Non-trivial Fixed Point of a \(\psi ^4_d\) Fermionic Theory, II: Anomalous Exponent and Scaling Operators

Abstract

We consider the Renormalization Group (RG) fixed-point theory associated with a fermionic \(\psi ^4_d\) model in \(d=1,2,3\) with fractional kinetic term, whose scaling dimension is fixed so that the quartic interaction is weakly relevant in the RG sense. The model is defined in terms of a Grassmann functional integral with interaction \(V^*\), solving a fixed-point RG equation in the presence of external fields, and a fixed ultraviolet cutoff. We define and construct the field and density scale-invariant response functions, and prove that the critical exponent of the former is the naive one, while that of the latter is anomalous and analytic. We construct the corresponding (almost-)scaling operators, whose two point correlations are scale-invariant up to a remainder term, which decays like a stretched exponential at distances larger than the inverse of the ultraviolet cutoff. Our proof is based on constructive RG methods and, specifically, on a convergent tree expansion for the generating function of correlations, which generalizes the approach developed by three of the authors in a previous publication (Giuliani et al. in JHEP 01:026, 2021. https://doi.org/10.1007/JHEP01(2021)026. arXiv:2008.04361 [hep-th]).