## Quantum gravity

We study the free and dressed propagators of physical and purely virtual particles in a finite interval of time $τ$ and on a compact space manifold $Ω$, using coherent states. In the free-field limit, the propagators are described by the entire function $(e^{z}-1-z)/z^{2}$, whose shape on the real axis is similar to the one of a Breit-Wigner function, with an effective width around $1/τ$. The real part is positive, in agreement with unitarity, and remains so after including the radiative corrections, which shift the function into the physical half plane. We investigate the effects of the restriction to finite $τ$ on the problem of unstable particles vs resonances, and show that the muon observation emerges from the right physical process, differently from what happens at $τ=\infty $. We also study the case of purely virtual particles, and show that, if $τ$ is small enough, there exists a situation where the geometric series of the self-energies is always convergent. The plots of the dressed propagators show testable differences: while physical particles are characterized by the usual, single peak, purely virtual particles are characterized by twin peaks.

We provide a diagrammatic formulation of perturbative quantum field theory in a finite interval of time $τ$, on a compact space manifold $Ω$. We explain how to compute the evolution operator $U(t_{\text{f}},t_{\text{i}})$ between the initial time $t_{\text{i}}$ and the final time $t_{\text{f}}=t_{\text{i}}+τ$, study unitarity and renormalizability, and show how to include purely virtual particles, by rendering some physical particles (and all the ghosts, if present) purely virtual. The details about the restriction to finite $τ$ and compact $Ω$ are moved away from the internal sectors of the diagrams (apart from the discretization of the three-momenta), and coded into external sources. So doing, the diagrams are as similar as possible to the usual $S$ matrix diagrams, and most known theorems extend straightforwardly. Unitarity is studied by means of the spectral optical identities, and the diagrammatic version of the identity $U^†(t_{\text{f}},t_{\text{i}})U(t_{\text{f}},t_{\text{i}})=1$. The dimensional regularization is extended to finite $τ$ and compact $Ω$, and used to prove, under general assumptions, that renormalizability holds whenever it holds at $τ=\infty $, $Ω=\mathbb{R}^{3}$. Purely virtual particles are introduced by removing the on-shell contributions of some physical particles, and the ghosts, from the core diagrams, and trivializing their initial and final conditions. The resulting evolution operator $U_{\text{ph}}(t_{\text{f}},t_{\text{i}})$ is unitary, but does not satisfy the more general identity $U_{\text{ph}}(t_{3},t_{2})U_{\text{ph}}(t_{2},t_{1})$ $=U_{\text{ph}}(t_{3},t_{1})$. As a consequence, $U_{\text{ph}}(t_{\text{f}},t_{\text{i}})$ cannot be derived from a Hamiltonian in a standard way, in the presence of purely virtual particles.

Quantum field theory is extended to include purely virtual “cloud sectors”, which allow us to define point-dependent observables, including a gauge invariant metric and gauge invariant matter fields, and calculate their off-shell correlation functions perturbatively in quantum gravity. Each extra sector is made of a cloud field, its anticommuting partner, a cloud function and a cloud Faddeev-Popov determinant. Thanks to certain cloud symmetries, the ordinary correlation functions and S matrix elements are unmodified. The clouds are rendered purely virtual, to ensure that they do not propagate unwanted degrees of freedom. So doing, the off-shell, diagrammatic version of the optical theorem holds and the extended theory is unitary. Every insertion in a correlation function can be dressed with its own cloud. The one-loop two-point functions of dressed scalars, vectors and gravitons are calculated. Their absorptive parts are positive, cloud independent and gauge independent, while they are unphysical if non purely virtual clouds are used. Renormalizability is proved to all orders by means of an extended Batalin-Vilkovisky formalism and its Zinn-Justin master equations. The purely virtual approach is compared to other approaches available in the literature.

