Weinberg Salam Model

Higgs field 　　

SU(2) gauge field 　　iW 3,2,1i U(1) gaugefield 　　B

complex scalar, SU(2) doublet 　Y=1 　

quark leptonSU(2) 　

U(1)hypercharge

1/3 -1 4/3 0-2/3 -2

Lagrangian density 　　　 22

4

1

4

1 BWL i

G --

)(|| VDL - 2 422 ||||)( V

YFG LLLLL

SU(2)×U(1)gauge symmetry

L 2

),/( 021

)/,( 210

R1R2

SU(3) 　3

13

Lorentzgroup

quark lepton

LL 21'

21

-- ii

F WgBYgiL

-

2

1iRiiR '

21

iBYgi

LRLR qdfqufL dc

uY†† h.c.LRLR leflf e

c ††

BYgiWgiD ii '21

21

SU(2)×U(1)gauge sym. is broken spontaneously 　　 /2-v2/00 vv.e.v.

redefinition　

V

12vv-

,0

21

v

U )( 0 iiieU

-

W3

W

W3

W

cos~sin~sin~cos~

WBZWBA gg /'tan W

mass of gauge fields ,2/gvMW ,2/'22 vggM Z 0AM

w Weinberg anglegauge field mixing 　

vM 2mass of W & Z get massive absorbing .

The electromagnetic U(1) gauge symmetry is preserved. 22 '/' gggge

WW cos'sin gge , electromagnetic coupling constant

h.c.)(2 LRLRLRLR

3

1,

jkekjjkkjjkdkjjkukjjk

eeffddfuufv

)( LRLRLR

3

1,jk

ekjjkkjjk

dkjjk

ukj

jkeeMMddMuuM

Yukawa interaction

fermion mass term

2

*)( ujkukjukj

ffvM

2

*)( djkdkjdkj

ffvM

2

*)( jkkjkj

ffvM

2

*)( ejkekjekj

ffvM

LRLR qdfqufL dc

uY†† h.c.LRLR leflf e

c ††

physuUu u physdUd d phys U physlUl l

uu

uu UMUM †phys d

dd

d UMUM †phys

UMUM †phys e

ee

e UMUM †phys

du UUV †CKM

eUUV †MNS

diagonalization

Cabibbo-Kobayashi-Maskawa matrix

Maki-Nakagawa-Sakata matrix

diagonal

jji

i qWWqgL

3

1,2 --

- physCKML

3

1,

phys

2 jijji

i dVWug

-

jji

i lWWlgL

3

1,2 --

- physMNSL

3

1,

phys

2 jijji

i eVWg

- +h.c.

+h.c.

