Chiral scalar-doublet Yukawa model (${\mathcal{L}}_Y=-g\bar{\psi }({\varphi }_1+i{\gamma }_5{\varphi }_2)\psi$)

The combined transformation $\psi \mapsto e^{-i\alpha {\gamma }_5/2}\psi$ and $({\varphi }_1+i{\varphi }_2)\mapsto e^{i\alpha }({\varphi }_1+i{\varphi }_2)$ preserves this interaction. The Dirac adjoint transforms with the same axial exponential as $\psi$. For potential $-{\mu }^2{r}^2/2+\lambda r^4/4$, $v=\mu /\sqrt{\lambda }$ gives a radial scalar of squared mass $2\lambda v^2$, a massless Goldstone boson and fermion physical mass $|g|v$. Cubic scalar vertices appear after the shift: $hhh$ has factor $-6i\lambda v$ and $h\pi \pi$ has factor $-2i\lambda v$.