The physics of fundamental interactions is going through a concerning, prolonged period of stagnation. The incredible success of the standard model of particle physics and the lack of new experimental data have frustrated our hopes in the future. On top of that, the scientific community shattered into a large number of isolated groups. Many mainstreams have consolidated, leaving not much room for the advancement of bright, original proposals. In frontier domains, like quantum gravity, most mainstreams have disavowed the inheritance of the glowing past and embarked on uncertain routes (string theory, loop quantum gravity and many others). It is time to make room for approaches that are really out of the box and can truly trigger a renaissance of particle physics. Yet, they can only be believable if they are solidly rooted in the successes of the past. This ERC project pursues a research line that does stem from the achievements of the past, but is radically new and has the potential to take us out of this dark period. It is based on the notion of purely virtual particle, which upgrades in a crucial way our understanding of fundamental interactions through quantum field theory. One of its key predictions in primordial cosmology could be confirmed experimentally within a decade. Nevertheless, the scientific community cannot afford another decade like the past ones, so it is imperative to act now. The new idea opens the door to unthinkable scenarios and has a huge amount of ramifications and applications to all areas of fundamental physics, with the potential to build bridges between quantum gravity, primordial cosmology and the phenomenology of particle physics beyond the standard model. More key predictions are expected to follow, together with crucial ideas for future colliders. Hopefully, they will trigger the breakthroughs that we need to make a U turn, activate a virtuous circle, reunite the scientific community and lead to the renaissance of particle physics.

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We review the concept of purely virtual particle and its uses in quantum gravity, primordial cosmology and collider physics. The fake particle, or “fakeon”, which mediates interactions without appearing among the incoming and outgoing states, can be introduced by means of a new diagrammatics. The renormalization coincides with one of the parent Euclidean diagrammatics, while unitarity follows from spectral optical identities, which can be derived by means of algebraic operations. The classical limit of a theory of physical particles and fakeons is described by an ordinary Lagrangian plus Hermitian, micro acausal and micro nonlocal self-interactions. Quantum gravity propagates the graviton, a massive scalar field (the inflaton) and a massive spin-2 fakeon, and leads to a constrained primordial cosmology, which predicts the tensor-to-scalar ratio r in the window 0.4≲1000r≲3.5. The interpretation of inflation as a cosmic RG flow allows us to calculate the perturbation spectra to high orders in the presence of the Weyl squared term. In models of new physics beyond the standard model, fakeons evade various phenomenological bounds, because they are less constrained than normal particles. The resummation of self-energies reveals that it is impossible to get too close to the fakeon peak. The related peak uncertainty, equal to the fakeon width divided by 2, is expected to be observable.

Symmetry 2022, 14(3), 521 | DOI: 10.3390/sym14030521

We reconsider the Lee-Wick (LW) models and compare their properties to the properties of the models that contain purely virtual particles. We argue against the LW premise that unstable particles can be removed from the sets of incoming and outgoing states in scattering processes. The removal leads to a non-Hermitian classical limit, besides clashing with the observation of the muon. If, on the other hand, all the states are included, the LW models have a Hamiltonian unbounded from below or negative norms. Purely virtual particles, on the contrary, lead to a Hermitian classical limit and are absent from the sets of incoming and outgoing states without implications on the observation of long-lived unstable particles. We give a *vademecum* to summarize the properties of most options to treat abnormal particles. We study a method to remove the LW ghosts only partially, by saving the physical particles they contain. Specifically, we replace a LW ghost with a certain superposition of a purely virtual particle and an ordinary particle, and drop only the former from the sets of the external states. The trick can be used to make the Pauli-Villars fields consistent and observable, without sending their masses to infinity, or to build a finite QED, by tweaking the original Lee-Wick construction. However, it has issues with general covariance, so it cannot be applied as is to quantum gravity, where a manifestly covariant decomposition requires the introduction of a massive spin-2 multiplet.

