228 CHAPTER 4. UNIFIED FIELD THEORY
whereλis a parameter,nf=6 is the flavor number of quarks, andq^2 is the transfer momen-
tum by the incidence electron.
In QCD, the coupling constantαsstands for the strength of the strong interactions be-
tween the quarks in a proton. It is clear that as the transfer momentumq^2 tends to infinitely
large,αstends to zero:
(4.5.52) αs→0 as q^2 →∞.
It means that the greater the incidence electron kinetics is, the more freedom the quarks
exhibit. The asymptotic freedom is based on the fact (4.5.51)-(4.5.52).
3.Strong interaction potential for asymptotic freedom. The strong interaction potentials
provide also a natural explanation for the asymptotic freedom phenomena. By (4.5.41) and
(4.5.44), the strong force between quarks is as
F=−gs(ρq)dΦq
dr
=g^2 s(ρq)[
1
r^2−
Aq
ρq
k^21 re−k^1 r]
,
which implies that there are two radiir 1 andr 2 such that
ρq<r 1 <r 1 <r 2 (r 1 = 1 /k 1 ),and the strong forceFsatisfies
(4.5.53) F
{
>0 for 0<r<r 1 ,
<0 forr 1 <r<r 2.By (4.5.53), we see that
F∼0 nearr=r 1.
This indicates that there is a free shell region inside a proton with radiusR≃r 1 , such that the
three quarks are free in this shell region.
When a lower energy electron collides with the proton, the electromagnetic force cause
the electron moving away, leading to the elastic scatteringas given by (4.5.49). However,
when a high speed electron collides with a proton, it can run into the inside of this proton,
interacting with one of the quarks. Since the quark was in a free shell region with no force
acting upon it, this particular quark will behavior as a freequark. As it moves into the
attracting region of the proton, the quark confinement will hold this quark, which, at the
same time, will collide with gluons, exchanging quarks, leading to the inelastic scatterings as
given in (4.5.50). This explains the asymptotic freedom.
Remark 4.20.The two phenomena of the quark confinement and the asymptoticfreedom
are consistent. On the one hand, the quarks in a proton are held tightly together by the strong
interaction to cause the quark confinement, and on the other hand, quarks look as if they are
free. The consistency is the direct consequence of the quarkpotential formula in (4.5.41).
4.5.5 Modified Yukawa potential
One of the mysteries of the strong interaction is the conflictcharacteristics exhibited in the
quark level and in the nucleon level. By the strong interaction potentials (4.5.41), we see that
(4.5.54) quark strong force is infinitely attractive atr= 10 −^16 cm;