internal symmetry :SU(3)c×SU(2)L×U(1)Y

standard model ( 標準模型 )

aG )8,,2,1( aiW )3,2,1( i B

b

t

s

c

d

u , ,

,,

ee

0

quarks

SU(3)c

leptons

L

Higgs scalar

Lorentzian invariance,locality,

renormalizability,

SU(3)c:color, SU(2)L:weak iso spin U(1)Y: hypercharge gauge symmetry

hypercharge

requirements:

fields: SU(3)c

: SU(2)L : U(1)Y

gauge bosons

3

fermions R

1

SU(2)L

2

1

1/3

1

1

2

2/3

4/3

1 2 1

L Rmatter fields

222

4

1

4

1

4

1 BWGL ia

G

)(|| VDL 22

22

2

1

vV ||)(

LL 2

1'

2

1

2

1qWgBYgGgiqL i

ia

asF

h.c.)( LRLRLRLR llflfqdfqufL cl

cduY

††††

2

1iRiiR '

2

1

2

1

i

aas qBYgGgiq

standard model ( 標準模型 )

YFG LLLLL

LL 2

1'

2

1lWgBYgil i

i

2

1iRiiR '

2

1

i

lBYgil

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 /2v

SB of Discrete Symmetry ( 離散的対称性 )

the lowest energy occurs at

model

with

signature change of : "discrete symmetry"

vv

V

/2vlowest energy at

VL 2)(2

1 422

4

1

2

1 V

vv

V

/2vthe lowest energy occurs at

VL 2)(2

1 422

4

1

2

1 V

potential

If 20

model

with

443222

4

1

4

1)(

2

1vvvL

22

2

1 m

vm 2

v

LULU † 00 U

lowest energy state = vacuum( 真空 ) 0

(v.e.v. 真空期待値 ) 000 vU: symmetry transformation

)( L

vv

V

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

VL 2)(2

1 422

4

1

2

1 V /2v

lowest energy at

so as to have 000

L and R components of fermions (Review)

2

1 5L

2

1 5R

RRLL LRRL

LR

),/( 021

)/,( 210

0L

0R

0

0i

ii

01

100

2},{ 0},{ 5 ,1)( 25

10

0132105 i

rep. of Lorentz group

Lorentz invariants

0 †

Dirac fermion

Chiral Symmetry of Fermions

The chiral symmetry forbids the fermion mass.

The kinetic term preserves chiral symmetry.

The mass term violates chiral symmetry.

LLLL ' U

RRRR ' U U '

R

L

0

0

U

UU

RRLL

LRRL

chiral transformation :

Chiral symmetries can be discrete or continuous.

10

015U5ieU

i

i

e

e

0

0

discrete chiral symmetry

continuous chiral symmetry

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 forbidden

If 20, the symmetry is broken spontaneously 　

v000 v

fvm mass of :m

LL i vf f

Fermion Mass Generation via SB of Discrete Chiral Sym.

Fermion mass term

vacuum expectation value　redefine the field　

000

The fermion mass is generated

signature change discrete chiral transformation

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

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 /|| 222

22

12 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

VL 22

21 )()(

2

1 22

22

12

22

1

2

)(4

)(2

V

v1 2 ])[(4

1 42222

21 vvV

v1 2 ])[(4

1 42222

21 vvV

VL 22

21 )()(

2

1 22

22

12

22

1

2

)(4

)(2

V

v1

])[(4

1)()(

2

1 42222

21

22

21 vvL

42222222

4

1)(

4

1)( vvv

22 )()(2

1 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. in the above model is a Nambu- Goldstone field.

])[(4

1 42222

21 vvV

4222 |||||| L

Lagrangian density

i

continuous chiral transformation

global U(1) transformation

)(2 LRRLf †

LL 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

fvm mass of :m The fermion mass is generated

vacuum expectation value　redefine the field　

000

Fermion Mass Generation via SB of Continuous Chiral Sym.

model of complex scalar field and U(1)gauge field A

Lagrangian density

symmetry

U(1) gauge invariance

)(' xie AAA '

42222

4

1||||||)(

DFL

igAD

　　

ie

ieigAD

ieg

iAig

1

Gauge Boson Mass Generation via SB -- Higgs mechanism

covariant derivative

g

iAA1

'

2|| D 2|'| igA 2222 )'()( Ag

Let , then

Let , then

　　　

42222

4

1||||||)(

DFL2|| D 2222 )'()( Ag

ie

g

iAA1

'

Lagrangian density 42222

4

1||||||)(

DFL

　　

ie

g

iAA1

'

2|| D 2|'| igA 2222 )'()( Ag

Let

, then

spontaneous breakdown 　 000 v

2/)( v

　　　

2|| D 222222222 )'(2

1)'()'(

2

1)(

2

1 AgAvgAvg

field redefinition

2

4

1)'( FL

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||||||)(

DFL2|| D 2222 )'()( Ag

ie

g

iAA1

'

The gauge boson becomes massive by absorbing NG boson .

