Spontaneous Breakdown (SB) of Symmetry (対称性の自発的破れ) 　　　 SB of Discrete Symmetry (離散的対称性)　　　 model real scalar field j with　　　　 Lagrangian density　　　 with potential　　 This is invariant under 　　 signature change of j : 　　 discrete group Z2 　　　 "discrete symmetry"　　　 微分　 m 2j +lj 3= 0 　　 m 2 =-lj 2 　　 If m2<0 the lowest energy occurs at j = v

lowest energy at model with potential　　 If m2<0 the lowest energy occurs at j = v

lowest energy at lowest energy state = vacuum(真空) If m2<0, the vacuum violates the symmetry, 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

lowest energy state = vacuum(真空) lowest energy at = = lowest energy state = vacuum(真空) (v.e.v. 真空期待値) U: symmetry transformation If m2<0, the vacuum violates the symmetry, while the Lagrangian is invariant. vacuum expectation value "spontaneous breakdown of the symmetry" 定数 係数 1 - - 2 + - 4 6 =1 redefine the field so as to have mass term constant = interaction terms mass of x : mx

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

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

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

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

Fermion Mass Generation via SB of Discrete Chiral Sym. Lorentz inv. chiral sym. kinetic term : allowed mass term : forbidden 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　 Fermion mass term　 is forbidden v.e.v. If m2<0, the symmetry is broken spontaneously. 　 mass term interaction terms kinetic term redefine the field　 mass of y : my The fermion mass is generated

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

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

Nambu- Goldstone field. Goldstone Theorem 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.

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　 kinetic term mass term interaction terms redefine the field　 mass of y : my The fermion mass is generated

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

Lagrangian density　 Let Let , then

spontaneous breakdown v.e.v. 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.

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

If a symmetry under continuous group is broken spontaneously, the system includes a massless field. Goldstone Theorem The massless particle is called Nambu- Goldstone field. +fermion y　　 chiral U(1)×U(1) symmetry 　 mass term : forbidden fermion mass generation by SB mass of y : Higgs mechanism　 complex scalar field f, U(1)gauge field Am　　 U(1) gauge symmetry　　　 v.e.v. SB　 field redefinition　　　 mass of A' The gauge boson mass is generated. mass of x The NG boson c is absorbed by A'.