Mathematical Principles of Theoretical Physics

4.6. WEAK INTERACTION THEORY 241

Hence, we deduce from (4.6.35) and (4.6.36) that

81

(

ρw

ρn

) 6

g^2 w=

8

√

2

(

mw

mp

) 2

× 10 −^5 ̄hc.

Then we derive the relation

(4.6.37) g^2 w=

8

81

√

2

(

mw

mp

) 2

× 10 −^5 ×

(

ρn

ρw

) 6

̄hc.

In a comparison with (4.5.66) and (4.6.37), we find that the strong chargegsand the weak

chargegwhave the same magnitude order. Direct computation shows that

g^2 w

g^2 s

≃ 0. 35 equivalently

gw

gs

≃ 0. 6.

4.6.4 PID mechanism of spontaneous symmetry breaking

In the derivation of equations (4.6.7) and (4.6.8) we have used the PID mechanism of sponta-
neous symmetry breaking. In this subsection we shall discuss the intermediate vector bosons
W±,Zand their dual scalar bosonsH±,H^0 , called the Higgs particles by using the PID mech-
anism of spontaneous symmetry breaking.
We know that the interaction field equations are oriented toward two directions: i) to
derive interaction forces, and ii) to describe the field particles and derive the particle transition
probability. The PID-PRI weak interaction field equations describing field particles are given
by (4.4.42)-(4.4.45). Here, for convenience, we write them again as follows:

∂νWν μa −

gw

̄hc

εbcagα βWα μbWβc−gwJμa= (∂μ−

1

4

(mHc

̄h

) 2

xμ+

gw

̄hc

(4.6.38) γbWμb)φa,

−∂μ∂μφa+

(m

Hc

h ̄

) 2

φa−

gw

̄hc

(4.6.39) γb∂μ(Wμbφa)

=

gw

̄hc

εbcagα β∂μ(Wα μbWβc)+gw∂μJaμ−

1

4

(

mHc

h ̄

)^2 xμ∂μφa,

wheremHis the Higgs particle mass, andWμa( 1 ≤a≤ 3 )describe the intermediate vector

bosons as follows

(4.6.40)

W±: Wμ^1 ±iWμ^2 ,

Z: Wμ^3 ,

andφadescribe the dual Higgs bosons as

(4.6.41)

H±: φ^1 ±iφ^2 ,

H^0 : φ^3.

By (4.4.46), equations (4.6.38)-(4.6.39) need to be supplemented with three gauge fixing

equations. According to physical requirement, we take these equations as

(4.6.42) ∂μWμa= 0 for 1≤a≤ 3.