6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 361
Remark 6.17.We need to explain that although each quark has three weak charges, due to
the weak force ranger≤ 10 −^16 cm and the distancer> 10 −^16 cm between the quarks in a
hadron, the main bound force of hadrons is strong interaction, and the weak forces between
the quarks an be ignored.
- Weak interaction potentials. In Section4.6.2, we deduced the weak interaction
potential (4.6.17). For convenience, here we again write this formulas as
(6.4.4)
W 0 =Ngw(ρ)e−r/r^0[
1
r−B
ρ(^1 +2 r
r 0 )e−r/r 0]
,
gw(ρ) =(
ρw
ρ) 3
gw,whereW 0 is the weak charge potential of a composite particle withNweak chargesgwand
with radiusρ, andρwis the weakton radius,Bthe weak interaction constants of this particle,
r 0 = 10 −^16 cm.
Based on (6.4.4), the weak potential energy generated by two particles withN 1 andN 2
weak charges and with radiiρ 1 ,ρ 2 is expressed as follows
(6.4.5) Vw=N 1 N 2 gw(ρ 1 )gw(ρ 2 )e−r/r^0
[
1
r−
B 12
ρ 12(
1 +
2 r
r 0)
e−r/r^0]
,
wheregw(ρ 1 )andgw(ρ 2 )are as in (6.4.4), andB 12 /ρ 12 depends on the types of the two
particles.
The basic weak chargegwsatisfies the relation (4.6.37), i.e.
(6.4.6) g^2 w= 5 × 10 −^3
(
ρn
ρw) 6
hc ̄.6.Strong interaction potentials. The strong interaction potential given by (4.5.39), reads
as
(6.4.7)
S 0 =gs(ρ)[
1
r−A
ρ(^1 +r
R)e−r/R]
,
gs(ρ) =(
ρw
ρ) 3
gs,whereRis as
R={
10 −^16 cm forw∗and quarks,
10 −^13 cm for hadrons.
By (6.4.7), the strong potential energy for two particles withN 1 ,N 2 strong charges and
with radiiρ 1 ,ρ 2 is given by
(6.4.8) Vs=N 1 N 2 gs(ρ 1 )gs(ρ 2 )
[
1
r−
A 12
ρ 12( 1 +
r
R)e−r/R]
,
wheregs(ρ 1 )andgs(ρ 2 )are as in (6.4.7), andA 12 /ρ 12 depends on the types of the two
particles.
The basic strong chargegssatisfies the relation (4.5.66), i.e.
(6.4.9) g^2 s= 2 × 10 −^2
(
ρn
ρw) 6
g^2 (g^2 ≃ 1 hc ̄).