Spontaneous Breakdown (SB) of Symmetry ( 対称性の自発的破れ )

1. 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 This is invariant under signature change of j : discrete group Z2 "discrete symmetry" 微分　 m2j+lj3=0 m2=-lj2 the lowest energy If m2<0 occurs atj = v

2. lowest energy at model potential with the lowest energy If m2<0 occurs atj = v

3. 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

4. lowest energy at = = lowest energy state = vacuum(真空) the vacuum violates the symmetry, If m2<0, 定数 while the Lagrangian is invariant. 係数 "spontaneous breakdown of the symmetry" 1 - - 2 + - 4 6 =1 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

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

6. Lorentz inv. mass term kinetic term Lorentz invariants kinetic term mass term

7. Lorentz inv. chiral sym. mass term kinetic term : allowed LR別々に変換　　 chiral transformation : Chiral symmetry can be discrete or continuous. discrete chiral sym. continuous chiral sym. = The kinetic term preserves chiral symmetry, & is allowed.

8. Lorentz inv. LR別々に変換　　 chiral transformation : chiral sym. mass term kinetic term : forbidden : allowed Chiral symmetry can be discrete or continuous. discrete chiral sym. discrete chiral sym. continuous chiral sym. continuous chiral sym. = The kinetic term preserves chiral symmetry, & is allowed. ≠ The fermion mass term violates chiral symmetry. is forbidden by the chiral symmetry.

9. Lorentz inv. chiral sym. mass term kinetic term : forbidden : allowed Fermion Mass Generation via SB of Discrete Chiral Sym. model of real scalar j and fermion y require symmetry under simultanous transformations signature change & chiral transformation invariant Lagrangian density is forbidden Fermion mass term v.e.v. If m2<0, the symmetry is broken spontaneously. interaction terms mass term kinetic term redefine the field mass of y :my The fermion mass is generated

10. 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

12. potential minimum 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

14. 代入 代入 kinetic term mass term interaction terms interaction terms masses of x, c : mx ,mc c: massless field Nambu- Goldstone field. 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.

15. 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 代入 vacuum expectation value mass term interaction terms kinetic term redefine the field mass of y :my The fermion mass is generated

16. Gauge Boson Mass Generation via SB -- Higgs mechanism model of complex scalarfield f and U(1)gaugefield Am symmetry U(1) gauge invariance transformation Lagrangian density covariant derivative ∂m(e-igaf)+ig (Am+∂ma)e-igaf = Dmf'= ∂mf+igAmf = e-iga ' '' Dmf = e-iga∂mf-ig∂mae-iaf 代入 , then Let Let , then

17. Lagrangian density Let Let , then

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

19. Spontaneous breakdown (SB) of symmetry real scalarj v.e.v. Z2 symmetry SB mass of x : field redefinition +fermion y chiral symmetry mass term :forbidden mass of y : fermion mass generation by SB complex scalarfield f v.e.v. SB global U(1) symmetry field redefinition c : Nambu- Goldstone boson masses of x, c :

20. 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. +fermiony mass term : forbidden chiral U(1)×U(1) symmetry mass of y : fermion mass generation by SB Higgs mechanism complex scalar field f, U(1)gauge field Am v.e.v. U(1) gauge symmetry SB field redefinition mass of A' The gauge boson mass is generated. mass of x The NG boson c is absorbed byA'.