Abstract
We explicitly compute the tree-level on-shell four-graviton amplitudes in four, five and six dimensions for local and weakly nonlocal gravitational theories that are quadratic in both, the Ricci and scalar curvature with form factors of the d’Alembertian operator inserted between. More specifically we are interested in renormalizable, super-renormalizable or finite theories. The scattering amplitudes for these theories turn out to be the same as the ones of Einstein gravity regardless of the explicit form of the form factors. As a special case the four-graviton scattering amplitudes in Weyl conformal gravity are identically zero. Using a field redefinition, we prove that the outcome is correct for any number of external gravitons (on-shell n−point functions) and in any dimension for a large class of theories. However, when an operator quadratic in the Riemann tensor is added in any dimension (with the exception of the Gauss-Bonnet term in four dimensions) the result is completely altered, and the scattering amplitudes depend on all the form factors introduced in the action.
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References
Y. Wang and X. Yin, Constraining Higher Derivative Supergravity with Scattering Amplitudes, arXiv:1502.03810 [INSPIRE].
B.S. DeWitt, Quantum Theory of Gravity. 3. Applications of the Covariant Theory, Phys. Rev. 162 (1967) 1239 [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
F.A. Berends and R. Gastmans, On the High-Energy Behavior in Quantum Gravity, Nucl. Phys. B 88 (1975) 99 [INSPIRE].
M.T. Grisaru, P. van Nieuwenhuizen and C.C. Wu, Gravitational Born Amplitudes and Kinematical Constraints, Phys. Rev. D 12 (1975) 397 [INSPIRE].
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
E. Tomboulis, Renormalizability and Asymptotic Freedom in Quantum Gravity, Phys. Lett. B 97 (1980) 77 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [INSPIRE].
X. Calmet, The Lightest of Black Holes, Mod. Phys. Lett. A 29 (2014) 1450204 [arXiv:1410.2807] [INSPIRE].
X. Calmet, D. Croon and C. Fritz, Non-locality in Quantum Field Theory due to General Relativity, arXiv:1505.04517 [INSPIRE].
T.R. Morris, Elements of the continuous renormalization group, Prog. Theor. Phys. Suppl. 131 (1998) 395 [hep-th/9802039] [INSPIRE].
R. Percacci, Asymptotic Safety, arXiv:0709.3851 [INSPIRE].
D.F. Litim, Fixed Points of Quantum Gravity and the Renormalisation Group, arXiv:0810.3675 [INSPIRE].
I.G. Avramidi, The Covariant technique for the calculation of the heat kernel asymptotic expansion, Phys. Lett. B 238 (1990) 92 [INSPIRE].
D.F. Litim, Critical exponents from optimized renormalization group flows, Nucl. Phys. B 631 (2002) 128 [hep-th/0203006] [INSPIRE].
A. Codello, R. Percacci, L. Rachwal and A. Tonero, Computing the Effective Action with the Functional Renormalization Group, arXiv:1505.03119 [INSPIRE].
L. Modesto, Super-renormalizable Quantum Gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [INSPIRE].
L. Modesto, Super-renormalizable Multidimensional Quantum Gravity, Astron. Rev. 8.2 (2013) 4 arXiv:1202.3151 [INSPIRE].
L. Modesto, Multidimensional finite quantum gravity, arXiv:1402.6795 [INSPIRE].
L. Modesto, Super-renormalizable Higher-Derivative Quantum Gravity, arXiv:1202.0008 [INSPIRE].
L. Modesto and L. Rachwal, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [arXiv:1407.8036] [INSPIRE].
L. Modesto and L. Rachwal, Universally Finite Gravitational & Gauge Theories, arXiv:1503.00261 [INSPIRE].
F. Briscese, L. Modesto and S. Tsujikawa, Super-renormalizable or finite completion of the Starobinsky theory, Phys. Rev. D 89 (2014) 024029 [arXiv:1308.1413] [INSPIRE].
