# Multi-Critical Multi-Field Models: A CFT Approach to the Leading Order

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## Abstract

**:**

## 1. Introduction

## 2. Review of the Single Scalar Field Case

#### 2.1. Relation to the FPRG Approach

#### 2.2. Example: The Universality Class of the Critical Ising Model in $d<4$

## 3. Multiple Scalar Field Case

- The scaling scalar fields arise from a mixing which is manifest in general in the splitting induced by different anomalous dimensions ${\gamma}_{i}$, eigenvalues of the anomalous dimension matrix.
- There are many more composite operators (even without derivatives) at LO, defined in general as a superposition of all monomials of the scaling fields ${\mathcal{S}}_{k}={S}_{{i}_{1}\cdots {i}_{k}}{\varphi}_{{i}_{1}}\cdots {\varphi}_{{i}_{k}}$ with scaling dimension ${\Delta}_{k}^{S}=k\phantom{\rule{0.166667em}{0ex}}\delta +{\gamma}_{k}^{S}$. The analysis leads to recurrence relations which can be solved to give secular equations for the LO anomalous dimensions ${\gamma}_{k}^{S}$ (eigenvalues) and the tensors ${S}_{{i}_{1}\cdots {i}_{k}}$ (eigenvectors).
- Structure constants (involving some composite operators ${\mathcal{S}}_{k}$) are obtained just as in the single field case.
- For unitary models with even interactions and models with cubic interactions2 (${d}_{c}=6$) one can obtain as in the single field case the criticality conditions and see that the relation with the FPRG approach.

#### 3.1. Field Anomalous Dimensions

#### 3.2. Anomalous Dimensions for Composite Operators

#### 3.3. Structure Constants

#### 3.4. Criticality Conditions

#### 3.5. Example: The Universality Class of the Critical $O\left(2\right)$ Heisenberg Model in $d<4$

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RG | Renormalization group |

FPRG | Functional perturbative renormalization group |

QFT | Quantum field theory |

CFT | Conformal field theory |

SDE | Schwinger–Dyson equations |

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1 | An operator is said primary when, taken in the origin, commutes with, or “is annihilated by”, the special conformal generator. |

2 | In the multi-field case critical models with cubic interactions can be either unitary or non unitary in perturbation theory. |

**Figure 1.**Wick contraction counting of a three point correlator. The vertices are labelled by $i=1,2,3$, the order of the i-th vertex is ${n}_{i}$, and there are ${l}_{ij}$ lines connecting two distinct vertices i and j.

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**MDPI and ACS Style**

Vacca, G.P.; Codello, A.; Safari, M.; Zanusso, O.
Multi-Critical Multi-Field Models: A CFT Approach to the Leading Order. *Universe* **2019**, *5*, 151.
https://doi.org/10.3390/universe5060151

**AMA Style**

Vacca GP, Codello A, Safari M, Zanusso O.
Multi-Critical Multi-Field Models: A CFT Approach to the Leading Order. *Universe*. 2019; 5(6):151.
https://doi.org/10.3390/universe5060151

**Chicago/Turabian Style**

Vacca, Gian Paolo, Alessandro Codello, Mahmoud Safari, and Omar Zanusso.
2019. "Multi-Critical Multi-Field Models: A CFT Approach to the Leading Order" *Universe* 5, no. 6: 151.
https://doi.org/10.3390/universe5060151