1. 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

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

3. 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

4. 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

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.
kinetic term
mass term
Lorentz invariants
kinetic term
mass term

7. = 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. 　　

8. = ≠ 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. 　　

9. 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

10. 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　　　

12. 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　

14. 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.

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

16. 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

17. Lagrangian density　
Let
Let
, then

18. 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.

19. 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 :

20. 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'.