1. Spontaneous breakdown (SB) of symmetry
real scalarj 　　　
Z2 symmetry 　　　
SB　
v.e.v.
field redefinition　　　
mass of x :
+fermion y　　
chiral symmetry　
mass term
:forbidden
fermion mass generation by SB
mass of y :
complex scalar field f　　　
global U(1) symmetry　　　
SB　
v.e.v.
field redefinition　　　
masses of x, c :
c : Nambu-Goldstone boson

2. If a symmetry under continuous group is
broken spontaneously, the system includes a massless field.
Goldstone Theorem
The massless particle is called Nambu- Goldstone field.
+fermion y　　
chiral U(1)×U(1) symmetry 　
mass term
: forbidden
fermion mass generation by SB
mass of y :
Higgs mechanism　
complex scalar field f, U(1)gauge field Am　　
U(1) gauge symmetry　　　
v.e.v.
SB　
field redefinition　　　
mass of A'
The gauge boson mass is generated.
mass of x
The NG boson c is absorbed by A'.

3. Non-Abelian Gauge Theory
SU(2) gauge field　　
SU(2) gauge symmetry　　　
transformation　　
invariant Lagrangian density　　　

4. Spontaneous Breakdown of Non-Abelian Gauge Symmetry
SU(2) gauge field　　
SU(2) doublet
complex scalar　　　
real field　　　
SU(2) gauge symmetry　　　
transformation　　
(t i ： Pauli matrix)　
invariant Lagrangian density　　　 V
potential　　 V = +

5. V
potential　　 V + =

6. 微分　
2m 2|f| +4l|f|3= 0 　　
If m2<0
the lowest energy state
(the vacuum state) occurs at
The vacuum violates SU(2) gauge symmetry spontaneously.
vacuum expectation value (v.e.v.)
redefinition　
x, c i : real
Then

7. redefinition　
x, c i : real
Then

12. (v+x)2 　　
[　　
gt jW'mj gt iW'jm　
]22 　　

16. mass of W'
The gauge boson mass is generated.
mass of x
The gauge boson becomes massive by absorbing NG boson c.

17. Weinberg Salam Model
SU(2)×U(1)gauge symmetry
SU(2) gauge field　　
U(1) gaugefield 　　
Higgs field　　
complex scalar, SU(2) doublet 　Yf=1　
quark lepton
Lorentz
group
SU(3) 　
SU(2) 　
U(1)hypercharge
quark lepton
quark lepton 3 1 2
1/3 -1
4/3 3 1 1
-2/3 -2
Lagrangian density　　　

18. Lagrangian density　　　

19. If m2<0
the vacuum is at
SU(2)×U(1)gauge sym. is broken spontaneously 　　
v.e.v.

21. =
gauge field mixing　
Weinberg angle
mass of gauge fields
mass of x
W & Z get massive absorbing c.

22. Am:electromagnetic field
The electromagnetic U(1) gauge symmetry is preserved.
, electromagnetic coupling constant =
gauge field mixing　
Weinberg angle
mass of gauge fields
mass of x
W & Z get massive absorbing c.

23. Am:electromagnetic field
The electromagnetic U(1) gauge symmetry is preserved.
, electromagnetic coupling constant
electroweak boson kinetic terms and self-interactions

24. gauge-boson fermion interaction terms

25. gauge-boson fermion interaction terms (QCD含む) 　　

26. Yukawa interaction
fermion mass term

27. diagonalization
diagonal
Cabibbo-Kobayashi-Maskawa matrix
Maki-Nakagawa-Sakata matrix
+h.c.
+h.c.