Spontaneous Breakdown (SB) of Symmetry ( 対称性の自発的破れ )
VL 2)(2
1
real scalar field with 422
4
1
2
1 V
Lagrangian density potential
This is invariant under
'
discrete group Z2 02 02
vv
V
V
If 20 /2
SB of Discrete Symmetry ( 離散的対称性 )
the lowest energy
model
with
signature change of : "discrete symmetry"
2
30
微分
occurs at v
2
2
/2vlowest energy at
VL 2)(2
1 422
4
1
2
1 V
VL 2)(2
1 422
4
1
2
1 V
potential
model
with vv
V
vv
V
If 20 /2the lowest energy occurs at v
v
LULU † 00 U
lowest energy state = vacuum( 真空 ) 0
(v.e.v. 真空期待値 ) 000 vU: symmetry transformation
If 20, the vacuum violates the symmetry,while the Lagrangian is invariant.
vacuum expectation value
"spontaneous breakdown of the symmetry"
redefine the field
/2vlowest energy at
so as to have 000
VL 2)(2
1 422
4
1
2
1 V
vv
V
22
2
1 m
vm 2
v
)( L
LULU † 00 U
lowest energy state = vacuum( 真空 ) 0
(v.e.v. 真空期待値 ) 000 vU: symmetry transformation
If 20, the vacuum violates the symmetry,while the Lagrangian is invariant.
vacuum expectation value
"spontaneous breakdown of the symmetry"
redefine the field
mass terminteraction terms
constant
mass of :
m
/2vlowest energy at
so as to have 000
VL 2)(2
1 422
4
1
2
1 V
vv
V
L
v ( 2)
定数
2)( 2
1
v v2
1( )2
2 v22 v 4
1 ( )4
2
2
1v )2( 22 vv
4
1 )464( 432234 vvvv
1 2
4
6 1
22v
係数
3v 4 4
1 4v
4
1
L and R components of fermions (Review)
2
1 5L
2
1 5R
RRLL
LRRL
LR
),/( 021
)/,( 210
0
L
0R
0
0i
ii
01
100
2},{ 0},{ 5 ,1)( 25
10
0132105 i
rep. of Lorentz group
Lorentz invariants
0 †
Dirac fermion
kinetic term
mass term
,RRLL LRRL Lorentz inv.
RRLL
LRRL Lorentz invariants kinetic term
mass term
kinetic term mass term
,RRLL LRRL Lorentz inv.
kinetic term mass term
The kinetic term preserves chiral symmetry,
LLL ' RRR ' ,' U ,
0
0
U
chiral transformation :
Chiral symmetry can be discrete or continuous.
10
015U5ieU
i
i
e
e
0
0discrete chiral sym.
continuous chiral sym.
RRLL ' ' ' '
1||||
' '
LR 別々に変換
LL *'
RR *'
* L L2|| RR2||
LRRL ' ' ' '' '
=
& is allowed.
chiral sym. : allowed
,RRLL LRRL Lorentz inv.
kinetic term mass term
,' U , 0
0
U
chiral transformation :
Chiral symmetry can be discrete or continuous. discrete chiral sym.
continuous chiral sym.
1||||
10
015U5ieU
i
i
e
e
0
0discrete chiral sym.
continuous chiral sym.
LLL ' RRR '
LR 別々に変換
LL *'
RR *'
RRLL ' ' ' '' ' L L2|| RR2||
LRRL ' ' ' '' ' * L R LR* ≠ is forbidden by the chiral symmetry.
The fermion mass term violates chiral symmetry.
=
& is allowed.
The kinetic term preserves chiral symmetry,
chiral sym. : allowed : forbidden
f f
4222
4
1
2
1)(
2
1 L
model of real scalar and fermion
invariant Lagrangian density
'require symmetry under simultanous transformations
i
5'
f
is forbiddenIf 20, the symmetry is broken spontaneously. v
v00
mmass of :m
L i
Fermion Mass Generation via SB of Discrete Chiral Sym.
Fermion mass term
v.e.v.
redefine the field
000
The fermion mass is generated
signature change
& chiral transformation
L
mass term interaction termskinetic term
fvv
,RRLL LRRL Lorentz inv.
kinetic term mass termchiral sym. : allowed : forbidden
VL 2
complex scalar field
422 |||| V
Lagrangian density potential
2/)( 21 i
invariant under ie '
sincos ' 2111
cossin' 2122
VL 22
21 )()(
2
1
global U(1) symmetry
: real
21 ,
21 , in terms of
invariant under
222
21
22
21
2
)(4
)(2
Vpotential
Lagrangian density
global O(2) symmetry
SB of Continuous Symmetry ( 連続的対称性 ) model:
continuous symmetry
V
VL 22
21 )()(
2
1 22
22
12
22
1
2
)(4
)(2
V
VL 22
21 )()(
2
1
222
21
22
21
2
)(4
)(2
V
potential
02 02
V
1
V
2
12
v1
000 1 v 000 2 2
v.e.v.
redefine the fields
If 20
/|| 2222
21
2 v
the lowest energy (vacuum state) occurs at
The vacuum violates U(1) ( O(2)) symmetry spontaneously.
