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We propose a novel extension to the recently developed non-commutative Riemannian foliated branch-cut quantum gravity (BCQG). Based on an extended Faddeev-Jackiw symplectic deformation of the conventional Poisson algebra, we investigate non-commutativity effects on a symplectic topological manifold that provides a natural isomorphic setting composed by a triad of canonically conjugate scalar complex fields which comprise quantum complementary dualities. Based on a complementary analytically continued Friedmann-type equation, combined with a quantum approach based on the Ho\v{r}awa-Lifshitz quantum gravity, we describe the dynamic evolution of the universe's wave function, unfolding unprecedented predictions for the cosmic evolution and inflation. The non-commutative foliated quantum gravity approach offers a new perspective on explaining the accelerated cosmic expansion of the universe, strongly suggesting that non-commutative algebra induces the late accelerated growth of both the universe's wave function and the corresponding scale factor, along with their quantum counterparts. In contrast to the conventional inflationary model, where inflation requires a remarkably fine-tuned set of initial conditions in a patch of the universe, non-commutative foliated quantum gravity, analytically continued to the complex plane, captures short and long scales of spacetime, leading to an evolutionary cosmic dynamic through a topological reconfiguration of the primordial cosmic matter and energy content. This result introduces new speculative framework elements regarding the reconfiguration of matter and energy due to an underlying non-commutative spatio-temporal structure as a driver of spacetime cosmic acceleration.

In this work we explore the possible scenario of strange stars admixed with fermionic or bosonic dark matter. For the description of the ``visible'' sector, we use a specific phenomenological quark model that takes into account in-medium effects for the quark masses through a suitable baryonic density dependence, in which the free parameters are chosen from the analysis of the stellar matter stability window (parametrizations presenting lower energy per baryon than iron nuclei). In the case of the dark sector, we investigate the predictions of fermionic and bosonic models. In the former we consider a spin $1/2$ dark particle, and the latter is described by a dark scalar meson. Both models present a repulsive vector interaction, particularly important for the bosonic model since it helps avoiding the collapse of the star due to the lack of the degeneracy pressure. Our results point out to possible stable strange stars configurations in agreement with data from PSR J0030+0451, PSR J0740+6620, NICER and HESS J1731-347. As another feature, we find stars with dark matter halo configurations for lower values of the dark particle mass., Comment: More references added. Some figures updated

The decays of the $B$ meson into vector mesons, observed at the LHCb experiment, provide an ideal laboratory to investigate particle physics phenomena with quantum information theory methods. In this article, we focus on the decays yielding a pair of $\it{\Phi}$ mesons to investigate the presence of entanglement in the spin correlations of the system and quantify the amount of Bell inequality violation it entails. Our results show that the present LHCb data allows access to entanglement and to the Bell inequality violation with a significance exceeding the 5$\sigma$ threshold in both the cases. This demonstrates that the strong and electroweak interactions responsible for the $B$ meson decay act as a source of entanglement and the quantum mechanics nature of high-energy phenomena. Particular attention is paid to the assessment of loopholes: deficiencies in the experimental setup which could invalidate the results of the Bell test., Comment: review article to appear in Symmetry MDPI, 16 pages

The experimental data on the helicity amplitudes of charmonium decays allow us to measure entanglement in final state spin correlations and test possible violations of the Bell inequality. We find that the Bell inequality is violated with a significance of 5$\sigma$ or more in the decays $\eta_{c}, \, \chi_{c}^{0}, \, J/\psi \to \Lambda +\bar \Lambda$ $J/\psi \to \Xi^- +\bar \Xi^+,\; \Xi^0 +\bar \Xi^0, \;\Sigma^{-} +\bar \Sigma^{+},\; \Sigma^{0} +\bar \Sigma^{0}$, $\psi(3686) \to \Xi^- +\bar \Xi^+,\; \Sigma^- +\bar \Sigma^+,\; \Sigma^{0} +\bar \Sigma^{0}$, $\chi^{0}_{c}, \, \chi_{c}^{1} \to \phi+ \phi$ . The decays $\psi(3686) \to \Lambda +\bar \Lambda$ and $\Xi^0 +\bar \Xi^0$ show the same violation but with less significance. The decay $\psi(3686) \to \Omega^- + \bar \Omega^+$ displays entanglement. These results firmly establish the presence of entanglement and quantum non-separability at high energies, in a setting with particles of different spins and interacting through electroweak and strong interactions. In addition, the relatively long lifetime of some of the strange baryons produced in the decays provides a natural probe to test whether quantum spin correlations remain after the particles have interacted with the beam pipe and the first few layers of the detector., Comment: 31 pages, 12 figures, version sent to journal

