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The physical gauge fields are two charged fields, W ± , and two neutral gauge bosons, Z and γ . The masses of the gauge bosons are given by: 

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Since the massless photon must couple with electromagnetic strength, e , the coupling con- stants define the weak mixing angle θ$_{W}$ , e = g sin θ$_{W}$ and e = g ′ cos θ$_{W}$ . 

One can also include fermions, let us consider the electron and its neutrino as an example. The fermions in terms of their left-and right-handed projections, ψ$_{L,R}$ = 1 $_{2}$(1 ∓ γ$_{5}$ ) ψ . We need to couple the fermions with the SU (2)$_{L}$ doublets, therefore: 

The known matter content of the SM has the following charges: Q$_{L}$ = (3 , 2 , 1 / 6) , ( B = 1 / 3 , L = 0), u$_{R}$ = ( ¯ 3 , 1 , − 2 / 3) , ( B = − 1 / 3 , L = 0), d$_{R}$ = ( ¯ 3 , 1 , 1 / 3) , ( B = − 1 / 3 , L = 0), L$_{L}$ = (1 , 2 , − 1 / 2) , ( B = 0 , L = 1), e$_{R}$ = (1 , 1 , 1) , ( B = 0 , L = − 1), H = (1 , 2 , 1 / 2) , ( B = 0 , L = 0), using the notation ( n$_{3}$, n$_{2}$, Y ), with color SU (3), weak SU (2) and hypercharge U ( Y ), respectively. In MSSM, anomaly cancelation will require additional Higgs with an opposite hypercharge, (1 , 2 , − 1 / 2). 

Using the hypercharge assignments of the fields, the leptons can be coupled in a gauge invariant manner to the SU (2)$_{L}$ × U (1)$_{Y}$ gauge fields, 

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All of the known fermions can be accommodated in the Standard Model in an identical manner as was done for the leptons. 

A fermion mass term, L mass = − mψψ = − m ( ψ$_{L}$ψ$_{R}$ + ψ$_{R}$ψ$_{L}$ ), is not gauge invariant under SU (2)$_{L}$ and U (1)$_{Y}$ . The gauge invariant Yukawa coupling of the Higgs boson to the up and down quarks is given by, L d ∼ − λ$_{d}$Q$_{L}$Hd$_{R}$ + h.c. : 

which provides mass term for the down quark if λ$_{d}$ = $_{m$_{d}$}$√$_{2}$$_{/v}$$_{, while the up quark mass is}$ determined by: L u = − λ$_{u}$Q$_{L}$ Φ $^{c}$u$_{R}$ + h.c. , since fact that H c ≡ − iτ$_{2}$H ∗ ( τ is a Paui matrix) 

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