We show that Josephson's relation, $\ensuremath{\upsilon}=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$, connecting the critical exponent $\ensuremath{\upsilon}$, describing the vanishing of the helicity modulus (superfluid density), to other thermodynamic and correlation exponents is an exact scaling law within the renormalization-group framework. We use the standard isotropic n-vector model with continuous spins; when d<4 hyperscaling presumably holds, and we have $\ensuremath{\upsilon}=(d\ensuremath{-}2)\ensuremath{\nu}$. Secondly, we derive the first two leading terms of the helicity modulus in an $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion; this allows second-order evaluation of universal amplitude ratios associated with the superfluid transition.
Scaling with a parameter, in contrast to ordinary scaling theory, makes predictions concerning amplitude functions as well as critical-point exponents. We provide the first test of these predictions, by examining the dependence of the amplitude on the anisotropy parameter R=JxyJz for the susceptibility and the second moment of the simple cubic and fcc Ising models with directional anisotropy. The "double-power-law" behavior found strongly supports the scaling predictions for thermodynamic functions and the two-spin-correlation function. Our analysis provides a measure of the ranges of R over which parameter scaling appears to hold for the simple cubic and fcc lattices. The relative domains of validity on the two lattices are interpretable in terms of the respective lattice structures.
There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.