The $6-\epsilon$ Expansion for Long-Range Lee--Yang and Percolation Criticality

The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the ${\epsilon }^{\mathrm{\prime }}=3\sigma -d$ expansion fixes the anomalous dimension at $\eta =2-\sigma$, whereas SR percolation has ${\eta }_{\mathrm{S}\mathrm{R}}<0$ near $d=6$. Sak’s matching condition then places the crossover above $\sigma =2$, outside the regime in which the LR interaction dominates. In spatial dimension $d=6-\epsilon$, we formulate a perturbative expansion for the LR ${\phi }^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<\delta <\epsilon \mathrm{/}3$, where $\delta =2-\sigma$. We derive the one-loop corrections to the critical exponents $\eta$ and $\nu$, which acquire nontrivial dependence on $\epsilon$ and $\delta$. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-\epsilon$ results as $\sigma \to 2$. These results support a crossover threshold ${\sigma }_{\ast }=2$ and remove the apparent discontinuity of $\eta$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee–Yang universality class and $q$-state Potts universality classes with $q<2$.