefano Gandolfi Los Alamos National Laboratory (LANL) National Nuclear Physics Summer School Massachusetts Institute of Technology (MIT) July 18-29, 2016 Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 1 / 33

2 Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 2 / 33

3 The Quantum Ladder??? Living Organisms, Man-made Structures Molecules Atomic Nuclei Quarks and Leptons Galaxy clusters Galaxies Atoms Superstrings? Stars Planets Cells, Materials Baryons and mesons subatomic atomic macroscopic reduction complexity Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 3 / 33

4 QCD Complex Systems Cosmos Quantum manybody physics Physics of Nuclei Nuclei in the Universe Atomic and Molecular Physics Condensed Matter Physics Materials Science Quantum Chemistry Fundamental interactions Particle Physics Cosmology Astrophysics Cosmology Astronomy Gravitation Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 4 / 33

5 The big picture of the microscopic world Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 5 / 33

6 Neutron stars Neutron star is a wonderful natural laboratory, rich of physics! D. Page Atmosphere: atomic and plasma physics Crust: physics of superfluids (neutrons, vortex), solid state physics (nuclei) Inner crust: deformed nuclei, pasta phase Outer core: nuclear matter Inner core: hyperons? quark matter? π or K condensates?...? Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 6 / 33

7 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

8 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

9 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

10 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

11 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

12 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

13 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

14 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

15 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

16 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Nuclear astrophysics: What s the relation between nuclear physics and neutron stars? What are the composition and properties of neutron stars? How do supernovae explode? How are heavy elements formed? Very incomplete list... Many questions will arise during these lectures! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 7 / 33

17 Fundamental questions in nuclear physics Let s start! Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 8 / 33

18 How do we describe nuclear systems? Degrees of freedom? LQCD quark models ab initio scale separation Resolution How are nuclei made? Origin of elements, isotopes Hot and dense quark-gluon matter Hadron structure Hadron-Nuclear interface CI DFT collective models Effective Field Theory Nuclear structure Nuclear reactions New standard model Applications of nuclear science Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 9 / 33

19 Tom Banks: only a fool would imagine that one should try to understand the properties of waves in the ocean in terms of Feynman-diagram calculations in the standard model, even if the latter understanding is possible in principle. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 10 / 33

20 Tom Banks: only a fool would imagine that one should try to understand the properties of waves in the ocean in terms of Feynman-diagram calculations in the standard model, even if the latter understanding is possible in principle. Weinberg s Laws of Progress in Theoretical Physics From: Asymptotic Realms of Physics (ed. by Guth, Huang, Jaffe, MIT Press, 1983). Third Law: 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! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 10 / 33

21 Tom Banks: only a fool would imagine that one should try to understand the properties of waves in the ocean in terms of Feynman-diagram calculations in the standard model, even if the latter understanding is possible in principle. Weinberg s Laws of Progress in Theoretical Physics From: Asymptotic Realms of Physics (ed. by Guth, Huang, Jaffe, MIT Press, 1983). Third Law: 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! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 10 / 33

22 Quantum chromodynamics (QCD) is THE theory. this (unrealistic) picture is already complicated. Calculations even more! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 11 / 33

23 Lattice QCD calculations of single hadron mass spectrum, A. S. Kronfeld, arxiv: H H * H s H s * B c B c * Andreas Kronfeld/Fermi Natl Accelerator Lab. (MeV) π ρ K K η η ω φ N Λ Σ Ξ Σ Ξ Ω Great predictive power, excellent agreement with experiments! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 12 / 33

24 Nucleon-nucleon binding energy from lattice QCD as a function of m π Bd (MeV ) NPLQCD, anisotropic Yamazaki et al. NPLQCD, isotropic Bnn (MeV) NPLQCD, anisotropic Yamazaki et al. NPLQCD, isotropic m π (MeV) Deuteron binding energy m π (MeV) Di-neutron binding energy K. Orginos, et. al, Phys. Rev. D 92, (2015). The problem is the sign problem. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 13 / 33

25 One possible solution: change the degrees of freedom! The goal: use effective nucleon dof s systematically. Seek model independence and theory error estimates Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities or at high momentum transfers, where phase transitions happen Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 14 / 33

