4.6. WEAK INTERACTION THEORY 247
Here
Jaμ=LγμσaL 1 ≤a≤ 3 ,
JLμ=νLγμνL+lLγμlL,
JRμ=RγμR, γμ=gμ αγα.- Masses are generated at the ground states. It is clear thatthe following state
φ=a,Wμa= 0 ,Bμ= 0 ,L= 0 ,R= 0 ,is a solution of (4.6.60)-(4.6.62), called a ground state. We take the translation transformation
φ→φ+a, Wμa→Wμa,Bμ→Bμ,L→L,R→R,then the massless equations (4.6.60) become massive, written as
∂ν
Wν μ^1
Wν μ^2
Wν μ^3
Bν μ
−M
Wμ^1
Wμ^2
Wμ^3
Bμ
(4.6.63)
=
g 1 gα βεab^1 Wα μaWβb−g 21 Jμ^1 + 21 g^21 (φ^2 + 2 aφ)Wμ^1
g 1 gα βεab^2 Wα μaWβb−g 21 Jμ^2 + 21 g^21 (φ^2 + 2 aφ)Wμ^2
g 1 gα βεab^3 Wα μaWβb−g 21 Jμ^3 +^12 g 1 (φ^2 + 2 aφ)(g 1 Wμ^3 −g 2 Bμ)
1
2 g^2 JL
μ+g^2 J
R
μ+
1
2 g^2 (φ(^2) + 2 aφ)(g 2 Bμ−g 1 W 3
μ)
,
whereMis the mass matrix given by
(4.6.64) M=
c^2
h ̄^2
m^21000
0 m^2100
0 0 m^21 −m^23
0 0 −m^23 m^22
,
and
m 1 c
h ̄
=
g 1 a
√
2,
m 2 c
h ̄=
g 2 a
√
2,
m 3 c
̄h=
√
g 1 g 2 a
√
2.
- The massesmWandmZcan be derived from (4.6.63) and (4.6.64) as follows. Accord-
ing to the IVB theory for particle transition, theW±particles are characterized as
W±:Wμ^1 ±iWμ^2.Hence we need the following transformation forWμ^1 andWμ^2 :
(4.6.65)
(
Wμ+
Wμ−)
=
1
√
2
(
1 i
1 −i)(
Wμ^1
Wμ^2