L. Modesto and S. Tsujikawa, Non-local massive gravity, Phys. Lett. B 727 (2013) 48 [arXiv:1307.6968] [INSPIRE].
M. Piva, Nonlocal theories of quantum gravity and gauge fields, Master Thesis, University of Pisa, Italy (2014).
E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146 [INSPIRE].
M. Eran, Higher-derivative Gauge And Gravitational Theories (supersymmetry), PhD thesis, University of California, Los Angeles (1998).
N.V. Krasnikov, Nonlocal gauge theories, Theor. Math. Phys. 73 (1987) 1184 [INSPIRE].
Yu. V. Kuzmin, The convergent nonlocal gravitation (in Russian), Sov. J. Nucl. Phys. 50 (1989) 1011 [INSPIRE].
D. Anselmi, Quantum gravity and renormalization, Mod. Phys. Lett. A 30 (2015) 1540004.
D. Anselmi, Background field method, Batalin-Vilkovisky formalism and parametric completeness of renormalization, Phys. Rev. D 89 (2014) 045004 [arXiv:1311.2704] [INSPIRE].
J.W. Moffat, Ultraviolet Complete Quantum Gravity, Eur. Phys. J. Plus 126 (2011) 43 [arXiv:1008.2482] [INSPIRE].
N.J. Cornish, Quantum nonlocal gravity, Mod. Phys. Lett. A 7 (1992) 631 [INSPIRE].
L. Modesto, J.W. Moffat and P. Nicolini, Black holes in an ultraviolet complete quantum gravity, Phys. Lett. B 695 (2011) 397 [arXiv:1010.0680] [INSPIRE].
C. Bambi, D. Malafarina and L. Modesto, Non-singular quantum-inspired gravitational collapse, Phys. Rev. D 88 (2013) 044009 [arXiv:1305.4790] [INSPIRE].
C. Bambi, D. Malafarina and L. Modesto, Terminating black holes in asymptotically free quantum gravity, Eur. Phys. J. C 74 (2014) 2767 [arXiv:1306.1668] [INSPIRE].
G. Calcagni, L. Modesto and P. Nicolini, Super-accelerating bouncing cosmology in asymptotically-free non-local gravity, Eur. Phys. J. C 74 (2014) 2999 [arXiv:1306.5332] [INSPIRE].
B. Craps, T. De Jonckheere and A.S. Koshelev, Cosmological perturbations in non-local higher-derivative gravity, JCAP 11 (2014) 022 [arXiv:1407.4982] [INSPIRE].
A.S. Koshelev and S.Y. Vernov, Cosmological Solutions in Nonlocal Models, Phys. Part. Nucl. Lett. 11 (2014) 960 [arXiv:1406.5887] [INSPIRE].
A.S. Koshelev, Stable analytic bounce in non-local Einstein-Gauss-Bonnet cosmology, Class. Quant. Grav. 30 (2013) 155001 [arXiv:1302.2140] [INSPIRE].
T. Biswas, A.S. Koshelev, A. Mazumdar and S. Yu. Vernov, Stable bounce and inflation in non-local higher derivative cosmology, JCAP 08 (2012) 024 [arXiv:1206.6374] [INSPIRE].
A.S. Koshelev and S.Y. Vernov, On bouncing solutions in non-local gravity, Phys. Part. Nucl. 43 (2012) 666 [arXiv:1202.1289] [INSPIRE].
A.S. Koshelev, Modified non-local gravity, Rom. J. Phys. 57 (2012) 894 [arXiv:1112.6410] [INSPIRE].
S. Yu. Vernov, Nonlocal Gravitational Models and Exact Solutions, Phys. Part. Nucl. 43 (2012) 694 [arXiv:1202.1172] [INSPIRE].
A.S. Koshelev and S. Yu. Vernov, Cosmological perturbations in SFT inspired non-local scalar field models, Eur. Phys. J. C 72 (2012) 2198 [arXiv:0903.5176] [INSPIRE].