26 One possible solution: change the degrees of freedom! The goal: use effective nucleon dof s systematically. Seek model independence and theory error estimates Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities or at high momentum transfers, where phase transitions happen Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 14 / 33

27 One possible solution: change the degrees of freedom! The goal: use effective nucleon dof s systematically. Seek model independence and theory error estimates Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities or at high momentum transfers, where phase transitions happen Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 14 / 33

28 One possible solution: change the degrees of freedom! The goal: use effective nucleon dof s systematically. Seek model independence and theory error estimates Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities or at high momentum transfers, where phase transitions happen Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 14 / 33

29 One possible solution: change the degrees of freedom! The goal: use effective nucleon dof s systematically. Seek model independence and theory error estimates Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities or at high momentum transfers, where phase transitions happen Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 14 / 33

30 One possible solution: change the degrees of freedom! The goal: use effective nucleon dof s systematically. Seek model independence and theory error estimates Future: Use lattice QCD to match via low-energy constants Need quark dof s at higher densities or at high momentum transfers, where phase transitions happen Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 14 / 33

31 Nucleon-nucleon interaction not fundamental (c.f. Lennard-Jones for noble gases) Range 1/m π 1.4 fm vs nucleon rms 0.9 fm Nucleon s wave functions overlap Nucleon s composite objects: expect three- and many-body forces important Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 15 / 33

32 Nucleon-nucleon interaction not fundamental (c.f. Lennard-Jones for noble gases) Range 1/m π 1.4 fm vs nucleon rms 0.9 fm Nucleon s wave functions overlap Nucleon s composite objects: expect three- and many-body forces important Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 15 / 33

33 Nucleon-nucleon interaction not fundamental (c.f. Lennard-Jones for noble gases) Range 1/m π 1.4 fm vs nucleon rms 0.9 fm Nucleon s wave functions overlap Nucleon s composite objects: expect three- and many-body forces important Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 15 / 33

34 Nucleon-nucleon interaction not fundamental (c.f. Lennard-Jones for noble gases) Range 1/m π 1.4 fm vs nucleon rms 0.9 fm Nucleon s wave functions overlap Nucleon s composite objects: expect three- and many-body forces important Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 15 / 33

35 Nucleon-nucleon interaction not fundamental (c.f. Lennard-Jones for noble gases) Range 1/m π 1.4 fm vs nucleon rms 0.9 fm Nucleon s wave functions overlap Nucleon s composite objects: expect three- and many-body forces important But, many nucleon-nucleon data available! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 15 / 33

36 Let s describe the (many-body) system using a non-relativistic Hamiltonian. The d.o.f. are nucleons, described as interacting point-like particles: H = 2 2m A i=1 2 i + i<j v ij + i<j<k V ijk +... The kinetic energy can corrected to account for the proton vs neutron mass difference v ij is an effective two-nucleon potential including the strong interaction and Coulomb force (with corrections due to the spin of nucleons, form factors, etc.) V ijk is a three-nucleon force, whose role and need will be clear later +... can include anything missing (four- five-...-body forces) Assumption: all the nucleon s form factors, their excitations, and other properties can be included in the potentials, and this description is valid until nucleons overlap too much (that means reasonably low densities and momenta), i.e. their structure don t change much. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 16 / 33

37 Let s describe the (many-body) system using a non-relativistic Hamiltonian. The d.o.f. are nucleons, described as interacting point-like particles: H = 2 2m A i=1 2 i + i<j v ij + i<j<k V ijk +... The kinetic energy can corrected to account for the proton vs neutron mass difference v ij is an effective two-nucleon potential including the strong interaction and Coulomb force (with corrections due to the spin of nucleons, form factors, etc.) V ijk is a three-nucleon force, whose role and need will be clear later +... can include anything missing (four- five-...-body forces) Assumption: all the nucleon s form factors, their excitations, and other properties can be included in the potentials, and this description is valid until nucleons overlap too much (that means reasonably low densities and momenta), i.e. their structure don t change much. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 16 / 33

