1. 1

5. Internal symmetries isospin symmetry => nuclear physics SU(3) – symmetry =>hadrons chiral summetry => pions color symmetry =>quarks electroweak symmetry => SU(2)xU(1) model >

6. Internal symmetries: broken by interaction ( electromagnetism breaks isospin ) broken by explicit symmetry breaking ( SU(3) – symmetry of hadrons ) unbroken ( color symmetry of quarks ) broken by spontaneous symmetry breaking ( chiral symmetry and electroweak symmetry)

7. Rutherford: He suggested in 1919 that there must exist a neutral partner of the proton. helium nucleus: charge: 2 x proton mass: 4 x proton

8. 1932: discovery of the neutron (J. Chadwick) atomic nuclei are composed of protons and neutrons

9. 9

10. nucleons: doublet of SU(2)

11. Lawrence Berkeley Nat. Lab

12. 1953 pion nucleus

14. delta: quadruplet ( 1230 MeV )

15. pions: triplet eta: singlet

16. 16

17. 17

19. U(n): group of complex unitary n x n matrices SU(n): n x n matrices with det U = 1

20. U = exp (iH) H: Hermitean n x n matrix

21. det U = exp i (trH) SU(n): det U = 1 tr H = 0

22. SU(n): (n x n - 1) generators SU(2): 3 SU(3): 8 SU(4): 15 SU(5): 24

31.  quarks triplet  fundamental representation

33.  hypercharge

35. quark triplet

37. irreducible representations choose state with maximal value of t(3) – proceed into the U, T and V directions to the left, until it stops

38. steps p and q External line of representation

39. each state is described by 3 numbers:

46. 46

47. 47

48. 0 1 2 3 0 1 3 6 10 1 3* 8 15 24 2 6* 15* 27 42 3 10* 24* 42* 64

49. direct product of representations

50. invariant operator e.g. for angular momentum

52. 1  0 3,3*  4/3 6,6*  10/3 8  3 10,10*  6 27  8

59. Bevatron in Berkeley

60. K-mesons: 1947 => Eta-meson: 1961

64. 64

65. 65

66. 66

68. 68

69. breaking of SU(3): much larger than the breaking of isospin symmetry

70. 70 940 MeV 1190 MeV 1318 MeV 1116 MeV

71. 71 ??? 1232 MeV 1530 MeV 1385 MeV

73. Physics given by a(t) - the various matrix elements => Clebsch-Gordan coefficients

76.  f - coupling  d - coupling Wigner-Eckart theorem -- SU(3)

77. Susumu Okubo (Rochester)

82. 82 1236 MeV 1672 MeV ? 1232 MeV 1530 MeV 1385 MeV

83. 83

84. 84

85. 85 496 MeV 138 MeV 958 MeV548 MeV 496 MeV

87. mixing changes the masses lower state  lower higher state  higher Experiment: mixing angle about 16 degrees

90. Why pi mesons have a small mass? Gell-Mann, Oakes, Renner (1968) Chiral Symmetry SU(3) => SU(3,L) x SU(3,R)

92. Chiral symmetry breaking: all eight mesons acquire masses

93. SU(3,L) x SU(3,R) SU(2,L) x SU(2,R) SU(2) K-mesons and eta meson massive pions massless pions massive

94. Why chiral symmetry?  QCD