# Harmonic Superspace Approach to the Effective Action in Six-Dimensional Supersymmetric Gauge Theories

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## Abstract

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## Contents

1 | Introduction | 2 | |

2 | Harmonic Superspace Formulation of 6D Supersymmetric Gauge Theories | 3 | |

3 | Quantum Corrections in 6D, $\mathcal{N}$ = (1, 0) Supersymmetric Electrodynamics | 6 | |

3.1 | Quantization, Feynman Rules, and Ward Identities in the Abelian Case . . . . . . . . . . . . . | 6 | |

3.2 | One-Loop Divergences and Their Gauge Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 9 | |

4 | Quantum Corrections in Non-Abelian 6D, $\mathcal{N}$ = (1, 0) and $\mathcal{N}$ = (1, 1) Supersymmetric | ||

Theories | 12 | ||

4.1 | Quantization of Non-Abelian 6D Gauge Theories in the Harmonic Superspace by the | ||

Background Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 12 | ||

4.2 | One-Loop Divergences in Harmonic Superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 15 | |

4.3 | Two-Loop Divergent Part of the Hypermultiplet Two-Point Green Function of 6D SYM | ||

Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 18 | ||

4.4 | Manifestly Gauge Covariant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 20 | |

4.5 | Low-Energy Effective Action . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 22 | |

5 | Conclusions | 25 | |

References | 26 |

## 1. Introduction

## 2. Harmonic Superspace Formulation of 6D Supersymmetric Gauge Theories

## 3. Quantum Corrections in 6D, $\mathcal{N}=(\mathbf{1},\mathbf{0})$ Supersymmetric Electrodynamics

#### 3.1. Quantization, Feynman Rules, and Ward Identities in the Abelian Case

#### 3.2. One-Loop Divergences and Their Gauge Dependence

## 4. Quantum Corrections in Non-Abelian $\mathbf{6}\mathit{D}$, $\mathcal{N}=(\mathbf{1},\mathbf{0})$ and $\mathcal{N}=(\mathbf{1},\mathbf{1})$ Supersymmetric Theories

#### 4.1. Quantization of Non-Abelian $6D$ Gauge Theories in the Harmonic Superspace by the Background Field Method

#### 4.2. One-Loop Divergences in Harmonic Superspace

#### 4.3. Two-Loop Divergent Part of the Hypermultiplet Two-Point Green Function of $6D$ SYM Theories

#### 4.4. Manifestly Gauge Covariant Analysis

#### 4.5. Low-Energy Effective Action

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**The harmonic supergraph representing the one-loop contribution to the three-point gauge-hypermultiplet function.

**Figure 6.**The lines (1), (2), (3), and (4) denote the propagators of the gauge, hypermultiplet, Faddeev–Popov, and Nielsen–Kallosh ghost superfields.

**Figure 7.**Harmonic supergraphs representing the one-loop two-point Green function of the background gauge superfield.

**Figure 8.**These two harmonic supergraphs determine the three-point gauge-hypermultiplet function in the one-loop approximation.

**Figure 9.**Supergraphs representing the two-point hypermultiplet Green function in the two-loop approximation.

**Figure 10.**In Figure 9, the gray circle corresponds to the one-loop polarization operator, which is given by the sum of the harmonic supergraphs depicted here.

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Buchbinder, I.; Ivanov, E.; Merzlikin, B.; Stepanyantz, K.
Harmonic Superspace Approach to the Effective Action in Six-Dimensional Supersymmetric Gauge Theories. *Symmetry* **2019**, *11*, 68.
https://doi.org/10.3390/sym11010068

**AMA Style**

Buchbinder I, Ivanov E, Merzlikin B, Stepanyantz K.
Harmonic Superspace Approach to the Effective Action in Six-Dimensional Supersymmetric Gauge Theories. *Symmetry*. 2019; 11(1):68.
https://doi.org/10.3390/sym11010068

**Chicago/Turabian Style**

Buchbinder, Ioseph, Evgeny Ivanov, Boris Merzlikin, and Konstantin Stepanyantz.
2019. "Harmonic Superspace Approach to the Effective Action in Six-Dimensional Supersymmetric Gauge Theories" *Symmetry* 11, no. 1: 68.
https://doi.org/10.3390/sym11010068