Spinor inner products The spinor inner product in particle physics is given by $\overline{\psi} \psi = \psi^{\dagger} \gamma_0 \psi $, where I take the convention that the zeroth gamma matrix is hermitian while the rest are anti-hermitian. This is invariant under spin group transformations, $\psi \rightarrow e^{\omega_{ab} S^{ab} }\psi$, with $\omega_{ab}$ real parameters and $S^{ab} = \frac{1}{4}[\gamma^a, \gamma^b]$. 
However, there is a second invariant inner product given by $\psi^{\dagger} \gamma_0 \gamma_5 \psi$, with $\gamma_5 = i\gamma_0 \gamma_1 \gamma_2 \gamma_3$. My question is, why not the other one? Is it down to the fact that there are observable differences between the two, and one is favoured by experiment?
 A: Well the $\gamma^5$ term does appear in the standard model.
As you duly note, both $\bar{\psi}\psi$ and $\bar{\psi} \gamma^5 \psi$ are Lorentz invariant, so the question to ask is, how are they different?
It turns out that the difference lies in their transformations under a Parity transformation. Without proof, I claim that under $P:(x,y,z) \to (-x,-y,-z)$, $\bar{\psi}\psi \to \bar{\psi}\psi$, while $\bar{\psi}\gamma^5\psi \to -\bar{\psi}\gamma^5\psi$.
So the first object is a true scalar while the second object is a pseudoscalar (i.e. it picks up a minus sign under $P$). Thus, if we have the second term in the Lagrangian, it breaks Parity symmetry.
It turns out (experimentally) that $P$ is indeed broken in the weak interaction. So we do have a weak interaction vertex (for a electron + neutrino $ \to W^-$ interaction, for example) that looks like
\begin{align}
\sim -\frac{-i g_{\mu\nu}}{2\sqrt{2}}\gamma^\mu(1-\gamma^5).
\end{align}
This is an axial vector coupling, which is not parity invariant.
However the pseudoscalar $\bar{\psi}\gamma^5\psi$ doesn't appear in the SM because the object $(1-\gamma^5)/2 = \mathbb{P}_L$ is the projector onto the left-handed component of a spinor, so it can only appear if you want a theory in which the mass term of the right-handed and left-handed spinors are different, which we don't observe experimentally.
