# Wilson-Fisher fixed points for any dimension

## TL;DR

This paper investigates non-local scalar field theories with tunable interactions, computes renormalization effects, and explores fixed points and critical exponents across various dimensions, revealing unitarity violations in higher dimensions.

## Contribution

It introduces a non-local scalar field theory with a real power Laplacian interaction, analyzes its renormalization and fixed points, and computes critical exponents in multiple dimensions.

## Key findings

- Non-trivial IR fixed points are found for all dimensions.
- Field renormalization is required at different loop levels depending on the dimension.
- Critical exponent η becomes negative for dimensions greater than four, indicating unitarity violation.

## Abstract

The critical behavior of a non-local scalar field theory is studied. This theory has a non-local quartic interaction term which involves a real power -\beta of the Laplacian. The parameter \beta can be tuned so as to make that interaction marginal for any dimension. The lowest order Feynman diagrams corresponding to coupling constant renormalization, mass renormalization, and field renormalization are computed. In all cases a non-trivial IR fixed point is obtained. Remarkably, for dimensions different from 4, field renormalization is required at the one-loop level. For d=4, the theory reduces to the usual local \phi^{4} field theory and field renormalization is required starting at the the two-loop level. The critical exponents \nu and \eta are computed for dimensions 2,3,4 and 5. For dimensions greater than four, the critical exponent \eta turns out to be negative for \epsilon>0, which indicates a violation of the unitarity bounds.