38 Let s describe the (many-body) system using a non-relativistic Hamiltonian. The d.o.f. are nucleons, described as interacting point-like particles: H = 2 2m A i=1 2 i + i<j v ij + i<j<k V ijk +... The kinetic energy can corrected to account for the proton vs neutron mass difference v ij is an effective two-nucleon potential including the strong interaction and Coulomb force (with corrections due to the spin of nucleons, form factors, etc.) V ijk is a three-nucleon force, whose role and need will be clear later +... can include anything missing (four- five-...-body forces) Assumption: all the nucleon s form factors, their excitations, and other properties can be included in the potentials, and this description is valid until nucleons overlap too much (that means reasonably low densities and momenta), i.e. their structure don t change much. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 16 / 33

39 Let s describe the (many-body) system using a non-relativistic Hamiltonian. The d.o.f. are nucleons, described as interacting point-like particles: H = 2 2m A i=1 2 i + i<j v ij + i<j<k V ijk +... The kinetic energy can corrected to account for the proton vs neutron mass difference v ij is an effective two-nucleon potential including the strong interaction and Coulomb force (with corrections due to the spin of nucleons, form factors, etc.) V ijk is a three-nucleon force, whose role and need will be clear later +... can include anything missing (four- five-...-body forces) Assumption: all the nucleon s form factors, their excitations, and other properties can be included in the potentials, and this description is valid until nucleons overlap too much (that means reasonably low densities and momenta), i.e. their structure don t change much. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 16 / 33

40 Let s describe the (many-body) system using a non-relativistic Hamiltonian. The d.o.f. are nucleons, described as interacting point-like particles: H = 2 2m A i=1 2 i + i<j v ij + i<j<k V ijk +... The kinetic energy can corrected to account for the proton vs neutron mass difference v ij is an effective two-nucleon potential including the strong interaction and Coulomb force (with corrections due to the spin of nucleons, form factors, etc.) V ijk is a three-nucleon force, whose role and need will be clear later +... can include anything missing (four- five-...-body forces) Assumption: all the nucleon s form factors, their excitations, and other properties can be included in the potentials, and this description is valid until nucleons overlap too much (that means reasonably low densities and momenta), i.e. their structure don t change much. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 16 / 33

41 Let s describe the (many-body) system using a non-relativistic Hamiltonian. The d.o.f. are nucleons, described as interacting point-like particles: H = 2 2m A i=1 2 i + i<j v ij + i<j<k V ijk +... The kinetic energy can corrected to account for the proton vs neutron mass difference v ij is an effective two-nucleon potential including the strong interaction and Coulomb force (with corrections due to the spin of nucleons, form factors, etc.) V ijk is a three-nucleon force, whose role and need will be clear later +... can include anything missing (four- five-...-body forces) Assumption: all the nucleon s form factors, their excitations, and other properties can be included in the potentials, and this description is valid until nucleons overlap too much (that means reasonably low densities and momenta), i.e. their structure don t change much. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 16 / 33

42 Nucleon-nucleon interaction The force between electrons (Coulomb) is the same in spin singlet and triplet. Only the (Fermi) statistics makes a distinction. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 17 / 33

43 Nucleon-nucleon interaction The force between electrons (Coulomb) is the same in spin singlet and triplet. Only the (Fermi) statistics makes a distinction. The force between nucleons strongly depends upon the spin and the isospin. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 17 / 33

44 Nucleon-nucleon interaction Let s consider the isospin (for the spin is the same story): { 1 T = 0 T z = 0 2 ( np pn ) T z = 1 pp T = 1 T z = ( np + pn ) T z = 1 nn Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 18 / 33

45 Nucleon-nucleon interaction Let s consider the isospin (for the spin is the same story): { 1 T = 0 T z = 0 2 ( np pn ) T z = 1 pp T = 1 T z = ( np + pn ) T z = 1 nn The NN interaction in T = 0 and T = 1 are very different! Example? The deuteron (T = 0) is bound, the dineutron (T = 1) is unbound! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 18 / 33