1,1 -- ntnx

2,2

tx

Path Integral Quantization

fields x̂

if ,,iftt xx

xxxx ˆeigenstate

1 xxxxdcompleteness

probability amplitude

f,f

tx

i,itx

, , dxn

xn xntn tn

, , dx1x1 x1

t1 t1x1

xn

ix̂iiii xxxx ˆ

cf. coordinate

1 iiiixxdx

1xx n

nd x 1xd

f,f

tx

i,itx

nn ttnn,, xx

11 ,,11

tt xx

, , dxixi xi

ti tixi

iti,x1,

1 itix

if

1,1 -- ntnx

2,2

tx

Path Integral Quantization

fields x̂ xxxx ˆeigenstate

1 xxxxdcompleteness

i,itx , , dx1

x1 x1t1 t1x1

ix̂iiii xxxx ˆ

cf. coordinate

1 iiiixxdx

1xx n

nd x 1xd

f,f

tx

i,itx

nn ttnn,, xx

11 ,,11

tt xx

, , dxixi xi

ti tixi

iti,x1,

1 itix

provability amplitude

if ,,iftt xx f,

ftx , , d

xnxn xn

tn tn xn

if

itHi

i teti

ii

,, ˆ

1 xx -

iti,

1x itHi te

ii ,ˆ

x-

H : Hamiltonianii ttii,, 11 xx

1,1 -- ntnx

2,2

tx

1xx n

nd x 1xd f,

ftx

i,itx

nn ttnn,, xx

11 ,,11

tt xx iti,x1,

1 itix

itHi

i teti

ii

,, ˆ

1 xx -

iti,

1x itHi te

ii ,ˆ

x-

H : Hamiltonianii ttii,, 11 xx

xxxx ˆ

1 xxxxd

xyyx i]ˆ,ˆ[

xx

- iˆ iiii

iii ett xx

xx ,,

: canonical conjugate of x̂

eigenstate

completeness

x̂

xxxx ˆ

1 xxxxd

xyyx i]ˆ,ˆ[

xx

- iˆ iiii

iii ett xx

xx ,,

: canonical conjugate of x̂

eigenstate

completeness

x̂

ii ttdiii

i,, xxxx

i

id xx iti

,1x iti

,x iti,x iti

,x itie - O((ti)2)H

iid xx iie x ) ( 1ix ix- itiHe -

ti

ti

xi

edi

ixx

ii x ix

・itH - ( )

1,1 -- ntnx

2,2

tx

1xx n

nd x 1xd f,

ftx

i,itx

nn ttnn,, xx

11 ,,11

tt xx iti,x1,

1 itix

itHi

i teti

ii

,, ˆ

1 xx -

iti,

1x itHi te

ii ,ˆ

x-

H : Hamiltonianii ttii,, 11 xx

ii ttdiii

i,, xxxx

i

id xx iti

,1x iti

,x iti,x iti

,x itie - O((ti)2)H

iid xx iie x ) ( 1ix ix- itiHe - ed

ii

xx ii x ix itH - ( )

itiLNe L : Lagrangian

xi

22V

Lii

ii

ied xxxx

( it)Vi 2/2

x- -

2/2/)( 22 -- ii xxed

ii

xx ( i 2/2/)( 22 -- ii xx itV - )

-

2/( 2i

ii

ied xxx

it)L

11

xxxx dd

nn

e itiLSN'

1,1 -- ntnx

2,2

tx

Path Integral Quantization

fields x̂ xxxx ˆeigenstate

1 xxxxdcompleteness

i,itx , , dx1

x1 x1t1 t1x1

ix̂iiii xxxx ˆ

cf. coordinate

1 iiiixxdx

1xx n

nd x 1xd

f,f

tx

i,itx

nn ttnn,, xx

11 ,,11

tt xx

, , dxixi xi

ti tixi

iti,x1,

1 itix

provability amplitude

if ,,iftt xx f,

ftx , , d

xnxn xn

tn tn xn

if

itHi

i teti

ii

,, ˆ

1 xx -

iti,

1x itHi te

ii ,ˆ

x-

ii ttdiii

i,, xxxx

i

id xx iti

,1x iti

,x iti,x iti

,x itie - O((ti)2)H

iid xx iie x ) ( 1ix ix- itiHe - ed

ii

xx ii x ix itH - ( )

itiLNe

H : Hamiltonian

L : Lagrangian

iii

i

ied xxxx

( it)Vi 2/2

x- - edi

ixx

( i 2/2/)( 22 -- ii xx itV - )

it)-

2/( 2i

ii

ied xxx

L

ii ttii,, 11 xx

11

xxxx dd

nn

e itiLSN'

Ldti

1,1 -- ntnx

2,2

tx

Path Integral Quantization

fields x̂ xxxx ˆeigenstate

1 xxxxdcompleteness

i,itx , , dx1

x1 x1t1 t1x1

ix̂iiii xxxx ˆ

cf. coordinate

1 iiiixxdx

1xx n

nd x 1xd

f,f

tx

i,itx

nn ttnn,, xx

11 ,,11

tt xx

, , dxixi xi

ti tixi

iti,x1,

1 itix

provability amplitude

if ,,iftt xx f,

ftx , , d

xnxn xn

tn tn xn

if

11

xxxx dd

nn

e itiLSN'

: Lagrangian densityD

if xdieN

4

'LD 1

1xxxx

ddn

n

xx

d 'N exx

d

L

xdi 4L 'N eD

xt ),( x

D11

xxxx dd

nn

xx

d: Lagrangian densityLxt ),( x

2,2

tx

1,1 -- ntnx

if ,ˆ,iftt

j xxx

nn ttdnnn

n,, xxxx

11 ,,111

1

ttd xxxx

i)(ˆf x

jj ttdjjj

j,, xxxx

f,

ftx

i,itx

jx̂

1 xxxxd

x j

operator

eigenvalue

xxxxd xxxxd xxxxd

nd x 1xd

1xx n f,

ftx nn tt

nn,, xx

11 ,,11

tt xx i,itx

x j

if xdieN

4

'LD

D xdie4L

'N (x)

