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 \chi satisfies \chi = \chi^c = i\sigma^2 \chi^*. (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 \psi^\dag … 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 (\gamma^\mu \partial_\mu) and a mass term which does not:

Dirac: \bar\psi (\gamma^\mu\partial_\mu + m_D) \psi
Majorana: \psi^\dag (\gamma^\mu\partial_\mu + m_M) \psi

Under an infinitesimal chiral transformation \psi \rightarrow (1+\alpha \gamma_5)\psi. Since anticommuting the \gamma_5 past a \gamma^\mu 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 \bar\psi has a \gamma^0 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.

5 Responses to “Correction: Majorana fermions and chiral symmetry”

  1. In order to complete the topic, perhaps we’d need a modern version (a highfalutin version?) of the argument “a majorana fermion, if massive, must be neutral”. What does “neutral” mean? To be C (and then PT) invariant? To do not couple with any vector current?
    Another subtopic, if susy is to enter, is the concept of Chiral Superfield, as in Weinberg vol III page 69.

  2. 2 Mark

    What you haven’t said is why we should care about respecting chiral symmetry… I look forward to the upcoming post on anomalies!

  3. Hi Mark! The point of respecting chiral symmetry was in my previous post. One can use chiral symmetry (even if it is broken by mass terms) to explain why there is no linear divergence in the fermion mass term. That is to say that chiral symmetry prevents us from having a “fermion hierarchy problem” where we expect the fermion masses to live at the scale of new physics. (Is there a deeper reason why chiral symmetry is important? You string theorists seem to make a big deal about it from time to time…)

    I’m still poking around for something interesting to say about anomalies. Maybe in the next few weeks. 🙂

    Hope all is well in France,

  4. New update: please note my corrections to this post (yes, they’re corrections to a correction). The bottom line is this: chiral symmetry protects fermion masses whether they are Dirac or Majorana. Most of what I’ve said about Majorana fermions is either naive or wrong… but in a nutshell, one can write a Majorana fermion as a “Dirac fermion with restrictions,” and hence the Lagrangian takes the same form and the protection works in the same way. Sorry for all the confusion!

  5. 5 Mark

    Hi Flip, France is fine I think, or at least I am. I hope Durham is warming up. Anyway, the reason I mentioned the above is that we might not care that chiral symmetry is preserved – why can’t we just put in an arbitrary mass term for the fermions and forget it? Of course, we can’t do so without breaking gauge invariance – and that’s also why we have to worry about chiral anomaly cancellation… sorry, I was just keen to throw my tuppence in!

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