Abstract
The “fakeon” is a fake degree of freedom, i.e. a degree of freedom that does not belong to the physical spectrum, but propagates inside the Feynman diagrams. Fakeons can be used to make higher-derivative theories unitary. Moreover, they help us clarify how the Lee-Wick models work. In this paper we study the fakeon models, that is to say the theories that contain fake and physical degrees of freedom. We formulate them by (nonanalytically) Wick rotating their Euclidean versions. We investigate the properties of arbitrary Feynman diagrams and, among other things, prove that the fakeon models are perturbatively unitary to all orders. If standard power counting constraints are fulfilled, the models are also renormalizable. The S matrix is regionwise analytic. The amplitudes can be continued from the Euclidean region to the other regions by means of an unambiguous, but nonanalytic, operation, called average continuation. We compute the average continuation of typical amplitudes in four, three and two dimensions and show that its predictions agree with those of the nonanalytic Wick rotation. By reconciling renormalizability and unitarity in higher-derivative theories, the fakeon models are good candidates to explain quantum gravity.
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References
T.D. Lee and G.C. Wick, Negative metric and the unitarity of the S-matrix, Nucl. Phys. B 9 (1969) 209 [INSPIRE].
T.D. Lee and G.C. Wick, Finite theory of quantum electrodynamics, Phys. Rev. D 2 (1970) 1033 [INSPIRE].
R.E. Cutkosky, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, A non-analytic S-matrix, Nucl. Phys. B 12 (1969) 281 [INSPIRE].
T.D. Lee, A relativistic complex pole model with indefinite metric, in Quanta: Essays in Theoretical Physics Dedicated to Gregor Wentzel, Chicago University Press, Chicago, U.S.A. (1970), p. 260.
N. Nakanishi, Lorentz noninvariance of the complex-ghost relativistic field theory, Phys. Rev. D 3 (1971) 811 [INSPIRE].
B. Grinstein, D. O’Connell and M.B. Wise, Causality as an emergent macroscopic phenomenon: The Lee-Wick O(N) model, Phys. Rev. D 79 (2009) 105019 [arXiv:0805.2156] [INSPIRE].
D. Anselmi and M. Piva, A new formulation of Lee-Wick quantum field theory, JHEP 06 (2017) 066 [arXiv:1703.04584] [INSPIRE].
D. Anselmi and M. Piva, Perturbative unitarity of Lee-Wick quantum field theory, Phys. Rev. D 96 (2017) 045009 [arXiv:1703.05563] [INSPIRE].
R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].
M.J.G. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29 (1963) 186 [INSPIRE].
D. Anselmi, Aspects of perturbative unitarity, Phys. Rev. D 94 (2016) 025028 [arXiv:1606.06348] [INSPIRE].
D. Anselmi, Algebraic cutting equations, arXiv:1612.07148 [INSPIRE].
G. ’t Hooft, Renormalization of massless Yang-Mills fields, Nucl. Phys. B 33 (1971) 173 [INSPIRE].
G. ’t Hooft, Renormalizable lagrangians for massive Yang-Mills fields, Nucl. Phys. B 35 (1971) 167 [INSPIRE].
K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
J. Julve and M. Tonin, Quantum gravity with higher derivative terms, Nuovo Cim. B 46 (1978) 137 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [INSPIRE].
I.G. Avramidi and A.O. Barvinsky, Asymptotic freedom in higher derivative quantum gravity, Phys. Lett. 159B (1985) 269 [INSPIRE].
D. Anselmi, On the quantum field theory of the gravitational interactions, JHEP 06 (2017) 086 [arXiv:1704.07728] [INSPIRE].
B. Grinstein, D. O’Connell and M.B. Wise, The Lee-Wick standard model, Phys. Rev. D 77 (2008) 025012 [arXiv:0704.1845] [INSPIRE].
C.D. Carone and R.F. Lebed, Minimal Lee-Wick extension of the Standard Model, Phys. Lett. B 668 (2008) 221 [arXiv:0806.4555] [INSPIRE].
J.R. Espinosa and B. Grinstein, Ultraviolet properties of the Higgs Sector in the Lee-Wick standard model, Phys. Rev. D 83 (2011) 075019 [arXiv:1101.5538] [INSPIRE].
C.D. Carone and R.F. Lebed, A higher-derivative Lee-Wick standard model, JHEP 01 (2009) 043 [arXiv:0811.4150] [INSPIRE].
B. Grinstein and D. O’Connell, One-loop renormalization of Lee-Wick gauge theory, Phys. Rev. D 78 (2008) 105005 [arXiv:0801.4034] [INSPIRE].
C.D. Carone, Higher-derivative Lee-Wick unification, Phys. Lett. B 677 (2009) 306 [arXiv:0904.2359] [INSPIRE].
E. Tomboulis, 1/N expansion and renormalization in quantum gravity, Phys. Lett. 70B (1977) 361 [INSPIRE].
E. Tomboulis, Renormalizability and asymptotic freedom in quantum gravity, Phys. Lett. 97B (1980) 77 [INSPIRE].
L. Modesto and I.L. Shapiro, Superrenormalizable quantum gravity with complex ghosts, Phys. Lett. B 755 (2016) 279 [arXiv:1512.07600] [INSPIRE].
L. Modesto, Super-renormalizable or finite Lee-Wick quantum gravity, Nucl. Phys. B 909 (2016) 584 [arXiv:1602.02421] [INSPIRE].
U.G. Aglietti and D. Anselmi, Inconsistency of Minkowski higher-derivative theories, Eur. Phys. J. C 77 (2017) 84 [arXiv:1612.06510] [INSPIRE].
F. Bloch and A. Nordsieck, Note on the radiation field of the electron, Phys. Rev. 52 (1937) 54 [INSPIRE].
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
For details, see R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge, U.K., (1966).
A. Salvio and A. Strumia, Agravity up to infinite energy, Eur. Phys. C 78 (2018) 124 [arXiv:1705.03896] [INSPIRE].
A. Salvio and A. Strumia, Agravity, JHEP 06 (2014) 080 [arXiv:1403.4226] [INSPIRE].
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Anselmi, D. Fakeons and Lee-Wick models. J. High Energ. Phys. 2018, 141 (2018). https://doi.org/10.1007/JHEP02(2018)141
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DOI: https://doi.org/10.1007/JHEP02(2018)141