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Another stupid question: Consider a Yukawa coupling [itex]\lambda \bar{\psi}_1 \psi_2 \phi[/itex] where [itex]\phi[/itex] is a scalar field in the [itex](2,-\frac{1}{2})[/itex] representation and [itex]\psi_1[/itex] and [itex]\psi_2[/itex] are lh. Weyl fields in the [itex](2,-\frac{1}{2})[/itex] and [itex](1,1)[/itex] representation of [itex]\mathrm{SU}(2) \times \mathrm{U}(1)[/itex]. Why does the occurrence of the singlet [itex](1,0)[/itex] on the rhs of

[tex](2,-\frac{1}{2}) \otimes (2,-\frac{1}{2}) \otimes (1,1) = (1,0) \oplus (3,0)[/tex]

imply that this term is gauge-invariant? What about the [itex](3,0)[/itex] part? I just can't see it.

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# Singlets and gauge invariance

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