Gaussian-to-Heisenberg crossover behavior of the specific heat in renormalized perturbation theory

A cutoff-dependent formulation of renormalized perturbation theory is used to study the Gaussian-to-Heisenberg crossover, describing the changeover from tricritical to critical-line behavior, for the specific heat, to second order in $\epsilon =4-d$, in extension of earlier work by Bruce and Wallace for the susceptibility. The role of the nonhomogeneous part of the renormalization-group equation for the bare vertex function ${\Gamma }^{\mathrm{}(2,0)\mathrm{}}$ - proportional to the specific heat - with two ${\phi }^2$ insertions is discussed in detail in an isotropic ${\phi }^4$ field model. An explicit crossover scaling function for the specific heat is calculated and the effective exponent ${\alpha }_{\mathrm{eff}}$ is found to differ considerably from an earlier first-order calculation.