Crossover exponents, fractal dimensions and logarithms in Landau–Potts field theories

We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry in d=6−epsilon (Landau–Potts field theories) and d=4−epsilon (hypertetrahedral models) up to three loops. We use our results to determine the epsilon-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests (q→0), and bond percolations (q→1). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of q upon analytic continuation, which are linked to logarithmic corrections of CFT correlators, and use the -expansion to determine the universal coefficients of such logarithms.