1. 1 Nuclear Physics Overview & Introduction Lanny Ray, University of Texas at Austin, Fall 2015 I.Nucleon+Nucleon System II.Nuclear Phenomenology III.Effective Interaction Theory IV.Nuclear Structure V.Nuclear Reactions VI.Scattering Theory Applications

2. 2 I. The Nucleon + Nucleon System Topics to be covered include, but are not necessarily limited to: Quantum numbers, symmetries, the deuteron One-pion exchange potential Phenomenological models Meson exchange potentials Effective chiral field theory models Scattering – amplitudes, phase shifts, observables Relativistic amplitudes

3. 3 Quantum numbers and symmetries: up quark down quark spin ½ ; isospin ½; parity = + I 3 = ½ I 3 =  ½ Isospin is an observed symmetry among most hadrons, e.g. the similarity in masses of protons and neutrons,        kaons, etc. and derives from the near equivalence of mass of the up and down quarks. The flavor independence of QCD together with the approximate up/down mass equivalence results in an isospin invariance in the nuclear interaction. Isospin symmetry is a BIG DEAL in nuclear physics! (particle physics sign convention) The lowest energy configuration for the nucleon is zero orbital angular momentum, spin ½, isospin ½, I 3 = +½ for protons and – ½ for neutrons, with parity +. M proton = 938.28 MeV; M neutron = 939.57 MeV where the small mass difference is due to Coulomb repulsion and u-d quark mass difference.

4. 4 Lowest mass nucleon resonances: N*(1470): spin-parity ½+; isospin ½  (1232): spin-parity 3/2+; isospin 3/2 which is a  S =1,  I=1 excitation of the nucleon Pion: The lowest energy configuration is zero orbital angular momentum, spin 0, parity =  L  isospin 1 Symmetries: The wave function for identical Fermions must be anti-symmetric and for hadrons (quarks) includes the spatial, spin, isospin (flavor), (and color) components. For two nucleons or two quarks interchange of labels 1,2 must therefore change the sign of the wave function. In this course we focus on the wave functions of nucleons and mesons and will ignore their internal (color & flavor) QCD structures.

5. 5 The deuteron – the only nucleon+nucleon bound state N P Orbital ang. mom = 0 Spin = 1 Parity = + (even) J  = 1 + Isospin = 0 (  1) L+S+I =  1 B.E. = 2.226 MeV Perhaps the reason there is no di-proton bound state is that the Coulomb replusion overcomes the nuclear attraction. If so then why isn’t there a bound di-neutron? Next we will derive the nuclear potential between two nucleons due to the exchange of one pion. We will see that even this simple exchange leads to spin & isospin dependent forces. When these interactions are combined from many meson exchanges we will see what accounts for the absence of a di-neutron in Nature.

6. 6 Spin & Isospin review

7. 7

8. 8 Nuclear interaction invariances The nuclear force is invariant wrt to: spatial rotation – total angular momentum conservation, but orbital angular momentum conservation spatial reflection – conservation of parity time reversal identical particle exchange – by including the isospin d.o.f. protons and neutrons are treated as identical fermions (EM effects are the exception) and hence the wave function must be antisymmetric wrt interchange of particle labels.

9. 9 One-Pion Exchange Potential (OPEP): N1N1 N2N2 

10. 10 Work out this integral Show

11. 11 N+N phenomenological potentials The earliest idea for the nuclear force originated with Hideki Yukawa’s paper in Proc. Phys. Math. Soc. Japan 17, 48 (1935) which showed that the exchange of a massive, spin 0 particle (meson) would generate an exponential potential of the form exp(-mr)/r and that a mass of about 100 MeV would do the job. The discovery of the muon in 1937 caused many to believe that the muon was the carrier of the nuclear interaction which turned out to be wrong. The pi-meson or pion was not discovered until 1947. N  N

12. 12 Generally, the early phenomenological NN potential models included a minimum number of spin-dependent terms which could account for the existence of a deuteron But no di-neutron, and the limited scattering data, e.g. central, spin-orbit and tensor, and they may or may not have included the theoretical OPEP. They only described the deuteron properties (B.E., magnetic dipole moment, electric quadrupole moment, d-state fraction) and N+N scattering data (scattering lengths and phase shifts) up to about 350 MeV lab collision energy where single pion production begins, the inelastic scattering threshold. An early, accurate model was introduced by R. Reid, Ann. Phys. (N.Y.) 50, 411 (1968). It is often still used as a bench mark test for codes because it is relatively simple and is local, ie. V = V(r) with no explicit momentum dependence. Starting the 1970s meson exchange based theoretical models appeared and I will summarize three – the Paris, Bonn and Nijmegen models. Then in the 80s-90s effective chiral symmetry based models started appearing; I will summarize the one I had the privilege of working on with Weinberg’s student and post-doc. N+N phenomenological potentials Show

