Wilson-Fisher fixed points for any dimension

Abstract

The critical behavior of a nonlocal scalar field theory is studied. This theory has a nonlocal quartic interaction term which involves a power-β of the Laplacian. The power-β is tuned so as to make that interaction marginal for any dimension. This leads to integer or half-integer values for β, depending on the space dimension. Introducing an auxiliary field, it is shown that the theory can be renormalized by means of local counterterms in the fields. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. In all cases, a nontrivial IR fixed point is obtained. Remarkably, for dimensions other than 4, field renormalization is required at the one-loop level. For d=4, the theory reduces to the usual local φ4 field theory, and field renormalization is required starting at the two-loop level. The critical exponents ν and η are computed for dimensions 2, 3, 4, and 5. For dimensions greater than 4, the critical exponent η turns out to be negative for ϵ>0, which indicates a violation of the unitarity bounds.