Mathematical Principles of Theoretical Physics

222 CHAPTER 4. UNIFIED FIELD THEORY

whereAis a constant given by

A=

k 02 cτ κ

4

with physical dimension

1

L

.

AssumeΦs=Φs(r)is radially symmetric, then (4.5.20) becomes

−

1

r^2

d

dr

(r^2

d

dr

)Φs=gsβ δ(r)+

gsA

ρw

1

r

e−k^0 r,

whose solution takes the form

(4.5.21) Φs=gs

[

β

r

−

A

ρw

φ(r)e−k^0 r

]

,

whereφsolves

(4.5.22) φ′′+ 2

(

1

r

−k 0

)

φ′−

(

2 k 0

r

−k^20

)

φ=

1

r

.

Assume that the solutionφof (4.5.22) is given by

(4.5.23) φ=

∞

∑

k= 0

αkrk.

Insertingφin (4.5.22) and comparing the coefficients ofrk, then we obtain the following
relations

α 1 =k 0 α 0 +

1

2

,

α 2 =

1

2

k^20 α 0 +

1

3

k 0 ,
..
(4.5.24).

αk=

2 k 0

k+ 1

αk− 1 −k^20 αk− 2 , ∀k≥ 2.

whereα 0 is a free parameter with dimensionL. Hence

φ(r) =α 0 ( 1 +k 0 r+

r

2 α 0

+o(r)),

and often it is sufficient to take a first-order approximation.

Step 3. Strong interaction potential of w∗-weakton.The formula (4.5.21) with (4.5.23)-
(4.5.24) provides an accurate strong interaction potential for thew∗-weaktons:

(4.5.25) Φ 0 =gsβ

[

1

r

−

A 0

ρw

φ ̃(r)e−k^0 r

]

,

whereA 0 =Aα 0 /βis a dimensionless parameter, andφ ̃(r)is

φ ̃= 1 +k 0 r+

r

2 α 0

+o(r),