spontaneous breakdown (sb) of symmetry
DESCRIPTION
Spontaneous breakdown (SB) of symmetry. real scalar j. v.e.v. Z 2 symmetry. SB. mass of x :. field redefinition. +fermion y. chiral symmetry. mass term. :forbidden. mass of y :. fermion mass generation by SB. complex scalar field f. global U(1) symmetry. v.e.v. SB. - PowerPoint PPT PresentationTRANSCRIPT
Spontaneous breakdown (SB) of symmetry
real scalar Z2 symmetry
v.e.v. 000 vSB vm 2mass of : v
4222
4
1
2
1)(
2
1 sL
+fermion sLL i f
mass of :mass term :forbiddenchiral symmetry
fvm
v00 SB
2/)( iv
complex scalar field
4222
cs |||| L
global U(1) symmetry
,2 vm masses of , : 0m
fermion mass generation by SB
field redefinition
v.e.v.
field redefinition
: Nambu-Goldstone boson
vv
V
02
Goldstone Theorem
+fermion
csLL mass termchiral U(1)×U(1) symmetry
i )(2 LRRLf †
Higgs mechanism
complex scalar field , U(1)gauge field A
42222 ||||||)(4
1 DFL
2/00 v
2/)( iev g
iAA1
'
If a symmetry under continuous group is broken spontaneously, the system includes a massless field.
The massless particle is called Nambu- Goldstone field.
: forbidden
mass of : fvm fermion mass generation by SB
SB U(1) gauge symmetry
v.e.v.
field redefinition gvmA '
vm 2
mass of A' The gauge boson mass is generated.
mass of The NG boson is absorbed by A'.
Non-Abelian Gauge Theory
2)(4
1 iWL
invariant Lagrangian density
transformation
SU(2) gauge symmetry iijkjki gWW
SU(2) gauge field
iW 3,2,1i
kjijkii WWgWW iW
)]([Tr'iiiiiiii iTiTiTiiiTjj eieeWTgeTW
jklklj iT )(
Spontaneous Breakdown of Non-Abelian Gauge Symmetry
22 ||)(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 ||||
(i ： Pauli matrix)
iijkjki gWW
iig
i2
SU(2) gauge field
iW 3,2,1i
iiWgi
2
1
kjijkii WWgWW
422 |||| 22 || 4||V V iW
D 2||
potential
22 || 4||V kjijkii WWgWW iW
iiWgi
2
1D 2
22
1 |||| 2||
potential
VDWL i 22)(
4
1 42 |||| V
iiWgiD
2
1kjijkiii WWgWWW
22
21
2 ||||||
22 ||)(4
1 DWL i V
2/2
V
12
]4/)2/|[(| 4222 vv
vU 02
1
vacuum expectation value (v.e.v.) 02/00 v
redefinition i : real
V
If 20 the lowest energy state
The vacuum violates SU(2) gauge symmetry spontaneously.
Then
(the vacuum state) occurs at
V
000 000 i
iiieU
kjijkiii WWgWWW
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
vv
/2
iiWgiD
2
1
22
21
2 ||||||
22||4||
30
微分
2/2v
v
44322
4
1
4
1vvv
redefinition Then
kjijkiii WWgWWW
vU 02
1 44322
4
1
4
1vvvV iiieU
iiieU i : real
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
vU 02
1
44322
4
1
4
1vvv V
jjW '
44322
4
1
4
1vvvV iiieU
kjijkiii WWgWWW
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
2
1
v0
)( UU iiWgi
2
1 U
2
1
v0
U )( U
†U gi2
1 †U UW ii
UWU ii† )( UU
† g/i2
iiWgi 2
1
vU 02
1
vU 0
2
1
vU 02
1
j jW 'jTr [ ]
2
1
2
1giU
v0
D
i2/
44322
4
1
4
1vvvV iiieU
kjijkiii WWgWWW
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
vU 02
1
D
vWgiU jj 0'
2
1
2
1
]/)(2/[Tr' gUiUUWUW iijj ††
[Tr' jjW iiW i2/ ( ]/) g
iiie iiie iiie iiie
44322
4
1
4
1vvvV iiieU
kjijkiii WWgWWW
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
vU 02
1
jjW '
2
1
2
1giU
v0
D
vWgiU jj 0'
2
1
2
1
]/)(2/[Tr' gUiUUWUW iijj ††
UWU ii† )( UU
† g/jW 'jTr [ ]i2/
44322
4
1
4
1vvvV iiieU
kjijkiii WWgWWW
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
vU 02
1
jjW '
2
1
2
1giU
v0
D
vWgiU jj 0'
2
1
2
1
]/)(2/[Tr' gUiUUWUW iijj ††
UWU ii† )( UU
† g/jW 'jTr [ ]i2/
) ( ) (
D
vWgiU jj 0'
2
1
2
1
44322
4
1
4
1vvvV iiieU
kjijkiii WWgWWW
2| | v02
1
jjWgi '
2
1�
iiWgi '
2
1
v0
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
vU 02
1
D
2
1v v ) 0( v jjWg '
iiWg '
v0
8
1
2)(2
1 gjW'j
giW'j
(v)2
[
]22
8
1 2)( v 2)'( iWg
]/)(2/[Tr' gUiUUWUW iijj ††
D
vWgiU jj 0'
2
1
2
1
44322
4
1
4
1vvvV iiieU
kjijkiii WWgWWW
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
vU 02
1
222 )'()(8
1)(
2
1 iWgv 2| | D
2)(2
1 8
1 2)( v 2)'( iWg
]/)(2/[Tr' gUiUUWUW iijj ††
D
vWgiU jj 0'
2
1
2
1
44322
4
1
4
1vvvV iiieU
kjijkiii WWgWWW
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
iiWgiD
2
1
vU 02
1
222 )'()(8
1)(
2
1 iWgv
44322
4
1
4
1vvv
kjijkiii WWgWWW '''''
D 2| | 222 )'()(8
1)(
2
1 iWgv iW
L 2 ) (
4
1 iW '
]/)(2/[Tr' gUiUUWUW iijj ††
2)'(4
1 iWL 222 )'()(8
1)(
2
1 iWgv
44322
4
1
4
1vvv
kjijkiii WWgWWW '''''
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
222 )'()(8
1)(
2
1 iWgv
44322
4
1
4
1vvv
kjijkiii WWgWWW ''''' iW
L 2 ) (
4
1 iW '
gvMW 2
1'
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 .
