Crossover functions by renormalization-group matching:O(ε2)results

By considering the relationship of the matching techniques of Bruce and Wallace to the differential renormalization-group generators, we find that a restatement of the former gives improved results with the same number of perturbative terms. In particular, the vertex functions and specific heat of a $n$-component spin system are given exactly in the spherical limit $n\to \infty$ even at first order in perturbation theory ($T>\mathrm{}{T}_c$). The nature of the nonlinear scaling variables is clarified, and the results are generally expressed in a more compact form. The general $n$-component disordered phase functions are rederived to $O\left({\epsilon }^2\right)$, where $\epsilon \equiv 4-d$. The cross-over equations for the $n=1\mathrm{}$ Ising-like case are derived for the Helmholtz potential $A\mathrm{}\left(M\right)\mathrm{}$, the magnetic field $\frac{h}{M}$, the inverse susceptibility ${\Gamma }_2$, and the correlation length $\xi$ to $O\left({\epsilon }^2\right)$.