4.5. STRONG INTERACTION POTENTIALS 219
be the strong charge potential of this particle. Then the strong force between two elementary
particles carrying strong charges is
F=−gs∇Φ 0.However, the strong interactions are layered, i.e. the strong forces only act on the same
level of particles, such as the quark level, the hadron level. Hence, the strong interaction
potentials are also layered. In fact, in the next subsectionwe shall show that for a particle
withNstrong chargesgsof the elementary particles, its strong interaction potential is given
by
(4.5.2)
Φs=Ngs(ρ)[
1
r−
A
ρ( 1 +kr)e−kr]
,
gs(ρ) =(
ρw
ρ) 3
gs,whereρwis the radius of the elementary particle (i.e. thew∗weakton),ρis the particle radius,
k>0 is a constant withk−^1 being the strong interaction attraction radius of this particle, and
Ais the strong interaction constant, which depends on the type of particles. Thus, the strong
force between such two particles is
(4.5.3) F=−Ngs(ρ)∇Φs,
wheregs(ρ)andΦsare as in (4.5.2).
In particular for thew∗-weakton, which possesses one strong chargegs, the formula
(4.5.2) becomes
(4.5.4) Φ 0 =gs
[
1
r−
A 0
ρw
( 1 +k 0 r)e−k^0 r]
,
where 1/k 0 is the attraction radius of the strong interaction for the elementary particles, i.e.
thew∗-weakton. According to the physical observation, we take the quantitative order
(4.5.5) k 0 = 1018 cm−^1.
In this subsection, we shall deduce formula (4.5.4) for thew∗-weaktons from the strong
interaction field equations (4.4.39)-(4.4.41). Taking inner product of the field equation (4.4.39)
and (4.4.40) withρk= (ρ 1 ,···,ρ 8 ), we derive that
∂νSν μ−gs
hc ̄
λijgα βSiα μSβj−gsQμ=[
∂μ−1
4
k^20 xμ+gsδ
hc ̄
Sμ]
(4.5.6) φs,
(
1
c^2∂^2
∂t^2−∆
)
φs+k^20 φs+1
4
(4.5.7) k 02 xμ∂μφs
=gs∂μQμ+gs
̄hc∂μ(λijgα βSiα μSβj−δSμφs),