# Leading CFT constraints on multi-critical models in d>2

## TL;DR

This paper uses conformal field theory constraints and Schwinger-Dyson equations to analyze multi-critical scalar quantum field theories in dimensions greater than two, deriving operator dimensions and structure constants.

## Contribution

It provides a unified CFT-based approach to compute critical exponents and structure constants for multi-critical models across various dimensions and potential types.

## Key findings

- Computed scaling dimensions for even potentials and Lee-Yang class.
- Derived structure constants using conformal invariance and expansions.
- Connected multi-critical models to minimal models in 2D.

## Abstract

We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial $\phi^{m}$ below their upper critical dimensions $d_c=\frac{2m}{m-2}$, and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers $m \ge 4$ these theories coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in $d=2$, while for odd $m$ the theories are non-unitary and start at $m=3$ with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators $\phi^k$ and of some families of structure constants in either the coupling's or the $\epsilon$-expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling's expansion.