Consider an electron emitting a single photon. There is exactly one gamma matrix (corresponding to the photon vertex) between the outgoing spinor $\bar{u}$ and incoming spinor $u$. This implies that both $\bar{u}$ and $u$ have to be right handed electrons (or both left handed).
In other words the non-zero amplitude corresponds to $u^{\dagger}(\frac{1 + \gamma^5}{2}) \gamma^0 \gamma^{\mu} (\frac{1 + \gamma^5}{2}) u$.
The amplitude of $u^{\dagger}(\frac{1 - \gamma^5}{2}) \gamma^0 \gamma^{\mu} (\frac{1 + \gamma^5}{2}) u$ would be 0.
Lets ignore the mass of the electron giving the helicities to be frame independent.
This means that both outgoing and incoming electrons have to have either spin +1/2 or spin -1/2.
The question is: when the +1/2 electron emits a photon (of spin +1) then, should not the outgoing electron have spin -1/2; so that spin is conserved? What/where is the flaw in my analysis?