The tetrahexahedric Calogero model

We consider the spherical reduction of the rational Calogero model (of type A _n-1, without the center of mass) as a maximally superintegrable quantum system. It describes a particle on the (n = 2)-sphere in a very special potential. A detailed analysis is provided of the simplest non-separable case, n = 4, whose potential blows up at the edges of a spherical tetrahexahedron, tesselating the two-sphere into 24 identical right isosceles spherical triangles in which the particle is trapped. We construct a complete set of independent conserved charges and of Hamiltonian intertwiners and elucidate their algebra. The key structure is the ring of polynomials in Dunkl-deformed angular momenta, in particular the subspaces invariant and antiinvariant under all Weyl reflections, respectively.     

supersymmetric 3-particles Calogero model

We constructed the most general N=4 superconformal 3-particles systems with translation invariance. In the basis with decoupled center of mass the supercharges and Hamiltonian possess one arbitrary function which defines all potential terms. We have shown that with the proper choice of this function one may describe the standard, A₂ Calogero model as well as G₂ and BC₂ Calogero models, which, by construction, possess N=4 superconformal symmetry. The main property of all these systems is that even with the coupling constant equal to zero they still contain nontrivial interactions in the fermionic sector. In other words, there are infinitely many non-equivalent N=4 supersymmetric extensions of the free action depending on one arbitrary function. We also considered quantization and explicitly showed how the supercharges and Hamiltonian are modified. In the quantum case the constructed systems exhibit only invariance with respect to N=4 Poincaré supersymmetry.     

Fun and frustration with Calogero model

We review some algebraical (oscillator) aspects of N-body single-species and multispecies Calogero models in one dimension. We treat them as a particular cases of deformed harmonic oscillators and discuss the corresponding Fock spaces. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Constants of motion of the four-particle Calogero model

We present the explicit expressions of the complete set of constants of motion of four-particle Calogero model with excluded center of mass, i.e. of the A ₃ rational Calogero model. Then we find the constants of motion of its spherical part, defining two-dimensional 12-center spherical oscillator, with the force centers located at the vertexes of cuboctahedron.     

Quantization and conformal properties of a generalized Calogero model

We analyze a generalization of the quantum Calogero model with the underlying conformal symmetry, paying special attention to the two-body model deformation. Owing to the underlying SU(1,1) symmetry, we find that the analytic solutions of this model can be described within the scope of the Bargmann representation analysis, and we investigate its dynamical structure by constructing the corresponding Fock space realization. The analysis from the standpoint of supersymmetric quantum mechanics (SUSYQM), when applied to this problem, reveals that the model is also shape invariant. For a certain range of the system parameters, the two-body generalization of the Calogero model is shown to admit a one-parameter family of self-adjoint extensions, leading to inequivalent quantizations of the system. PACS 02.30.Ik; 03.65.Fd; 03.65.-w     

Density waves in the Calogero model - revisited

The Calogero model bears, in the continuum limit, collective excitations in the form of density waves and solitary modulations of the density of particles. This sector of the spectrum of the model was investigated, mostly within the framework of collective-field theory, by several authors, over the past 15 years or so. In this work we shall concentrate on periodic solutions of the collective BPS-equation (also known as “finite amplitude density waves”), as well as on periodic solutions of the full static variational equations which vanish periodically (also known as “large amplitude density waves”). While these solutions are not new, we feel that our analysis and presentation add to the existing literature, as we explain in the text. In addition, we show that these solutions also occur in a certain two-family generalization of the Calogero model, at special points in parameter space. A compendium of useful identities associated with Hilbert transforms, including our own proofs of these identities, appears in Appendix A. In Appendix B we also elucidate in the present paper some fine points having to do with manipulating Hilbert-transforms, which appear ubiquitously in the collective field formalism. Finally, in order to make this paper self-contained, we briefly summarize in Appendix C basic facts about the collective field formulation of the Calogero model.     

