### 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)

## Papers

Here are all papers published on Renormalization.COM

A more compact single-page list can be found at this link

We formulate a new quantization principle for perturbative quantum field theory, based on a minimally non time-ordered product, and show that it gives the theories of physical particles and purely virtual particles. Given a classical Lagrangian, the quantization proceeds as usual, guided by the time-ordered product, up to the common scattering matrix $S$, which satisfies a unitarity or a pseudounitarity equation. The physical scattering matrix $S_{\text{ph}}$ is built from $S$, by gluing $S$ diagrams together into new diagrams, through non time-ordered propagators. We classify the most general way to gain unitarity by means of such operations, and point out that a special solution “minimizes” the time-ordering violation. We show that the scattering matrix $S_{\text{ph}}$ given by this solution coincides with the one obtained by turning the would-be ghosts (and possibly some would-be physical particles) into purely virtual particles (fakeons). We study tricks to descend and ascend in a unique way among diagrams, and illustrate them in several examples: the ascending chain from the bubble to the hexagon, at one loop; the box with diagonal, at two loops; other diagrams, with more loops.

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J. High Energy Phys. 12 (2022) 088 | DOI: 10.1007/JHEP12(2022)088

arXiv: 2210.14240 [hep-th]

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.

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arXiv: 2207.12401 [hep-th]

We extend quantum field theory by including purely virtual “cloud” sectors, which allow us to define physical off-shell correlation functions of gauge invariant quark and gluon fields. Thanks to certain “cloud symmetries”, the new sectors do not change the fundamental physics. In particular, the ordinary correlation functions and the S matrix amplitudes remain the same. Each cloud sector is made of a cloud field, its anticommuting partner, a cloud function and a cloud Faddeev-Popov determinant. Every field insertion in a correlation function can be made gauge invariant by dressing it with an independent cloud. The cloud sectors are rendered purely virtual, to ensure that they do not propagate extra degrees of freedom. The off-shell, diagrammatic version of the optical theorem holds, and the extended theory is unitary. The one-loop two-point functions of the dressed quarks and gluons are calculated. Their absorptive parts are gauge independent, cloud independent and positive (while they are cloud dependent and possibly negative, if the clouds are defined by means of the Feynman prescription). A gauge/cloud duality simplifies the computations and shows that the gauge choice is just a particular cloud. Renormalizability is proved to all orders by means of an extended Batalin-Vilkovisky formalism and its Zinn-Justin master equations. We compare the purely virtual approach with the Coulomb nonlocal dressing of Dirac for QED, and the one of Lavelle and McMullan for non-Abelian gauge theories. We also comment on the use of Wilson lines and ‘t Hooft composite fields.

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arXiv: 2207.11271 [hep-ph]

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 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.

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

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.

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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.

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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.

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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]

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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