4.6. WEAK INTERACTION THEORY 239
Note that the dimensions ofBandβ 0 inφ(r)are 1/LandL. The parameterB=Bβ 0 is
dimensionless. Physically, we can only measure the value ofB. Therefore we takeφin its
second-order approximation as follows
φ=β 0 ( 1 +k 0 r).In addition, by (4.6.16) we take
k 1 =k, k 0 = 2 k, k= 1016 cm−^1.Thus, the formula (4.6.29) is written as
Φs=gwe−kr[
1
r−
B
ρw( 1 + 2 kr)e−kr]
.
This is the weak interaction potential of a weakton, which isas given in (4.6.19).
In the same fashion as used in the layered formula (4.5.39) for the strong interaction, for
a particle with radiusρandNweak chargesgw, we can deduce the layered formula of weak
interaction potentials as in the form given by (4.6.17).
4.6.3 Physical conclusions for weak forces
As mentioned earlier, the layered weak interaction potential formula (4.6.17) plays the same
fundamental role as the Newtonian potential for gravity andthe Coulomb potential for elec-
tromagnetism. Hereafter we explore a few direct physical consequences of the weak interac-
tion potentials.
- Short-range nature of weak interaction. By (4.6.17) it is easy to see that for all
particles, their weak interaction force-range is as
r=1
k= 10 −^16 cm,which is consistent with experimental data.
2.Weak force formula. For two particles with radiiρ 1 ,ρ 2 , and withN 1 ,N 2 weak charges
gw. Their weak charges are given by
(4.6.31) gw(ρj) =Nj
(
ρw
ρj) 3
gw forj= 1 , 2.The weak potential energy for the two particles is
(4.6.32) V=gw(ρ 1 )gw(ρ 2 )e−kr
[
1
r−
B
ρ( 1 + 2 kr)e−kr]
,
wheregw(ρ 1 )andgw(ρ 2 )are as in (4.6.31),ρis a radius depending onρ 1 andρ 2 , andBis a
constant depending on the types of these two particles.