The scalar garlands in the boson star

In the paper, we introduce the formalism to examine the impact of the hidden scalar sector to the conformal and the electroweak symmetries breaking. The novel approach to the scalar boson star (BS) naturalness is considered. The BS is presented by the local scalar field containing the Higgs boson field and the garland-like scalar dilaton fields of the conformal field theory. We show that taking into account the repulsive self-interactions and the flatness degree in the dark scalar sector prevents instability of the BS related to the black hole formation. We study in details how electroweak symmetry breaking affects the hidden sector by breaking its conformal symmetry and generating a mass gap to avoid the infra-red divergence. We apply our formalism to determine the modification of the Higgs quartic coupling away from its standard model (SM) value within the influence of the hidden sector. We have estimated the rate of the deviation from the SM with production of leptonic pairs due to decay of the dilaton and the dark photon. The latter is the subject of the dilaton decay first. The effect of new physics should be visible at the maximal energies of the LHC and the FCC. Introduction. Any phenomena in the Universe are governed by the principles of symmetry. Since the triumph of the Higgs discovery [1,2] which is based on the electroweak (EW) symmetry and its breaking [3,4], there are unexplained rates of both the Higgs mass and the vacuum expectation value (VEV) v where the latter is the measure of the mass scales in the standard model (SM) of elementary particles. The SM is widely regarded as an effective theory below the EW scale ∼ TeV, set by v. New physics (NP) at multi-TeV is currently actively explored at the LHC and in astroparticle experiments to find a hint for physics beyond the SM, though no any clear signals of NP have been found so far. Recent evidence, the theoretical and the phenomenological studies have emphasised the potentially crucial role played by evolving scalar fields, called the ”dilaton” fields, in understanding the cosmological processes in the early Universe. There is the possibility that various new vector bosons in hidden sector, e.g., dark photons (DP) dynamically related with dilatons, comprise the cosmological missing mass. The scale invariance and the breaking of the invariance are related with a new scalar field. This means one may have in the spectrum the CP even scalar field that can be identified as the pseudoGoldstone boson associated with the spontaneous breaking of conformal symmetry at the scale f ≥ v. The model with the effective Lagrangian density (LD) should still be scale-invariant, and the effective scalar potential should preserve the pattern of the classical potential. We shall study the model containing both the Higgs boson and the dilaton, where all the scales are induced by spontaneous breakdown of scale invariance. The light dilaton itself decoupled from the SM sector can be associated with the dark matter (DM) or even with dynamical dark energy in the early Universe where the dynamical breaking of scale symmetry by the Higgs can explain a mechanism for the inflation [5]. Like in the Higgs case, the mass of the dilaton, is proportional to f where the coefficient before f is the parameter defined by the dynamics of the dilaton and indicates the rate of deviations from exact scale invariance [6-16]. If the SM is embedded in the conformal field theory (CFT), the dilaton is an important example that can allow to go beyond the SM. In some cases, the dilaton may have the mixing with the SM Higgs boson at energies below f resulting with two physical scalar states each with production, e.g., by the colliders or due to an annihilation of spin 1/2 DM in case of the dilatons. In the paper [15], the authors introduced a new approach to the Higgs naturalness problem where the main experimental prediction is a dilaton in the 0.1 10 GeV range that mixes with the Higgs. The mixing angle is proportional to the ratio between the masses squared of the dilaton and the Higgs boson. As to the collider physics, there is the region of parameter space where the existence of a light dilaton is compatible with the LHC exclusion limits and where the hierarchy between the dilaton mass and f is of the order ∼ O(1 − 10%). This represents the new physics that dedicated the collider searches for light new scalar states, e.g., the long-lived particles (LLP), in the LHC Run 3 that will be able to further explore and test. In the paper [17], it has been shown that in the range of the dilaton mass [60 160] GeV the LHC direct searches in the di-jets and the di-photon channels are dominant and a lower bound for f is 3 TeV. The upper limit of f is estimated in [18]: f < 5.33 TeV if the dilaton is lighter than the top quark, or f < 4.87 TeV otherwise. The experimental and theoretical results, together with several strong constraints in flavour physics and charged lepton physics indicate that there might be no NP with direct and sizeable couplings to SM sector up to energy MNP ∼ 10 TeV [19] unless specific symmetries and breaking down are assumed or/and postulated.