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

U(1) symmetry of phase transformations

Symmetry Groups

U(1) group of multiplications by U e i with real .

f →f ' e

i f

The transformations are commutative. Abelian group

Fields are transformed as

Infinitesimal transformation

the only irreducible representation is one dimensional.

f →f ' f i f f i f

f †f is invariant under global and gauge transformation.

f †→f †' e

i f † Then

f †f →f †' f ' e

i f †e i f

f †f ∴

called commutative group or

Invariants under all the infinitesimal transformation is invariant under the whole connected part of the group

Because the transformations are commutative.

Example 2: O(3) symmetry of space rotations

X: infinitesimal rotation

angular momentum

kijkji JiJJ ˆ]ˆ,ˆ[

iJ )3,2,1( igenerator ijkjki iJ

kijkji JiJJ ],[commutators

representations on Fock space iJˆ

commutatorsirreducible representations are specified by a harf integer j

2j +1dimensional representation

O(3)group of 3×3 real matrices A with AAt 1(orthogonal)

o(3) Lie Algebra of X such that Ae iX

∊ O(3) rr A' space rotation of o(3)

The transformations are not commutative. non-Abelian group

non-commutative group

Example 3: SU(2) isospin symmetry

isospin

kijkji IiII ˆ]ˆ,ˆ[

)3,2,1( igenerator 2/iiI

kijkji IiII ],[commutators

representations on Fock space iI

commutators

irreducible representations are specified by a harf integer i 2i +1dimensional representation

SU(2)group of 2×2 complex matrices U withUU †=1 (unitary) & det U = 1 (special)

SU(2) is homomorphic to O(3)

su(2) Lie Algebra of X such that e iX∊ SU(2) i Pauli matrices

non-commutative group

Example 4: SU(3) unitary symmetry

unitary spin

kijkji FifFF ˆ]ˆ,ˆ[

)8,,2,1( igenerator 2/iiF

kijkji if ],[commutators

representations on Fock space iF

commutators

irreducible representations are specified by two integers

SU(3)group of complex 3×3 matrices U withUU †=1 (unitary) & det U = 1 (special)

su(3) Lie Algebra of X such that e iX∊ SU(3)

i Gell-mann matrices

0 1 2 3

0 1 3 6 10

1 3* 8 15

2 6* 15* 27

3 10* ......

non-commutative group

Example: U(1) symmetry

Global Symmetry and Gauge Symmetry

transformations are

f →f ' e i f

f †f is invariant under global and gauge transformation.

Fields are transformed as

the spacetime coordinates.

independent ofdependent on

GlobalGauge

but not invariant under global transformation, because

∂f →∂f ' e i ∂f i ∂e if

∂f ' †∂f ' (∂f if ∂ )(∂f if ∂) ≠ ∂f †∂f

e i (∂f if ∂)

∂ f †∂ f is invariant under global transformation,

f †→f †' e

i f †

4222 |||||| mAieQL

feff fig '

fAigffD f )(

gauge invariant Lagrangian 　

covariant derivative

gauge field 　

DeD ig'

21f

figef fig f )('

AAA '

)( mAeQi

2

1i i i ii

'fD

fige fig f )(

''' fAigf f

fig A( ) fe fig

2 1 QQQ

fD

fig ige f ( fig A ) f fig fige

DeD ig'

ige' ige'

sistem with 21 and,spinors,scalar f ff eQg

ieQeU cf

AA ' cf

iiigTeU

iiGgTG

kijkji TifTT ],[

iijkjki fgGG

AeQG cf

U' †U '

UU†''

UU †'' ))(( UU †

)(')( GG iUUiUU

†† UUiUU )(' GG

iiigTe' ieQe' cf

' )( UU U

'D UD

)( GiD

GG iUiU ')(

kjijks

iii GGfgGGG jkijki GgfG

24

1 iG GL

mGgTiL iiF

4222|||| mGgTiL ii

S

iijkjki fgGG

iiigT

iiigT

)( iiigT

D

DTgD ii

iiGgTi

D

DTgD ii

iiGgTi

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 leflfqdfqufL cu

cuuY

††††

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