Mathematical Principles of Theoretical Physics

4.5. STRONG INTERACTION POTENTIALS 221

whereδ(r)is the Dirac delta function,θμj is a constant tensor, inversely proportional to the
volume of the particle. Hence

ρkSμjλijkθiμ=Sμjθjμδ(r),

whereS
j
μ∼S

j

μ(^0 )takes the following average value

(4.5.14) S

j

μ=

1

|Bρw|

∫

Bρw

Sμjdv.

Hereρwis the radius of aw∗-weakton. Later, we shall see that

S

j

μ∼

1

r

asr→ 0.

Hence we deduce from (4.5.14) that

S

j

μ=ξ

j

μρ

− 1

w (ξ

j

μ is a constant tensor).

Thus, (4.5.13) becomes

(4.5.15) ∂μQμ=−
κ
ρw

δ(r) (ρwis the radius of aw∗-weakton),

whereκis a parameter given by

(4.5.16) κ=

2 gs

hc ̄

ξμjθ

μ

j,

andκis inversely proportional to the volume ofw∗-weakton.
Therefore, equation (4.5.12) is rewritten as

(4.5.17) −∆φs+k^20 φs=−
gsκ
ρw

δ(r),

whose solution is given by

(4.5.18) φs=−
gsκ
ρw

1

r

e−k^0 r.

Step 2. Solution of (4.5.11).The quantitygsQ=−gsQ 0 is the strong charge density of a
w∗-weakton, and without loss of generality, we assume that

(4.5.19) Q=β δ(r),

andβ>0 is a constant, inversely proportional to the volume of thew∗-weakton. Hence
(4.5.11) can be rewritten as

(4.5.20) −∆Φs=gsβ δ(r)+

gsA

ρw

1

r

e−k^0 r,