Scaling critical behavior of superconductors at zero magnetic field

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.