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

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)

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

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

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nucleons: doublet of SU(2)

Lawrence Berkeley Nat. Lab

1953 pion nucleus

delta: quadruplet ( 1230 MeV )

pions: triplet eta: singlet

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

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

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

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

 quarks triplet  fundamental representation

 hypercharge

quark triplet

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

steps p and q External line of representation

each state is described by 3 numbers:

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* * 15* * 24* 42* 64

direct product of representations

invariant operator e.g. for angular momentum

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

Bevatron in Berkeley

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

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

MeV 1190 MeV 1318 MeV 1116 MeV

71 ??? 1232 MeV 1530 MeV 1385 MeV

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

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

Susumu Okubo (Rochester)

MeV 1672 MeV ? 1232 MeV 1530 MeV 1385 MeV

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MeV 138 MeV 958 MeV548 MeV 496 MeV

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

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

Chiral symmetry breaking: all eight mesons acquire masses

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

Why chiral symmetry?  QCD