Generalized uncertainty principles are effective changes to the Heisenberg uncertainty principle that emerge in several quantum gravity models. In the present letter, we study the consequences that two classes of these modifications yield on the physics of neutrinos. Besides analyzing the change in the oscillation probabilities that the generalized uncertainty principles entail, we assess their impact on the neutrino coherence length and their possible interpretation as nonstandard neutrino interactions. Constraints cast by present and planned neutrino experiments on the generalized uncertainty principles parameters are also derived., Comment: 9 pages, matches published version

We propose a novel test to assess the unitarity of the Cabibbo-Kobayashi-Maskawa matrix, $V_{\scriptscriptstyle{\rm CKM}}$, at present and future collider experiments. Our strategy makes use of the $W^+W^-$ production cross section to directly probe the $V_{\scriptscriptstyle{\rm CKM}}^{\dagger} V_{\scriptscriptstyle{\rm CKM}}$ product, which regulates the high-energy behavior of the observable. The violation of unitarity is signalled by an anomalous behavior of the cross section that grows quadratically with the $W^+W^-$ invariant mass with respect to the Standard Model prediction. By using the recent ATLAS measurements of the $W^+W^-$ cross section we are able to constrain the maximal unitarity violation allowed by current data, producing a bound complementary to the results of flavor physics experiments. Forecasts for the high luminosity phase of the LHC and for the future 100 TeV hadron collider are also discussed., Comment: 13 pages, 2 figures

The Future Circular Collider (FCC) -- in its first incarnation as a lepton collider -- will produce, according to the proposed design, more than 100 billion pairs of $\tau$ leptons after working for four years at the energy of the $Z$-boson resonance. The $\tau$ lepton is special because its relatively long lifetime makes it possible to reconstruct the momenta of neutrinos emitted in the single pion decay mode. The resulting large number of events is an ideal source for a full quantum tomography of the process that will test quantum entanglement and the violation of Bell inequality with unprecedented precision. In addition, the study of polarizations and spin correlations can provide a competitive determination of the Weinberg angle $ \theta_W$ and constrain possible anomalous couplings in the neutral electroweak current. We utilize analytic results and Monte Carlo simulations to explore to what extent these goals might be accomplished., Comment: 21 pages; matches the published version

The study of entanglement in particle physics has been gathering pace in the past few years. It is a new field that is providing important results about the possibility of detecting entanglement and testing Bell inequality at colliders for final states as diverse as top-quark, $\tau$-lepton pairs and $\Lambda$-baryons, massive gauge bosons and vector mesons. In this review, after presenting definitions, tools and basic results that are necessary for understanding these developments, we summarize the main findings -- as published by the beginning of year 2024 -- including analyses of experimental data in $B$ meson decays and top-quark pair production. We include a detailed discussion of the results for both qubit and qutrits systems, that is, final states containing spin one-half and spin one particles. Entanglement has also been proposed as a new tool to constrain new particles and fields beyond the Standard Model and we introduce the reader to this promising feature as well., Comment: Review article, published version in Progress in Particle and Nuclear Physics

