19S1 D. Anselmi
Theories of gravitation




D. Anselmi
From Physics To Life

A journey to the infinitesimally small and back

In English and Italian

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Recent Papers


Here are all papers published on Renormalization.COM

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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:

arXiv: 2109.06889 [hep-th]

Extensions to the Standard Model that use strictly off-shell degrees of freedom – the fakeons – allow for new measurable interactions at energy scales usually precluded by the constraints that target the on-shell propagation of new particles. Here we employ the interactions between a new fake scalar doublet and the muon to explain the recent Fermilab measurement of its anomalous magnetic moment. Remarkably, unlike in the case of usual particles, the experimental result can be matched for fakeon masses below the electroweak scale without contradicting the stringent precision data and collider bounds on new light degrees of freedom. Our analysis, therefore, demonstrates that the fakeon approach offers unexpected viable possibilities to model new physics naturally at low scales.


Phys. Rev. D 104 (2021) 035009 | DOI: 10.1103/PhysRevD.104.035009

arXiv: 2104.03249 [hep-ph]

We introduce a new way of modeling the physics beyond the Standard Model by considering fake, strictly off-shell degrees of freedom: the fakeons. To demonstrate the approach and exemplify its reach, we re-analyze the phenomenology of the Inert Doublet Model under the assumption that the second doublet is a fakeon. Remarkably, the fake doublet avoids the most stringent $Z$-pole constraints regardless of the chosen mass scale, thereby allowing for the presence of new effects well below the electroweak scale. Furthermore, the absence of on-shell propagation prevents fakeons from inducing missing energy signatures in collider experiments. The distinguishing features of the model appear at the loop level, where fakeons modify the Higgs boson $h\rightarrow\gamma\gamma$ decay width and the Higgs trilinear coupling. The running of Standard Model parameters proceeds as in the usual Inert Doublet Model case. Therefore, the fake doublet can also ensure the stability of the Standard Model vacuum. Our work shows that fakeons are a valid alternative to the usual tools of particle physics model building, with the potential to shape a new paradigm, where the significance of the existing experimental constraints towards new physics must necessarily be reconsidered.


J. High Energy Phys. 10 (2021) 132 | DOI: 10.1007/JHEP10(2021)132

arXiv: 2104.02071 [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.


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.


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.


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.


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

arXiv: 2007.15023 [hep-th]

We formulate quantum field theories of massive fields of arbitrary spins. The presence of both physical and fake particles, organized into multiplets, makes it possible to fulfill the requirements of locality, unitarity and renormalizability at the same time. The theories admit cubic and quartic self-interactions and can be coupled to quantum gravity and gauge fields. The simplest irreducible bosonic and fermionic multiplets are made of towers of alternating physical and fake particles. Their mass spectrum is constrained by RG invariant relations and depends on just one or two masses. The fixed points of the renormalization-group flow are scale invariant, but not necessarily conformal invariant. The Palatini version of quantum gravity with fakeons is equivalent to the non-Palatini one coupled to a peculiar multiplet of order 3. As a consequence, it is equally renormalizable and unitary.


J. High Energy Phys. 07 (2020) 176 | DOI: 10.1007/JHEP07(2020)176

arXiv: 2006.01163 [hep-th]


We derive the predictions of quantum gravity with fakeons on the amplitudes and spectral indices of the scalar and tensor fluctuations in inflationary cosmology. The action is $R+R^{2}$ plus the Weyl-squared term. The ghost is eliminated by turning it into a fakeon, that is to say a purely virtual particle. We work to the next-to-leading order of the expansion around the de Sitter background. The consistency of the approach puts a lower bound ($m_{\chi }>m_{\phi }/4$) on the mass $m_{\chi }$ of the fakeon with respect to the mass $m_{\phi }$ of the inflaton. The tensor-to-scalar ratio $r$ is predicted within less than an order of magnitude ($4/3 < N^{2}r<12$ to the leading order in the number of $e$-foldings $N$). Moreover, the relation $r\simeq -8n_{T}$ is not affected by the Weyl-squared term. No vector and no other scalar/tensor degree of freedom is present.


J. High Energy Phys. 07 (2020) 211 | DOI: 10.1007/JHEP07(2020)211

arXiv: 2005.10293 [hep-th]


The search for purely virtual quanta has attracted interest in the past. We consider various proposals and compare them to the concept of fake particle, or “fakeon”. In particular, the Feynman-Wheeler propagator, which amounts to using the Cauchy principal value inside Feynman diagrams, violates renormalizability, unitarity and stability, due to the coexistence of the prescriptions $\pm i\epsilon $. We contrast the Feynman, fakeon and Feynman-Wheeler prescriptions in ordinary as well as cut diagrams. The fakeon does not have the problems of the Feynman-Wheeler propagator and emerges as the correct concept of purely virtual quantum. It allows us to make sense of quantum gravity at the fundamental level, and places it on an equal footing with the standard model. The resulting theory of quantum gravity is perturbative up to an incredibly high energy.


J. High Energ. Phys. 03 (2020) 142 | DOI: 10.1007/JHEP03(2020)142

arXiv: 2001.01942 [hep-th]

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14B1 D. Anselmi

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

Last update: May 9th 2015, 230 pages

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1. Functional integral
2. Renormalization
3. Renormalization group
4. Gauge symmetry
5. Canonical formalism
6. Quantum electrodynamics
7. Non-Abelian gauge field theories
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