46 Nucleon-nucleon interaction Let s consider the isospin (for the spin is the same story): { 1 T = 0 T z = 0 2 ( np pn ) T z = 1 pp T = 1 T z = ( np + pn ) T z = 1 nn The NN interaction in T = 0 and T = 1 are very different! Example? The deuteron (T = 0) is bound, the dineutron (T = 1) is unbound! In general: V NN = V S=0,T =0 + V S=1,T =0 + V S=0,T =1 + V S=1,T =1 Note: V ST have different contributions in different relative angular momenta! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 18 / 33

47 Traditional approach (credit D. Furnsthal, T. Papenbrock) Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 19 / 33

48 Example: Argonne v 18 (Wiringa, Stoks, Schiavilla, PRC (1995)) includes an electromagnetic, one-pion exchange (long), and intermediate and a short range part v ij = v γ ij + vij π + vij I + vij S = p v p (r ij )O p ij The operators depend on relative states of the two nucleons. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 20 / 33

49 Example: Argonne v 18 (Wiringa, Stoks, Schiavilla, PRC (1995)) includes an electromagnetic, one-pion exchange (long), and intermediate and a short range part v ij = v γ ij + vij π + vij I + vij S = p v p (r ij )O p ij The operators depend on relative states of the two nucleons. There are charge independent (CI): O CI ij = [ 1, σ i σ j, S ij, L S, L 2, L 2 (σ i σ j ), (L S) 2] [1, τ i τ j ] Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 20 / 33

50 Example: Argonne v 18 (Wiringa, Stoks, Schiavilla, PRC (1995)) includes an electromagnetic, one-pion exchange (long), and intermediate and a short range part v ij = v γ ij + vij π + vij I + vij S = p v p (r ij )O p ij The operators depend on relative states of the two nucleons. There are charge independent (CI): O CI ij = [ 1, σ i σ j, S ij, L S, L 2, L 2 (σ i σ j ), (L S) 2] [1, τ i τ j ] And charge dependent (CD) and charge symmetry breaking (CSB) terms: O CD ij = [1, σ i σ j, S ij ] T ij, O CSB ij = τ zi + τ zj. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 20 / 33

51 Example: Argonne v 18 (Wiringa, Stoks, Schiavilla, PRC (1995)) includes an electromagnetic, one-pion exchange (long), and intermediate and a short range part v ij = v γ ij + vij π + vij I + vij S = p v p (r ij )O p ij The operators depend on relative states of the two nucleons. There are charge independent (CI): O CI ij = [ 1, σ i σ j, S ij, L S, L 2, L 2 (σ i σ j ), (L S) 2] [1, τ i τ j ] And charge dependent (CD) and charge symmetry breaking (CSB) terms: O CD ij = [1, σ i σ j, S ij ] T ij, O CSB ij = τ zi + τ zj. where S ij =3σ i ˆr ij σ j ˆr ij σ i σ j is the tensor L ij = 1 2i (r i r j ) ( i j ) is the relative angular momentum S ij = 1 2 (σ i + σ j ) is the total spin of the pair and T ij = 3τ zi τ zj τ i τ j is the isotensor operator. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 20 / 33

52 Argonne v 18 (Wiringa, Stoks, Schiavilla, PRC (1995)) For example, the long-range part (one pion exchange) has the form where v 5,6 [Y (m π r ij )σ i σ j + T (m π r ij )S ij ] [1, τ i τ j ] Y (x) = e x x ξ(r) is the Yukawa function, T (x) = ( x + 3 x 2 ) Y (x)ξ(r) is the tensor function, and ξ(r) = 1 exp( cr 2 ) is a cutoff. The intermediate part has similar form: and the short range is v I ij = 18 p=1 I p T 2 (µr ij )O p ij v S ij = 18 p=1 [ P p + Q p r + R p r 2] W (r)o p ij where W (r) is a Wood-Saxon potential, and there are 42 free parameters I p, P p, Q p and R p. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 21 / 33

53 Argonne v 18 (Wiringa, Stoks, Schiavilla, PRC (1995)) V central V στ V tensor-τ V(r) (MeV) Argonne v r (fm) Example: radial functions v 1, v 4 and v 6 that multiply respectively the operators 1, σ i σ j τ i τ j, and S ij τ i τ j. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 22 / 33