Spontaneous Breakdown of Non-Abelian Gauge Symmetry

42222 ||||)(4

1 DWL i

SU(2) doubletcomplex scalar

invariant Lagrangian density

2221

1211

2

1

2

1

i

i

transformation

SU(2) gauge symmetry

real field

ij2,1, ji

22

21

2 ||||||

(i ： Pauli matrix) 　

iijkjki gWW

iig

i2

SU(2) gaugefield

iW 3,2,1i

D iiWgi

2

1

kjijkiii WWgWWW

42222 ||||)(4

1 DWL i

42222 ||||)(4

1 DWL i

invariant Lagrangian density

V

12

]4/)2/|[(| 4222 vv

/2v422 |||| V

veiii 0

2

1

vacuum expectation value 000 v

redefinition

i : real field 44322

4

1

4

1vvvV

If 20

/|||||| 22222

221

212

211

22

21

2 v

the lowest energy (vacuum state) occurs at

The vacuum violates SU(2) gauge symmetry spontaneously.

Then

veiii 0

2

142222 ||||)(4

1 DWL i

veiii 0

2

142222 ||||)(4

1 DWL i

44322

4

1

4

1vvv 422 |||| V

422 |||| V

veiii 0

2

1redefinition

44322

4

1

4

1vvvV Then

D

iiWgi

2

1

iiWgi

2

1

veiii 0

2

1

iiWgi '

2

1iiie

2

1

v

0

iiW ' geeeWeiiiiiiii iiiiii /)(

22 )'()( ii WW

veiii 0

2

142222 ||||)(4

1 DWL i

222 )'()(8

1)(

2

1 iWgv

2|| D v02

1

iiWgi '

2

1�

iiWgi '

2

1

v

0

44322

4

1

4

1vvv 422 |||| V

222 )'()(8

1)(

2

1 iWgv 2|| D

veiii 0

2

142222 ||||)(4

1 DWL i

44322

4

1

4

1vvv 422 |||| V

222 )'()(8

1)(

2

1 iWgv

2|| D v02

1

iiWgi '

2

1�

iiWgi '

2

1

v

0

222 )'()(8

1)(

2

1 iWgv

2)'(4

1 iWL 222 )'()(8

1)(

2

1 iWgv

44322

4

1

4

1vvv

2|| D

veiii 0

2

142222 ||||)(4

1 DWL i

44322

4

1

4

1vvv 422 |||| V

2)'(4

1 iWL 222 )'()(8

1)(

2

1 iWgv

44322

4

1

4

1vvv

veiii 0

2

142222 ||||)(4

1 DWL i

2)'(4

1 iWL 222 )'()(8

1)(

2

1 iWgv

44322

4

1

4

1vvv

2)'(4

1 iWL 222 )'()(8

1)(

2

1 iWgv

44322

4

1

4

1vvv

veiii 0

2

142222 ||||)(4

1 DWL i

gvMW 2

1'

kjijkiii WWgWWW '''''

0'' 2' i

Wi WMW

0' iW

22'

2

')'(

2

1)''(

4

12

iW

iiW

WMWWL

vm 2

mass of W' The gauge boson mass is generated.

mass of The gauge boson becomes massive by absorbing NG boson .

Spontaneous breakdown (SB) of symmetry

real scalar Z2 symmetry

v.e.v. 000 vSB vm 2mass of

m

v

4222

4

1

2

1)(

2

1 sL

fermion sLL i f

y の質量 my

生成

質量項 は禁止chiral 対称性 対称性の自発的破

れ

fvm

v00 対称性の自発的破 れ

2/)( iv

複素 scalar 場 f

4222

cs |||| L

global U(1) 対称 性

vm 2x, h の質量mx ,mc

0m

連続群が自発的な破れるとき、質量 0 の粒子が現れる

Goldstone の定理

この粒子を南部 - Goldstone 粒子というfermion 　　

csLL

の質量 m 生成

質量項 は禁止chiral U(1) 対称性　対称性の自発的破れ

　fvm

i )(2 LRRLf †

Higgs 機構　　　

複素 scalar 場と U(1)gauge 場 A

の模型 　　 42222

4

1||||||)(

DFL

対称性の自発的破れ　

v00 2/)( iev

gvmA '

vm 2

A' の質量　 gauge 場質量の生成

の質量　

g

iAA1

'

非可換群 gauge 対称性の自発的破れ

42222 ||||)(4

1 DWL i

SU(2)doublet複素 scalar場　　　

Lagrangian 密度　　　

2221

1211

2

1

2

1

i

i

変換　　

SU(2) gauge 対称性　　　

実場　　　

ij2,1, ji

22

21

2 ||||||

(i ： Pauli 行列 ) 　

iijkjki gWW

iig

i2

SU(2) gauge 場　　iW 3,2,1i

D iiWgi

2

1

kjijkiii WWgWWW

対称性の自発的破れ　　 V

12

]4/)2/|[(| 4222 vv

02 /2v とおく422 |||| V

veiii 0

2

1

真空期待値 000 v

場の再定義　

v は実数

i は実数場

44322

4

1

4

1vvvV