Mathematical Principles of Theoretical Physics

218 CHAPTER 4. UNIFIED FIELD THEORY

1.Duality between W±,Z Bosons and Higgs Bosons H±,H^0 .The three massive vector
bosons, denoted byW±,Z^0 , has been discovered experimentally. The field equations (4.4.42)
give rise to a natural duality:

(4.4.47) Z^0 ↔H^0 , W±↔H±,

whereH^0 ,H±are three dual scalar bosons, called the Higgs particles. The neutral HiggsH^0 ,
discovered by LHC in 2012, and the charged HiggsH±, to be discovered experimentally.
In Section4.6, we shall introduce the dual bosons (4.4.47) by applying the field equations
(4.4.42)-(4.4.45) with gauge equations (4.4.46).

2.Weak force.If we consider the weak interaction force, we have to use the equations
decoupled from (4.3.31)-(4.3.35):

∂νWν μa −

gw

̄hc

εbcagα βWα μbWβc−gwJaμ=

[

∂μ−

1

4

k^2 wxμ+

gw

̄hc

γWμ

]

(4.4.48) φwa,

∂μ∂μφwa−k^2 φw^2 +

gw

hc ̄

γ ∂μ(Wμφwa)−

1

4

(4.4.49) k^2 xμ∂μφwa

=−gw∂μJaμ−

gw

hc ̄

εbcagα β∂μ(Wα μbWβc),

iγμ(∂μ+i

gw

̄hc

Wμaσa)ψ−

mc

̄h

(4.4.50) ψ= 0 ,

whereγ,kware constants,Wμis as in (4.3.41).
In Section4.6, we shall deduce the layered formulas of weak interaction potentials by
applying the equations (4.4.48)-(4.4.50).

Remark 4.17.The duality of four fundamental interactions is very important, and is a direct
consequence of PID. With this duality, and with the PRI invariant potentialsWμandSμgiven
by (4.3.41), we obtain explanations for a number of physical problems such as the dark matter
and dark energy phenomena, the weak and strong forces and potential formulas, the quark
confinement, the asymptotic freedom, the strong potentialsof nucleons etc. Also this study
leads to the needed foundation for the weakton model of elementary particles and the energy
level theory of subatomic particles, and gives rise to a mechanism for subatomic decays and
scatterings.

4.5 Strong Interaction Potentials

4.5.1 Strong interaction potential of elementary particles

Thanks to PRI, the strong interaction potential takes the linear combination of the eightSU( 3 )
gauge potentials as follows

(4.5.1) Sμ=ρkSkμ,

whereρk= (ρ 1 ,···,ρ 8 )is theSU( 3 )tensor as given in (4.3.29).
Letgsbe the strong charge of an elementary particle, i.e. thew∗weakton introduced in
Chapter 5 , and
Φ 0 =S 0 the time-component of (4.5.1)