 O(4) symmetry of a complex scalar doublet

Two canonically normalized complex scalar fields with potential $V=m^2{\Phi }^{†}\Phi +\lambda ({\Phi }^{†}\Phi {)}^2$ 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 ${\varphi }_1=(q_1+i{q}_2)/\sqrt{2}$ and ${\varphi }_2=(q_3+i{q}_4)/\sqrt{2}$ makes the kinetic term ${\sum }_A(\partial q_A{)}^2/2$ and the potential a function of ${\sum }_A{q}_A^2$. 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.