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Vacuum Effective Actions and Mass-Dependent Renormalization in Curved Space^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Mass-Dependent Schemes

## 3. Heat Kernel Representation of the Effective Action in Curved Space

## 4. Renormalized Action in Two Dimensions

#### 4.1. Non-Minimally Coupled Scalar Field in Two Dimensions

#### 4.2. Dirac Field in Two Dimensions

#### 4.3. Proca Field in Two Dimensions

## 5. Renormalized Action in Four Dimensions

#### 5.1. Non-Minimally Coupled Scalar Field in Four Dimensions

#### 5.2. Dirac Field in Four Dimensions

#### 5.3. Proca Field in Four Dimensions

## 6. Comments on the UV Structure of the Effective Action

## 7. Scheme Dependence and Quantum Gravity

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Non-Local Expansion of the Heat Kernel

## Appendix B. Further Mathematical Details

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1. | This happens because the scale $\mu $ of dimensional regularization, which we use to subtract the poles, can be interpreted as a very high energy scale which is bigger than any other scale in the theory and in particular bigger than the electron’s mass. |

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**MDPI and ACS Style**

Franchino-Viñas, S.A.; de Paula Netto, T.; Zanusso, O.
Vacuum Effective Actions and Mass-Dependent Renormalization in Curved Space. *Universe* **2019**, *5*, 67.
https://doi.org/10.3390/universe5030067

**AMA Style**

Franchino-Viñas SA, de Paula Netto T, Zanusso O.
Vacuum Effective Actions and Mass-Dependent Renormalization in Curved Space. *Universe*. 2019; 5(3):67.
https://doi.org/10.3390/universe5030067

**Chicago/Turabian Style**

Franchino-Viñas, Sebastián A., Tibério de Paula Netto, and Omar Zanusso.
2019. "Vacuum Effective Actions and Mass-Dependent Renormalization in Curved Space" *Universe* 5, no. 3: 67.
https://doi.org/10.3390/universe5030067