Mathematical Principles of Theoretical Physics

16 CHAPTER 1. GENERAL INTRODUCTION

1.6 Theory of Strong Interactions

The strong interaction is responsible for the nuclear forcethat binds protons and neutrons
(nucleons) together to form the nucleus of an atom, and is theforce that holds quarks together
to form protons, neutrons, and other hadron particles. The current theory of strong interaction
is the quantum chromodynamics (QCD), based on anSU( 3 )non-Abelian gauge theory.
Two most important observed basic properties of the strong interaction are the asymptotic
freedom and the quark confinement. The theoretical understanding of these properties are still
lacking. There have been many attempts such as the lattice QCD, which is developed in part
for the purpose of understanding the quark confinement.
Modern theory of QCD leaves a number of key problems open for along time, including

1) (Quark confinement) Why is there no observed quark?

2) Why is there asymptotic freedom?

3) What is the strong interaction potential?

4) Can one derive the Yukawa potential from first principles?

5) Why is the strong force short-ranged?

These are longstanding problems, and the new field theory forstrong interaction based on
PID and PRI completely solves these open problems. We now present some basic ingredients
of this new development, and we refer the interested readersto (Ma and Wang,2014c) and
Section4.5for details.

PIDSU( 3 )gauge field equations

The strong interaction field equations decoupled from the unified field model are

Gk js

[

∂μSμ νj −gsΛcdjgα βScα νSdβ

]

−gsQνk=

[

∂ν+δk^2 Skν−

1

4

m^2 sxν

]

(1.6.1) φks,

[

iγμ

(

∂μ+igsSkμτk

)

−mq

]

(1.6.2) ψ= 0.

1.Gluons and dual scalar gluons.For the strong interaction, the field equations induce
a duality between the eight spin-1 massless gluons, described by theSU( 3 )gauge fields
{Skμ|k= 1 ,···, 8 }, and the eight spin-0 dual gluons, described by the dual fields{φks|a=
1 ,···, 8 }. As we shall see from the explianation of quark confinements gluons and their dual
gluons are confined in hadrons.

2.Attracting and repulsive behavior of strong force. As indicated before and from the
strong interaction potentials derived hereafter, strong interaction can display both attracting
and repulsive behavior. The repulsive behavior is due mainly to the spin-1 gluon fields, and
the attraction is caused by the spin-0 dual gluon fields.

4.5 Strong Interaction Potentials

Different from gravity and electromagnetic force, strong interaction is of short-ranged
with different strengths in different levels. For example,in the quark level, strong interaction