2,2

tx

1,1 -- ntnx

if ,ˆ,iftt

j xxx

nn ttdnnn

n,, xxxx

11 ,,111

1

ttd xxxx

i)(ˆf x

jj ttdjjj

j,, xxxx

f,

ftx

i,itx

1 xxxxd

x j

eigenvalue

f,f

tx nn ttnn,, xx

11 ,,11

tt xx i,itx

x j

if xdieN

4

'LD

D xdie4L

'N (x)

nd x 1xd

1xx n

xdiexN

4

)('LD i)(f x

nn ttdnnn

n,, xxxx

11 ,,111

1

ttd xxxx

aa ttdaaa

a,, xxxx

bb ttd

bbbb

,, xxxx

if ,ˆˆ,iftt

ba xxxx i))(ˆ)(ˆ(f ba xxT

ba tt f,

ftx

ax̂

i,itx

bx̂

1 xxxxd

xa

xb

nd x 1xd

1xx n xa

xb

2,2

tx

1,1 -- ntnxf,

ftx nn tt

nn,, xx

11 ,,11

tt xx i,itx

if xdieN

4

'LD

xdiexN4

)('LD i)(f x

D (xa) (xb) xdie4L'N

xdi

ba exxN4

)()('LD i))()((f ba xxT

nn ttdnnn

n,, xxxx

11 ,,111

1

ttd xxxx

aa ttdaaa

a,, xxxx

bb ttd

bbbb

,, xxxx

if ,ˆˆ,iftt

ba xxxx i))(ˆ)(ˆ(f ba xxT

ba tt f,

ftx

i,itx

1 xxxxd

xa

xb

nd x 1xd

1xx n xa

xb

2,2

tx

1,1 -- ntnxf,

ftx nn tt

nn,, xx

11 ,,11

tt xx i,itx

D (xa) (xb) xdie4L'N

xdi

ba exxN4

)()('LD i))()((f ba xxT

xdi

nn exxNxxT4

)()(i))()((f 11LD

xdi

ba exxN4

)()('LD i))()((f ba xxT

xdi

nn exxNxxT4

)()(i))()((f 11LD

xdi

nn exxNxxT4

)()(i))()((f 11LD

generating functionalfunctional derivative

hxJZyxhxJZ

yJxJZ

h

)]([)]()([lim)()]([

0

--

xdi

nn exxNxxT4

)()(i))()((f 11LD

J][JZ D L(ie xd 4)

cf. partial derivative

hxfhxf

xxf jijj

hi

j })({})({lim

})({0

-

xdie4) ( L

J)(yJ

xdie4) ( L

J)(yJ

xdi 4) ( L J

1 (x) xd 4

xdie4) ( L

J

(y) xdie4) ( L

J

0))()((0 1 nxxT

0limh h h(x-y)i i

01 )()(

Jn

n

xJxJ

)]([ xJZ

)0(Z

ni)(-

][JZ

xdJie

4)(

LD

0))()((0 1 nxxT 01 )()(

Jn

n

xJxJ

)]([ xJZ

)0(Z

ni)(-

J][JZ D L(ie xd 4)

