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SU(2) in Standard Model and SUSY extensions

  1. Jun 24, 2009 #1
    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:

    [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?
  2. jcsd
  3. Jun 24, 2009 #2


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

    Now, for SU(2), we can also raise indices with the Levi-Civita symbol, so we could define a different object [itex]{\cal H}^i = \varepsilon^{ij}H_j[/itex]. Can we take this to be the same as [itex]H^{\dagger i}[/itex]? The answer is no. Suppose we try a relation of the form [itex]{\cal H}^i = \eta H^{\dagger i}[/itex], where [itex]\eta[/itex] is a numerical factor. You should be able to show that both components of this equation hold if and only if [itex]|\eta|^2 = -1[/itex], which is not possible. So [itex]\varepsilon^{ij}H_j[/itex] and [itex]H^{\dagger i}[/itex] must be different objects.
    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.
  4. Jun 26, 2009 #3
    What's really confusing is eqn. (97.11) in Srednicki. Basically, it defines as the complex anti-fundamental representation of SU(5): [tex]\psi^i=(\overline{d^r} , \overline{d^b} , \overline{d^g} , e, -\nu)[/tex].

    The last part, the one that transforms under the SU(2) subgroup, [tex] (e,-\nu) [/tex], is just the normal SU(2) doublet [tex]\psi_i=(\nu, e)[/tex] 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.
  5. Jun 29, 2009 #4


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    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.
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