Scaling function for two-point correlations. II. Expansion to order1n

Abstract

The zero-field two-point correlation function of an $n$-vector system in $d$ dimensions is calculated to order $\frac{1}{n}$ for $T\ge T_c$ and $2<d<\mathrm{}4\mathrm{}$. The critical-point scaling function is obtained as a closed cutoff-independent integral. As $t=\frac{(T-T_c)}{T_c}\to 0$ at fixed wave vector $q$, the variation of the Fourier transform of the correlation function is ${\hat{E}}_1\mathrm{}\left(q\right)\mathrm{}{t}^{1-\alpha }+{\hat{E}}_2\mathrm{}\left(q\right)t+{\hat{E}}_3\mathrm{}\left(q\right)\mathrm{}{t}^2+\cdots$, where $\alpha$ is the specific-heat exponent and ${\hat{E}}_2$, ${\hat{E}}_3$ are of order $\frac{1}{n}$. At $d=3\mathrm{}$ the term $t^2$ is modified by logarithmic corrections. As $q\to 0$ at fixed nonzero $t$, deviations of the scaling function from the Ornstein-Zernike form ${(1+{\xi }^2{q}^2)}^{-1\mathrm{}}$ are of order $\frac{{10}^{-3\mathrm{}}{\xi }^4{q}^4}{n}$. Results are compared with high-temperature series-expansion estimates and with the $\epsilon$-expansion results of previous work.