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This article gives the potential energy function of quark in the gluon field, derives the wave function of quark in stationary state and the quark confinement and asymptotic freedom, shows that a quark is composed of two different color gluons, expounds the formation mechanism of the quark confinement and asymptotic freedom and the physical substance of “colors” of quark, and discusses the stability of hadrons in the end.

According to the analysis of experiments, quantum chromodynamics tends to think that quarks inside hadron have two important features―the quark confinement and asymptotic freedom [

Without missing generality, we discuss the proton as an example since the proton is the only stable hadron. Assume that there is a spherical symmetrical gluon field of the radius R in the proton where the gluons are point particles. In order to have the confinement and asymptotic freedom for the valence quarks

where r is the distance from the center to the quark

is the interactional mass of proton within the range of radius r, where the factor 3 is from that the quark has three kinds of “colors” in the proton whose mass is

is the interaction mass of quark and running, the constant

usually taken approximately as one third of nuclear mass;

dius of quark, and the constant

It is not difficult to see that from Equations (2) and (3) since

there are

The above two equations show the gluon field is within the proton and limited.

It is interesting that the mass distribution in the proton can tell us intuitively why there is the quark confinement and asymptotic freedom. From equation (2) we obtain the mass density in the proton

where

where

and it satisfies the normalization condition:

So the wave function of quark with flavor f and color c in stationary state is

where_{fc} is the free energy of quark and its potential energy

Thus the probability density of quark distribution

where

struction of proton is independent of energy of quarks, and this is commonly referred to as the scaling. From the

above equation we can know that the probability density

at the boundary and zero at the center, as shown in the

Substitute Equations (2) and (3) into Equation (1) which could be written as

where the constant

Take

Note

so the position of that the net force is zero

The function curve of the potential energy

It is not difficult to see that from Equation (13)

this is the quark confinement; and

this is the asymptotic freedom. Thus, we have derived both the quark confinement and the asymptotic freedom from Equation (1). In order to find out the physical mining of the quark confinement and asymptotic freedom, let us discuss the coupling coefficient of quark with the gluon field.

Rewrite the net force on quark (see the Equation (13)) into Coulomb type:

where

is the coupling coefficient of quark

this is identical with equation (15) in physical meaning and denotes the quark confinement, too. And

this shows that the net force on quark in the depths of gluon field is repulsion. The above equation is coincident

with Equation (16) in physical meaning, and the reason the Equation (16) equals zero is that

As shown in Equation (4) there is

this shows that the mass of quark turns out to be zero near the boundary of the proton. What does that imply? Since Equation (1), from which the quark confinement and asymptotic freedom have been derived, is credible and

We know the gluons have no mass according to the quantum gauge theory. Thereby, the logical explanation of the above equation is that when a quark enters the region of

Firstly, if we believe that quark consists of gluons, the gluons should be charged because quarks are charged. Otherwise, the conservation law of charge will be violated. In fact, quantum gauge theory did not conclude that gluons are not charged. It is inappropriate to equate the gluon and photon. However, a quark cannot be composed of two gluons with equal and opposite electric charges. If so, quark is electrically neutral.

Secondly, the gluons as quanta of non-Abelian gauge field should be fermions rather than bosons. In fact, because the quarks are fermions of spin 1/2, the gluons to constitute a quark should be fermions, too. Therefore, two gluons of the same color are repulsive to each other and cannot constitute a quark according to Pauli exclusion principle. The gluons to constitute a quark can only be different gluons in color.

From the above analysis we could put eight kinds of gluons into two groups according to the electric charge of gluons. The gluon group consists of four kinds

the anti-gluon group consists of four kinds

where the color index

Assuming that only the two different color gluons or anti-gluons can constitute a quark or anti-quark, the two groups constitute justly 6 kinds of quarks or anti-quarks shown in the following, respectively.

Quarks:

Anti-quarks:

If the electric charge or spin of a quark equals the sum of that of its two constituent gluons, the charge of 6 kinds of quarks is

where

and the spin of every quark or anti-quark is

The gluons and anti-gluons could annihilate into different particles in different interactional courses when they met. For example, the meson

According to the above composition rules of quarks, the formation mechanism of the quark confinement and asymptotic freedom could be explained clearly.

A quark composed of two different color gluons is repulsed by same quarks (or gluons) according to Pauli exclusion principle, and at the same time attracted due to the exchange of gluons with the rest quarks (or gluons). Since the number of the same gluons in color is only one eighth of the total gluons and anti-gluons, the probability of a quark to be attracted is greater than its probability to be repulsed in the general case. But when a quark is in the region of

As said previously, gluons are fermions. So a quark composed of two different color gluons has three kinds of wave functions, or three kinds of “colors”. If the probability of the quark in each “color” state is equal, the hadron observed is in the singlet color state, or colorless state. This could be proved in the following.

Since a quark

where

where

which is the number of states of that two spins equal

Substituting the above equation into Equation (28) and then into Equation (10), we see that the wave function

Notice the probability density

This is Equation (11) used previously, it shows that the valence quark

A lot of experiments show that the pseudoscalar mesons and baryons are unstable besides proton. According to the above composition rules of quarks, it is not difficult to see that from the equations (33) and (34) [

are unstable is that they contain gluons and anti-gluons annihilated each other in meeting, and the reason the baryons are unstable besides proton is that they contain the same gluons repulsive to each other. The reason the proton is stable is that it is a lightest baryon in

As for the so-called particles in

The above analysis shows that the potential energy function of quark and the assumption of that a quark which is composed of two different color gluons put forward in this article are in accordance with a lot of experiments, and so are reasonable and credible.