Mathematical Principles of Theoretical Physics

224 CHAPTER 4. UNIFIED FIELD THEORY

which is the strong attraction radius of nucleons, and

Qkμ=qγμτkq (τk=τk).

Let the strong interaction nucleon potentialΦknand its dual potentialφnbe defined by

Φn=ρkSk 0 , φn=ρkφnk.

In the same spirit as for deriving the weakton potential equations (4.5.11) and (4.5.12), for
Φnandφnwe deduce that

(4.5.33)

−∆Φn=gsQn−

1

4

k^21 cτ 1 φn,

−∆φn+k^21 φn=gs∂μQμ,

wherecτ 1 is the wave length ofφn, and

Qμ=ρkQkμ= (Q 0 ,Q 1 ,Q 2 ,Q 3 )

represents the quark current density inside a nucleon. Similar to (4.5.15) and (4.5.19), forQn
and∂μQμwe have

(4.5.34) ∂μQμ=−
κn
ρn

δ(r), Qn=βnδ(r),

whereρnis the radius of a nucleon,βnandκnare constants, inversely proportional to the
volume of nucleons. Hence, in the same fashion as in deducing(4.5.25), from (4.5.33) and
(4.5.34) we derive the following strong nucleon potential as

(4.5.35) Φn=βngs

[

1

r

−

An

ρn

φ(r)e−k^1 r

]

,

whereφ(r)is as

(4.5.36) φ(r) = 1 +k 1 r+

r

2 α 0

+o(r), α 0 as in (4.5.24),

andAnis a dimensionless parameter:

An=

κnk^21 cτ 1

4 βn

.

Note thatβnis inversely proportional to the volumeVnof a nucleon. Hence we have

(4.5.37)

βn

β

=

NV 0

Vn

=N

(

ρw

ρn

) 3

(N= 3 ),

whereNis the number of strong charges in a nucleon,βis the parameter as in (4.5.25) and
(4.5.26), andV 0 the volume of aw∗-weakton. Sincegsβin (4.5.25) is taken asgs, by (4.5.37)
the formula (4.5.35) is expressed as

(4.5.38)

Φn=gs(ρn)

[

1

r

−

An

ρn

φ(r)e−knr

]

,

gs(ρn) = 3

(

ρw

ρn

) 3

gs,