Scaling function for two-point correlations. I. Expansion near four dimensions

A detailed calculation to order ${\epsilon }^2$ of the two-point correlation function in the zero-field critical region ($T\ge T_c$) is presented for an $n$-vector system in $d=4-\epsilon$ dimensions. Scaling behavior is verified over the full parameter range and the scaling function is obtained as a universal explicit convergent integral (independent of any cutoffs). Study of the scaling function as $T\to T_c$ in the finite ($q\ne 0$) momentum limit, confirms the presence of terms varying as $t^{1-\alpha }$ and as $t$ [with $t=\frac{(T-T_c)}{T_c}$]; explicit evaluation of their amplitudes reliably determines the form of the scattering intensity at fixed $q$ and locates the corresponding maximum accurately.