Universal Ratios Among Correction-to-Scaling Amplitudes on the Coexistence Curve

Correction-to-scaling amplitudes are calculated to second order in $\epsilon =4-d$ by renormalized perturbation theory. It is shown that ratio of any two such amplitudes is universal, and that thermodynamic scaling laws among effective exponents break down beyond zeroth order in $\epsilon$. Values found are $\frac{{a_{\xi }}^{+}}{{a_{\chi }}^{+}}=0.65\mathrm{} (n=1, d=3) \mathrm{and} 0.63 (n=2, d=3)$ in good agreement with high-temperature series, and $\frac{{a_m}^{-}}{{a_{\chi }}^{+}}=0.85\mathrm{} (n=1, d=3)$.