13. 13 N+N phenomenological potentials Empirical knowledge: the nuclear force depends on everything it can as allowed by the underlying symmetries of QCD. it is short range, ~few fm (10 -15 m) N+N cross sections are ~4 fm 2 = 40 mb nuclear forces are very strongly repulsive at short distances less than the proton radius, ~ 0.7 fm only p+n forms a bound state and it is I=0, J  = 1 + Recent review article: R. Machleidt and D. R. Entem, Phys. Rep. 503, 1-75 (2011)

14. 14 N+N phenomenological potentials 3 S 1 phase shift 1 S 0 phase shift Potential appears attractive at lower energies, but becomes repulsive at higher energies; the increasing p.s. at low energy also indicates a bound or nearly bound state. E I3I3  1 n+n 0 p+n 1 p+p I=1 I=0 (-2.2 MeV) Few MeV unbound Low-lying N+N states

15. 15 N+N phenomenological potentials Consider a spherical square well with a weakly bound s-wave state: Now consider s-wave scattering from this potential at low kinetic energy E 0 where: r V(r) 0 V0V0 R EBEB E0E0

16. 16 N+N phenomenological potentials r V(r) 0 V0V0 R EBEB E0E0 Re Im EBEB kBkB complex energy & momentum plane

17. 17 N+N phenomenological potentials E (lab K.E.) 0 Increases for attractive V(r) For weakly bound state For unbound state close to zero (depends on details of potential)

18. 18 Reid Soft-Core NN Potential (1968) OPEP + empirical sum of Yukawa potentials in the form: Notation for N+N states: Clebsch-Gordan coefficient

19. 19 Reid soft-core NN potential in MeV: Isospin = 1

20. 20 Reid soft-core NN potential in MeV: Isospin = 0

21. 21 Reid soft-core and hard-core deuteron radial wave functions for L = 0 and 2 s state d state

22. 22 For low energy p+p, n+n states in L = 0, I = 1, S = 0 This is the only nuclear interaction for the 1 S 0, I=1 state. It is insufficient to bind p+p or n+n. Far too weak to bind p+n The tensor potential is non-central (deformed) and couples |L-L’|=2 states. It’s strong attraction, acting through the L=2 p+n state ( 3 D 1 ) allows the p+n to have a stable bound state – deuteron!  exchange plus empirical terms Paris NN Potential For low energy p+n states in L=0, I = 0, S = 1 ( 3 S 1 ) the central and tensor potentials are:

23. 23 Paris NN Potential

24. 24 Paris NN Potential

25. 25 Bonn NN Potential In Advances in Nuclear Physics, Vol. 19, p. 189, (1989). Meson exchanges included

26. 26 Bonn NN Potential Pseudo-scalar interaction Tensor interaction

27. 27 Bonn NN Potential

28. 28 Bonn NN Potential

29. 29 Relevant mesons for N+N interactions

30. 30 Nijmegen NN potential The model includes  ‘  S* and the J=0 parts of the Pomeron, f, f’ and A 2 mesons. Spin operators

31. 31 Nijmegen NN potential

32. 32 Nijmegen NN potential

33. 33 Nijmegen NN potential

34. 34 An example of modern N-N interaction models based on QCD using effective field theories as pioneered by S. Weinberg in the 70’s Effective Chiral Lagrangians Ordonez, Ray, van Kolck, Phys. Rev. C 53, 2086 (1996) (low energy modes of QCD – pion, nucleon,  resonance fields)

35. 35 Effective chiral field theory model

36. 36 Effective chiral field theory model

37. 37 The total list of spin-isospin operators in the model: Effective chiral field theory model

38. 38 Effective chiral field theory model the whole enchilada

39. 39 Effective chiral field theory model The 25 fitting parameters of the model  N  coupling  N derivative couplings NN contact interaction effective couplings

40. 40 Effective chiral field theory model

41. 41 Scattering: Schrodinger eq., boundary conditions, phase shifts, scattering amplitudes, observables First, consider the scattering of two neutral, spin 0 particles which interact by a spherically symmetric, finite range potential V(r): Solve the Sch.Eq. numerically in this region Match to asymptotic boundary conditions at any large r where V(r) vanishes

42. 42

43. 43 Beam (particles/area/time) Target (nuclei/area) Beam area A B detector solid angle subtended by detector:  =A D /R 2 R detector acceptance area A D Scattering definitions for the Differential Cross Section In nuclear physics typical units are mb/sr (milli-barns/steradian) where 1 barn = 10  24 cm 2 ; 1 mb = 10  27 cm 2 = 0.1 fm 2

44. 44 Scattering observables from the asymptotic incoming and scattered wave functions Z axis We must solve the Schrodinger eq., match to the above boundary conditions to obtain f  and then we can directly compare to the experimental d  /d .

