Why is the term $\bar{L}\phi R$ invariant under $SU(2)_L \times U(1)_Y$ where $\phi$ is the Higgs field?
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In the explanation for the fermion masses, the term
L¯ϕRL¯ϕR
is added to the Lagrangian where LL is a left-handed SU(2)SU(2) doublet, ϕϕ is the Higgs left-handed SU(2)SU(2) doublet and RR is a SU(2) singlet. The L¯ϕRL¯ϕRis said to be invariant under SU(2)L×U(1)YSU(2)L×U(1)Ysymmetry gauge transformations. I understand that L¯ϕL¯ϕ is invariant under SU(2)LSU(2)L but if we include RR as above then under its gauge transformation under U(1)YU(1)Y, won't we be left with a term of the form eiYθ(x)eiYθ(x) that we cannot cancel hence breaking the gauge invariance
L¯ϕRL¯ϕR
is added to the Lagrangian where LL is a left-handed SU(2)SU(2) doublet, ϕϕ is the Higgs left-handed SU(2)SU(2) doublet and RR is a SU(2) singlet. The L¯ϕRL¯ϕRis said to be invariant under SU(2)L×U(1)YSU(2)L×U(1)Ysymmetry gauge transformations. I understand that L¯ϕL¯ϕ is invariant under SU(2)LSU(2)L but if we include RR as above then under its gauge transformation under U(1)YU(1)Y, won't we be left with a term of the form eiYθ(x)eiYθ(x) that we cannot cancel hence breaking the gauge invariance
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in the standard model, the left handed leptons have hypercharge −1/2−1/2, the right handed leptons have hypercharge −1−1, the left handed quarks have hypercharge 1/61/6, the right handed 'up' quarks have hypercharge 2/32/3, the right handed 'down' quarks have hypercharge −1/3−1/3, and the Higgs has hypercharge 1/21/2. Does this help? Arguably the trickiest term to write down is that coupling the Higgs to the right handed 'up' quarks to the left handed quarks .
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