A.S. Koshelev, Non-local SFT Tachyon and Cosmology, JHEP 04 (2007) 029 [hep-th/0701103] [INSPIRE].
L. Modesto, T. de Paula Netto and I.L. Shapiro, On Newtonian singularities in higher derivative gravity models, JHEP 04 (2015) 098 [arXiv:1412.0740] [INSPIRE].
Y.-D. Li, L. Modesto and L. Rachwal, Exact solutions and spacetime singularities in nonlocal gravity, arXiv:1506.08619 [INSPIRE].
V.P. Frolov, Mass-gap for black hole formation in higher derivative and ghost free gravity, Phys. Rev. Lett. 115 (2015) 051102 [arXiv:1505.00492] [INSPIRE].
V.P. Frolov, A. Zelnikov and T. de Paula Netto, Spherical collapse of small masses in the ghost-free gravity, JHEP 06 (2015) 107 [arXiv:1504.00412] [INSPIRE].
V.P. Frolov, Do Black Holes Exist?, arXiv:1411.6981 [INSPIRE].
V.P. Frolov, Information loss problem and a ’black hole‘ model with a closed apparent horizon, JHEP 05 (2014) 049 [arXiv:1402.5446] [INSPIRE].
V.P. Frolov and G.A. Vilkovisky, Spherically Symmetric Collapse in Quantum Gravity, Phys. Lett. B 106 (1981) 307 [INSPIRE].
V.P. Frolov and G.A. Vilkovisky, Quantum Gravity Removes Classical Singularities And Shortens The Life Of Black Holes, IC-79-69.
D. Anselmi, Properties Of The Classical Action Of Quantum Gravity, JHEP 05 (2013) 028 [arXiv:1302.7100] [INSPIRE].
D. Anselmi, Renormalization and causality violations in classical gravity coupled with quantum matter, JHEP 01 (2007) 062 [hep-th/0605205] [INSPIRE].
D. Anselmi, Absence of higher derivatives in the renormalization of propagators in quantum field theories with infinitely many couplings, Class. Quant. Grav. 20 (2003) 2355 [hep-th/0212013] [INSPIRE].
N. Marcus and A. Sagnotti, The Ultraviolet Behavior of N = 4 Yang-Mills and the Power Counting of Extended Superspace, Nucl. Phys. B 256 (1985) 77 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Annal. Poincare Phys. Theor. A 20 (1974) 69.
M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].
S. Deser and A.N. Redlich, String Induced Gravity and Ghost Freedom, Phys. Lett. B 176 (1986) 350 [Erratum ibid. 186B (1987) 461] [INSPIRE].
T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249] [INSPIRE].
T. Biswas, T. Koivisto and A. Mazumdar, Nonlocal theories of gravity: the flat space propagator, arXiv:1302.0532 [INSPIRE].
S. Alexander, A. Marciano and L. Modesto, The Hidden Quantum Groups Symmetry of Super-renormalizable Gravity, Phys. Rev. D 85 (2012) 124030 [arXiv:1202.1824] [INSPIRE].
F. Briscese, A. Marcianó, L. Modesto and E.N. Saridakis, Inflation in (Super-)renormalizable Gravity, Phys. Rev. D 87 (2013) 083507 [arXiv:1212.3611] [INSPIRE].
J. Khoury, Fading gravity and self-inflation, Phys. Rev. D 76 (2007) 123513 [hep-th/0612052] [INSPIRE].
G. Calcagni and L. Modesto, Nonlocal quantum gravity and M-theory, Phys. Rev. D 91 (2015) 124059 [arXiv:1404.2137] [INSPIRE].
L. Modesto, Towards a finite quantum supergravity, arXiv:1206.2648 [INSPIRE].
G. Calcagni, M. Montobbio and G. Nardelli, Localization of nonlocal theories, Phys. Lett. B 662 (2008) 285 [arXiv:0712.2237] [INSPIRE].
