Mathematical Principles of Theoretical Physics

236 CHAPTER 4. UNIFIED FIELD THEORY

wherecτis the wave length ofφw,Qw=−Jw 0 , andJwμis as in (4.6.5). The two dual equations
(4.6.13) and (4.6.14) for the weak interaction potentialsΦwandφware coupled with the Dirac
equations (4.4.50), written as

(4.6.15) iγμ

(

∂μ+i

gw

hc ̄

Wμaσa

)

ψ−

mc

h ̄

ψ= 0.

In addition, by (4.6.9) and (4.6.10) we can determine values of the parametersk 0 andk 1 as
follows

(4.6.16) k 0 = 2 k 1 , k 1 = 1016 cm−^1.

The parameters 1/k 0 and 1/k 1 represent the attracting and repulsive radii of weak interaction
forces.
In the next subsection, we shall apply the equations (4.6.13)-(4.6.15) to derive the layered

4.6.2 Layered formulas of weak forces.

4.6.2 Layered formulas of weak forces

Now we deduce from (4.6.13)-(4.6.16) the following layered formulas for the weak interac-
tion potential:

(4.6.17)

Φw=gw(ρ)e−kr

[

1

r

−

B

ρ

( 1 + 2 kr)e−kr

]

,

gw(ρ) =N

(

ρw

ρ

) 3

gw,

whereΦwis the weak force potential of a particle with radiusρand carryingNweak charges
gw(gwis the unit of weak charge for each weakton, an elementary particle),ρwis the weakton
radius,Bis a parameter depending on the particles, and

(4.6.18)

1

k

= 10 −^16 cm,

represents the force-range of weak interaction.
To derive the layered formulas (4.6.17), first we shall deduce the following weak interac-
tion potential for a weakton

(4.6.19) Φ^0 w=gse−kr

[

1

r

−

B 0

ρw

( 1 + 2 kr)e−kr

]

.

To derive the solutionφwof (4.6.14), we need to compute the right-hand term of (4.6.14).
By (4.6.5) we have
∂μJwμ=ωa∂μψ γμσa+ωaψ γμσa∂μψ.

Due to the dirac equation (4.6.15),

∂μψ γμσaψ=−i

gw

hc ̄

Wμbψ γμσbσaψ+i

mc

̄h

ψ σaψ,

ψ γμσa∂μψ=i

gw

̄hc

Wμbψ γμσaσbψ−i

mc

̄h

ψ σaψ.