4.2. PHYSICAL SUPPORTS TO PID 197
which are the variational equations of the Yang-Mills action
(4.2.22) LYM=
∫
−1
4
Fμ νaFμ νa+LFdx,whereLFis as in (4.2.18) withDμ=∂μ+igAaμτa, and
∂μFμ νa =−Aaν−∂ν(∂μAaμ)+o(A).Thus, the gauge field equations (4.2.21) are reduced to the bosonic field equations (4.2.20).
In other words, the gauge fieldsAaμ( 1 ≤a≤N^2 − 1 )satisfying (4.2.21) representN^2 − 1
massless bosons with spin-1 because eachAaμis a vector field.
We are now in position to introduce the Higgs mechanism. Physical experiments show
that the weak interacting fields should beSU( 2 )gauge fields with masses, representing 3
massive bosonic particles. However, as mentioned in (4.2.21), the gauge fields satisfying
theSU( 2 )Yang-Mills theory are 3(=N^2 − 1 )massless bosons. To overcome this difficulty,
(Higgs, 1964 ;Englert and Brout, 1964 ;Guralnik, Hagen and Kibble, 1964 ) suggested to add
a scalar fieldφinto the Yang-Mills action (4.2.22) to create masses. In fact, we cannot add a
massive termmAaμAμainto the Yang-Mills action (4.2.22); otherwise this action will violate
the gauge symmetry. But the Higgs mechanism can ensure the gauge invariance for the Yang-
Mills action, and spontaneously break the gauge symmetry infield equations at a ground state
of the Higgs fieldφ.
For clearly revealing the essence of the Higgs mechanism, weonly take one gauge field
(there are four gauge fields in the GWS theory). In this case, the Yang-Mills action density is
in the form
(4.2.23) LYM=−
1
4
gμ αgν β(∂μAν−∂νAμ)(∂αAβ−∂βAα)+ψ(iγμDμ−m)ψ,wheregμ νis the Minkowski metric, and
(4.2.24) Dμψ= (∂μ+igAμ)ψ.
It is clear that the action (4.2.23) is invariant under the followingU( 1 )gauge transformation
(4.2.25) ψ→eiθψ, Aμ→Aμ−
1
g∂μθ.The variation equations of (4.2.23) are
(4.2.26)
Aμ+∂μ(∂νAν)+gJμ= 0 ,
(iγμDμ−m)ψ= 0 ,
Jμ=ψ γμψ,which are invariant under the gauge transformation (4.2.25). It is clear that the bosonic parti-
cleAμin (4.2.26) is massless.
To generate mass forAμ, we add a Higgs sectorLHto the Yang-Mills action (4.2.23):
(4.2.27)
LH=−
1
2
gμ ν(Dμφ)†(Dμφ)+1
4
(φ†φ−ρ^2 )^2 ,Dμφ= (∂μ+igAμ)φ,
(Dμφ)†= (∂μ−igAμ)φ†,