The photon field is the non chiral piece of SU(2)xU(1), independently of symmetry breaking or not, isn't it?

But before symmetry breaking, each gauge boson has only a chiral gaugino as superpartner. Is it still possible, and correct, to arrange two of them in order to form an electrically charged "Dirac" fermion, able to couple to the photon field?

Probably this is a textbook question, but all the textbooks I have seed do first break susy, then electroweak, then this start point is never seen.

  • $\begingroup$ To clarify: I am worried about the couplig gauge-gaugino, and perhaps including the kinetics. The doubt comes because I can not see how a non-Dirac fermion could interact via the exchange of a photon, so I wondered if two gauginos in a massless SU(2)xU(1) theory need to be arranged to form a Dirac fermion. $\endgroup$ – user135 Oct 1 '11 at 19:50

No, the photon is not "the non-chiral piece" of $SU(2) \times U(1)_Y$ before symmetry breaking. The photon is the $SU(2) \times U(1)_Y$ gauge boson that is invariant under $Q_{\rm elmg}$, the electric charge, which is given by $$ Q_{\rm elmg} = \frac{Y}{2} + T_3 $$ where the first term is the hypercharge, the generator of $U(1)_Y$, and the second term is the $z$-component among the three generators of the $SU(2)$ factor of the gauge group. The photon is the gauge boson which is preserved by $Q_{\rm elmg}$ i.e. carries $Q_{\rm elmg}=0$, much like the Higgs condensate which also carries $Q_{\rm elmg}=0$; that's what makes the combination $Q_{\rm elmg}$ special (this is what makes the Higgs condensate electrically neutral and the corresponding gauge field, the electromagnetic field, remains a long-range force mediated by a massless particle) which shows that the particular combination is only picked as preferred by the symmetry breaking.

Obviously, one would get equivalent physics if $T_3$ were replaced by the component in any other direction of the 3-dimensional isospin space but the Higgs condensate would have to have the same new direction as well. Electromagnetism inevitably involves non-chiral physics because the chiral couplings only appear at and above the electroweak scale (the Higgs vev). If you assume that the lightness of the Higgs is explained naturally, this rule may be understood as well: all the masses above the electroweak scale are cancelled by some chiral physics when they contribute to the Higgs mass via loops; the lower-mass contributions don't have to be canceled so they may be non-chiral.

The parity-symmetric nature of $U(1)_{\rm elmg}$ (and all conjugate groups etc.) may also be seen from the grand unified starting point. One may see that the 2-spinors in GUT (or Pati-Salam) representations come in pairs with the same electric charge. What you seem to be missing is that the chirality or non-chirality of the gauge interactions is mostly a property of the fermions in the cubic vertex, not just the gauge boson. Interactions and dynamics of gauge bosons themselves is always non-chiral in 4 dimensions; only the fermions may bring the chirality.

Concerning the photino, there are several misconceptions underlying your description. First of all, photino is a neutral particle so it is more appropriate to think about it as a real Majorana fermion, not a chiral Weyl fermion. Second, and this is related, such Majorana fermions generically have masses that don't allow them to be combined into Dirac fermions with a uniform mass: the eigenvalues of the two Majorana pieces in the would-be Dirac fermion almost always differ.

Third, and this is related to the second point, it is unnatural to talk about photino because the kinetic and mass terms are generally mixed with other electrically neutral spin-1/2 superpartners, namely the "zino" and the (two) neutral Higgsinos. One must look at these four Majorana spinors, there are general mass terms and one may derive 4 different Majorana neutralinos out of them with 4 different mass eigenvalues.

  • $\begingroup$ The question does not even mention the photino; the description could give opportunity to some misconceptions, and it is good that you alert about them. But I asked for the "SU(2)xU(1) gauginos", and I was really more interested about if any of these, before symmetry breaking -and thus still massless, all of them-, could interact via an exchange of photons. $\endgroup$ – user135 Oct 1 '11 at 17:14
  • $\begingroup$ Dear @Alejandro, electrically charged particles interact via the exchange of a photon. Electrically neutral, $Q=0$, particles have a vanishing cubic coupling to the photons. This statement is totally independent of any supersymmetry or electroweak symmetry. So if you're asking about the coupling of particles to the photon field, your whole discussion of superpartners and the electroweak symmetry is just an irrelevant distraction. Your combination of these topics at different levels of complexity simply has no discernible logic; it seems just as a meaningless sequence of buzzwords. $\endgroup$ – Luboš Motl Oct 1 '11 at 18:56
  • $\begingroup$ I am interested in the particular case gauge-gaugino, where the vertex comes from the purely supersymmetric theory, no the gauge-matter vertex. I am sorry you have misread the question, but you could have asked in the comment section before entering an answer that you need to defend agressively (yep, I perceive your previous comment as agressive). Even more constructively, could you try to produce some questions yourself? We need new questions to pass the beta phase. $\endgroup$ – user135 Oct 1 '11 at 19:53
  • $\begingroup$ Dear @Alejandro, gauge vertices (photon-A-A) that come from a supersymmetric theory are the same vertices as in a nonsupersymmetric theory. Supersymmetric theories are just special examples of theories. Concerning the questions, I really don't believe this server needs questions like this one, and I wanted to ask you to refrain from asking a large number of questions of this type and quality. $\endgroup$ – Luboš Motl Oct 1 '11 at 20:48
  • $\begingroup$ Dear @Lubos, from the look of the discussion, I could agree that the question can be decomposed in simpler ones. As a minimum, it seems that I am mixing at least three questions: the origin of the gaugino-gauge vertex, the chirality of the gauginos, and the non-chirality of the electromagnetic sector. $\endgroup$ – user135 Oct 4 '11 at 11:24

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