Validity of the long-range expansion in the $n$-vector model

The critical behavior of an $n$-component system in $d$ dimensions with long-range (LR) interactions decaying as $\frac{1}{{\left|\overrightarrow{\mathrm{r}}\right|}^{d+\sigma }}$, for $\sigma >0\mathrm{}$, is reexamined near the crossover to short-range (SR) behavior, where $\sigma =2\mathrm{}$, by means of renormalized perturbation theory in ${\epsilon }^{\prime }=2\mathrm{}\sigma -d$. It is pointed out that $2-\sigma$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\sigma =2\mathrm{}$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\sigma <2\mathrm{}$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\epsilon =4-d$ breaks down under a weak LR perturbation.