Random Field Ising Model and Parisi-Sourlas Supersymmetry. Part II. Renormalization Group

We revisit perturbative RG analysis in the replicated Landau-Ginzburg description of the Random Field Ising Model near the upper critical dimension 6. Working in a field basis with manifest vicinity to a weakly-coupled Parisi-Sourlas supersymmetric fixed point (Cardy, 1985), we look for interactions which may destabilize the SUSY RG flow and lead to the loss of dimensional reduction. This problem is reduced to studying the anomalous dimensions of "leaders" -- lowest dimension parts of $S_n$-invariant perturbations in the Cardy basis. Leader operators are classified as non-susy-writable, susy-writable or susy-null depending on their symmetry. Susy-writable leaders are additionally classified as belonging to superprimary multiplets transforming in particular $\textrm{OSp}(d | 2)$ representations. We enumerate all leaders up to 6d dimension $\Delta = 12$, and compute their perturbative anomalous dimensions (up to two loops). We thus identify two perturbations (with susy-null and non-susy-writable leaders) becoming relevant below a critical dimension $d_c \approx 4.2$ - $4.7$. This supports the scenario that the SUSY fixed point exists for all $3 < d \leq 6$, but becomes unstable for $d < d_c$.