Abstract
We consider the one-loop renormalization of a real scalar field interacting with a Dirac spinor field in curved spacetime. A general Yukawa interaction is considered which includes both a scalar and a pseudoscalar coupling. The scalar field is assumed to be non-minimally coupled to the gravitational field and to have a general quartic self-interaction potential. All of the one-loop renormalization group functions are evaluated and in the special case where there is no mass scale present in the classical theory (apart from the fields) we evaluate the one-loop effective action up to and including order R2 in the curvature. In the case where the fermion is massive we include a pseudoscalar mass term in γ5 and we show that although the γ5 term can be removed by a redefinition of the spinor field an anomaly in the effective action arises that is related to the familiar axial current anomaly.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.W. Gibbons, Quantum field theory in curved spacetime, in General relativity: an Einstein centenary survey, S.W. Hawking and W. Israel eds., Cambridge University Press, (1979), pp. 639-679.
N.D. Birrell and P.C.W. Davies, Quantum fields in Curved Space, Cambridge University Press, (1982).
S.A. Fulling, Aspects of Quantum Field Theory in Curved Spacetime, Cambridge University Press, (1989).
I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective Action in Quantum Gravity. IOP Publishing, (1992).
L.H. Ford, Quantum field theory in curved space-time, in Particles and fields. Proceedings, 9th Jorge Andre Swieca Summer School, Campos do Jordao, Brazil, February 16-28, 1997, pp. 345-388, gr-qc/9707062 [INSPIRE].
L.E. Parker and D.J. Toms, Quantum Field Theory in Curved Spacetime, Cambridge University Press, (2009).
B.S. DeWitt, Dynamical Theory of Groups and Fields. Gordon and Breach, (1965).
B.S. DeWitt, Quantum Field Theory in Curved Space-Time, Phys. Rept. 19 (1975) 295 [INSPIRE].
L. Parker and D.J. Toms, Renormalization Group Analysis of Grand Unified Theories in Curved Space-time, Phys. Rev. D 29 (1984) 1584 [INSPIRE].
I.L. Shapiro, Asymptotic Behavior of Effective Yukawa Coupling Constants in Quantum R 2 Gravity With Matter, Class. Quant. Grav. 6 (1989) 1197 [INSPIRE].
S.D. Odintsov and I.L. Shapiro, General relativity as the low-energy limit in higher derivative quantum gravity, Class. Quant. Grav. 9 (1992) 873 [INSPIRE].
E. Elizalde and S.D. Odintsov, Renormalization group improved effective potential for interacting theories with several mass scales in curved space-time, Z. Phys. C 64 (1994) 699 [hep-th/9401057] [INSPIRE].
E. Elizalde and S.D. Odintsov, The Higgs-Yukawa model in curved space-time, Phys. Rev. D 51 (1995) 5950 [hep-th/9503111] [INSPIRE].
E. Elizalde, S.D. Odintsov and A. Romeo, Improved effective potential in curved space-time and quantum matter, higher derivative gravity theory, Phys. Rev. D 51 (1995) 1680 [hep-th/9410113] [INSPIRE].
E. Elizalde and S.D. Odintsov, A renormalization group improved nonlocal gravitational effective Lagrangian, Mod. Phys. Lett. A 10 (1995) 1821 [gr-qc/9508041] [INSPIRE].
F. Sobreira, B.J. Ribeiro and I.L. Shapiro, Effective Potential in Curved Space and Cut-Off Regularizations, Phys. Lett. B 705 (2011) 273 [arXiv:1107.2262] [INSPIRE].
M. Herranen, T. Markkanen, S. Nurmi and A. Rajantie, Spacetime curvature and the Higgs stability during inflation, Phys. Rev. Lett. 113 (2014) 211102 [arXiv:1407.3141] [INSPIRE].
O. Czerwinska, Z. Lalak and L. Nakonieczny, Stability of the effective potential of the gauge-less top-Higgs model in curved spacetime, JHEP 11 (2015) 207 [arXiv:1508.03297] [INSPIRE].
T. Markkanen, S. Nurmi, A. Rajantie and S. Stopyra, The 1-loop effective potential for the Standard Model in curved spacetime, arXiv:1804.02020 [INSPIRE].
T. Prokopec and R.P. Woodard, Production of massless fermions during inflation, JHEP 10 (2003) 059 [astro-ph/0309593] [INSPIRE].
B. Garbrecht and T. Prokopec, Fermion mass generation in de Sitter space, Phys. Rev. D 73 (2006) 064036 [gr-qc/0602011] [INSPIRE].
B. Garbrecht, Ultraviolet Regularisation in de Sitter Space, Phys. Rev. D 74 (2006) 043507 [hep-th/0604166] [INSPIRE].
S.-P. Miao and R.P. Woodard, Leading log solution for inflationary Yukawa, Phys. Rev. D 74 (2006) 044019 [gr-qc/0602110] [INSPIRE].
L.D. Duffy and R.P. Woodard, Yukawa scalar self-mass on a conformally flat background, Phys. Rev. D 72 (2005) 024023 [hep-ph/0505156] [INSPIRE].
