Pseudoscalar fermion mass Is it possible to add to a lagrangian a pseudoscalar mass term for the fermion: 
$$i M \bar{\psi} \gamma_5 \psi$$
The $i$ makes it hermitian. Would this cause any inconsistency in the field theory? If not, how would such a fermion differ from a regular fermion? For example, how would the propagator look like?
Obviously this term breaks parity, but I don't see what it may not be included in a parity violating theory.
 A: In fact, as long as you consider just the kinetic term plus your pseudo-mass term, there is no violation of parity. Indeed, your mass term can be written as 
$$
iM\bar{\psi}_L\psi_R+\mathrm{h.c.}
$$
and one can clearly re-absorbe the $i$ into, say, $\psi_R$ that is $\psi_R\rightarrow -i\psi_R$  given that the kinetic term is left invariant under this transformation. So, despite the look, the mass you wrote is just a genuine Dirac masss term that respects parity. 
A: I was thinking about a similar questions these days. I am not an expert in the field, but my conclusion is that is theoretically possible, at least at first sight, as an effective Yukawa interaction. I am not sure if it will introduces inconsistencies. 
A quick review of my thoughts: A Lagrangian is a scalar that must respect the symmetries of the spacetime, i.e: must be Lorentz-invariant and must respect CPT invariance. This leads to consider $\mathcal{L}=\mathcal{L}^\dagger$. 
To answer the question properly lets generalize and consider a lagrangian for fermions $\psi$ of mass $M_\psi$ with two types of Yukawa interaction: a scalar interaction with the field $\phi$ and a scalar interaction with the field $\rho$. Consider those two fields real, for sake of simplicity.
$$
\mathcal{L}=\mathcal{L}_{Dirac} + \mathcal{L}_{KG, \phi} + \mathcal{L}_{KG, \rho} + a\phi\bar{\psi}\psi + b\rho\bar{\psi}\gamma^5\psi
$$
Where, $\mathcal{L}_{Dirac}$ is the usual kinetic term for fermions and $\mathcal{L}_{KG}$ the usual kinetic term for mesons $\phi, \rho$.
In order to achieve $\mathcal{L}=\mathcal{L}^\dagger$ is required that $a\in\mathbb{R}$ and $b=ic, c\in\mathbb{R}$. We have a Yukawa-type interaction for the scalar messon with coupling $a$ and for the pseudoscalar meson of coupling $ic$. Consider those mesons to have mass $M_\phi$ and $M_\rho$ respectively. If we restrict ourselves to energies much lower than $M_\rho$ or $M_\phi$ we obtain an effective theory where the fields $\phi,\rho$ are non-dynamic, and we can set the value of those $a\phi=\Phi>0$ and $b\rho=iP$. 
Then you achieve an effective theory of a fermion with a scalar mass term $(M_\psi+\Phi)\bar{\psi}\psi$ and a pseudoscalar mass term $iP\bar{\psi}\gamma^5\psi$ that violates parity. 
