# 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

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