Scaling function for two-point correlations with long-range interactions to order 1/η

The two-point correlation function Ĝ (q, ξ) is calculated in the critical region of momentum space q in terms of a suitable correlation lenght ξ, by means of perturbation expansion to order 1/n, for an n-vector system with long-range interactions decaying as |R/a|−(d + σ), for |R/a| å 1, where a is the spacing on a d-dimensional lattice, σ < d < 2σ and 0 < σ ⩽ 2 − ηSR. The calculations are done in zero field for T ⩾ Tc. Explicit expansions for long-range propagators are developed for σ « 1 and for the neighborhood of σ ⪅ 2 − ηSR, in terms of which a universal, cut-off independent scaling function is obtained over the whole range of x = |q| ξ, and it is shown that the amplitude of the correlation-length dependence of the susceptibility becomes a universal parameter. Both the exponents and the coefficients of the expansion for fixed q as (T − Tc)Tc→0 are calculated explicitly. The former are shown to require the validity of the operator-product expansion and explicit logarithmic correction terms are obtained for d = d∗ = 3σ/2. For these and other dimensionalities, the coefficients are shown to be finite functions of d and σ. The correction to the Ornstein-Zernike form is given explicitly, with non-integer powers of x that have finite coefficients.