45. 45 Solving the Schrodinger Eq. for spin 0, neutral particles scattering from a spherically symmetric potential We have in mind potentials that are too strong to allow perturbation expansions but require the S.E. to be solved. In general, analytic solutions are not available. Solutions for the phase shifts, scattering amplitude and diff. Xsec must be obtained numerically. Show Partial Wave Expansion

46. 46 Boundary conditions at large r, i.e. well outside the short-range nuclear potential:

47. 47 Boundary conditions at large r, i.e. well outside the short-range nuclear potential:

48. 48 Consider scattering of two, spin 0 charged particles with combined short-range nuclear interaction plus a Coulomb interaction. The Coulomb interaction is infinite range, so the asymptotic waves become Coulomb distorted plane waves and Coulomb distorted spherical waves: Point-like Coulomb: V C (r)=Z 1 Z 2 e 2 /r Coulomb potential for finite charge distribution Solve Sch.Eq. numerically With potential V N +V C Match numerical w.f. to regular and irregular Coulomb wave functions which are solutions of the radial Sch.Eq. With V C (r)=Z 1 Z 2 e 2 /r Next, include the Coulomb interaction:

49. 49

50. 50 Next, include spin, e.g. a spin ½ particle scattering from a potential, or a proton scattering from a J  = 0 + nucleus x y z scattering plane

51. 51 Spin ½ particle scattering from a spin-dependent potential Show

52. 52 Spin ½ particle scattering from a spin-dependent potential Ref. Rodberg and Thaler, “Introduction to The Quantum Theory of Scattering,” (1967), Ch.11, pg. 278-285.

53. 53 Spin ½ particle scattering from a spin-dependent potential Show

54. 54 We are almost done, but we still need to see how to calculate observables from the scattering amplitudes f, g. For this we need to introduce the spin density matrix formalism which makes calculating any and all observables a breeze. i th columnj th row Spin Density Matrix

55. 55 Spin Density Matrix The spin density matrix has the same dimensions as the spin operators, (2s+1). It should be possible to express the former in terms of the latter,

56. 56 Spin Observables Now let’s use this fancy machinery to calculate observables. For spin ½ + spin 0 systems recall that the differential cross section is (see slide 50): beam target  polarization yield beam target  polarization yield Show

57. 57 Spin Observables beam target  polarization Rotation of P in scattering plane High Resolution Spectrometer Los Alamos Meson Physics Facility (LAMPF)

58. 58 Spin Observables George Igo Jerry Hoffmann

59. 59 Finally, we can treat the N+N scattering problem which is spin ½ x ½ x y z scattering plane

60. 60 Spin ½ x ½ scattering Kinematic Tensors 1 K.P P.N N.K K N P K i K j P i P j N i N j K i P j +K j P i K i N j +K j N i P i N j +P j N i Tensors of the Spin operators 1 p 1 t  p.  t  p +  t  p  t  p x  t  i,p  j,p +  j,p  i,t Tensor Rank 0 1 2 The N+N scattering amplitude is constructed from these spin and kinematic tensors such that rotational, parity and time-reversal symmetries are satisfied. The most general form for spin ½ x ½ is: Show

61. 61 Spin ½ x ½ scattering

62. 62 Spin ½ x ½ scattering

63. 63 Spin ½ x ½ scattering

64. 64 Spin ½ x ½ scattering (using Bystricky’s amplitude definition) Moravcsik a b c m g h Bystricky (a+b)/2 f/2 e/2 (a-b)/2 c/2 -d/2 Bystricky a b c d e f Moravcsik a+m a  m 2g -2h 2c 2b

65. 65 Dirac representation of the N+N scattering amplitudes The above forms for the N+N scattering amplitudes are given in terms of the Pauli spin operators and are referred to as the Pauli representation. We can also represent this same physical information using Dirac matrices in the so-called Dirac representation. This new form provides the essential input for relativistic nuclear structure and scattering theories. N+N c.m. initial final

66. 66 Dirac representation of the N+N scattering amplitudes Show

67. 67 Dirac representation of the N+N scattering amplitudes [from McNeil, Ray, Wallace, Phys. Rev. C 27, 2123 (1983)] This matrix eqn. is derived for p + nucleus scattering. To use it for N+N scattering in the c.m. omit row 4, use N+N c.m. k, E cm and set A=1.

68. 68 Dirac representation of the N+N scattering amplitudes [from McNeil, Ray, Wallace, Phys. Rev. C 27, 2123 (1983)] For N+N scattering use N+N c.m. k, E cm and set A=1.

69. 69 This concludes Chapter 1: The Nucleon+Nucleon System