If you have doublet [tex]Q=(u,d) [/tex], and want to give the u-quark mass, you have to connect it to the Higgs VEV [tex]H=(\nu,0)[/tex] doublet through the adjoint opertion:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]H^{\dagger i}Q_i [/tex]

Connecting H and Q through the Levi-Civita symbol [tex]e_{ij} [/tex]:

[tex]e^{ji} H_{ i}Q_j [/tex]

results in d-quark mass, not u-quark mass.

But SU(2) is special because it's pseudo-real, meaning that its complex conjugate representation is equivalent to the original representation. Or in other words, the adjoint of H is not unique from H. In the mathematical physics books, it says you don't have to worry about up or down indices in SU(2), because the Levi-Civita symbol, being 2-dimensional, can raise or lower stuff for you. So does it make sense to raise H by taking the complex conjugate representation instead of using the Levi-Civita symbol?

The H field has hypercharge -1/2 (this depends on convention but the convention I use is -1/2). So [tex]H^{\dagger} [/tex] would have hypercharge +1/2. In supersymmetry, instead of [tex]H^{\dagger} [/tex], two different Higgs field are defined. One Higgs field has hypercharge -1/2, and the other +1/2 hypercharge. This seems to be conceptually different from using the adjoint operation/complex representation to get a quantity with +1/2 hypercharge. In Srednicki's book, for example, the 3rd term of (96.1) is the same term as the 2nd term of 89.5, except a new Higgs field is used instead of the daggered Higgs field. I realize in supersymmetry that daggering a field has consequences such as changing a left chiral superfield into a right one, consequences absent in non-supersymmetric theories. But can't you build a superpotential out of both left and right chiral superfields, and use one Higgs field (and it's adjoint) instead of two separate Higgs fields?

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# SU(2) in Standard Model and SUSY extensions

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