198 CHAPTER 4. UNIFIED FIELD THEORY
whereρ 6 =0 is a constant. Obviously, the following action
(4.2.28) L=
∫
(LYM+LH)dx,and its variational equations
δL
δAμ=∂ν(∂νAμ−∂μAν)−gJμ−ig
2(φ(Dμφ)†−φ†Dμφ) = 0 ,δL
δ ψ(4.2.29) = (iγμDμ−m)ψ= 0 ,
δL
δ φ
=DμDμφ+ (φ^2 −ρ^2 )φ= 0 ,are invariant under the gauge transformation
(4.2.30) (ψ,φ)→(eiθψ,eiθφ), Aμ→Aμ−
1
g∂μθ.Equations (4.2.29) are still massless. However, we note that( 0 , 0 ,ρ)is a solution of
(4.2.29), which represents a ground state in physics, i.e. a vacuum state. Consider a transla-
tion forΦ= (A,ψ,φ)atΦ 0 = ( 0 , 0 ,ρ)as
Φ=Φ ̃+Φ 0 , Φ ̃= (A ̃,ψ ̃,φ ̃),then the equations (4.2.29) become
(4.2.31)
∂ν(∂νA ̃μ−∂μA ̃ν)−g^2 ρ^2 A ̃μ−gJ ̃μ+gJ ̃μ(φ ̃) = 0 ,
(iγμDμ−m)ψ ̃= 0 ,
DμDμ(φ ̃+ρ)− 2 ρ^2 φ ̃+ (φ ̃+ρ)φ ̃^2 = 0 ,where
J ̃μ(φ) =i
2(φ ̃(Dμφ ̃)†−φ ̃†Dμφ ̃).We see thatA ̃μ attains its massm=gρin (4.2.31), but equations (4.2.31) break the
invariance for the gauge transformation (4.2.30). The process that masses are created by the
spontaneous gauge-symmetry breaking is called the Higgs mechanism. Meanwhile, the field
φ ̃in (4.2.31), called the Higgs boson, is also obtain its massm=
√
2 ρ.
In the following, we show that PID provides a new mechanism for generating masses,
drastly different from the Higgs mechanism.
In view of (4.1.34) and (4.1.35), based on PID, the variational equations of the Yang-Mills
action (4.2.23) with the divA-free constraint are in the form
(4.2.32)
∂ν(∂νAμ−∂μAν)−gJμ=[
∂μ−1
4
(mc̄h) 2
xμ+λAμ]
φ,(iγμDμ−mf)ψ= 0 ,