# Critical Properties of Three-Dimensional Many-Flavor QEDs

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Fermionic and Bosonic QEDs

#### 1.2. Supersymmetric QEDs

#### 1.3. Reduced QEDs

#### 1.4. Outline of the Review

## 2. General QED Model and Conventions

#### 2.1. The General gQED${}_{3}$ Model

#### 2.2. Feynman Rules

- Each matter-field loop ($\psi $ and $\varphi $ field charge flow) gives a factor of $2{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}$, i.e., graphically
- Each fermion loop ($\psi $ and $\lambda $ field fermion flow) gives a factor $(-1)$ and a trace over the spinorial indices, i.e., graphically

#### 2.3. Numerator Algebra

#### 2.4. Renormalization Setup

#### 2.5. The Large-${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}$ Expansion

- (1)
- Compute the one-loop polarization using bare Feynman rules and compute the LO-softened photon by resumming the one-loop polarization.
- (2)
- Compute the other diagrams of the theory at $\mathrm{O}(1/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f})$ using the LO-softened photon only.

- (3)
- Compute the two-loop polarizations using the LO-softened photon propagator and compute the new NLO-softened photon propagator by resumming the two-loop polarization.
- (4)
- Compute the other diagrams of the theory at NLO, i.e., $\mathrm{O}(1/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}^{2})$ using both the LO and NLO-softened photon propagators.

## 3. Perturbative Calculations up to NLO in gQED${}_{3}$

#### 3.1. Gauge-Multiplet Polarizations at LO

#### 3.1.1. Photon Polarization at LO

#### 3.1.2. $\epsilon $-Scalar Polarization at LO

#### 3.1.3. Photino Polarization at LO

#### 3.1.4. IR-Softened Gauge Multiplet at LO

#### 3.2. Matter-Multiplet Self-Energies at LO

#### 3.2.1. Electron Self-Energy at LO

#### 3.2.2. Selectron Self-Energy at LO

#### 3.3. Vanishing Contributions and Generalized Furry Theorem

#### 3.4. Gauge-Multiplet Polarizations at NLO

#### 3.4.1. Photon Polarization at NLO

#### 3.4.2. $\epsilon $-Scalar Polarization at NLO

#### 3.4.3. Photino Polarization at NLO

#### 3.4.4. IR-Softened Gauge Multiplet at NLO

#### 3.5. Matter-Multiplet Self-Energies at NLO

#### 3.5.1. Electron Self-Energy at NLO

#### 3.5.2. Selectron Self-Energy at NLO

## 4. Critical Exponents and Observables

#### 4.1. Results for Fermionic QED${}_{3}$

(114a) | |

(114b) | |

(114c) | |

(114d) |

#### 4.2. Results for $\mathcal{N}=1$ SQED${}_{3}$

(116a) | |

(116b) | |

(116c) | |

(116d) | |

(116e) | |

(116f) |

#### 4.3. Results for Bosonic QED${}_{3}$

(119a) | |

(119b) | |

(119c) | |

(119d) |

#### 4.4. Results for Reduced QED${}_{4,3}$ (Graphene)

(124a) | |

(124b) | |

(124c) | |

(124d) |

#### 4.5. Results for Reduced $\mathcal{N}=1$ SQED${}_{4,3}$ (Super-Graphene)

(136a) | |

(136b) | |

(136c) | |

(136d) | |

(136e) | |

(136f) |

#### 4.6. Stability of the IR Fixed Point

## 5. Dynamical (Matter) Mass Generation

#### 5.1. The (Semi-Phenomenological) Gap Equation

#### 5.2. Results for (S)QED${}_{3}$

#### 5.3. Results for (S)QED${}_{4,3}$ (Graphene and Super-Graphene)

#### 5.4. Meta-Analysis of the Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Appendix on Multiloop Massless Techniques

#### Appendix A.1. One Loop

#### Appendix A.2. Two Loop

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**Figure 2.**Vanishing one-loop matter triangles in gQED${}_{3}$. (

**a**) ${T}_{1}$, (

**b**) ${T}_{2}$, (

**c**) ${T}_{3}$, (

**d**) ${T}_{4}$, (

**e**) ${T}_{5}$, (

**f**) ${T}_{6}$, (

**g**) ${T}_{7}$, (

**h**) ${T}_{8}$.

**Figure 3.**Phase diagrams for dynamical mass generation in: (

**a**) Graphene (QED${}_{4,3}$) from (160) and (

**b**) super-graphene (SQED${}_{4,3}$) from (162). Note that the relevant case for (super-)graphene is ${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}=2$. Insulator refers to an excitonic insulating phase, while metal refers to a semimetallic phase.

**Figure 4.**All results obtained in this article for the critical number of fermion flavors below which a dynamical mass is generated in various QED models. The darker it is, the more likely the corresponding model is massive for a given ${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}$. Note that the case of interest is generally ${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}=1$ for all the QED${}_{3}$ variants. For graphene (QED${}_{4,3}$) and super-graphene (SQED${}_{4,3}$), the case of interest is usually ${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}=2$.

