Mean-field expansion for spin models with medium-range interactions

We study the critical crossover between the Gaussian and the Wilson-Fisher fixed point for general O(N)-invariant spin models with medium-range interactions. We perform a systematic expansion around the mean-field solution, obtaining the universal crossover curves and their leading corrections. In particular we show that, in three dimensions, the leading correction scales as $R^{-3}, R$ being the range of the interactions. We compare our results with the existing numerical ones obtained by Monte Carlo simulations and present a critical discussion of other approaches.