= O(4) symmetry of a complex scalar doublet
{c}
Two canonically normalized <complex scalar fields> with potential $V=m^2\Phi^\dagger\Phi+\lambda(\Phi^\dagger\Phi)^2$ are four real fields with a radial potential. Consequently the isolated action is invariant under the <orthogonal group> $O(4)$, not only under complex-linear $U(2)$ transformations. Writing $\phi_1=(q_1+iq_2)/\sqrt2$ and $\phi_2=(q_3+iq_4)/\sqrt2$ makes the kinetic term $\sum_A(\partial q_A)^2/2$ and the potential a function of $\sum_Aq_A^2$. The six real-plane rotations give currents $j_{AB}^\mu=q_B\partial^\mu q_A-q_A\partial^\mu q_B$. Gauge or Yukawa couplings can break this larger symmetry, so it should not be asserted for an arbitrary interacting doublet.
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