Phys. Rev. D 105 (2022) 125017 | DOI: 10.1103/PhysRevD.105.125017

We study the resummation of self-energy diagrams into dressed propagators in the case of purely virtual particles and compare the results with those obtained for physical particles and ghosts. The three geometric series differ by infinitely many contact terms, which do not admit well-defined sums. The peak region, which is outside the convergence domain, can only be reached in the case of physical particles, thanks to analyticity. In the other cases, nonperturbative effects become important. To clarify the matter, we introduce the energy resolution $\Delta E$ around the peak and argue that a “peak uncertainty” $\Delta E\gtrsim \Delta E_{\text{min}}\simeq \Gamma _{\text{f}}/2$ around energies $E\simeq m_{\text{f}}$ expresses the impossibility to approach the fakeon too closely, $m_{\text{f}}$ being the fakeon mass and $\Gamma _{\text{f}}$ being the fakeon width. The introduction of $\Delta E$ is also crucial to explain the observation of unstable long-lived particles, like the muon. Indeed, by the common energy-time uncertainty relation, such particles are also affected by ill-defined sums at $\Delta E=0$, whenever we separate their observation from the observation of their decay products. We study the regime of large $\Gamma _{\text{f}}$, which applies to collider physics (and situations like the one of the $Z$ boson), and the regime of small $\Gamma _{\text{f}}$, which applies to quantum gravity (and situations like the one of the muon).

J. High Energy Phys. 06 (2022) 058 | DOI: 10.1007/JHEP06(2022)058

We prove spectral optical identities in quantum field theories of physical particles (defined by the Feynman $i\epsilon $ prescription) and purely virtual particles (defined by the fakeon prescription). The identities are derived by means of purely algebraic operations and hold for every (multi)threshold separately and for arbitrary frequencies. Their major significance is that they offer a deeper understanding on the problem of unitarity in quantum field theory. In particular, they apply to “skeleton” diagrams, before integrating on the space components of the loop momenta and the phase spaces. In turn, the skeleton diagrams obey a spectral optical theorem, which gives the usual optical theorem for amplitudes, once the integrals on the space components of the loop momenta and the phase spaces are restored. The fakeon

prescription/projection is implemented by dropping the thresholds that involve fakeon frequencies. We give examples at one loop (bubble, triangle, box, pentagon and hexagon), two loops (triangle with “diagonal”, box with diagonal) and arbitrarily many loops. We also derive formulas for the loop integrals with fakeons and relate them to the known formulas for the loop integrals with physical particles.

J. High Energy Phys. 11 (2021) 030 | DOI: https://doi.org/10.1007/JHEP11(2021)030

We study primordial cosmology with two scalar fields that participate in inflation at the same time, by coupling quantum gravity (i.e., the theory $R+R^2+C^2$ with the fakeon prescription/projection for $C^2$) to a scalar field with a quadratic potential. We show that there exists a perturbative regime that can be described by an asymptotically de Sitter, cosmic RG flow in two couplings. Since the two scalar degrees of freedom mix in nontrivial ways, the adiabatic and isocurvature perturbations are not RG invariant on superhorizon scales. It is possible to identify the correct perturbations by using RG invariance as a guiding principle. We work out the resulting power spectra of the tensor and scalar perturbations to the NNLL and NLL orders, respectively. An unexpected consequence of RG invariance is that the theory remains predictive. Indeed, the scalar mixing affects only the subleading corrections, so the predictions of quantum gravity with single-field inflation are confirmed to the leading order.

J. Cosmol. Astropart. Phys. 07 (2021) 037 | DOI: 10.1088/1475-7516/2021/07/037

We study inflation as a “cosmic” renormalization-group flow. The flow, which encodes the dependence on the background metric, is described by a running coupling $\alpha $, which parametrizes the slow roll, a de Sitter free, analytic beta function and perturbation spectra that are RG invariant in the superhorizon limit. Using RG invariance as a guiding principle, we classify the main types of flows according to the properties of their spectra, without referring to their origins from specific actions or models. Novel features include spectra with essential singularities in $\alpha $ and violations of the relation $r+8n_{\text{t}}=0$ to the leading order. Various classes of potentials studied in the literature can be described by means of the RG approach, even when the action includes a Weyl-squared term, while others are left out. In known cases, the classification helps identify the models that are ruled out by data. The RG approach is also able to generate spectra that cannot be derived from standard Lagrangian formulations.

Class. Quantum Grav. 38 (2021) 225011 | DOI: 10.1088/1361-6382/ac2b07