### Course

19S1 D. Anselmi
Theories of gravitation

Program

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### Book

D. Anselmi
From Physics To Life

A journey to the infinitesimally small and back

In English and Italian

Available on Amazon:
US: book | ebook  (in EN)
IT: book | ebook  (in IT)

## Quantum gravity

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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The techical part of the application can be viewed here

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.

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

arXiv: 2203.02516 [hep-th]

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.

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

arXiv: 2202.10483 [hep-th]

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).

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

arXiv: 2201.00832 [hep-ph]

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.

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

arXiv: 2105.05864 [hep-th]

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.

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

arXiv: 2103.01653 [hep-th]

We compute the inflationary perturbation spectra and the quantity $r+8n_{T}$ to the next-to-next-to-leading log order in quantum gravity with purely virtual particles (which means the theory $R+R^{2}+C^{2}$ with the fakeon prescription/projection for $C^{2}$). The spectra are functions of the inflationary running coupling $\alpha (1/k)$ and satisfy the cosmic renormalization-group flow equations, which determine the tilts and the running coefficients. The tensor fluctuations receive contributions from the spin-2 fakeon $\chi _{\mu \nu }$ at every order of the expansion in powers of $\alpha \sim 1/115$. The dependence of the scalar spectrum on the $\chi _{\mu \nu }$ mass $m_{\chi }$, on the other hand, starts from the $\alpha^{2}$ corrections, which are handled perturbatively in the ratio $m_{\phi}/m_{\chi }$, where $m_{\phi }$ is the inflaton mass. The predictions have theoretical errors ranging from $\alpha ^{4}\sim 10^{-8}$ to $\alpha^{3}\sim 10^{-6}$. Nontrivial issues concerning the fakeon projection at higher orders are addressed.

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J. Cosmol. Astropart. Phys. 02 (2021) 029 | DOI: 10.1088/1475-7516/2021/02/029

arXiv: 2010.04739 [hep-th]

We study the running of power spectra in inflationary cosmology as a renormalization-group flow from the de Sitter fixed point. The beta function is provided by the equations of the background metric. The spectra of the scalar and tensor fluctuations obey RG evolution equations with vanishing anomalous dimensions in the superhorizon limit. By organizing the perturbative expansion in terms of leading and subleading logs, we calculate the spectral indices, their runnings, the runnings of the runnings, etc., to the next-to-leading log order in quantum gravity with fakeons (i.e., the theory $R+R^2+C^2$ with the fakeon prescription/projection for $C^2$). We show that these quantities are related to the spectra in a universal way. We also compute the first correction to the relation $r=−8n_T$ and provide a number of quantum gravity predictions that can be hopefully tested in the forthcoming future.

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J. Cosmol. Astropart. Phys. 01 (2021) 048 | DOI: 10.1088/1475-7516/2021/01/048

arXiv: 2007.15023 [hep-th]

Testable predictions of quantum gravity with fakeons on the spectra of the CMB radiation

Based on the paper 20A3 Renorm (arXiv: 2006.01163 [hep-th])

Talk given online by M. Piva for the Tokyo Institute of Technology, on Jun 16th, 2020

Talk given online by Marco Piva for the University of Sussex, on July 20th, 2020

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Testable predictions of quantum gravity with fakeons on the spectra of the CMB radiation

Talk given by M. Piva at the National Institute of Chemical Physics and Biophysics, Tallinn, Estonia, on Jun 9th, 2020

Based on the paper 20A3 Renorm (arXiv: 2006.01163 [hep-th])

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Quantum Gravity

### Book

14B1 D. Anselmi
Renormalization

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)

Last update: May 9th 2015, 230 pages

Avaibable on Amazon:

Contents:
Preface
1. Functional integral
2. Renormalization
3. Renormalization group
4. Gauge symmetry
5. Canonical formalism
6. Quantum electrodynamics
7. Non-Abelian gauge field theories
Notation and useful formulas
References

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