Mathematical Principles of Theoretical Physics

4.6. WEAK INTERACTION THEORY 245

1. The fields in the electroweak theory are as follows:
   SU( 2 )gauge fields: Wμ^1 ,Wμ^2 ,Wμ^3 ,
   U( 1 )gauge field: Bμ,

Dirac spinor doublets: L=

(

νL

lL

)

,

Dirac spinor singlet: R=lR,

Higgs scalar doublet: φ=

(

φ+

φ^0

)

,

whereνLandlLare the left-hand neutrino and lepton,lRis the right-hand lepton,(φ+,φ^0 )
are scalar fields with electric charge (1,0).

2. The Lagrangian action of the electroweak theory is given by

(4.6.55) LWS=LG+LD+LH,

whereLGis the gauge sector,LDis the Dirac sector, andLHis the Higgs sector:

(4.6.56)

LG=−

1

4

Wμ νaWμ νa−

1

4

Bμ νBμ ν,

LD=iLγμDμL+iRγμDμR,

LH=

1

2

(Dμφ)†(Dμφ)+

λ

4

(φ†φ−a^2 )^2 −Gl(LφR+Rφ†L).

Hereλ,a,Glare constants,

(4.6.57)

Wμ νa =∂μWνa−∂νWμa+g 1 εbcaWμbWνc,

Bμ ν=∂μBν−∂νBμ,

DμR= (∂μ+ig 2 Bμ)R,

DμL= (∂μ+i

g 2

2

Bμ−i

g 1

2

Wμaσa)L,

Dμφ= (∂μ−i

g 2

2

Bμ−i

g 1

2

Wμaσa)φ,

g 1 ,g 2 are coupling constants ofSU( 2 )andU( 1 )gauge fields, andσa( 1 ≤a≤ 3 )are the
Pauli matrices.

3. The action (4.6.55)-(4.6.57) is invariant under the followingSU( 2 )andU( 1 )gauge
   transformations:

• SU( 2 )gauge transformation:

L→e

i

2 θaσaL,

φ→e

i 2 θaσa

φ,

R→R,

Wμa→Wμa−

2

g 1

∂μθa+εbcaθbWμc, i.e. as in (2.4.38),

Bμ→Bμ.