000
v00 iv
000
])[(4
1 42222
21 vvV
VL 22
21 )()(
2
1 22
22
12
22
1
2
)(4
)(2
V
minimum
VL 22
21 )()(
2
1 22
22
12
22
1
2
)(4
)(2
V
v1 2 ])[(4
1 42222
21 vvV
2 v1
])[(4
1 42222
21 vvV
VL 22
21 )()(
2
1 22
22
12
22
1
2
)(4
)(2
V
v1
42222222
4
1)(
4
1)( vvv
L
2
vm 2masses of : m,m0m
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.
])[(4
1 42222
21 vvV
mass term interaction terms
kinetic term
interaction terms
22 )()(2
1
: massless field Nambu- Goldstone field.
代入代入
4222 |||||| L
Lagrangian density
i
continuous chiral transformation
global U(1) transformation
)(2 LRRLf †
L i vf )( 5if
ie2' 5
' ie
000 1 v
model of complex scalar and fermion require symmetry under the simultaneous transformations
is forbiddenfermion mass term If 20, the symmetry is broken spontaneously
2/)( iv
000 v
mmass of :m The fermion mass is generated
vacuum expectation value redefine the field
000
Fermion Mass Generation via SB of Continuous Chiral Sym.
L
mass term interaction termskinetic term
fv
代入
model of complex scalar field and U(1)gauge field A
Lagrangian density
symmetry
U(1) gauge invariance ,' )( xige )(' xAAA
42222
4
1||||||)(
DFL
igAD
ie
ieigAD
ieg
iAig
1
Gauge Boson Mass Generation via SB -- Higgs mechanism
covariant derivative
'A
2|| D 2|'| igA 2222 )'()( Ag
Let , then
Let , then
transformation
'D
D '
∂ igA
∂(eig)ig (A∂ )eig eig∂ig∂ei
eig ' ' ' D
De xig )(
g
iA1
代入
42222 ||||||)(4
1 DFL
2|| D 2222 )'()( Ag
ie
g
iAA1
'
Lagrangian density 42222
4
1||||||)(
DFL
ie
g
iAA1
'
2|| D 2|'| igA 2222 )'()( Ag
Let
Let , then
ieigAD
ieg
iAig
1
spontaneous breakdown 000 v2/)( v
2|| D 222222222 )'(2
1)'()'(
2
1)(
2
1 AgAvgAvg
field redefinition
222222222 )'(2
1)'()'(
2
1)(
2
1 AgAvgAvg
43224
4
1
4
1 vvv
gvmA '
vm 2
mass of A' The gauge boson mass is generated.mass of
000 v.e.v.
42222 ||||||)(4
1 DFL
2|| D 2222 )'()( Ag
ie
g
iAA1
'
The gauge boson becomes massive by absorbing NG boson .
mass term interaction terms
L 2)'(4
1 F
Spontaneous breakdown (SB) of symmetry
real scalar Z2 symmetry
v.e.v. 000 vSB vm 2mass of : v
4222
4
1
2
1)(
2
1 sL
+fermion sLL i f
mass of :
mass term :forbiddenchiral symmetry fvm
v00 SB
2/)( iv
complex scalar field
4222
cs |||| L
global U(1) symmetry
,2 vm masses of , : 0m
fermion mass generation by SB
field redefinition
v.e.v.
field redefinition : Nambu-
Goldstone boson
Goldstone Theorem
+fermion
csLL mass termchiral U(1)×U(1) symmetry
i )(2 LRRLf †
Higgs mechanism
complex scalar field , U(1)gauge field A
42222
4
1||||||)(
DFL
v00
2/)( iev g
iAA1
'
If a symmetry under continuous group is broken spontaneously, the system includes a massless field.
The massless particle is called Nambu- Goldstone field.
: forbidden
mass of : fvm fermion mass generation by SB
SB U(1) gauge symmetry
v.e.v.
field redefinition gvmA '
vm 2
mass of A' The gauge boson mass is generated.
mass of The NG boson is absorbed by A'.