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.
Highlights
To overcome these difficulties, further prescriptions were supplemented later
We conclude this section by mentioning other integral representations of the average continuation, which will be useful for the proof of perturbative unitarity
We take a unique LW threshold P and assume that it is located on the real axis
Summary
We study the Lee-Wick models by nonanalytically Wick rotating their Euclidean versions. In D = 2 there is no singularity, because the pinching just occurs at the boundaries γ, γdef of the regions AP , AdPef. We view the result of the calculation in AdPef as a function of the k integration domain Dkdef. The loop integral is both Lorentz invariant and analytic in (a neighborhood of) OP , even before making the domain deformation We check these properties explicitly in the examples of section 5. When p → 0 the corresponding regions Ai squeeze onto curves with endpoints at the thresholds The calculations beyond such thresholds are performed with a procedure analogous to the one described above: first, we evaluate the loop integral inside a region Ai; we deform the k integration domain till Ai gets squeezed onto a curve; we take such a curve as the final region Ai, or enlarge it to some neighborhood of it by analytic continuing the result found in it. We arrange each Ai so as to make it Lorentz invariant for real external momenta
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