The Belle II collaboration has recently observed the rare decay $B^+\to K^+ \nu \bar{\nu}$, finding an excess with respect to the Standard Model prediction. We explore the possibility that the data entails long-distance interactions induced by a massless dark photon, $\gamma_{\scriptscriptstyle{D}}$. This couples at the tree-level to an invisible, dark sector and to the Standard Model via higher-dimensional operators, such as the chromomagnetic-dipole coupling that we use to explain the excess. As the process $B^+\to K^+ \gamma_{\scriptscriptstyle{D}}$ is forbidden by angular momentum conservation, the transition mediated by the off-shell dark photon yields a three-body final state comprising a pair of dark fermions that show as a missing energy continuum in the detector, faking the neutrino signature. We show that the Belle II data is explained for perturbative values of the parameters of the model. This scenario predicts new contributions to the neutral $B$ meson decays $B^0\to K^* \gamma_{\scriptscriptstyle{D}}$, in which the emission of a on-shell dark photon is allowed, yielding a monochromatic missing energy signature. Analogously, an excess due to the emission of a dark photon is predicted for the $B^0_s\to \phi + E_{\rm miss}$ decay that could be scrutinized next at the LHCb experiments., Comment: 17 pages, 3 figures

Branch-cut gravity (BCG) is an extended version of the ontological domain of General Relativity, which is analytically continued to the complex plane. When combined with the Hawking-Hertog multiverse conception, BCG successfully addresses the issue of the primordial singularity. It consistently portrays the early Universe as a Riemannian foliation in which the singularities of the multiverse merge, giving rise to a smooth branching topological structure that resembles continuously connected Riemann surfaces. This structure introduces a new cosmic scale factor that is analytically continued into the complex plane. In this contribution, we start with the recently developed Wheeler DeWitt-Horava-Lifshitz non-commutative BCG formulation of quantum gravity. We investigate the impact of a non-commutative mini-superspace of variables obeying Poisson algebra on the accelerated behavior of the branch-cutting cosmic scale factor. drive spacetime acceleration, offering a new perspective on explaining the accelerating expansion rate of our Universe. As far as primordial relic gravitational waves are concerned, our predictions reveal an intricate transition between the two phases of the branched Universe: a contraction phase preceding the conventional concept of a primordial singularity and a subsequent expansion phase whose transition region is characterized by a Riemannian topological foliation structure. Furthermore, this transition is characterized by asymmetric distributions of gravitational wave intensities., Comment: 35 pages, 17 figures

The Generalized Uncertainty Principle (GUP) is motivated by the premise that spacetime fluctuations near the Planck scale impose a lower bound on the achievable resolution of distances, leading to a minimum length. Inspired by a semiclassical method that integrates the GUP into the partition function by deforming its phase space, we induce a modification on the thermodynamic quantities of the MIT bag model that we propose serves as an effective semiclassical description of deconfined quark matter in a space with minimal length. We investigate the consequences of this deformation on the zero-temperature limit, revealing a saturation limit for the energy density, pressure and baryon number density and an overall decrease of the thermodynamic quantities which suggests an enhanced stability against gravitational collapse. These findings extend existing research on GUP-deformed Fermi gases. Ultimately, our description introduces effects of quantum gravity in the equations of state for compact stars in a mathematically simple manner, suggesting potential for extension to more complex systems.

The large number of $\tau$ leptons available at the Belle II experiment makes it possible to study their properties and the extent of their compositeness. Our strategy relies on three observables defined in terms of elements of the polarization density matrix of the produced $\tau$ pairs, which we obtain via quantum tomography. The observables are able to explore values of the magnetic dipole moment down to $6.3 \times 10^{-4}$, which is two order of magnitude better than the current experimental limit, constrain the electromagnetic radius below $4.3 \times 10^{-3}$ fm and exclude an electric dipole moment larger than $1.7 \times 10^{-17}$ e cm. The quoted limits are at the $95\%$ confidence level and are obtained for a benchmark integrated luminosity of 1 ab$^{-1}$., Comment: 9 pages, 2 figures

A novel first order action principle has been proposed as the possible foundation for a more fundamental theory of General Relativity and the Standard Model. It is shown in this article that the proposal consistently incorporates gravity and matter fields, and guides to a new and robust path towards unification of fundamental interactions., Comment: 6 pages and references

