= Solution
The scalar kinetic and potential terms depend only on the rotation-invariant combinations $\partial_\mu\phi_1\partial^\mu\phi_1+\partial_\mu\phi_2\partial^\mu\phi_2$ and $\phi_1^2+\phi_2^2$. They are unchanged by the constant <SO(2)> rotation. The Yukawa term is invariant by <3/a/ii>. Finally, anticommutation gives $U\gamma^\mu=\gamma^\mu U^{-1}$, so
$$
\bar\psi'i\gamma^\mu\partial_\mu\psi'
=\bar\psi U i\gamma^\mu U\partial_\mu\psi
=\bar\psi i\gamma^\mu\partial_\mu\psi.
$$
No derivative of $U$ appears because the parameter is global. Thus the entire classical <Lagrangian density> has the combined <chiral symmetry>.
An explicit <Dirac mass term> instead changes as
$$
-m\bar\psi\psi\longmapsto
-m\bar\psi e^{-i\alpha\gamma_5}\psi
=-m\cos\alpha\,\bar\psi\psi+im\sin\alpha\,\bar\psi\gamma_5\psi.
$$
It is not invariant for arbitrary $\alpha$ unless $m=0$. For the concrete choice $\alpha=\pi$, the original <mass term> reverses sign. The invariant Yukawa term can nevertheless generate a <mass> after a <vacuum> is selected, because the scalar field transforms along with the <fermions>; this is spontaneous rather than explicit breaking.
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