G. Calcagni and G. Nardelli, Non-local gravity and the diffusion equation, Phys. Rev. D 82 (2010) 123518 [arXiv:1004.5144] [INSPIRE].
G.V. Efimov, Nonlocal Interactions (in Russian), Nauka, Moscow (1977).
V.A. Alebastrov and G.V. Efimov, A proof of the unitarity of S matrix in a nonlocal quantum field theory, Commun. Math. Phys. 31 (1973) 1 [INSPIRE].
V.A. Alebastrov and G.V. Efimov, Causality in the quantum field theory with the nonlocal interaction, Commun. Math. Phys. 38 (1974) 11 [INSPIRE].
G.V. Efimov, Amplitudes in nonlocal theories at high energies, Theor. Math. Phys. 128 (2001) 1169 [INSPIRE].
D. Anselmi, Functional integration measure in quantum gravity, Phys. Rev. D 45 (1992) 4473 [INSPIRE].
D. Anselmi, On delta(0) divergences and the functional integration measure, Phys. Rev. D 48 (1993) 680 [INSPIRE].
D. Anselmi, Covariant Pauli-Villars regularization of quantum gravity at the one loop order, Phys. Rev. D 48 (1993) 5751 [hep-th/9307014] [INSPIRE].
D. Anselmi, Background field method, Batalin-Vilkovisky formalism and parametric completeness of renormalization, Phys. Rev. D 89 (2014) 045004 [arXiv:1311.2704] [INSPIRE].
D. Anselmi, Weighted power counting and chiral dimensional regularization, Phys. Rev. D 89 (2014) 125024 [arXiv:1405.3110] [INSPIRE].
D. C. Dunbar and P. S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [INSPIRE].
J.F. Donoghue and T. Torma, Infrared behavior of graviton-graviton scattering, Phys. Rev. D 60 (1999) 024003 [hep-th/9901156] [INSPIRE].
T. Biswas and N. Okada, Towards LHC physics with nonlocal Standard Model, Nucl. Phys. B 898 (2015) 113 [arXiv:1407.3331] [INSPIRE].
I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective action in quantum gravity, IOP Publishing, U.K. (1992).
M. Asorey, J.L. Lopez and I.L. Shapiro, Some remarks on high derivative quantum gravity, Int. J. Mod. Phys. A 12 (1997) 5711 [hep-th/9610006] [INSPIRE].
A. Accioly, A. Azeredo and H. Mukai, Propagator, tree-level unitarity and effective nonrelativistic potential for higher-derivative gravity theories in D dimensions, J. Math. Phys. 43 (2002) 473 [INSPIRE].
F.d.O. Salles and I.L. Shapiro, Do we have unitary and (super)renormalizable quantum gravity below the Planck scale?, Phys. Rev. D 89 (2014) 084054 [arXiv:1401.4583] [INSPIRE].
R.J. Rivers, Lagrangian Theory for Neutral Massive Spin-2 Fields, Nuovo Cim. 34 (1964) 386.
P.D. Mannheim, Making the Case for Conformal Gravity, Found. Phys. 42 (2012) 388 [arXiv:1101.2186] [INSPIRE].
P. Van Nieuwenhuizen, On ghost-free tensor lagrangians and linearized gravitation, Nucl. Phys. B 60 (1973) 478 [INSPIRE].
B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156 (1985) 315 [INSPIRE].
D. Hochberg and T. Shimada, Ambiguity in Determining the Effective Action for String Corrected Einstein Gravity, Prog. Theor. Phys. 78 (1987) 680 [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
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Donà, P., Giaccari, S., Modesto, L. et al. Scattering amplitudes in super-renormalizable gravity. J. High Energ. Phys. 2015, 38 (2015). https://doi.org/10.1007/JHEP08(2015)038
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DOI: https://doi.org/10.1007/JHEP08(2015)038