# Six dimensional Landau-Ginzburg-Wilson theory

## TL;DR

This paper performs a three-loop renormalization of a six-dimensional cubic Landau-Ginzburg-Wilson theory with $O(N) imes O(m)$ symmetry, establishing its universality class and analyzing its fixed point structure.

## Contribution

It provides the first three-loop renormalization of the six-dimensional $O(N) imes O(m)$ symmetric theory and connects it to four-dimensional models, estimating the conformal window.

## Key findings

- Critical exponents agree with large $N$ $d$-dimensional results.
- Fixed point structure analyzed to estimate the conformal window.
- Renormalization group functions confirm universality class.

## Abstract

We renormalize the six dimensional cubic theory with an $O(N)$ $\times$ $O(m)$ symmetry at three loops in the modified minimal subtraction (MSbar) scheme. The theory lies in the same universality class as the four dimensional Landau-Ginzburg-Wilson model. As a check we show that the critical exponents derived from the three loop renormalization group functions at the Wilson-Fisher fixed point are in agreement with the large $N$ $d$-dimensional critical exponents of the underlying universal theory. Having established this connection we analyse the fixed point structure of the perturbative renormalization group functions to estimate the location of the conformal window of the $O(N)$ $\times$ $O(2)$ model.