Nuclear Physics Overview & Introduction Lanny Ray, University of Texas at Austin, Fall 2015

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

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. Quantum numbers and symmetries: up quark down quark spin ½ ; isospin ½; parity = + I3= ½ I3= -½ (particle physics sign convention) Isospin is an observed symmetry among most hadrons, e.g. the similarity in masses of protons and neutrons, p+,-,0, S+,-,0, X+,-, 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! The lowest energy configuration for the nucleon is zero orbital angular momentum, spin ½, isospin ½, I3 = +½ for protons and – ½ for neutrons, with parity +. Mproton = 938.28 MeV; Mneutron = 939.57 MeV where the small mass difference is due to Coulomb repulsion and u-d quark mass difference.

4. Lowest mass nucleon resonances: N*(1470): spin-parity ½+; isospin ½ D(1232): spin-parity 3/2+; isospin 3/2 which is a DS =1, DI=1 excitation of the nucleon The lowest energy configuration is zero orbital angular momentum, spin 0, parity = (+)(-)(-1)L = -, isospin 1 Pion: 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. The deuteron – the only nucleon+nucleon bound state Orbital ang. mom = 0 Spin = 1 Parity = + (even) Jp = 1+ Isospin = 0 (-1)L+S+I = -1 B.E. = 2.226 MeV P N 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. Spin & Isospin review

7. Spin & Isospin review

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. N1 p N2 One-Pion Exchange Potential (OPEP):

10. Work out this integral Show

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 p

12. N+N phenomenological potentials 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. Show

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 (10-15m) N+N cross sections are ~4 fm2 = 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, Jp = 1+ Recent review article: R. Machleidt and D. R. Entem, Phys. Rep. 503, 1-75 (2011)

14. N+N phenomenological potentials 3S1 phase shift 1S0 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. Low-lying N+N states E Few MeV unbound I=1 I=0 (-2.2 MeV) -1 n+n 0 p+n 1 p+p I3

15. V(r) E0 r 0 EB V0 R 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 E0 where:

16. Im V(r) E0 r 0 kB EB Re V0 R EB N+N phenomenological potentials complex energy & momentum plane

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

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

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

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

22. Paris NN Potential p, 2p, wexchange plus empirical terms For low energy p+p, n+n states in L = 0, I = 1, S = 0 This is the only nuclear interaction for the 1S0, I=1 state. It is insufficient to bind p+p or n+n. For low energy p+n states in L=0, I = 0, S = 1 (3S1) the central and tensor potentials are: 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 (3D1) allows the p+n to have a stable bound state – deuteron!

23. Paris NN Potential

24. Paris NN Potential

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

26. Bonn NN Potential Pseudo-scalar interaction Tensor interaction

27. Bonn NN Potential

28. Bonn NN Potential

29. Relevant mesons for N+N interactions

30. Nijmegen NN potential The model includes p,h,h‘,r,w,f,d,e,S* and the J=0 parts of the Pomeron, f, f’ and A2 mesons. Spin operators

31. Nijmegen NN potential

32. Nijmegen NN potential

33. Nijmegen NN potential

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, D resonance fields)

35. Effective chiral field theory model

36. Effective chiral field theory model

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

38. Effective chiral field theory model the whole enchilada

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

40. Effective chiral field theory model

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): Match to asymptotic boundary conditions at any large r where V(r) vanishes Solve the Sch.Eq. numerically in this region

43. Scattering definitions for the Differential Cross Section solid angle subtended by detector: DW=AD/R2 detector detector acceptance area AD R Beam (particles/area/time) Beam area AB Target (nuclei/area) In nuclear physics typical units are mb/sr (milli-barns/steradian) where 1 barn = 10-24 cm2; 1 mb = 10-27 cm2 = 0.1 fm2

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(q), and then we can directly compare to the experimental ds/dW.

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. Partial Wave Expansion Show

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

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

48. Next, include the Coulomb interaction: 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: VC(r)=Z1Z2e2/r Coulomb potential for finite charge distribution Match numerical w.f. to regular and irregular Coulomb wave functions which are solutions of the radial Sch.Eq. With VC(r)=Z1Z2e2/r Solve Sch.Eq. numerically With potential VN+VC

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