Scaling behavior of second-order phase transitions

Using the Landau Hamiltonian for the description of second-order phase transitions, we give a proof of scaling for any continuous number of dimensions below four. The proof is based on a summation of diagrams having a power-law divergence and standard renormalization-group methods. The proof is constructive in that it leads to an unambiguous calculation for the critical exponents $\eta$ and $\gamma$. We present in this paper a detailed discussion of the proof; we also compare our method with the $\epsilon$ expansion leading to an interesting aspect of that theory: We find that the contribution to the critical exponents of order $\epsilon$ can be gotten without any calculation of diagrams. In this paper we have only made a lowest-order calculation in three dimensions. To this order we are of course unable to locate the relevant fixed point, but it leads to a relation between $\eta$ and $\gamma$—also to lowest order—which is such that if $\gamma$ is fixed to be 1.25, then $\eta$ turns out to be 0.12.