  We consider the scaling behavior in the critical domain of superconductors at
zero external magnetic field. The first part of the paper is concerned with the
Ginzburg-Landau model in the zero magnetic field Meissner phase. We discuss the
scaling behavior of the superfluid density and we give an alternative proof of
Josephson's relation for a charged superfluid. This proof is obtained as a
consequence of an exact renormalization group equation for the photon mass. We
obtain Josephson's relation directly in the form $\rho_{s}\sim t^{\nu}$, that
is, we do not need to assume that the hyperscaling relation holds. Next, we
give an interpretation of a recent experiment performed in thin films of
$YBa_{2}Cu_{3}O_{7-\delta}$. We argue that the measured mean field like
behavior of the penetration depth exponent $\nu'$ is possibly associated with a
non-trivial critical behavior and we predict the exponents $\nu=1$ and
$\alpha=-1$ for the correlation lenght and specific heat, respectively. In the
second part of the paper we discuss the scaling behavior in the continuum dual
Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau
model, we discuss the continuum dual version by considering a family of
scalings characterized by a parameter $\zeta$ introduced such that
$m_{h,0}^2\sim t^{\zeta}$, where $m_{h,0}$ is the bare mass of the magnetic
induction field. We discuss the difficulties in identifying the renormalized
magnetic induction mass with the photon mass. We show that the only way to have
a critical regime with $\nu'=\nu\approx 2/3$ is having $\zeta\approx 4/3$, that
is, with $m_{h,0}$ having the scaling behavior of the renormalized photon mass.