Quantum dissipative Higgs model

By using a continuum of oscillators as a reservoir, we present a classical and a quantum-mechanical treatment for the Higgs model in the presence of dissipation. In this base, a fully canonical approach is used to quantize the damped particle on a spherical surface under the action of a conservative central force, the conjugate momentum is defined and the Hamiltonian is derived. The equations of motion for the canonical variables and in turn the Langevin equation are obtained. It is shown that the dynamics of the dissipative Higgs model is not only determined by a projected susceptibility tensor that obeys the Kramers–Kronig relations and a noise operator but also the curvature of the spherical space. Due to the gnomonic projection from the spherical space to the tangent plane, the projected susceptibility displays anisotropic character in the tangent plane. To illuminate the effect of dissipation on the Higgs model, the transition rate between energy levels of the particle on the sphere is calculated. It is seen that appreciable probabilities for transition are possible only if the transition and reservoir’s oscillators frequencies to be nearly on resonance.     

The spherical sector of the Calogero model as a reduced matrix model

We investigate the matrix-model origin of the spherical sector of the rational Calogero model and its constants of motion. We develop a diagrammatic technique which allows us to find explicit expressions of the constants of motion and calculate their Poisson brackets. In this way we obtain all functionally independent constants of motion to any given order in the momenta. Our technique is related to the valence-bond basis for singlet states.     

Deformed Heisenberg algebras, a Fock-space representation and the Calogero model

We describe generally deformed Heisenberg algebras in one dimension. The condition for a generalized Leibniz rule is obtained and solved. We analyze conditions under which deformed quantum-mechanical problems have a Fock-space representation. One solution of these conditions leads to a q-deformed oscillator already studied by Lorek et al., and reduces to the harmonic oscillator only in the infinite-momentum frame. The other solution leads to the Calogero model in ordinary quantum mechanics, but reduces to the harmonic oscillator in the absence of deformation. Received: 27 April 2000 / Published online: 8 September 2000     

Quantum hydrodynamics, the quantum benjamin-ono equation, and the Calogero model

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.     

Unitarity in Dirichlet Higgs model

We show that a five-dimensional Universal Extra Dimension model, compactified on a line segment, is consistently formulated even when the gauge symmetry is broken solely by non-zero Dirichlet boundary conditions on a bulk Higgs field, without any quartic interaction. We find that the longitudinal W ⁺ W ⁻ elastic scattering amplitude, under the absence of the Higgs zero mode, is unitarized by exchange of infinite towers of KK Higgs bosons. Resultant amplitude scales linearly with the scattering energy μ ?s\propto\sqrt{s}, exhibiting five-dimensional nature. A tree-level partial-wave unitarity condition is satisfied up to 6.7 (5.7) TeV for the KK scale m _KK=430 (500) GeV, favored by the electroweak data within 90% CL.     

Permutation invariant algebras, a Fock space realization and the Calogero model

We study permutation invariant oscillator algebras and their Fock space representations using three equivalent techniques, i.e. (i) a normally ordered expansion in creation and annihilation operators, (ii) the action of annihilation operators on monomial states in Fock space and (iii) Gram matrices of inner products in Fock space. We separately discuss permutation invariant algebras which possess hermitean number operators and permutation invariant algebras which possess non-hermitean number operators. The results of a general analysis are applied to the -extended Heisenberg algebra, underlying the M-body Calogero model. Particular attention is devoted to the analysis of Gram matrices for the Calogero model. We discuss their structure, eigenvalues and eigenstates. We obtain a general condition for positivity of eigenvalues, meaning that all norms of states in Fock space are positive if this condition is satisfied. We find a universal critical point at which the reduction of the physical degrees of freedom occurs. We construct dual operators, leading to the ordinary Heisenberg algebra of free Bose oscillators. From the Fock-space point of view, we briefly discuss the existence of a mapping from the Calogero oscillators to the free Bose oscillators and vice versa. Received: 26 July 2001 / Revised version: 9 January 2002 / Published online: 12 April 2002     

Collective field formulation of the multi-species Calogero model and its duality symmetries

We study the collective field formulation of a restricted form of the multi-species Calogero model, in which the three-body interactions are set to zero. We show that the resulting collective field theory is invariant under certain duality transformations, which interchange, among other things, particles and antiparticles, and thus generalize the well known strong-weak coupling duality symmetry of the ordinary Calogero model. We identify all these dualities, which form an Abelian group, and study their consequences. We also study the ground state and small fluctuations around it in detail, starting with the two-species model, and then generalizing to an arbitrary number of species.     