(y) xdie4) ( L

J

xdie4) ( L

J)(yJ

i

xdie4) ( L

J)(yJ

(y) xdie4) ( L

Ji

xdie4) ( L

J)(yJ

4

421 -- K

22 K

422

4221

--L

422

4221

---

0))()((0 1 nxxT 01 )()(

Jn

n

xJxJ

)]([ xJZ

)0(Z

ni)(-

D iexd 4

J4

421 -- K

xdJie

4)(

LD][JZ

D e xdi 4

4

- 4e xdi 4

K

21

- J

D e - xdi 4

4 4

e

- JKxdi

214

Ji

44

4

- Ji

xdie

- JKxdi

e

214

D

4

421 -- K

22 K

422

4221

--L

422

4221

--- D iexd 4

J4

421 -- K

xdJie

4)(

LD][JZ

D e xdi 4

4

- 4e xdi 4

K

21

- J

D e - xdi 4

4 4

e

- JKxdi

214

Ji

44

4

- Ji

xdie

- JKxdi

e

214

D

--- --- JJKJKKJKxdi

e1114 )()(

21

44

4

- Ji

xdie

D

44

4

- Ji

xdie

D xdie4

JJK 1

21 - xdie

4

)()(21 11

JKKJK--

---

][JZ

- - JJKxdiJi

xdiee

144

4

21

4

44

4

- Ji

xdie

JJK 1

21 - xdie

4

22 K

22 K

][JZ

- - JJKxdiJi

xdiee

144

4

21

4

-

-

244

44

421

41

Jixdi

Jixdi

2

114

114 )(

21

21)(

211

-- JyJKxdiJyJKxdi

-

3

114 )(

21

61 JyJKxdi

0))()((0 1 nxxT 01 )()(

Jn

n

xJxJ

)]([ xJZ

)0(Z

ni)(-

22 K

0],[ ji cc

0},{ ji

kk iiiiii

221

1

02 i

02

2

j

F

0 id ijjid 0},{ jid 0},{ ji dd

commuting c- 数

anti-commuting c- 数

ic

ijji -(Grassman 数 )0],[ jic

微分

i

積分

NNNjjjjNN

NNAAN 111

2/)1(11

!)1( --

)exp(2121 jijiNN AddddddI

21

21jiji cBc

NedcdcdcJ-

BN det/2/

NNNN AN 11

2/)1( det!)1( --

cf

Ae yyxAxydxdi det)(),()(44

DD

)exp( jijiA Njijijiji A

NA )(

!1)(1

I

N

ii

N

ii dd

11

)(!

1 NjijiAN

)( NjijiA

I 2/)1()1( -- NN Adet

Lxdie4

DDD],,[Z

04 Lxdie

iiixdi

e,,1

4 L

-

mixdi

e11

21

224

scalar と fermion の系generating functional

],,[ Z DDD L 10 LL

0L

1L

)(21)(

21 222 mi --

g-- 4

41),,(

(4 xdie ) 10 LL ),,(

DDD xdie4 (4 xdie),,(1 L 0L )

DDD

iiixdi

e,,1

4 L

ii JGxdi L4exp

)()(41 2 mDiG i --L

2G0 )(

41 ii GG --L jiji GG

-- 2

21

- 2Kneed gauge fixing

gauge theory

is inappropriate because

and does not have inverse.

kjijkiii GGgfGGG -- )( iiGigTD

)( 1-K

- 1KK

generating functional

gauge boson と fermion の系iG

],,[ JZ DDDG

gauge fixing ai BG

1)(][][ )( - iUix

i BGdUG

)(][ )(

,

iUiiix

i BGdG -

iiddU ][

KG i det/][

))(()(

))((, xG

yK

iUijyjxi

KG i det][

)()( 42 yxGfg kijks

ij --

Ke lnTr )ln(Tr 2 ijije -

nij

n nCe

-

21

11Tr

kijks

ij Gfg

JiGiSi eGJZ *][ D xdJGJG ii 4*

iijkjki gGG

iSiiUi

ix

i eGBGdUGZ ][)(]0[ )(

,

-DiSiii

ix

i eGBGdUG ][)(,

-D

iSiiii eGBGGZ ][)(]0[

-DxdBiiSiiiii

i

eeGBGBGZ42)(

21

][)(]0[ --

DD

xdGiiSiii

eeGG42)(

21

][

- D)ln(Tr 2 ijije - kijk

sij Gfg

Faddeev Popov ghost

jiji yyxKxydxdK )(),(*)(expdet 44 *DD)(*)()(),(*)( 444 xDxxdiyyxKxydxd jijijiji

- )( kijkijij GgfD

KG i det][

=1

DDDDD *],,,[ * GJZ

iiiiii JGxdi **exp 4 L

FFPGFG LLLLL

2G )(

41 iG-L kjijkiii GGgfGGG --

2GF )(

21 iG

-L

jiji D *)(FP L

)(F mDi -L

jkikjijjij GgfD )(

)( iiGigTD

2/iiT

10 LLL F0

FP0

GF0

G00 LLLLL

22GF0

G0 )(

21)(

41 iii GGG

---LLii

*FP0L

)(F0 mi -L

kjiiijkG GGGGfg )(2

31 -L

lkjiklmijm GGGGffg4

24G1 -L

kjiijk Ggf *)(G1 L

iiGTg-G

1L

G1

G1

4G1

3G11

LLLLL