# Complex masses for Dirac and Weyl spinors

I'm trying understand how to rotate Dirac fields to absorb complex phases in masses. I have a few related questions:

1. With Weyl spinors, I understand, $$\mathcal{L} = \text{kinetic} + |M|e^{i\theta}\xi\chi + \textrm{h.c.}$$ The phase removed by separate left- and right-handed rotations, e.g. $\xi \to e^{-i\theta/2}\xi$ and $\chi \to e^{-i\theta/2}\chi$. These phases cancel in the kinetic terms.

Is it correct that with Dirac spinors, $$\mathcal{L} = \bar\psi |M|e^{i\theta\gamma_5} \psi = \text{Re}(M)\bar\psi\psi +i\text{Im}(M)\bar\psi\gamma_5\psi$$ and the phase is removed by $\psi \to e^{-i\theta\gamma_5/2}\psi$? The appearance of the $\gamma_5$ in the phase troubles me a little - I suppose this is telling us that Weyl spinors are a more suitable basis than Dirac spinors?

2. If the field is Majorana, $\xi = \chi$, and the field can still absorb a phase? I think I must be making a trivial mistake.

For example, Majorana neutrino fields cannot absorb phases, leading to extra CP violation.

And in SUSY, the gaugino Majorana soft-breaking masses are e.g. $M_1e^{i\theta}$. Can their phases be re-absorbed via a field redefinition? I don't think they can. So I must have a mistake.

• The appearance of $\gamma_5$ signifies that you're doing an axial transformation. In theories with an axial anomaly, such transformations are not a symmetry of the quantum theory, so I wonder if you're not allowed to do that. – Siva May 23 '13 at 21:46
• @Siva, you are allowed. At an extra price, however. E.g. one way to solve the Schwinger model is to cancel the interaction between $A$ and $\psi$ by making approriate chiral transformation, and poperly accounting for the anomaly. – Peter Kravchuk May 23 '13 at 21:51
• @Siva Yes, this will ultimately lead to strong CP/chiral anomaly, something I want to eventually understand... – innisfree May 23 '13 at 21:53
• @innisfree, the equality involving $\gamma_5$ is not clear to me, where the $\theta$ has gone? Also, why do you write the 'complex phase in mass' with $\gamma_5$ at the very beginning? – Peter Kravchuk May 23 '13 at 21:55
• @PeterKravchuk $Me^{i\theta\gamma_5} = M\cos\theta + Mi\gamma_5\sin\theta = \text{Re}(M)\ldots$, because $\gamma_5^2=1$. I wrote it with $\gamma_5$ in at the beginning because of something I read in Dine's Supersymmetry book. – innisfree May 23 '13 at 22:04

Having looked at a few more sources, I think I know the answer now, but please do correct me if I'm wrong.

For 1., I still find the appearance of the $\gamma_5$ matrix in Dirac spinor description surprising. I suppose the resolution is that the Weyl spinors are a more intuitive description in some cases.

For 2., I think the source of my confusion is some ambiguous statements in the literature. I've heard that Majorana fields "cannot absorb a phase," which I've misunderstood to mean cannot absorb a phase from a complex mass term, whereas it means that Majorana fields must be singlets in the fundamental representation/must have no non-zero quantum numbers, because the symmetry would violate the Majorana condition itself.

The statement that Majorana fields cannot absorb phases from complex masses is true only in certain circumstances, namely, if there are interaction terms in the Lagrangian which are not invariant under the field redefinition. In the case of the Majorana neutrinos, because one no longer has the freedom to rotate right-handed fields only to avoid complications with left-handed weak interactions, fewer phases can be absorbed.

For the gauginos, I think there are F-terms, which prevent the total removal of the phases, leaving physical phases e.g. $\text{Arg}(M_i \mu)$.