54 Two slides on scattering theory (1/2) At large distances before the scattering, ψ e ikz after the scattering, ψ e ikz + f (θ) r e ikr (let s assume that k k ) f (θ) is the scattering amplitude, and it is directly related to the differential cross-section: dσ = f (θ) 2 dω For a central potential f (θ) can be expanded as f (θ) = 1 (2l + 1) ( e 2iδ l 1 ) P l (cos θ) 2ik where δ l are the phase shifts, and σ tot = 4π k 2 (2l + 1) sin 2 δ l (k) l l Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 23 / 33

55 Two slides on scattering theory (2/2) In the limit of slow particles (low momenta) k cot δ(k) 1 a r ek where a is the scattering length and r e the effective range. Very schematically (but pedagogical...), let s consider a two-body system with attractive interaction: a < 0 a = a > 0 u(r) u(r) u(r) v(r) v(r) v(r) No bound states 0 r Bound state with E b =0 0 r E b 0 r 2 2m a 2 The nucleon-nucleon 3 S 1 channel (deuteron) has a 5.5 fm and is slightly bound. The 1 S 0 (two neutrons) has a 18 fm and is slightly unbound. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 24 / 33

56 Argonne v 18 (Wiringa, Stoks, Schiavilla, PRC (1995)) The parameters are fit to nucleon-nucleon scattering data up to lab energies of 350 MeV with very high precision, χ 2 1. Phase shifts: δ (deg) Argonne v 18 np Argonne v 18 pp Argonne v 18 nn SAID 7/06 np δ (deg) S1 Argonne v 18 np SAID 7/06 np S E lab (MeV) E lab (MeV) δ (deg) Argonne v 18 np Argonne v 18 pp Argonne v 18 nn SAID 7/06 np 3 P E lab (MeV) δ (deg) D1 Argonne v 18 np SAID 7/06 np E lab (MeV) Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 25 / 33

57 There are also other (simpler) versions of Argonne potentials, AV 8, AV 6,..., but also others like those of the Nijmegen group, CD-Bonn potentials, and many others. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 26 / 33

58 There are also other (simpler) versions of Argonne potentials, AV 8, AV 6,..., but also others like those of the Nijmegen group, CD-Bonn potentials, and many others. Another more recent approach, consists in developing nucleon-nucleon interactions within the framework of chiral effective field theory. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 26 / 33

59 Chiral EFT interactions Main idea of EFT: identify scales of the problem that are different, and expand in the ratio. Ideal systematic improvements possible! Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 27 / 33

60 Chiral EFT interactions Main idea of EFT: identify scales of the problem that are different, and expand in the ratio. Ideal systematic improvements possible! In the nucleon-nucleon interaction, what are the d.o.f. involved? Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 27 / 33

61 Chiral EFT interactions Main idea of EFT: identify scales of the problem that are different, and expand in the ratio. Ideal systematic improvements possible! In the nucleon-nucleon interaction, what are the d.o.f. involved? Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 27 / 33

62 Chiral EFT interactions Main idea of EFT: identify scales of the problem that are different, and expand in the ratio. Ideal systematic improvements possible! In the nucleon-nucleon interaction, what are the d.o.f. involved? Pretend that m π ( 140MeV ) 0 (soft scale) and m N ( 939MeV ) (hard scale). In the low-energy (low-momentum) limit, we can expand the interaction in powers of (Q/Λ) ν, where Q soft scale, and Λ hard scale. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 27 / 33

63 Chiral EFT interactions One possible power counting (Weinberg) ν in (Q/Λ) ν is given by ( V i d i + n ) i 2 2 ν = 4 + 2N + 2L + i where N=nucleons involved in the process, L=pion loops, V i =vertices of type i, d i =derivatives and or insertions of m π, n i =nucleonic fields operators. Note: Adding one nucleon increases one order in Q/Λ Expect many-body forces! Natural expectation that V 2 V 3 V 4... Some coupling constants from experiments, some need to be fit Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 28 / 33

