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'.