**Table 1.**Parameter values used to recover the different large-${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}$ models from the gQED${}_{3}$ action (6).

Model | S | n | $\mathcal{E}$ |
---|---|---|---|

$\mathcal{N}=1$ SQED${}_{3}$ | 1 | 2 | 1 |

Bosonic QED${}_{3}$ | 1 | 0 | 0 |

Fermionic QED${}_{3}$ | 0 | 2 | 0 |

fQED${}_{3}$ | ${\gamma}_{{m}_{\psi}}=1.0808/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}+0.1174/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}^{2}+O(1/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}^{3})$ |

SQED${}_{3}$ | ${\gamma}_{{m}_{\psi}}=0.4053/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}-0.1696/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}^{2}+\mathrm{O}(1/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}^{3})$ |

bQED${}_{3}$ | ${\gamma}_{{m}_{\varphi}}=0.5404/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}-1.2553/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}^{2}+\mathrm{O}(1/{N}_{\phantom{\rule{-1.13809pt}{0ex}}f}^{3})$ |

**Table 3.**Reproduced from [30] and updated. D$\chi $SB in fQED${}_{3}$: some values of ${N}_{\phantom{\rule{-0.56905pt}{0ex}}c}$ obtained over the years with different methods. The value obtained with our method is grayed. Note that recent analytical methods (including ours) converge to a value of ${N}_{\phantom{\rule{-0.56905pt}{0ex}}c}$ in the range $]2,3[$ such that a dynamical mass is generated for ${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}\le 2$. On the other hand, results from lattice simulations are inconsistent. This may partly be due to the fact that, as ${N}_{\phantom{\rule{-1.13809pt}{0ex}}f}=2$ is close to ${N}_{\phantom{\rule{-0.56905pt}{0ex}}c}$, the dynamically generated mass is so small (see estimate and discussion in [15]) that it is difficult to extract from lattice simulations.

${\mathit{N}}_{\phantom{\rule{-0.56905pt}{0ex}}\mathit{c}}$ in fQED${}_{3}$ | Method | Year |
---|---|---|

∞ | Schwinger–Dyson (LO) | 1984 [3] |

∞ | Schwinger–Dyson (non-pert., Landau gauge) | 1990, 1992 [8,134] |

∞ | RG study | 1991 [135] |

∞ | Lattice simulations | 1993, 1996 [136,137] |

$<4.4$ | F-theorem | 2015 [138] |

$(4/3)(32/{\pi}^{2})=4.32$ | Schwinger–Dyson (LO, resum.) | 1989 [6] |

$4.422$ | RG study (one loop) (${N}_{c}^{\mathrm{conf}}\approx 6.24$) | 2016 [139] |

4 | Functional RG ($4.1<{N}_{c}^{\mathrm{conf}}<10.0$) | 2014 [140] |

$3<{N}_{c}<4$ | RG study | 2001 [141] |

$3.5\pm 0.5$ | Lattice simulations | 1988, 1989 [142,143] |

$3.31$ | Schwinger–Dyson (NLO, Landau gauge) | 1993 [9,10] |

$3.29$ | Schwinger–Dyson (NLO, Landau gauge) | 2016 [16] |

$32/{\pi}^{2}\approx 3.24$ | Schwinger–Dyson (LO, Landau gauge) | 1988 [5] |

3.0084–3.0844 | Schwinger–Dyson (NLO, resum.) | 2016 [17] |

$2.89$ | RG study (one loop) | 2016 [144] |

$2.85$ | Schwinger–Dyson (NLO, resummation, $\forall \xi $) | 2016 [15,17] |

$1+\sqrt{2}=2.41$ | F-theorem | 2016 [145] |

$2.27$ | Effective gap equation (NLO, $\forall \xi $, double resum.) | 2022 [115] |

$<9/4=2.25$ | RG study (one loop) | 2015 [146] |

$<3/2$ | Free energy constraint | 1999 [147] |

$1<{N}_{c}<4$ | Lattice simulations | 2004, 2008 [148,149] |

0 | Schwinger–Dyson (non-pert., Landau gauge) | 1990 [7] |

0 | Lattice simulations | 2015, 2016 [150,151] |

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Metayer, S.; Teber, S.
Critical Properties of Three-Dimensional Many-Flavor QEDs. *Symmetry* **2023**, *15*, 1806.
https://doi.org/10.3390/sym15091806

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Metayer S, Teber S.
Critical Properties of Three-Dimensional Many-Flavor QEDs. *Symmetry*. 2023; 15(9):1806.
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Metayer, Simon, and Sofian Teber.
2023. "Critical Properties of Three-Dimensional Many-Flavor QEDs" *Symmetry* 15, no. 9: 1806.
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