Leading order CFT analysis of multi-scalar theories in d>2

TL;DR

This paper analyzes multi-scalar conformal field theories in dimensions greater than two using CFT constraints and Schwinger-Dyson equations, providing leading order anomalous dimensions, structure constants, and critical equations without assuming symmetry.

Contribution

It offers a general leading order analysis of multi-scalar theories in d>2, including non-symmetric models and those with $S_q$ symmetry, extending previous methods with new critical equations and computations.

Findings

Computed anomalous dimensions for fields and operators.

Derived critical equations matching RG methods.

Analyzed models with $S_q$ symmetry in various dimensions.

Abstract

We investigate multi-field multicritical scalar theories using CFT constraints on two- and three-point functions combined with the Schwinger-Dyson equation. This is done in general and without assuming any symmetry for the models, which we just define to admit a Landau-Ginzburg description that includes the most general critical interactions built from monomials of the form $\phi_{i_1} \cdots \phi_{i_m}$. For all such models we analyze to the leading order of the $\epsilon$-expansion the anomalous dimensions of the fields and those of the composite quadratic operators. For models with even $m$ we extend the analysis to an infinite tower of composite operators of arbitrary order. The results are supplemented by the computation of some families of structure constants. We also find the equations which constrain the nontrivial critical theories at leading order and show that they coincide with the ones obtained with functional perturbative RG methods. This is done for the case $m=3$ as well as for all the even models. We ultimately specialize to $S_q$ symmetric models, which are related to the $q$-state Potts universality class, and focus on three realizations appearing below the upper critical dimensions $6$, $4$ and $\frac{10}{3}$, which can thus be nontrivial CFTs in three dimensions.