We perform an improvement in a thermodynamical consistent model with density dependent quark masses ($m'_{u,d,s}$) by introducing effects of quark confinement/deconfinement phase transition, at high density regime and zero temperature, by means of the traced Polyakov loop ($\Phi$). We use realistic values for the current quark masses, provided by the Particle Data Group, and replace the constants of the interacting part of $m'_{u,d,s}$ by functions of $\Phi$, leading to a first order phase transition structure, for symmetric and stellar quark matter, with $\Phi$ being the order parameter. We show that the improved model points out the direction of the chiral symmetry restoration due to the emergence of a deconfined phase. In another application, we construct quark stars mass-radius profiles, obtained from this new model, and show to be possible to satisfy recent astrophysical observational data coming from the LIGO and Virgo Collaboration, and the NICER mission concerning the millisecond pulsars PSR J0030+0451, and PSR J0740+6620., Comment: 11 pages, 13 figures

We consider a scale-invariant inverse seesaw model with dynamical breaking of gauge symmetry and lepton number. In some regions of the parameter space, the Majoron - the pseudo-Goldstone of lepton number breaking - is a viable dark matter candidate. The bound on the Majoron decay rate implies a very large dilaton vacuum expectation value, which also results in a suppression of other dark matter couplings. Because of that, the observed dark matter relic abundance can only be matched via the freeze-in mechanism. The scalar field which gives mass to heavy neutrinos can play the role of the inflaton, resulting in a tensor-to-scalar ratio $r \lesssim 0.01$ for metric inflation and $r \lesssim 0.21$ for Palatini gravity., Comment: 31 pages, 3 figures, version published in JHEP, typos corrected

We explore the phenomenology of quantum entanglement at collider experiments by computing the polarization density matrix of processes yielding two massive gauge bosons. After reviewing the formalism, we detail observables suitable to test the presence of entanglement and quantum correlations in the di-boson system. The implied violation of Bell inequalities can be observed with future data at the LHC in the decays of the Higgs boson to $Z$ boson pairs., Comment: Contribution to the 2023 Electroweak session of the 57th Rencontres de Moriond

The violation of the Bell inequality is one of the hallmarks of quantum mechanics and can be used to rule out local deterministic alternative descriptions. We utilize the data analysis published by the LHCb collaboration on the helicity amplitudes for the decay $B^0\to J/\psi \,K^*(892)^0$ to compute the entanglement among the polarizations of the final vector mesons and the violation of the Bell inequality that it entails. We find that quantum entanglement can be detected with a significance well above 5$\sigma$ (nominally 84$\sigma$) and Bell inequality is violated with a significance well above 5$\sigma$ (nominally 36$\sigma$) -- thereby firmly establishing these distinguishing feature of quantum mechanics at high energies in a collider setting and in the presence of strong and weak interactions. Entanglement is also present and the Bell inequality is violated in other decays of the $B$ mesons into vector mesons, but with lesser significance., Comment: 5 pages, improved text, a few results updated, same as published version in PRD Letter

We write down the teleparallel equivalent to Hassan-Rosen bigravity, which is written using a torsionful but curvature-free connection. The theories only differ by a boundary term. The equivalence was proven, both by using perturbation theory and Hamiltonian analysis. It is further shown how one can construct novel bigravity theories within the teleparallel framework. Some of those are analyzed through perturbation theory, and it is found that all of the considered novel bigravity theories suffer from pathologies. In particular, it is found that a construction with two copies of new general relativity leads to ghostly degrees of freedom which are not present in the single tetrad teleparallel corresponding theory. We demonstrate how the teleparallel framework allows to easily create theories with derivative interaction. However, it is shown through perturbation theory that the simplest model is not viable. Furthermore, we demonstrate some steps in the Hamiltonian analysis of teleparallel bigravity with two copies of new general relativity and some toy models. The results rule out some of the novel teleparallel bigravity theories, but also demonstrate techniques in perturbation theory and Hamiltonian analysis which could be further used for more profound theories in the future., Comment: 32 pages, 1 figure, matches published version

