Mathematical Principles of Theoretical Physics

4.6 Weak Interaction Theory

It is clear that the following state

(Wμa,φwa,ψ) = ( 0 ,φ 0 a, 0 ) withφ 0 abeing constants,

is a solution of (4.6.3) and (4.6.4), which is a ground state ofφwa. Leta 0 =φ 0 aωa, which is a
constant. Take the transformation

φw→φw+a 0 , Wμa→Wμa, ψ→ψ.

Then the equations (4.6.3) and (4.6.4) are rewritten as

∂νWν μ−k^21 Wμ−

gw

hc ̄

(4.6.7) κabgα βWα μaWβb−gwJwμ

=

[

∂μ−

1

4

k^20 xμ+

gw

̄hc

γWμ

]

φμ−

1

4

a 0 k^20 xμ,

[ 1

c^2

∂

∂t^2

−∆

]

(4.6.8) φw+k^20 φw−gw∂μJwμ+k^20 a 0

=

gw

hc ̄

∂μ

[

κabgα βWα μaWβb−γWμ(φμ+a 0 )

]

−

1

4

k^20 xμ∂μφw,

wherek 1 =

√

gwγa 0 / ̄hcrepresents mass.

Thus, (4.6.7) and (4.6.8) have masses as

(4.6.9) k 0 =mHc/h ̄, k 1 =mWc/ ̄h,

wheremHandmWare the masses of Higgs andW±bosons. Physical experiments measured
the values ofmHandmWas

(4.6.10) mH⋍160 GeV/c^2 , mW⋍80 GeV/c^2.

By (4.4.46), equations (4.6.7) and (4.6.8) need to add three gauge fixing equations. Based
on the superposition property of the weak charge forces, thedual potentialsW 0 andφwshould
satisfy linear equations, i.e. the time-componentμ=0 equation of (4.6.7) and the equation
(4.6.8) should be linear. Therefore, we have to take the three gaugefixing equations in the
following forms

(4.6.11)

κab

[

gα βWαa 0 Wβb−∂μ(WμaW 0 b)

]

+γW 0 φw−

̄hc

gw

a 0 k^20

4

x 0 = 0 ,

gw

hc ̄

∂μ

[

κabgα βWα μaWβb−γWμφw

]

−

k^20

4

xμ∂μφw−k^20 a 0 = 0 ,

∂μWμ= 0 ,

and with the static conditions

(4.6.12)

∂

∂t

Φw= 0 ,

∂

∂t

φw= 0 , (Φw=W 0 ).

With the equations (4.6.11) and the static conditions (4.6.12), the time-componentμ= 0
equation of (4.6.7) and its dual equation (4.6.8) become

−∆Φw+k^21 Φw=gwQw−

1

4

(4.6.13) k^20 cτ φw,

(4.6.14) −∆φw+k^20 φw=gw∂μJμw,