Correction: Majorana fermions and chiral symmetry
10 March 2008: Correction again. This post is also incorrect. I was bad and was using sloppy notation for my Majorana conjugate fermions. The correct definition is that a Majorana fermion satisfies . (Even this is kind of heuristic notation, but let’s leave specifics for another time… or perhaps to Wess and Bagger.) One can think of a Majorana fermion as a Dirac fermion with restrictions. In doing so, one can write down a Majorana 4-spinor (which is a Dirac spinor with restrictions) and then proceed to write the usual Dirac Lagrangian. All the man while, of course, one keeps in mind that the Majorana condition holds.
However, it is then manifest that the terms in the Majorana fermion Lagrangian transform in exactly the same way as the Dirac Lagrangian. And hence the application of chiral symmetry to protect the mass term from divergences works just as it did for the Dirac case.
Hence most of the things I’ve written below are incorrect. The most egregious error is my sloppy notation of writing the Majorana conjugate fermion as … what was I thinking???
Many thanks to those who discussed this with me today to fish out my errors.
[Original post below]
Piscator made a very good comment pointing out an error on my previous post. I had erroneously claimed that because Majorana mass terms are invariant under chiral-symmetry, they are not protected by chiral symmetry the same way Dirac masses are.
This appears not to be true. In fact, chiral symmetry does protect Majorana masses as well. This was used in the original split-SUSY paper (hep-th/0405159) to keep the gaugino masses light even though the SUSY breaking scale was pushed well above the TeV scale.
I couldn’t find a description of chiral symmetry for Majorana fermions in any textbooks, so here’s my own explanation — any further corrections would be especially appreciated. The point is that the quadratic part of the fermion Lagrangian contains a derivative term which contains a gamma matrix () and a mass term which does not:
Under an infinitesimal chiral transformation . Since anticommuting the past a gives us a minus sign, the derivative and mass terms will have opposite signs.
From a symmetry point of view, this is `bad’ because then the Lagrangian doesn’t transform nicely. This is fixed when we take the limit where the mass term vanishes. Then the [quadratic] Lagrangian only contains the derivative term, which gets an overall +/- sign. The Dirac conjugate fermion has a hidden in it, so the derivative-term-only Dirac Lagrangian gets an overall plus sign. On the other hand, the derivative-term-only Majorana Lagrangian gets an overall minus sign.
In highfalutin language we could say that under a chiral transformation the Dirac derivative term transforms as a scalar while the Majorana derivative term transforms as a pseudoscalar. Meanwhile, the Dirac mass term transforms as a pseudoscalar while the Majorana mass term transforms as a scalar. It’s not important whether the mass term is invariant (scalar), as I had mistakenly suggested in my last post, rather it is important whether the mass term transforms the same way as the derivative term—which it does not for either case.
So the key point is this: chiral symmetry protects all fermion masses because in the limit where the mass term goes to zero, the resulting derivative-term-only Lagrangian transforms as a well-defined representation of the chiral symmetry (scalar or pseudoscalar).
In light of this, one further amendment to my previous post is necessary: I had mistakenly said that because Majorana mass terms aren’t protected by chiral symmetry (we’ve shown here that they are), it makes sense that the Majorana mass term in the neutrino see-saw is much larger than the electroweak symmetry breaking scale. This is wrong. The reason why the Majorana mass term can be at such a different scale is simply that it has nothing to do with electroweak symmetry breaking and would have to be generated by a different mechanism living at the much higher scale. This has nothing to do with chiral symmetry.
Thanks to Piscator for pointing out this inconsistency of my last post! Any further corrections would be more than welcome.
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