Canonical Abelian Higgs shift ($\varphi =(v+h+ib)/\sqrt{2}$)

For $\varphi =\kappa (v+h+ib)$, the real fluctuation kinetic terms are $-{\kappa }^2[(\partial h{)}^2+(\partial b{)}^2]$. The same $v$ labels a Minimum of $V=\lambda (|\varphi {|}^2-v^2/2{)}^2/4$ and gives canonically normalized real fields exactly when ${\kappa }^2=1/2$. This condition also makes $G=\partial \cdot A-\xi gvb$ cancel the gauge-scalar derivative mixing: its coefficient is $(2{\kappa }^2-1)gvA\cdot \partial b$. Changing the normalization of the fields without changing the gauge function need not be a conventional diagonal $R_{\xi }$ gauge.