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I understand that Dirac masses for fermions are "technically natural" (in the 't Hooft sense of technical naturalness) because they break chiral symmetry. In limit where the Dirac mass goes to zero, chiral symmetry is restored, so one expects any loop corrections to the Dirac mass to be proportional to the Dirac mass itself. That is to say, there is no power law UV sensitivity in a Dirac mass parameter.

Is the same true for Majorana mass terms? Because the Majorana fermions are real, is it still true that the mass term breaks chiral symmetry? It seems so, since one may write a 4-component Majorana fermion in terms of a single Weyl fermion:

$\Psi_D = \begin{pmatrix} \psi_\alpha \\ \bar\psi^{\dot{\alpha}}\end{pmatrix}$,

where the $\alpha$ and $\dot{\alpha}$ are Weyl ($SL(2,C)$) indices. Then a Majorana mass term takes the form:

$M\bar\Psi_M \Psi_M = M(\psi^2 + \bar\psi^2)$,

where the Weyl spinor contractions use the $\epsilon$ tensor. These terms appear to break chiral symmetry as well.

Is this the correct picture?

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Because of the Majorana condition $\psi=\psi^C$, Majorana fermions are singlets with respect to gauge symmetries, including $SU(2)$. Furthermore, no chiral symmetry forbids a Majorana mass.

Thus the natural mass scale for Majorana fermions is the Planck mass.

The right-handed neutrino could be Majorana, with interesting implications. See the see-saw mechanism; the difference between the Planck and the electroweak scales results in the neutrino mass scale, $M_W^2/M_P$.

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  • $\begingroup$ I'm not so sure about this. I agree that the Majorana fermion is self conjugate and thus must be a gauge singlet, but chiral symmetry is a separate issue. Is it not true that in the absence of any mass term, the Majorana fermion has a U(1) symmetry, $\psi \to e^{i\alpha}\psi$? And Isn't it further true that this symmetry is broken by the Majorana mass term? Hence the mass term should be technically natural, no? (I should note that I'm writing $\psi$ as a Weyl spinor.) $\endgroup$ – Pengpeng Xu Jun 20 '13 at 1:32
  • $\begingroup$ I now think that I agree with you---there is a chiral symmetry (as noted above), but this is a global chiral symmetry. The global symmetry can be broken (in principle) by Planck scale gravitational effects. A diagrammatic way of seeing this is to draw the helicity arrows on the fermion line. A mass insertion flips the fermion chirality. Thus a left-helicity, left-chiral SU(2) doublet electron is flipped into the left-helicity component of a right-chiral SU(2) singlet electron. $\endgroup$ – Pengpeng Xu Feb 28 '14 at 23:28
  • $\begingroup$ [continuing] So an electron picks up a Higgs-vev induced mass. This mass term carries SU(2)xU(1) quantum numbers and these symmetries are protected even against gravitational corrections (e.g. black holes that can eat global symmetries). For the global chiral symmetry allowed to a Majorana fermion, on the other hand, one can imagine a "black hole" gravitational mass insertion which simply violates the global chiral charge and allows you to write a similar helicity flip. Heuristically (this is very hand-wavey, of course) we thus expect a Planck-scale quantum correction to the Majorana mass. $\endgroup$ – Pengpeng Xu Feb 28 '14 at 23:34
  • $\begingroup$ I now think this is wrong - don’t upvote it - what I write about chiral symmetry looks mistaken $\endgroup$ – innisfree Apr 10 at 14:55

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