weinberg salam model
DESCRIPTION
Weinberg Salam Model. SU(2)×U(1)gauge symmetry. SU(2) gauge field. U(1) gaugefield . Higgs field. complex scalar, SU(2) doublet Y f =1. Lorentz group. quark lepton. SU(3) . U(1)hypercharge. SU(2) . quark lepton. - 1 . 1/3. 2. 3. 0. 4/3. 3. 1. - 2. - 2/3. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/3.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/4.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/5.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/6.jpg)
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
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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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/8.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/9.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/10.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/11.jpg)
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)
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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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/13.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/14.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/15.jpg)
xdi
ba exxN4
)()('LD i))()((f ba xxT
xdi
nn exxNxxT4
)()(i))()((f 11LD
xdi
nn exxNxxT4
)()(i))()((f 11LD
![Page 16: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/16.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/17.jpg)
][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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/18.jpg)
(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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/19.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/20.jpg)
][JZ
- - JJKxdiJi
xdiee
144
4
21
4
44
4
- Ji
xdie
JJK 1
21 - xdie
4
22 K
22 K
![Page 21: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/21.jpg)
][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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/22.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/23.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/24.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/25.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/26.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/27.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/28.jpg)
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: Weinberg Salam Model](https://reader036.vdocuments.mx/reader036/viewer/2022081420/56815efd550346895dcdbaa0/html5/thumbnails/29.jpg)
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