Quantum tomography provides the full reconstruction of the density matrix of a state. We use it to study the Higgs boson decay into weak gauge bosons. Anomalous couplings beyond the Standard Model can be constrained by means of observables easily defined in terms of the polarization density matrix. We describe a strategy based on three observables that together provide the most stringent limits. Two of these observables are linked to the entanglement between the polarizations of the two gauge bosons, the other is based on CP-odd combinations of one momentum and two polarizations. We find for the $Z$ channel that this strategy offers, already with the available LHC data, limits competitive with the best available bounds. We argue that the inclusion of these observables in routine experimental analyses can lead to more stringent global fit limits., Comment: 14 pages, 4 figures, typos corrected in equations: 2.1, 3.3, 3.4, 3.8, 3.11, and 3.14-3.20

Quantum entanglement of weak interaction gauge bosons produced at colliders can be explored by computing the corresponding polarization density matrix. To this end, we consider the Higgs boson decays $H\to W W^*$ and $H\to Z Z^*$, in which $W^*$ and $Z^*$ are off-shell states, and the $WW$, $WZ$ and $ZZ$ di-boson production in proton collisions. The polarization density matrix of the di-boson state is determined by the amplitude of the production process and can be experimentally reconstructed from the angular distribution of the momenta of the final states into which the gauge bosons decay. We show that a suitable instance of the Bell inequality is violated in $H\to Z Z^*$ to a degree that can be tested at the LHC with future data. The same Bell inequality is violated in the production of $WW$ and $ZZ$ boson pairs for invariant masses above 900 GeV and scattering angles close to $\pi/2$ in the center of mass frame. LHC data in this case are not sufficient to establish the violation of the Bell inequality. We also analyze the prospects for detecting Bell inequality violations in di-boson final states at future $e^+e^-$ and muon colliders. A further observable that provides a lower bound on the amount of polarization entanglement in the di-boson system is computed for each of the examined processes. The analytic expressions for the polarization density matrices are presented in full in an Appendix. We also provide the unitary matrices required in the optimization procedure necessary in testing the Bell inequalities., Comment: 59 pages, 20 figures, same as published version

The inert doublet model is a well-motivated extension of the Standard Model that contains a dark matter candidate and modifies the dynamics of the electroweak symmetry breaking. In order to detail its phenomenology, we perform a comprehensive study of cosmic phase transitions and gravitational wave signals implied by the framework, accounting for the latest results of collider experiments. We require the neutral inert scalar to constitute, at least, a subdominant part of the observed dark matter abundance. While most of the phase transitions proceed through a single step, we identify regions of the parameter space where the electroweak vacuum is reached after multiple phase transitions. The resulting gravitational wave spectrum is generally dominated by single-step transitions and, in part of the parameter space, falls within the reach of future gravitational wave detectors such as LISA, BBO or DECIGO. We find that direct detection experiments efficiently probe the part of parameter space associated with multi-step phase transitions, which remain unconstrained only in the Higgs resonance region testable with future monojet searches. The implications of the new determination of the $W$ boson mass are also discussed., Comment: 27 pages, 17 figures. Version accepted by JCAP

We consider the classically scale invariant Higgs-dilaton model of dynamical symmetry breaking extended with an extra scalar field that plays the role of dark matter. The Higgs boson is light near a critical boundary between different symmetry breaking phases, where quantum corrections beyond the usual Gildener-Weinberg approximation become relevant. This implies a tighter connection between dark matter and Higgs phenomenology. The model has only three free parameters, yet it allows for the observed relic abundance of dark matter while respecting all constraints. The direct detection cross section mediated by the Higgs boson is determined by the dark matter mass alone and is testable at future experiments., Comment: 9 pages, 4 figures; for a webinar presentation of the paper see https://www.youtube.com/watch?v=ZTIdhPiI7hk

The standard description of particles and fundamental interactions is crucially based on a regular metric background. In the language of differential geometry, this dependence is encoded into the action via Hodge star dualization. As a result, the conventional forms of the scalar and Yang-Mills actions break down in a pregeometric regime where the metric is degenerate. This suggests the use of first order formalism, where the metric may emerge from more fundamental constituents and the theory can be consistently extended to the pregeometric phase. We systematically explore different realizations and interpretations of first order formalism, finding that a fundamental vector or spinor substructure brings about continuum magnetization and polarization as integration constants. This effect is analogous to the description of the cosmological dark sector in a recent self-dual formulation of gravity, and the similar form obtained for the first order Yang-Mills theory suggests new paths toward unification., Comment: 33 pages, no figures

