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

^{1}

^{2}

^{*}

^{†}

## 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

## References

- Appelquist, T.; Carazzone, J. Infrared Singularities and Massive Fields. Phys. Rev. D
**1975**, 11, 2856. [Google Scholar] [CrossRef] - Ribeiro, T.G.; Shapiro, I.L.; Zanusso, O. Gravitational form factors and decoupling in 2D. Phys. Lett. B
**2018**, 782, 324–331. [Google Scholar] [CrossRef] - Franchino-Viñas, S.A.; Netto, T.D.; Shapiro, I.L.; Zanusso, O. Form factors and decoupling of matter fields in four-dimensional gravity. Phys. Lett. B
**2019**, 790, 229–236. [Google Scholar] [CrossRef] - Gorbar, E.V.; Shapiro, I.L. Renormalization group and decoupling in curved space. JHEP
**2003**, 0302, 21. [Google Scholar] [CrossRef] - Gorbar, E.V.; Shapiro, I.L. Renormalization group and decoupling in curved space. 2. The Standard model and beyond. JHEP
**2003**, 0306, 4. [Google Scholar] [CrossRef] - Buchbinder, I.L.; de Berredo-Peixoto, G.; Shapiro, I.L. Quantum effects in softly broken gauge theories in curved space-times. Phys. Lett. B
**2007**, 649, 454–462. [Google Scholar] [CrossRef] - Barvinsky, A.O.; Vilkovisky, G.A. The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity. Phys. Rep.
**1985**, 119, 1–74. [Google Scholar] [CrossRef] - Barvinsky, A.O.; Vilkovisky, G.A. Beyond the Schwinger-Dewitt Technique: Converting Loops Into Trees and In-In Currents. Nucl. Phys. B
**1987**, 282, 163–188. [Google Scholar] [CrossRef] - Barvinsky, A.O.; Vilkovisky, G.A. Covariant perturbation theory. 2: Second order in the curvature. General algorithms. Nucl. Phys. B
**1990**, 333, 471–511. [Google Scholar] [CrossRef] - Codello, A.; Zanusso, O. On the non-local heat kernel expansion. J. Math. Phys.
**2013**, 54, 013513. [Google Scholar] [CrossRef][Green Version] - Shapiro, I.L.; Sola, J. Massive fields temper anomaly induced inflation. Phys. Lett. B
**2002**, 530, 10–19. [Google Scholar] [CrossRef] - Pelinson, A.M.; Shapiro, I.L.; Takakura, F.I. On the stability of the anomaly induced inflation. Nucl. Phys. B
**2003**, 648, 417–445. [Google Scholar] [CrossRef] - Shapiro, I.L. The Graceful exit from the anomaly induced inflation: Supersymmetry as a key. Int. J. Mod. Phys. D
**2002**, 11, 1159–1170. [Google Scholar] [CrossRef] - Starobinsky, A.A. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett. B
**1980**, 91, 99–102. [Google Scholar] [CrossRef] - Starobinsky, A.A. The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy. Sov. Astron. Lett.
**1983**, 9, 302. [Google Scholar] - Netto, T.D.P.; Pelinson, A.M.; Shapiro, I.L.; Starobinsky, A.A. From stable to unstable anomaly-induced inflation. Eur. Phys. J. C
**2016**, 76, 544. [Google Scholar] [CrossRef] - Shapiro, I.L.; Sola, J.; Stefancic, H. Running G and Lambda at low energies from physics at M(X): Possible cosmological and astrophysical implications. JCAP
**2005**, 0501, 012. [Google Scholar] [CrossRef] - Rodrigues, D.C.; Letelier, P.S.; Shapiro, I.L. Galaxy Rotation Curves from General Relativity with Infrared Renormalization Group Effects. arXiv, 2010; arXiv:1102.2188 [astro-ph.CO]. [Google Scholar]
- Shapiro, I.L.; Sola, J. On the possible running of the cosmological ‘constant’. Phys. Lett. B
**2009**, 682, 105–113. [Google Scholar] [CrossRef] - Fröb, A.; Roura, M.B.; Verdaguer, E. One-loop gravitational wave spectrum in de Sitter spacetime. JCAP
**2012**, 1208, 009. [Google Scholar] [CrossRef] - Nelson, B.L.; Panangaden, P. Scaling Behavior Of Interacting Quantum Fields In Curved Space-time. Phys. Rev. D
**1982**, 25, 1019. [Google Scholar] [CrossRef] - Buchbinder, I.L. Renormalization Group Equations In Curved Space-time. Theor. Math. Phys.
**1984**, 61, 1215. [Google Scholar] [CrossRef] - Buchbinder, I.L.; Odintsov, S.D.; Shapiro, I.L. Effective Action in Quantum Gravity; IOP: Bristol, UK, 1992; 413p. [Google Scholar]
- Maggiore, M.; Mancarella, M. Nonlocal gravity and dark energy. Phys. Rev. D
**2014**, 90, 023005. [Google Scholar] [CrossRef] - Codello, A.; Jain, R.K. On the covariant formalism of the effective field theory of gravity and leading order corrections. Class. Quant. Grav.
**2016**, 33, 225006. [Google Scholar] [CrossRef][Green Version] - Codello, A.; Jain, R.K. On the covariant formalism of the effective field theory of gravity and its cosmological implications. Class. Quant. Grav.
**2017**, 34, 035015. [Google Scholar] [CrossRef][Green Version] - Knorr, B.; Saueressig, F. Towards reconstructing the quantum effective action of gravity. Phys. Rev. Lett.
