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internal symmetry :SU(3) c ×SU(2) L ×U(1) Y

standard model ( 標準模型 ). renormalizability,. requirements:. Lorentzian invariance,. locality,. internal symmetry :SU(3) c ×SU(2) L ×U(1) Y. gauge symmetry. SU(3) c :color, SU(2) L :weak iso spin U(1) Y : hypercharge. fields. gauge bosons. : SU(3) c. : U(1) Y. : SU(2) L. SU(2) L.

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internal symmetry :SU(3) c ×SU(2) L ×U(1) Y

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  1. standard model(標準模型) renormalizability, requirements: Lorentzian invariance, locality, internal symmetry :SU(3)c×SU(2)L×U(1)Y gauge symmetry SU(3)c:color, SU(2)L:weak iso spin U(1)Y: hypercharge fields gauge bosons : SU(3)c : U(1)Y : SU(2)L SU(2)L hypercharge matter fields SU(3)c L R L R fermions quarks 4/3 1/3 3 1 2 -2/3 leptons 0 -1 1 1 2 -2 Higgs scalar 1 2 1

  2. standard model(標準模型)

  3. Spontaneous Breakdown (SB) ofSymmetry (対称性の自発的破れ) SB of Discrete Symmetry (離散的対称性) Lagrangian density real scalar field j with model potential with signature change of j : This is invariant under discrete group Z2 "discrete symmetry" the lowest energy occurs at If m2<0

  4. lowest energy at model potential with the lowest energy occurs at If m2<0

  5. lowest energy at lowest energy state = vacuum(真空) the vacuum violates the symmetry, If m2<0, while the Lagrangian is invariant. "spontaneous breakdown of the symmetry" U: symmetry transformation vacuum expectation value (v.e.v.真空期待値) redefine the field so as to have mass term constant = interaction terms mass of x : mx

  6. L and R components of fermions (Review) Dirac fermion rep. of Lorentz group y =yL+yR Lorentz invariants

  7. Chiral Symmetry of Fermions chiral transformation : Chiral symmetries can be discrete or continuous. The kinetic term preserves chiral symmetry. The mass term violates chiral symmetry. The chiral symmetry forbids the fermion mass. discrete chiral symmetry continuous chiral symmetry

  8. Fermion Mass Generation via SB of Discrete Chiral Sym. model of real scalar j and fermion y require symmetry under simultanous transformations signature change discrete chiral transformation invariant Lagrangian density is forbidden Fermion mass term If m2<0, the symmetry is broken spontaneously vacuum expectation value redefine the field mass of y :my The fermion mass is generated

  9. SB of Continuous Symmetry (連続的対称性) : real complex scalarfield model: Lagrangian density potential invariant under global U(1) symmetry continuous symmetry in terms of Lagrangian density potential invariant under global O(2) symmetry

  10. potential the lowest energy (vacuum state) occurs at If m2<0 The vacuum violates U(1) ( = O(2)) symmetry spontaneously. v.e.v. redefine the fields

  11. masses of x, c : mx ,mc Goldstone Theorem If a symmetry under continuous group is broken spontaneously, the system includes a massless field. The massless particle is called Nambu- Goldstone field. c in the above model is a Nambu- Goldstone field.

  12. Fermion Mass Generation via SB of Continuous Chiral Sym. model of complex scalar f and fermion y require symmetry under the simultaneous transformations global U(1) transformation continuous chiral transformation Lagrangian density fermion mass term is forbidden If m2<0, the symmetry is broken spontaneously vacuum expectation value redefine the field mass of y :my The fermion mass is generated

  13. Gauge Boson Mass Generation via SB -- Higgs mechanism model of complex scalarfield f and U(1)gaugefield Am symmetry U(1) gauge invariance Lagrangian density covariant derivative , then Let Let , then

  14. Lagrangian density Let , then

  15. v.e.v. spontaneous breakdown field redefinition mass of A' The gauge boson mass is generated. mass of x The gauge boson becomes massive by absorbing NG boson c.

  16. Spontaneous Breakdown of Non-Abelian Gauge Symmetry SU(2) gaugefield SU(2) doublet complex scalar real field SU(2) gauge symmetry transformation (t i ： Paulimatrix) invariant Lagrangian density

  17. invariant Lagrangian density

  18. the lowest energy (vacuum state) occurs at If m2<0 The vacuum violates SU(2) gauge symmetry spontaneously. vacuum expectation value x, c i : real field redefinition Then

  19. redefinition Then

  20. mass of W' The gauge boson mass is generated. mass of x The gauge boson becomes massive by absorbing NG boson c.

  21. Spontaneous breakdown (SB) of symmetry real scalarj Z2 symmetry v.e.v. mass of xmx SB fermion y は禁止 質量項 chiral 対称性　 対称性の自発的破れ　 yの質量my生成 複素scalar場f global U(1) 対称性　　　 対称性の自発的破れ　 x, hの質量mx ,mc

  22. Goldstoneの定理 連続群が自発的な破れるとき、質量0の粒子が現れる この粒子を南部- Goldstone粒子という fermion y は禁止 chiral U(1)対称性　 質量項 yの質量my生成 対称性の自発的破れ　 複素scalar場f とU(1)gauge場Amの模型 　　 Higgs 機構　　　 対称性の自発的破れ　 xの質量　 A' の質量　 gauge場質量の生成

  23. 非可換群 gauge対称性の自発的破れ SU(2) gauge場　　 実場　　　 SU(2)doublet 複素scalar場　　　 SU(2) gauge対称性　　　 変換　　 (t i ： Pauli行列) Lagrangian密度　　　

  24. 対称性の自発的破れ　　 とおく vは実数 真空期待値 x, c i は実数場 場の再定義　

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