The recently proposed multi-phase criticality principle in Coleman-Weinberg models can provide a new explanation for the hierarchy between the electroweak and new physics scales. When applied to the Standard Model, a Higgs boson as light as the pseudo-Goldstone boson of broken scale invariance occurs. The suppressed mixing between the two light fields still carries information about the large scale of symmetry breaking, albeit up to logarithmic corrections. In this work we probe this scenario with the present LHC data and assess the impact of future lepton and hadron colliders. Our results show that the multi-phase criticality can easily explain the apparent absence of new physics at the energy scales tested in current experiments., Comment: 9 pages, 4 figures

We study the evolution of light spectator fields in an asymmetric polynomial potential. During inflation, stochastic fluctuations displace the spectator field from the global minimum of its potential, populating the false vacuum state and thereby allowing for the formation of false vacuum bubbles. By using a lattice simulation, we show that these bubbles begin to contract once they re-enter the horizon and, if sufficiently large, collapse into black holes. This process generally results in the formation of primordial black holes, which, due to the specific shape of their mass function, are constrained to yield at most 1% of the total dark matter abundance. However, the resulting population can source gravitational wave signals observable at the LIGO-Virgo experiments, provide seeds for supermassive black holes or cause a transient matter-dominated phase in the early Universe., Comment: 25 pages, 8 figures

We explore the electroweak vacuum stability in the framework of a recently proposed paradigm for the origin of Yukawa couplings. These arise as low energy effective couplings radiatively generated by portal interactions with a hidden, or dark, sector at the one-loop level. Possible tree-level Yukawa couplings are forbidden by a new underlying symmetry, assumed to be spontaneously broken by the vacuum expectation value of a new scalar field above the electroweak scale. As a consequence, the top Yukawa interaction ceases to behave as a local operator at energies above the new sector scale and, therefore, cannot contribute to the running of the quartic Higgs coupling at higher energies. By studying two complementary scenarios, we explicitly show that the framework can achieve the stability of the electroweak vacuum without particular tuning of parameters. The proposed mechanism requires the existence of a dark sector and new portal messenger scalar interactions that, connecting the Standard Model to the dark sector fields, could be tested at the LHC and future collider experiments., Comment: 27 pages, 4 figures, new results and one figure added, improved text, revised version as published one

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., Comment: 7 pages, 5 figures, PRD version

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 $h\to\gamma\gamma$ 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., Comment: 12 pages, 7 figures, version accepted by JHEP

We show that the observed primordial perturbations can be entirely sourced by a light spectator scalar field with a quartic potential, akin to the Higgs boson, provided that the field is sufficiently displaced from vacuum during inflation. The framework relies on the indirect modulation of reheating, which is implemented without any direct coupling between the spectator field and the inflaton and does not require non-renormalisable interactions. The scenario gives rise to local non-Gaussianity with $f_{\rm NL}\simeq 5$ as the typical signal. As an example model where the indirect modulation mechanism is realised for the Higgs boson, we study the Standard Model extended with right-handed neutrinos. For the Standard Model running we find, however, that the scenario analysed does not seem to produce the observed perturbation., Comment: Figures modified, analysis extended, references added, matches published version

The Coleman-Weinberg mechanism can realise different phases of dynamical symmetry breaking. In each phase a combination of scalars, corresponding to the pseudo-Goldstone boson of scale invariance, has a loop-suppressed mass. We show that additional scalars, beyond the pseudo-Goldstone bosons, can become light at critical points in the parameter space where two different phases co-exist. We present a minimal implementation of the mechanism in multi-scalar models, detailing how loop-suppressed masses and mixings can be computed. We discuss realisations of the resulting multi-phase criticality principle and its relevance to the case of the Higgs boson., Comment: 10 pages, 2 figures, version accepted in PLB