 Solution

Put $U=e^{-i\alpha {\gamma }_5/2}$. The transformed Dirac adjoint is ${\bar{\psi }}^{\prime }={\psi }^{†}{U}^{†}{\gamma }^0=\bar{\psi }U$, since ${\gamma }^0$ anticommutes with ${\gamma }_5$. It is therefore incorrect to transform the two sides of a Yukawa bilinear with mutually inverse matrices.

Write $M(\varphi )={\varphi }_1+i{\gamma }_5{\varphi }_2$. The transformed Yukawa interaction is invariant exactly whenSince $M$ commutes with ${\gamma }_5$, this is $M({\varphi }^{\prime })=e^{i\alpha {\gamma }_5}M(\varphi )$. Use i and equate the coefficients of $1$ and $i{\gamma }_5$:This is the global SO(2) rotation of a chiral scalar-doublet Yukawa model. Equivalently, $\Phi ={\varphi }_1+i{\varphi }_2$ transforms as ${\Phi }^{\prime }=e^{i\alpha }\Phi$, while the two chiral spinor components transform with opposite half-angle phases. The sign of the scalar rotation follows from the negative sign in the specified fermion transformation.