London screening with an unnormalized complex condensate (${\lambda }_L^{-2}=2e^2{v}^2$)

For scalar kinetic term $|D\phi {|}^2$ and condensate magnitude $|\phi |=v$ without a factor $1/\sqrt{2}$, the canonical vector mass is $m_A^2=2e^2{v}^2$. In a constant-phase gauge the current is $\mathbf{j}=-2e^2{v}^2\mathbf{A}$. A static magnetic field obeys ${\nabla }^2\mathbf{B}=m_A^2\mathbf{B}$, so its penetration length is $1/(\sqrt{2}|e|v)$. This normalization differs from defining the vacuum amplitude as $v/\sqrt{2}$.