**2018**, 121, 161304. [Google Scholar] [CrossRef] [PubMed] - Codello, A.; Tetradis, N.; Zanusso, O. The renormalization of fluctuating branes, the Galileon and asymptotic safety. JHEP
**2013**, 1304, 036. [Google Scholar] [CrossRef] - Brouzakis, N.; Codello, A.; Tetradis, N.; Zanusso, O. Quantum corrections in Galileon theories. Phys. Rev. D
**2014**, 89, 125017. [Google Scholar] [CrossRef] - Codello, A.; Zanusso, O. Fluid Membranes and 2d Quantum Gravity. Phys. Rev. D
**2011**, 83, 125021. [Google Scholar] [CrossRef] - Avramidi, I.G. Covariant Studies of Nonlocal Structure of Effective Action. Sov. J. Nucl. Phys.
**1989**, 49, 735–739. (In Russian) [Google Scholar] - Hamber, H.W.; Toriumi, R. Cosmological Density Perturbations with a Scale-Dependent Newton’s G. Phys. Rev. D
**2010**, 82, 043518. [Google Scholar] [CrossRef] - Hamber, H.W.; Toriumi, R. Scale-Dependent Newton’s Constant G in the Conformal Newtonian Gauge. Phys. Rev. D
**2011**, 84, 103507. [Google Scholar] [CrossRef] - Asorey, M.; Gorbar, E.V.; Shapiro, I.L. Universality and ambiguities of the conformal anomaly. Class. Quant. Grav.
**2003**, 21, 163. [Google Scholar] [CrossRef] - Reuter, M. Nonperturbative evolution equation for quantum gravity. Phys. Rev. D
**1998**, 57, 971. [Google Scholar] [CrossRef] - Reuter, M.; Saueressig, F. Quantum Gravity and the Functional Renormalization Group; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Percacci, R. An Introduction to Covariant Quantum Gravity and Asymptotic Safety; Series 100 Years of General Relativity; World Scientific: Singapore, 2017; Volume 3. [Google Scholar]
- Donoghue, J.F.; Ivanov, M.M.; Shkerin, A. EPFL Lectures on General Relativity as a Quantum Field Theory. arXiv, arXiv:1702.00319 [hep-th].
- Codello, A.; Percacci, R.; Rachwał, L.; Tonero, A. Computing the Effective Action with the Functional Renormalization Group. Eur. Phys. J. C
**2016**, 76, 226. [Google Scholar] [CrossRef] - Goncalves, B.; de Berredo-Peixoto, G.; Shapiro, I.L. One-loop corrections to the photon propagator in the curved-space QED. Phys. Rev. D
**2009**, 80, 104013. [Google Scholar] [CrossRef] - Ruf, M.S.; Steinwachs, C.F. Renormalization of generalized vector field models in curved spacetime. Phys. Rev. D
**2018**, 98, 025009. [Google Scholar] [CrossRef] - Ruf, M.S.; Steinwachs, C.F. Quantum effective action for degenerate vector field theories. Phys. Rev. D
**2018**, 98, 085014. [Google Scholar] [CrossRef] - Brown, L.S.; Cassidy, J.P. Stress Tensor Trace Anomaly in a Gravitational Metric: General Theory, Maxwell Field. Phys. Rev. D
**1977**, 15, 2810. [Google Scholar] [CrossRef] - Barvinsky, A.O.; Nesterov, D.V. Nonperturbative heat kernel and nonlocal effective action. arXiv, 2004; arXiv:hep-th/0402043. [Google Scholar]
- Codello, A.; D’Odorico, G. Scaling and Renormalization in two dimensional Quantum Gravity. Phys. Rev. D
**2015**, 92, 024026. [Google Scholar] [CrossRef] - Zamolodchikov, A.B. Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory. JETP Lett.
**1986**, 43, 730. [Google Scholar] - Jack, I.; Osborn, H. Analogs for the c Theorem for Four-dimensional Renormalizable Field Theories. Nucl. Phys. B
**1990**, 343, 647–688. [Google Scholar] [CrossRef] - Osborn, H. Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories. Nucl. Phys. B
**1991**, 363, 486–526. [Google Scholar] [CrossRef] - Codello, A.; D’Odorico, G.; Pagani, C. Functional and Local Renormalization Groups. Phys. Rev. D
**2015**, 91, 125016. [Google Scholar] [CrossRef] - Jack, I.; Osborn, H. Background Field Calculations in Curved Space-time. 1. General Formalism and Application to Scalar Fields. Nucl. Phys. B
**1984**, 234, 331–364. [Google Scholar] [CrossRef] - Martini, R.; Zanusso, O. Renormalization of multicritical scalar models in curved space. arXiv, 2018; arXiv:1810.06395 [hep-th]. [Google Scholar]
- El-Menoufi, B.K. Quantum gravity of Kerr-Schild spacetimes and the logarithmic correction to Schwarzschild black hole entropy. JHEP
**2016**, 1605, 035. [Google Scholar] [CrossRef] - Donoghue, J.F.; El-Menoufi, B.K. Covariant non-local action for massless QED and the curvature expansion. JHEP
**2015**, 1510, 044. [Google Scholar] [CrossRef] - Codello, A. Large N Quantum Gravity. New J. Phys.
**2012**, 14, 015009. [Google Scholar] [CrossRef] - Eichhorn, A.; Gies, H.; Scherer, M.M. Asymptotically free scalar curvature-ghost coupling in Quantum Einstein Gravity. Phys. Rev. D
**2009**, 80, 104003. [Google Scholar] [CrossRef] - Groh, K.; Saueressig, F. Ghost wave-function renormalization in Asymptotically Safe Quantum Gravity. J. Phys. A
**2010**, 43, 365403. [Google Scholar] [CrossRef]

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. |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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