O. Zanusso, L. Zambelli, G.P. Vacca and R. Percacci, Gravitational corrections to Yukawa systems, Phys. Lett. B 689 (2010) 90 [arXiv:0904.0938] [INSPIRE].
A. Eichhorn, A. Held and J.M. Pawlowski, Quantum-gravity effects on a Higgs-Yukawa model, Phys. Rev. D 94 (2016) 104027 [arXiv:1604.02041] [INSPIRE].
A. Eichhorn and A. Held, Mass difference for charged quarks from quantum gravity, arXiv:1803.04027 [INSPIRE].
K.-y. Oda and M. Yamada, Non-minimal coupling in Higgs-Yukawa model with asymptotically safe gravity, Class. Quant. Grav. 33 (2016) 125011 [arXiv:1510.03734] [INSPIRE].
N. Christiansen, A. Eichhorn and A. Held, Is scale-invariance in gauge-Yukawa systems compatible with the graviton?, Phys. Rev. D 96 (2017) 084021 [arXiv:1705.01858] [INSPIRE].
A. Rodigast and T. Schuster, Gravitational Corrections to Yukawa and phi**4 Interactions, Phys. Rev. Lett. 104 (2010) 081301 [arXiv:0908.2422] [INSPIRE].
G. Narain, Exorcising Ghosts in Induced Gravity, Eur. Phys. J. C 77 (2017) 683 [arXiv:1612.04930] [INSPIRE].
S. González-Martín and C.P. Martin, Do the gravitational corrections to the β-functions of the quartic and Yukawa couplings have an intrinsic physical meaning?, Phys. Lett. B 773 (2017) 585 [arXiv:1707.06667] [INSPIRE].
S. González-Martín and C.P. Martin, Unimodular Gravity and General Relativity UV divergent contributions to the scattering of massive scalar particles, JCAP 01 (2018) 028 [arXiv:1711.08009] [INSPIRE].
S. González-Martín and C.P. Martin, Scattering of fermions in the Yukawa theory coupled to Unimodular Gravity, Eur. Phys. J. C 78 (2018) 236 [arXiv:1802.03755] [INSPIRE].
M.M. Anber, J.F. Donoghue and M. El-Houssieny, Running couplings and operator mixing in the gravitational corrections to coupling constants, Phys. Rev. D 83 (2011) 124003 [arXiv:1011.3229] [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, (1965).
I.G. Avramidi, Heat Kernel and Quantum Gravity, vol. 64., Springer, (2000).
D.V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
K. Kirsten, Spectral Functions in Mathematics and Physics, CRC Press, (2010).
T.S. Bunch and L. Parker, Feynman Propagator in Curved Space-Time: A Momentum Space Representation, Phys. Rev. D 20 (1979) 2499 [INSPIRE].
P.B. Gilkey, Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compos. Math. 38 (1979) 201.
P.B. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975) 601.
L. Parker and D.J. Toms, New Form for the Coincidence Limit of the Feynman Propagator, or Heat Kernel, in Curved Space-time, Phys. Rev. D 31 (1985) 953 [INSPIRE].
I. Jack and L. Parker, Proof of Summed Form of Proper Time Expansion for Propagator in Curved Space-time, Phys. Rev. D 31 (1985) 2439 [INSPIRE].
D.J. Toms, Renormalization of Interacting Scalar Field Theories in Curved Space-time, Phys. Rev. D 26 (1982) 2713 [INSPIRE].
M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, (1995).
D.J. Toms, Quantum gravitational contributions to quantum electrodynamics, Nature 468 (2010) 56 [arXiv:1010.0793] [INSPIRE].
D.J. Toms, Quadratic divergences and quantum gravitational contributions to gauge coupling constants, Phys. Rev. D 84 (2011) 084016 [INSPIRE].
G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].
S.R. Coleman and E.J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7 (1973) 1888 [INSPIRE].
D.J. Toms, The Effective Action and the Renormalization Group Equation in Curved Space-time, Phys. Lett. B 126 (1983) 37 [INSPIRE].
I.L. Buchbinder and S.D. Odintsov, Effective Potential and Phase Transitions Induced by Curvature in Gauge Theories in Curved Space-time, Yad. Fiz. 42 (1985) 1268 [INSPIRE].
K. Fujikawa, Path Integral Measure for Gauge Invariant Fermion Theories, Phys. Rev. Lett. 42 (1979) 1195 [INSPIRE].
K. Fujikawa, Comment on Chiral and Conformal Anomalies, Phys. Rev. Lett. 44 (1980) 1733 [INSPIRE].
K. Fujikawa, Path Integral for Gauge Theories with Fermions, Phys. Rev. D 21 (1980) 2848 [Erratum ibid. D 22 (1980) 1499] [INSPIRE].
D.J. Toms, Gauged Yukawa model in curved spacetime, arXiv:1805.01700 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1804.08350
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Toms, D.J. Effective action for the Yukawa model in curved spacetime. J. High Energ. Phys. 2018, 139 (2018). https://doi.org/10.1007/JHEP05(2018)139
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2018)139