64 Chiral EFT interactions One possible power counting (Weinberg) ν in (Q/Λ) ν is given by ( V i d i + n ) i 2 2 ν = 4 + 2N + 2L + i where N=nucleons involved in the process, L=pion loops, V i =vertices of type i, d i =derivatives and or insertions of m π, n i =nucleonic fields operators. Note: Adding one nucleon increases one order in Q/Λ Expect many-body forces! Natural expectation that V 2 V 3 V 4... Some coupling constants from experiments, some need to be fit Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 28 / 33

65 Chiral EFT interactions One possible power counting (Weinberg) ν in (Q/Λ) ν is given by ( V i d i + n ) i 2 2 ν = 4 + 2N + 2L + i where N=nucleons involved in the process, L=pion loops, V i =vertices of type i, d i =derivatives and or insertions of m π, n i =nucleonic fields operators. Note: Adding one nucleon increases one order in Q/Λ Expect many-body forces! Natural expectation that V 2 V 3 V 4... Some coupling constants from experiments, some need to be fit Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 28 / 33

66 Chiral EFT interactions One possible power counting (Weinberg) ν in (Q/Λ) ν is given by ( V i d i + n ) i 2 2 ν = 4 + 2N + 2L + i where N=nucleons involved in the process, L=pion loops, V i =vertices of type i, d i =derivatives and or insertions of m π, n i =nucleonic fields operators. Note: Adding one nucleon increases one order in Q/Λ Expect many-body forces! Natural expectation that V 2 V 3 V 4... Some coupling constants from experiments, some need to be fit Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 28 / 33

67 Chiral EFT interactions Example (I), one-pion exchange (N = 2, L = 0): Two identical vertices: V 1 =2, d 1 =1, n 1 =2, ν = 0 (LO) Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 29 / 33

68 Chiral EFT interactions Example (I), one-pion exchange (N = 2, L = 0): Two identical vertices: V 1 =2, d 1 =1, n 1 =2, ν = 0 (LO) Example (II), contact interactions (N = 2, L = 0): One vertex: V 1 =1, d 1 =0, n 1 =4, ν = 0 (LO) Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 29 / 33

69 Chiral EFT interactions Example (I), one-pion exchange (N = 2, L = 0): Two identical vertices: V 1 =2, d 1 =1, n 1 =2, ν = 0 (LO) Example (II), contact interactions (N = 2, L = 0): One vertex: V 1 =1, d 1 =0, n 1 =4, One vertex: V 1 =1, d 1 =2, n 1 =4, ν = 0 (LO) ν = 2 (NLO) Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 29 / 33

70 Chiral EFT interactions So at each order, draw all the possible diagrams, and that s it! Several versions available on the market up to N 3 LO (maybe higher). Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 30 / 33

71 Chiral EFT interactions Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 31 / 33

72 Chiral EFT interactions Other possible expansions are possible, i.e. pionless theory, delta-full,... In the same way also electroweak currents and other operators can be constructed that are consistent with the Hamiltonian. Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 32 / 33

73 Summary of this lecture: Introduction: nuclear physics in a big contest Questions in nuclear physics Degrees of freedom Nucleon-nucleon interactions Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 33 / 33

74 Summary of this lecture: Introduction: nuclear physics in a big contest Questions in nuclear physics Degrees of freedom Nucleon-nucleon interactions Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 33 / 33

75 Summary of this lecture: Introduction: nuclear physics in a big contest Questions in nuclear physics Degrees of freedom Nucleon-nucleon interactions Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 33 / 33

76 Summary of this lecture: Introduction: nuclear physics in a big contest Questions in nuclear physics Degrees of freedom Nucleon-nucleon interactions Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 33 / 33

77 Summary of this lecture: Introduction: nuclear physics in a big contest Questions in nuclear physics Degrees of freedom Nucleon-nucleon interactions Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 33 / 33

78 Summary of this lecture: Introduction: nuclear physics in a big contest Questions in nuclear physics Degrees of freedom Nucleon-nucleon interactions End for today... Stefano Gandolfi (LANL) - stefano@lanl.gov Introduction and nuclear interactions 33 / 33