Narrow Results

In this paper, we seek analytically checkable necessary and sufficient condition for copositivity of a three-dimensional symmetric tensor. We first show that for a general third-order three-dimensional symmetric tensor, checking copositivity is equivalent to solving a quartic equation and some quadratic equations. All of them can be solved analytically. Thus, we present an analytical way to check copositivity of a third-order three-dimensional symmetric tensor. Then, we consider a model of vacuum stability for ${\mathbb{Z}}_3$ scalar dark matter. This is a special fourth-order three-dimensional symmetric tensor. We show that an analytically expressed necessary and sufficient condition for this model bounded from below can be given, by using a result given by Ulrich and Watson in 1994.

After the gauge invariant gluon condensation, gluons remain as massless excitations. When the effective theory describing the condensation and the excitation is constructed, a constraint has to be imposed for the vacuum to be stable. The constraint implies the perfect dia-electricity and assures the solution of color flux tube when quarks are introduced. Thus the dia-electric model of Kogut–Susskind and 't Hooft is derived by the stability of the condensed vacuum.

The long-awaited Higgs particle H around 125 GeV has been observed at the LHC. Interpreting it as the Standard Model Higgs boson and if there is no new physics between electroweak and Planck scale, we then do not have a stable vacuum. Here, we give a brief review of the electroweak vacuum stability and some related theoretical issues in the Standard Model. Possible ways to save the stability are also discussed.

The vacuum stability condition of the Standard Model (SM) Higgs potential with mass in the range of 124–127 GeV puts an upper bound on the Dirac mass of the neutrinos. We study this constraint with the right-handed neutrino masses up to TeV scale. The heavy neutrinos contribute to ΔL = 2 processes like neutrinoless double beta decay and same-sign-dilepton (SSD) production in the colliders. The vacuum stability criterion also restricts the light-heavy neutrino mixing and constrains the branching ratio (BR) of lepton flavor-violating process, like μ→eγ mediated by the heavy neutrinos. We show that neutrinoless double beta decay with a lifetime ~10²⁵ years can be observed if the lightest heavy neutrino mass is <4.5 TeV. We show that the vacuum stability condition and the experimental bound on μ→e γ together put a constrain on heavy neutrino mass M_R>3.3 TeV. Finally we show that the observation of SSDs associated with jets at the LHC needs much larger luminosity than available at present. We have estimated the possible maximum cross-section for this process at the LHC and show that with an integrated luminosity 100 fb^-1 it may be possible to observe the SSD signals as long as M_R < 400 GeV.

We compute the quantum gravitational contributions to the Standard Model (SM) effective potential and analyze their effects on the Higgs vacuum stability in the framework of effective field theory. Einstein gravity necessarily implies the existence of higher dimension ${\varphi }^6$ and ${\varphi }^8$ operators with novel couplings ${\eta }_{1/2}$ in the Higgs sector. The beta functions of these couplings are established and the impact of the gravity induced contributions on electroweak vacuum stability is studied. We find that the true minimum of the SM effective potential now lies below the Planck scale for almost the entire parameter space $({\eta }_{1/2}(m_{\mathrm{\text{t}}})>0.01)$. In addition quantum gravity is shown to contribute to the minimal value of the SM next-to-leading order (NLO) effective potential at the percent level. The quantum gravity induced contributions yield a metastable vacuum for a large fraction of the parameter space in the flowing couplings ${\eta }_{1/2}$.

In order to address the observation of the neutrino oscillations and the metastability of the Standard Model (SM), we extend the fermion sector with two right-handed (i.e. sterile) neutrinos, and the scalar sector of the SM with a real scalar, the Hill field. The latter takes the role of a Majoron and generates the Majorana masses for the neutrino sector, such that the particle spectrum features two CP-even scalars h₁ and h₂, and also two heavy, mass degenerate neutrinos. When the h₁ is identified with the scalar resonance at $\sim$ 125 GeV and the condition is imposed that the h₁ self-coupling and its running vanish at the Planck scale, the scalar mixing and the vacuum expectation value of the Hill field are fixed by the h₂ mass.

The h₂ can be searched for at the LHC, and it has prospects of being discovered for the target integrated luminosities of the HL-LHC and the Future Circular hadron Collider (FCC-hh) when its mass is on the weak scale. The knowledge of the h₂ mass and its decay properties can yield an insight into its coupling to the heavy neutrinos, and thus also on the heavy neutrino mass scale. This yields an interesting connection between potentially detectable heavy scalars in high-energy proton collisions and the mass scale of the heavy neutrinos that is testable at the LHC and at future colliders.

In this review, we present a theory of cosmological constant and dark energy (DE), based on the topological structure of the vacuum. The multiple point principle (MPP) is reviewed. It demonstrates the existence of the two vacua into the SM. The Froggatt–Nielsen’s prediction of the top-quark and Higgs masses is given in the assumption that there exist two degenerate vacua in the SM. This prediction was improved by the next-order calculations. We also considered Sidharth’s theory of cosmological constant based on the noncommutative geometry of the Planck scale space–time, what gives an extremely small DE density providing the accelerating expansion of the Universe. Theory of two degenerate vacua — the Planck scale phase and electroweak (EW) phase — is also reviewed, topological defects in these vacua are investigated, also the Compton wavelength phase suggested by Sidharth is discussed. A general theory of the phase transition and the problem of the vacuum stability in the SM is reviewed. Assuming the existence of a new scalar $S$ bound state $6t+6\bar{t}$, earlier predicted by Froggatt, Nielsen and Laperashvili, we try to provide the vacuum stability in the SM and exact accuracy of the MPP.

In this paper, the vacuum energy density and generation of topological mass are investigated for a system of a real and complex scalar fields interacting with each other. In addition to that, it is also included the quartic self-interaction for each one of the fields. The condition imposed on the real field is the periodic condition, while the complex field obey a quasi-periodic condition. The system is placed in a scenario where the CPT-even aether-type Lorentz symmetry violation takes place. We allow that the Lorentz violation affects the fields with different intensities. The vacuum energy density, its loop correction, and the topological mass are evaluated analytically. It is also discussed the possibility of different vacuum states and their corresponding stability requirements, which depends on the conditions imposed on the fields, the interaction coupling constants and also the Lorentz violation parameters. The formalism used here to perform this investigation is the effective potential one, which is written as a loop expansion via path integral in quantum field theory.

I will discuss the prediction of the Higgs boson mass coming from asymptotic safety of the Standard Model (SM), and related lower bounds on it from Higgs inflation and absolute stability of SM vacuum.

After the discovery of a Higgs-like boson with a mass of m_h ≈ 125 GeV at the LHC, we can now attempt to draw conclusions about physics beyond the Standard Model. I argue that there are several hints towards new physics at intermediate scales Λ ≳ 10⁸ GeV. I review a class of stringy models with intermediate scale SUSY which relate the observed Higgs mass to symmetries of the Higgs sector. I then discuss radiative corrections to m_h, unification, dark matter and the possibility of classically unstable UV completions in these models.