Page 1
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
Page 2
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
Page 3
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 ††
Page 4
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.
Page 5
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
Page 6
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
Page 7
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 - ( )
Page 8
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'
Page 9
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'
Page 10
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
Page 11
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)
Page 12
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
Page 13
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
Page 14
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
Page 15
xdi
ba exxN4
)()('LD i))()((f ba xxT
xdi
nn exxNxxT4
)()(i))()((f 11LD
xdi
nn exxNxxT4
)()(i))()((f 11LD
Page 16
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)(-
Page 17
][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
Page 18
(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
Page 19
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--
---
Page 20
][JZ
- - JJKxdiJi
xdiee
144
4
21
4
44
4
- Ji
xdie
JJK 1
21 - xdie
4
22 K
22 K
Page 21
][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
Page 22
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
Page 23
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
Page 24
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
Page 25
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
Page 26
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
Page 27
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
Page 28
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
Page 29
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