Mathematical Principles of Theoretical Physics

238 CHAPTER 4. UNIFIED FIELD THEORY

Inserting (4.6.25) into (4.6.13) we get

(4.6.26) −∆Φw+k 12 Φw=gwQw+

gwB

ρw

1

r

e−k^0 r,

whereBis a parameter with dimension 1/L, given by

B=

1

4

κk^20 cτ, κis as in (4.6.24).

SincegwQw=−gwJ 0 wis the weak charge density, we have

Qw=β δ(r),

andβis a scaling factor. We can take proper unit forgssuch thatβ=1. Thus, putting
Qw=δ(r)in (4.6.26) we derive that

(4.6.27) −∆Φw+k^21 Φw=gwδ(r)+
gwB
ρw

1

r

e−k^0 r.

Let the solution of (4.6.27) be radially symmetric, then the equation (4.6.27) can be equiva-
lently written as

(4.6.28) −

1

r^2

d

dr

(

r^2

d

dr

)

Φw+k^21 Φw=gwδ(r)+

gwB

ρw

1

r

e−k^0 r.

The solution of (4.6.28) can be expressed as

(4.6.29) Φw=

gw

r

e−k^1 r−

gwB

ρw

φ(r)e−k^0 r,

whereφ(r)satisfies the equation

(4.6.30) φ′′+ 2

(

1

r

−k 0

)

φ′−

(

2 k 0

r

+k^21 −k^20

)

φ=

1

r

.

Letφbe in the form

φ=

∞

∑

n= 0

βnrn (the dimension ofφ isL).

Insertingφinto (4.6.30), comparing coefficients ofrn, we get

β 1 =k 0 β 0 +

1

2

,

β 2 =

1

2

(k^20 −k^21 )β 0 +

1

3

k 0 ,

..

.

βn=

2 k 0

n+ 1

βn− 1 −(k^20 −k^21 )βn− 2 , n≥ 2.