Generalized Calogero and Toda Models

JETP Letters - A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has...     

Probing the Higgs sector of $$$$ Higgs triplet model at LHC

In this paper, we investigate the Higgs Triplet Model with hypercharge \(Y_{\varDelta }=0\) (HTM0), an extension of the Standard model, caracterized by a more involved scalar spectrum consisting of two CP even Higgs \(h^0, H^0\) and two charged Higgs bosons \(H^\pm \). We first show that the parameter space of HTM0, usually delimited by combined constraints originating from unitarity and BFB as well as experimental limits from LEP and LHC, is severely reduced when the modified Veltman conditions at one loop are also imposed. Then, we perform an rigorous analysis of Higgs decays either when \(h^0\) is the SM-like or when the heaviest neutral Higgs \(H^0\) is identified to the observed 125 GeV Higgs boson at LHC. In these scenarios, we perform an extensive parameter scan, in the lower part of the scalar mass spectrum, with a particular focus on the Higgs to Higgs decay modes \(H^0 \rightarrow h^0h^0, H^\pm \,H^\mp \) leading predominantly to invisible Higgs decays. Finally, we also study the scenario where \(h^0, H^0\) are mass degenerate. We thus find that consistency with LHC signal strengths favours a light charged Higgs with a mass about 176–178 GeV. Our analysis shows that the diphoton Higgs decay mode and \(H \rightarrow Z \gamma \) are not always positively correlated as claimed in a previous study. Anti-correlation is rather seen in the scenario where h is SM like, while correlation is sensitive to the sign of the potential parameter \(\lambda \) when H is identified to 125 GeV observed Higgs.     

Higgs boson masses in the model with two Higgs doublets and spontaneous CP-violation

The renormalization group equations are investigated for two Higgs doublet extension of a standard model with spontaneous CP violation. Assuming the validity of perturbation theory up to unification energies, the restrictions for the Higgs boson masses are found.     

Two Higgs Doublet Model and model independent interpretation of neutral Higgs boson searches

Searches for the neutral Higgs bosons and are used to obtain limits on the Type II Two Higgs Doublet Model (2HDM(II)) with no CP–violation in the Higgs sector and no additional particles besides the five Higgs bosons. The analysis combines approximately 170 pb of data collected with the OPAL detector at GeV with previous runs at and GeV. The searches are sensitive to the and decay modes of the Higgs bosons. For the first time, the 2HDM(II) parameter space is explored in a detailed scan, and new flavour independent analyses are applied to examine regions in which the neutral Higgs bosons decay predominantly into light quarks or gluons. Model–independent limits are also given. Received: 11 July 2000 / Published online: 21 December 2000     

Enhancement of Higgs to diphoton decay width in non-perturbative Higgs model

We investigate a possibility if a loop diagram via Higgsino can enhance the Higgs to diphoton decay width in supersymmetric models with an extension of Higgs sector. A model with an additional non-renormalizable term of Higgs fields is firstly analyzed where the higher order term can introduce the Higgs coupling to Higgsinos as well as charged Higgs bosons. We point out that a choice of the Higgs coupling to obtain a significant size of enhancement of diphoton decay width reduces the Higgs mass and/or a size of non-renormalizable term needs to be large and a cutoff scale is around the weak scale. Another model in which the Higgsino mass term is generated by a non-perturbative instanton effect via a strong dynamics in a context of SUSY QCD is also suggested. It is shown that the sign of the Higgs coupling to fermions is opposite from perturbative models due to an operator including bosonic fields in the denominator and a constructive contribution to the diphoton decay amplitude can be easily obtained in this kind of model.