VDWL i 22)(
4
1 42 |||| V 2
22
12 ||||||
2)'(4
1 iWL 222 )'()(8
1)(
2
1 iWgv
44322
4
1
4
1vvv
kjijkiii WWgWWW '''''
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 02/3 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 2
1'
2
1
ii
F WgBYgiL
2
1iRiiR '
2
1
i
BYgi
LRLR qdfqufL uc
uY†† h.c.LRLR leflf u
c ††
BYgiWgiD ii '2
1
2
1
quark lepton1
1
)(|| VDL 2 422 ||||)( V 22
21
2 ||||||
Lagrangian density 22
4
1
4
1 BWL i
G
)(|| VDL 2 422 ||||)( V
YFG LLLLL
LL 2
1'
2
1
ii
F WgBYgiL
2
1iRiiR '
2
1
i
BYgi
LRLR qdfqufL uc
uY†† h.c.LRLR leflf u
c ††
BYgiWgiD ii '2
1
2
1
SU(2)×U(1)gauge sym. is broken spontaneously
/2v2/00 v
)(|| VDL 2 422 ||||)( V 22
21
2 ||||||
If 20 2/|||||| 222
21
2 the vacuum is at
v.e.v.V
12vv
U(1)SU(2)U
vU 02
1 )( 0 iiieU
vUBgYiWgi ii 0
2
1
2
1
2
1
iiii BgWg ~'
~)(2)'( UiUUBYgWgU ii
††
D
vBgYiWgiU ii 0~
2
1~
2
1
2
1
]/)(2/[Tr~
gUiUUWUW iijj ††
]'/)(2/[Tr~
gUiUUBYUB ††
iiii BgWg ~'
~)(2)'( UiUUBYgWgU ii
††
D
]/)(2/[Tr~
gUiUUWUW iijj ††
]'/)(2/[Tr~
gUiUUBYUB ††
vBgYiWgiU ii 0~
2
1~
2
1
2
1D
vUBgYiWgi ii 0
2
1
2
1
2
1
vBgYiWgiU ii 0~
2
1~
2
1
2
1
2|| D
D
8/])~
'~
[()(2/)( 22222
BYgWgv ii
2|| D v02
1 †UBgYiWgi jj
~
2
1~
2
1�
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 angle
gauge field mixing
vM 2mass of W & Z get massive absorbing .
vBgYiWgiU ii 0~
2
1~
2
1
2
1
vBgYiWgiU ii 0~
2
1~
2
1
23222212 )~
'~
()~
()~
( BYgWgWgWg
=
8/])~
'~
[()(2/)( 22222
BYgWgv ii
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 angle
gauge field mixing
vM 2mass of W & Z get massive absorbing .
23222212 )~
'~
()~
()~
( BYgWgWgWg
=
2|| D
The electromagnetic U(1) gauge symmetry is preserved.
A:electromagnetic field
22 '/' gggge
WW cos'sin gge
, electromagnetic coupling constant
22 )(4
1)(
4
1 BW
2][2
][][][ 4
1
4
1
2
1 ZAWW
)cot'( W ZAA
2][
2][][
2
][][][
4
1''
2
1'''
WWgWAWAe
WWAieWAWieWAWie
electroweak boson kinetic terms and self-interactions
The electromagnetic U(1) gauge symmetry is preserved.
A:electromagnetic field
22 '/' gggge
WW cos'sin gge
, electromagnetic coupling constant
32 IYQ /2/)( 21 i
FL
jQZg
W
2L3
W
sin2
1
cos
jWWg
L2
gauge-boson fermion interaction terms
AeQi
LL 2
1'
2
1
ii
F WgBYgiL
2
1iRiiR '
2
1
i
BYgi
jj
jl lAeQilL
3
1
jj
j lQZg
l
W
2L3
W
3
1
sin2
1
cos
jj
j lWWg
l
L
3
1 2
gauge-boson fermion interaction terms (QCD 含む )
jj
j qAeQiq
3
1
ji
is
jj qG
gq
2
3
1
jj
j qQZg
q
W
2L3
W
3
1
sin2
1
cos
jj
j qWWg
q
L
3
1 2
h.c.)(2
LRLRLRLR
3
1,
jkekjjkkjjkdkjjkukjjk
eeffddfuufv
)( LRLRLR
3
1,jk
ekjjkkjjk
dkjjk
ukj
jk
eeMMddMuuM
Yukawa interaction
fermion mass term
2
*)( ujkukjukj
ffvM
2
*)( djkdkjdkj
ffvM
2
*)( jkkjkj
ffvM
2
*)( ejkekjekj
ffvM
LRLR qdfqufL uc
uY†† h.c.LRLR leflf u
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 qWWqg
L
3
1,2
physCKML
3
1,
phys
2jij
jii dVWu
g
jji
i lWWlg
L
3
1,2
physMNSL
3
1,
phys
2jij
jii eVW
g
+h.c.
+h.c.