SU(2) in Standard Model and SUSY extensions

[RedX]

If you have doublet Q=(u,d), and want to give the u-quark mass, you have to connect it to the Higgs VEV H=(\nu,0) doublet through the adjoint opertion:

H^{\dagger i}Q_i

Connecting H and Q through the Levi-Civita symbol e_{ij}:

e^{ji} H_{ i}Q_j

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 H^{\dagger} would have hypercharge +1/2. In supersymmetry, instead of H^{\dagger}, 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?

 

[Avodyne]

So does it make sense to raise H by taking the complex conjugate representation instead of using the Levi-Civita symbol?

Yes. Suppose we had SU(N) instead of SU(2); then clearly (H_i)^\dagger H_i should be SU(N) invariant. So we must define hermitian conjugation as raising the index: (H_i)^\dagger = (H^\dagger)^i.

Now, for SU(2), we can also raise indices with the Levi-Civita symbol, so we could define a different object {\cal H}^i = \varepsilon^{ij}H_j. Can we take this to be the same as H^{\dagger i}? The answer is no. Suppose we try a relation of the form {\cal H}^i = \eta H^{\dagger i}, where \eta is a numerical factor. You should be able to show that both components of this equation hold if and only if |\eta|^2 = -1, which is not possible. So \varepsilon^{ij}H_j and H^{\dagger i} must be different objects.

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?

No. The theory is supersymmetric if and only if the superpotential is a function of left-chiral fields only. That's why a second Higgs field must be introduced in supersymmetric theories.

 

[RedX]

What's really confusing is eqn. (97.11) in Srednicki. Basically, it defines as the complex anti-fundamental representation of SU(5): \psi^i=(\overline{d^r} , \overline{d^b} , \overline{d^g} , e, -\nu).

The last part, the one that transforms under the SU(2) subgroup, (e,-\nu), is just the normal SU(2) doublet \psi_i=(\nu, e) raised with the Levi-Civita.

So the complex anti-fundamental representation of SU(5) has an unbroken fundamental non-complex representation of SU(2).

Anyways, another thing that is annoying is it should be RGB, not RBG, in analogy to computer terminology.

 

[Avodyne]

It's common for a complex rep under a group G to have pieces (or even the whole thing) that are real or pseudoreal under a subgroup H. For example, under the SO(N) subgroup of SU(N), the fundamental rep N is real.