Scaling theory and logarithmic singularities

It is shown that, within Kadanoff's scaling theory for the Ising model, the amplitudes of the singularities depend upon an analytic contribution to the free energy of the scaled model, which is the self-energy of the cells. In particular a logarithmic singularity of the specific heat occurs in this theory only if the self-energy is nonzero. In Wilson's renormalisation group theory amplitudes furthermore depend upon the second-order contribution in ε to the renormalised temperature.